The BayesMallows package implements methods for Bayesian preference learning with the Mallows rank model, as originally described in Vitelli et al. (2018), and further developed in Asfaw et al. (2016) and Crispino et al. (2018). This vignette describes the usage of the package, starting from the complete data cases, through top-\(k\) rankings, pairwise comparisons, and finally clustering. We refer to the above mentioned papers, as well as the review Liu et al. (2018) for a thorough description of the methods. The necessary methods for data preprocessing, tuning of algorithms, and assessment of the posterior distributions will be described along the way. For ways to compute the partition function of the Mallows rank model with various distance measures, please see the separate vignette.

Main functions in the `BayesMallows` package.
Function Name	Description
`compute_mallows`	Compute the posterior distribution of the Bayesian Mallows model. This is the main function of the package. Returns an object of class `BayesMallows`.
`compute_mallows_mixtures`	Compute multiple Mallows models with different number of mixture components. This is a convenience function for determining the number of mixtures to use.
`sample_mallows`	Sample from the Mallows model.
`plot.BayesMallows`	Quick plots of the posterior densities of \(\alpha\) and \(\rho\).
`assess_convergence`	Study the convergence of the Markov chain, in order to determine burnin and other algorithm parameters.
`plot_elbow`	Create an elbow plot for comparing models with different number of clusters.
`plot_top_k`	Plot the top-\(k\) rankings. Particularly relevant when the data is in the form of pairwise comparisons.
`assign_cluster`	Compute the cluster assignment of assessors.
`compute_map_consensus`	Compute the MAP estimate of the latent ranks.
`compute_cp_consensus`	Rank the items according to their CP consensus.
`compute_posterior_intervals`	Compute Bayesian posterior intervals for the parameters.
`generate_initial_ranking`	Generate an initial ranking, for the case of missing data or pairwise comparisons.
`generate_transitive_closure`	Generate the transitive closure for a set of pairwise comparisons.
`estimate_partition_function`	Estimate the partition function of the Mallows model using either importance sampling or an asymptotic approximation.

Example datasets in the `BayesMallows` package.
Dataset Name	Description
`beach_preferences`	Stated pairwise preferences between random subsets of 15 images of beaches, by 60 assessors.
`sushi_rankings`	Complete rankings of 10 types of sushi by 5000 assessors.
`potato_visual`	Complete rankings of 20 potatoes by weight, based on visual inspection, by 12 assessors.
`potato_weighing`	Complete rankings of 20 potatoes by weight, where the assessors were allowed to weigh the potatoes in their hands, by 12 assessors.
`potato_true_ranking`	Vector of true weight rankings for the 20 potatoes in the example datasets `potato_visual` and `potato_weighing`.

BayesMallows User Guide

Having described the model, we now show some case studies demonstrating the use of the BayesMallows package. To reduce the computing time, we have typically taken a fairly small number of posterior samples, and rather focused on demonstrating the different methods. In general, we recommend taking a larger number of samples than what we do here, in order to obtain good approximations of the posterior distributions.

Completely Ranked Data

BayesMallows comes with example data described in Liu et al. (2018). A total of 12 assessors were asked to rank 20 potatoes based on their weight. In the first round, the assessors were only allowed to study the potatoes visually, while in the second round, the assessors were also allowed to hold the potatoes in their hands in order to compare them. The data sets are named potato_visual and potato_weighing, respectively. The true ordering of the potatoes’ weights are stored in the vector potato_true_ranking.

The potato_visual dataset is shown below. The column names P1, …, P20 represent potatoes, and the row names A1, …, A12 represent assessors. The potato_weighing dataset has a similar structure.

Example dataset `potato_visual`.
	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	1	2	6	18	7	4	14	12	13
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	7	16	8	5	12	9	13
A9	8	16	18	19	12	13	6	20	5	3	7	1	4	2	17	10	9	15	14	11
A10	7	17	19	18	9	15	5	20	3	10	11	1	2	6	16	8	4	13	12	14
A11	12	16	19	15	13	18	7	20	3	5	11	1	2	6	17	10	4	14	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	13	10	11

Algorithm Tuning

The compute_mallows function is the workhorse of BayesMallows. It runs the Metropolis-Hastings algorithm and returns the posterior distribution of the scale parameter \(\alpha\) and the latent ranks \(\rho\) of the Mallows model. To see all its arguments, please run ?compute_mallows in the console.

We start by using all the default values of the parameters, so we only need to supply the matrix of ranked items. We use the potato_visual data printed above.

model_fit <- compute_mallows(potato_visual)

The argument returned is a list object of class BayesMallows, which contains a whole lot of information about the MCMC run.

class(model_fit)
names(model_fit)

The function assess_convergence produces plots for visual convergence assessment. We start by studying \(\alpha\), which is the default. The plot is shown below, and looks good enough, at least to begin with.

assess_convergence(model_fit)

Next, we study the convergence of \(\rho\). To avoid too complicated plots, we pick 5 items to plot. Again, you can read more about this function by running ?assess_convergence in the console.

assess_convergence(model_fit, type = "rho", items = 1:5)

Based on these plots, it looks like the algorithm starts to converge after around 1000 iterations. Discarding the first 1000 iterations as burn-in hence seems like a safe choice. We create a new list element for model_fit called burnin, in which this choice is saved. Having done this, we do not have to provide a burnin argument to subsequent analyses of the posterior distributions.

model_fit$burnin <- 1000

Posterior Distributions

Once we are confident that the algorithm parameters are reasonable, we can study the posterior distributions of the model parameters using the generic function plot.BayesMallows.

Scale Parameter \(\alpha\)

With a burnin of 1000, the original model_fit object from the previous subsection has 1000 MCMC samples. The default parameter of plot.BayesMallows is \(alpha\), so we can study the posterior distribution with the simple statement below.

plot(model_fit)

We can also get the posterior credible intervals for \(\alpha\):

intervals <- compute_posterior_intervals(model_fit, parameter = "alpha")

parameter	mean	median	conf_level	hpdi	central_interval
alpha	10.95	10.908	95 %	[9.759,12.241]	[9.718,12.229]

Latent Ranks \(\rho\)

Obtaining posterior samples from \(\rho\) is in general harder than for \(\alpha\). Some items tend to be very sticky. We start by plotting the model_fit object from above. We now have to tell plot.BayesMallows that we want a plot of type = "rho" and all the items. This gives us posterior the posterior density of all the items.

plot(model_fit, type = "rho", items = 1:20)

We can also find the posterior intervals of the latent ranks:

compute_posterior_intervals(model_fit, parameter = "rho")

item	parameter	mean	median	conf_level	hpdi	central_interval
P1	rho	10	10	95 %	[9,11]	[9,11]
P2	rho	17	17	95 %	[16,18]	[16,18]
P3	rho	19	19	95 %	[19]	[19]
P4	rho	16	16	95 %	[16,18]	[16,18]
P5	rho	10	10	95 %	[9,12]	[9,12]
P6	rho	15	15	95 %	[15]	[15]
P7	rho	6	6	95 %	[5,7]	[5,7]
P8	rho	20	20	95 %	[20]	[20]
P9	rho	3	3	95 %	[3]	[3]
P10	rho	4	4	95 %	[4]	[4]
P11	rho	11	11	95 %	[9,11]	[9,11]
P12	rho	1	1	95 %	[1]	[1]
P13	rho	2	2	95 %	[2]	[2]
P14	rho	7	7	95 %	[6,7]	[6,7]
P15	rho	18	18	95 %	[17,18]	[17,18]
P16	rho	8	8	95 %	[8]	[8]
P17	rho	5	5	95 %	[5,6]	[5,6]
P18	rho	14	14	95 %	[13,14]	[13,14]
P19	rho	12	12	95 %	[10],[12]	[10,12]
P20	rho	13	13	95 %	[13,14]	[13,14]

Jumping over \(\alpha\)

Updating \(\alpha\) in each step may not be necessary. With the alpha_jump argument, we can tell the MCMC algorithm to update \(\alpha\) only every alpha_jumpth iteration.

model_fit <- compute_mallows(potato_visual, alpha_jump = 10)

We should assess convergence again:

assess_convergence(model_fit, type = "alpha")

Thinning

Saving a large number of iterations of \(\rho\) gets quite expensive, so compute_mallows has a rho_thinning parameter. It specifies that only each rho_thinningth iteration of \(\rho\) should be saved to memory. We double the number of iterations, while setting rho_thinning = 2. This gives us the same number of posterior samples.

Please be careful with thinning. In this small data example it is definitely wasteful! Running the same number of iterations without thinning always gives a better approximation of the posterior distribution. Thinning might be useful when you need to run a large number of iterations to explore the space of latent ranks, and the latent ranks from all iterations do not fit in memory. (See, e.g., Gelman et al. (2004) for a discussion of thinning).

model_fit <- compute_mallows(potato_visual, alpha_jump = 10, rho_thinning = 2)

Varying the Distance Metric

We can try to use the Kendall distance instead of the footrule distance.

model_fit <- compute_mallows(potato_visual, metric = "kendall")

Validation of Input

It is also worth pointing out that compute_mallows checks if the input data are indeed ranks. Let us try this by manipulating the first row of potato_visual, giving rank 1 to the first two items:

potato_modified <- potato_visual
potato_modified[1, 1:2] <- 1

model_fit <- compute_mallows(potato_modified)
#> Error in compute_mallows(potato_modified): Not valid permutation.

Top-\(k\) Rankings

Encoding of Missing Ranks

Now imagine that the assessors in the potato experiment were asked to rank only the top-five heaviest potatoes. We generate these data by retaining only ranks 5 or higher in potato_visual, setting the rest to NA.

potato_top <- potato_visual * if_else(potato_visual > 5, NA_integer_, 1L)

In Vitelli et al. (2018) it is shown that the unranked items do not effect the MAP estimates of the ranked items in this top-k setting. In this case, there are 8 potatoes which have been ranked, and so the unranked potatoes should have uniform posterior distributions between 9 and 20. However, arriving at these uniform posteriors require a large number of MCMC iterations, so we instead remove these items:

item_ranked <- apply(potato_top, 2, function(x) !all(is.na(x)))
potato_top <- potato_top[, item_ranked, drop = FALSE]

We are now left with this 12 by 8 matrix:

Example dataset `potato_top`.
	P7	P9	P10	P12	P13	P14	P16	P17
A1	4	3	5	1	2	NA	NA	NA
A2	NA	4	3	1	2	NA	NA	5
A3	NA	NA	3	2	1	4	5	NA
A4	5	3	4	1	2	NA	NA	NA
A5	2	3	NA	1	4	5	NA	NA
A6	NA	3	NA	1	2	4	NA	5
A7	5	3	NA	1	2	NA	NA	4
A8	NA	4	3	1	2	NA	NA	5
A9	NA	5	3	1	4	2	NA	NA
A10	5	3	NA	1	2	NA	NA	4
A11	NA	3	5	1	2	NA	NA	4
A12	NA	3	4	1	2	NA	NA	5

Metropolis-Hastings Algorithm with Missing Ranks

The compute_mallows function automatically recognizes the NA values as missing ranks, and augments the data, as described in Section 4.1 of Vitelli et al. (2018). Let us try:

model_fit <- compute_mallows(potato_top, save_augmented_data = TRUE)

Looking at the returned object, we see that any_missing is TRUE, so compute_mallows has correctly detected that there are missing values. We set save_augmented_data = TRUE so we can get a diagnostic plot for the convergence of the data augmentation. After you are confident that the algorithm converges, you might want to use the default save_augmented_data = FALSE, because saving the data takes memory.

model_fit$any_missing
#> [1] TRUE

model_fit also has assessor-wise acceptance rates for the augmentation.

model_fit$aug_acceptance

assessor	acceptance_rate
1	0.56
2	0.56
3	0.90
4	0.58
5	0.59
6	0.59
7	0.58
8	0.54
9	0.56
10	0.56
11	0.51
12	0.55

Algorithm Tuning

The tuning of \(\alpha\) and \(\rho\) can be done exactly as described above for the full potato data. Let us now study the convergence of the data augmentation.

Looking back, we see that assessor 1 does not have values for potatoes 14, 16, and 17. Hence, we expect these to jump between ranks 6, 7, and 8, since the 5 ranked items are fixed to values 1,…,5. Below we have plotted potato 14, which fluctuates between 6 and 8. You can confirm that these same happens to 16 and 17.

assess_convergence(model_fit, type = "Rtilde", assessors = 1, 
                   items = "P14")

Assessor 1 has ranks for potatoes 7, 9, 10, 12, and 13, which are fixed. You can confirm that they are fixed by running the following command.

assess_convergence(model_fit, type = "Rtilde", assessors = 1, 
                   items = c("P7", "P9", "P10", "P12", "P13"))

Posterior Distributions

We can study the posterior distributions of \(\alpha\) and \(\rho\) just like we did above.

Other Distance Measures

Like for the complete ranks, we can vary the distance measure used in the Mallows model. Here are command for running with Cayley, Kendall, and Spearman distance.

model_fit <- compute_mallows(potato_top, metric = "cayley")
model_fit <- compute_mallows(potato_top, metric = "kendall")
model_fit <- compute_mallows(potato_top, metric = "spearman")

Ranks Missing at Random

If the ranks are missing at random, we cannot remove the unranked items as we did for top-\(k\) rankings above. Let us assume that 10 % of the data in potato_visual have disappeared due to a disk failure. We generate these in the code chunk below:

missing_indicator <- if_else(
  runif(nrow(potato_visual) * ncol(potato_visual)) < 0.1,
                            NA_real_, 1)
potato_missing <- potato_visual * missing_indicator

The data now look like the following:

Example dataset `potato_missing`.
	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	NA	19	NA	6	16	4	20	3	5	12	1	2	9	NA	8	7	14	13	11
A2	10	NA	19	17	11	NA	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	NA	12
A5	12	17	19	NA	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	NA	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	NA	2	6	18	7	4	14	12	NA
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	NA	16	8	5	12	9	13
A9	8	NA	18	NA	12	13	6	20	5	3	NA	1	4	NA	17	10	9	15	14	11
A10	7	17	19	18	NA	15	NA	20	3	10	11	1	2	6	16	8	NA	13	12	14
A11	NA	16	19	15	NA	18	7	20	3	NA	11	NA	2	6	17	10	4	NA	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	NA	10	NA

Algorithm Tuning

We supply potato_missing to compute_mallows as before:

model_fit <- compute_mallows(potato_missing)

We check the convergence with the following commands. The plots (not shown) would show good convergence before 1,000 iterations.

assess_convergence(model_fit)

assess_convergence(model_fit, type = "rho", items = 1:6)

Again, we can look at the acceptance rates for the augmented data. Now let us compare it to the number of missing ranks per assessor.

bind_cols(model_fit$aug_acceptance, 
          n_missing = apply(potato_missing, 1, function(x) sum(is.na(x))))

assessor	acceptance_rate	n_missing
1	0.61	3
2	0.66	2
3	1.00	0
4	1.00	1
5	1.00	1
6	1.00	1
7	0.55	2
8	1.00	1
9	0.19	4
10	0.45	3
11	0.11	5
12	0.99	2

We see that for assessors who have either 0 or 1 missing ranks, their augmentations are always accepted. In the case of 0 missing ranks, there is in fact no augmentation going on at all, but we use the convention that the assessor’s own complete data is accepted.

Posterior Distributions

The posterior distributions can be studied as shown above.

Pairwise Preferences

Handling of pairwise preferences in the Mallows rank model is described in Section 4.2 of Vitelli et al. (2018).

Introduction

Let us start by considering a toy example with two assessors and five items. Assessor 1 has stated a set of preferences \[ \mathcal{B}_{1} = \left\{A_{1} \prec A_{2}, A_{2} \prec A_{5}, A_{4} \prec A_{5} \right\} \] and assessor 2 has the set of preferences \[ \mathcal{B}_{2} = \left\{ A_{1} \prec A_{2}, A_{2} \prec A_{3}, A_{3} \prec A_{4} \right\}. \]

Data Model

Each time an assessor is asked to compare two objects, a measurement is made. Therefore, in order to keep the data tidy (Wickham (2014)), we define a dataframe in which each row corresponds to a pairwise comparison. The columns (variables) are the assessor, the bottom item, and the top item.

In the code snippet below, we define such a dataframe for the toy example presented above:

pair_comp <- tribble(
  ~assessor, ~bottom_item, ~top_item,
  1, 1, 2,
  1, 2, 5,
  1, 4, 5,
  2, 1, 2,
  2, 2, 3,
  2, 3, 4
)

knitr::kable(pair_comp, caption = "Dataset `pair_comp`.")

Dataset `pair_comp`.
assessor	bottom_item	top_item
1	1	2
1	2	5
1	4	5
2	1	2
2	2	3
2	3	4

Transitive Closure

Next, we need to find the transitive closure for the set of pairwise comparisons given by each user. BayesMallows comes with a function generate_transitive_closure to do just this.

pair_comp_tc <- generate_transitive_closure(pair_comp)

As we can see, pair_comp_tc has an additional row containing the relation \(A_{4} \prec A_{5}\) for assessor 1. For assessor 2, \[\text{tc}(\mathcal{B}_{2}) = \mathcal{B}_{2} \cup \left\{ A_{1} \prec A_{3}, A_{1} \prec A_{4}, A_{2} \prec A_{4}\right\},\] so three new rows have been added.

Dataframe `pair_comp_tc`.
assessor	bottom_item	top_item
1	1	2
1	1	5
1	2	5
1	4	5
2	1	2
2	1	3
2	2	3
2	1	4
2	2	4
2	3	4

The dataframe returned by generate_transitive_closure inherits from tibble, but has subclass BayesMallowsTC. The compute_mallows function uses this information to ensure that the object provided has been through the generate_transitive_closure function. If it has not, compute_mallows will do it for us, but this may lead to additional computing time when running several diagnostic runs and trying out different parameters, since the transitive closure will be recomputed each time.

Initial Ranking

We can also generate an initial ranking, consistent with the pairwise comparisons. Again, compute_mallows will do it for us, but we may save time by computing it once and for all before we starting running the algorithms.

initial_ranking <- generate_initial_ranking(pair_comp_tc)

Rather than digging deeper into this toy example, we go on with a real application.

Beach Preferences

The beach preference dataset is described in Section 6.2 of Vitelli et al. (2018), and is available in the dataframe beach_preferences in BayesMallows. In short, \(60\) assessors were each asked to perform a random set of pairwise comparisons between pictures of \(15\) beaches. The first few rows are listed below.

beach_preferences

Example dataset `beach_preferences`
assessor	bottom_item	top_item
1	2	15
1	5	3
1	13	3
1	4	7
1	5	15
1	12	6

Transitive Closures

We start by generating the transitive closure of the preferences.

beach_tc <- generate_transitive_closure(beach_preferences)

We can compare the dataframes before and after. We see that the number of rows has been approximately doubled, and that beach_tc has subclass BayesMallowsTC has it should.

str(beach_preferences)
#> Classes 'tbl_df', 'tbl' and 'data.frame':    1442 obs. of  3 variables:
#>  $ assessor   : num  1 1 1 1 1 1 1 1 1 1 ...
#>  $ bottom_item: num  2 5 13 4 5 12 14 7 8 14 ...
#>  $ top_item   : num  15 3 3 7 15 6 6 9 10 9 ...
str(beach_tc)
#> Classes 'BayesMallowsTC', 'tbl_df', 'tbl' and 'data.frame':  2921 obs. of  3 variables:
#>  $ assessor   : num  1 1 1 1 1 1 1 1 1 1 ...
#>  $ bottom_item: int  4 5 7 4 5 7 8 9 13 14 ...
#>  $ top_item   : int  1 1 1 3 3 3 3 3 3 3 ...

Initial Ranking

Next, we generate an initial ranking.

beach_init_rank <- generate_initial_ranking(beach_tc)

We can also take a look at the first 6 rows in it.

First 6 rows in `beach_init_rank`.
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
7	15	3	14	13	2	10	12	6	4	1	5	8	11	9
5	15	14	6	2	13	12	11	1	7	4	10	9	8	3
3	9	7	15	14	2	6	13	1	8	12	11	10	5	4
5	15	2	14	11	8	9	10	1	4	3	13	7	12	6
9	15	7	5	6	1	14	13	4	3	2	12	11	8	10
9	13	5	12	15	3	7	11	1	6	4	14	8	10	2

Let us add column names to the initial ranking. This will make the plots more informative. But be aware that if the names are too long, some plots will be filled with text. We therefore name them B1, B2, …, B15.

colnames(beach_init_rank) <- paste0("B", seq(from = 1, to = ncol(beach_init_rank), by = 1))

Algorithm Tuning

We can now check the convergence, using the same tools as before.

test_run <- compute_mallows(
  rankings = beach_init_rank, 
  preferences = beach_tc,
  save_augmented_data = TRUE
  )

The trace plots for \(\alpha\) and \(\rho\) show good convergence.

Convergence of Rtilde

When using assess_convergence, we can set type = "Rtilde" and choose assessors and items. We now illustrate this by going a bit deeper into some assessors. Referring to the mathematical notation in the top of the vignette, Rtilde corresponds to the augmented rankings \(\tilde{R}_{1}, \dots, \tilde{R}_{N}\).

Assessor 1

Let us look at Beach 2 for assessor 1.

beach_tc %>% 
  filter(assessor == 1, bottom_item == 2 | top_item == 2)

assessor	bottom_item	top_item
1	2	6
1	2	15

It is implied by the preferences of assessor 1 that \(\{B_{2} \prec B_{6}\}\) and \(\{B_{2} \prec B_{15}\}\).

assess_convergence(test_run, type = "Rtilde", 
                   assessors = 1, items = c(2, 6, 15))

This seems correct.

Next, no ordering is implied between beach 2 and 4 for assessor 1.

beach_tc %>% 
  filter(assessor == 1, bottom_item %in% c(2, 4), top_item %in% c(2, 4)) %>% 
  nrow()
#> [1] 0

The traces of item 2 and 4 do indeed cross.

assess_convergence(test_run, type = "Rtilde", 
                   assessors = 1, items = c(2, 4))

No ordering is implied between beach 5 and beach 8 either:

beach_tc %>% 
  filter(assessor == 1, bottom_item %in% c(5, 8), top_item %in% c(5, 8)) %>% 
  nrow()
#> [1] 0

The traces of beaches 5 and 8 do indeed cross.

assess_convergence(test_run, type = "Rtilde", 
                   assessors = 1, items = c(5, 8))

Assessor 2

We go on with assessor 2. Let us take a look at, say, beach 10.

beach_tc %>% 
  filter(assessor == 2, bottom_item == 10 | top_item == 10)

assessor	bottom_item	top_item
2	10	5
2	7	10

Beach 10 is preferred to beach 7, but disfavored to beach 5. The trace plot does indeed show this:

assess_convergence(test_run, type = "Rtilde", 
                   assessors = 2, items = c(5, 7, 10))

No ordering is implied between beach 1 and beach 15 for assessor 2.

beach_tc %>% 
  filter(assessor == 1, bottom_item %in% c(1, 15), top_item %in% c(1, 15)) %>% 
  nrow()
#> [1] 0

Their traces do indeed cross.

assess_convergence(test_run, type = "Rtilde", 
                   assessors = 2, items = c(1, 15))

We delete the test run before going on.

rm(test_run)

Posterior Distributions

Based on the convergence diagnostics, and being fairly conservative, we set burnin = 1000, and take an additional 5,000 samples. We still save the augmented data, because we are going to call the function plot_top_k below.

model_fit <- compute_mallows(
  rankings = beach_init_rank, 
  preferences = beach_tc,
  nmc = 6000,
  save_augmented_data = TRUE
  )

model_fit$burnin <- 1000

The posterior densities of \(\alpha\) and \(\rho\) can be studied as shown above.

CP Consensus

We can rank the beaches according to the cumulative probability (CP) consensus. This functionality is provided by compute_cp_consensus, which returns a dataframe.

compute_cp_consensus(model_fit)

ranking	item	cumprob
1	B9	0.85
2	B6	1.00
3	B3	0.79
4	B11	0.98
5	B15	0.99
6	B10	1.00
7	B1	1.00
8	B7	0.53
9	B5	0.72
10	B13	1.00
11	B4	0.54
12	B8	0.98
13	B12	0.64
14	B14	0.97
15	B2	1.00

MAP Consensus

The maximum a posterior (MAP) consensus ranking is the a posterior most likely value of \(\rho\).

compute_map_consensus(model_fit)

probability	item	map_ranking
0.17	B9	1
0.17	B6	2
0.17	B3	3
0.17	B11	4
0.17	B15	5
0.17	B10	6
0.17	B1	7
0.17	B7	8
0.17	B5	9
0.17	B13	10
0.17	B8	11
0.17	B4	12
0.17	B12	13
0.17	B14	14
0.17	B2	15

Posterior Intervals

We can compute posterior intervals for several parameters. Here we do it for \(\rho\).

compute_posterior_intervals(model_fit, parameter = "rho")

item	parameter	mean	median	conf_level	hpdi	central_interval
B1	rho	7	7	95 %	[7]	[7]
B2	rho	15	15	95 %	[14,15]	[14,15]
B3	rho	3	3	95 %	[3,4]	[3,4]
B4	rho	12	11	95 %	[11,13]	[11,13]
B5	rho	9	9	95 %	[8,10]	[8,10]
B6	rho	2	2	95 %	[1,2]	[1,2]
B7	rho	9	8	95 %	[8,9]	[8,10]
B8	rho	12	12	95 %	[11,12]	[11,12]
B9	rho	1	1	95 %	[1,2]	[1,2]
B10	rho	6	6	95 %	[6]	[6]
B11	rho	4	4	95 %	[3,4]	[3,4]
B12	rho	13	13	95 %	[13,15]	[12,15]
B13	rho	10	10	95 %	[8,10]	[8,10]
B14	rho	14	14	95 %	[12,15]	[12,15]
B15	rho	5	5	95 %	[5]	[5]

Posterior Probability of Being Ranked Top-\(k\)

We can also find the posterior probability for each beach of being ranked top-\(k\), both in \(\rho\) and among the assessors.

We can do a top-\(k\) plot with plot_top_k. By default, k = 3. It may be necessary to do some experimentation with the rel_widths argument to get a good looking plot.

plot_top_k(model_fit, rel_widths = c(1, 8))

Clustering of Assessors

In many situations, it is interesting to divide the assessors into clusters, in which each the assessors in each cluster have similar preferences. In Vitelli et al. (2018), Section 4.3, it is shown how the Bayesian Mallows model can be extended to do exactly this, with a mixture approach. BayesMallows supports this through the optional argument n_clusters to compute_mallows.

Introduction

To illustrate how to perform clustering, let us create some clusters in the potato_visual dataset. This dataset has 12 assessors. We leave the first 6 assessors as is, but for the last 6, we revert the rankings.

potato_manipulated <- potato_visual
potato_manipulated[7:12, ] <- 21 - potato_manipulated[7:12, ]

Dataset `potato_manipulated`.
	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	12	5	2	4	11	6	16	1	18	13	10	20	19	15	3	14	17	7	9	8
A8	7	3	1	2	10	6	15	4	17	18	11	20	19	14	5	13	16	9	12	8
A9	13	5	3	2	9	8	15	1	16	18	14	20	17	19	4	11	12	6	7	10
A10	14	4	2	3	12	6	16	1	18	11	10	20	19	15	5	13	17	8	9	7
A11	9	5	2	6	8	3	14	1	18	16	10	20	19	15	4	11	17	7	13	12
A12	7	6	2	5	9	3	13	1	18	17	12	20	19	14	4	15	16	8	11	10

Let us try first without clustering.

model_fit <- compute_mallows(rankings = potato_manipulated)

Looking at the convergence of \(\alpha\), we see that this model does not settle on a consensus ranking.

assess_convergence(model_fit)

Let us instead introduce clustering:

model_fit <- compute_mallows(
  rankings = potato_manipulated, 
  n_clusters = 2
  )

Algorithm Tuning

Convergence of \(\alpha\)

We can assess convergence in the usual way. Let us start with alpha. The assess_convergence method now shows one trace per cluster.

assess_convergence(model_fit)

Convergence of \(\rho\)

We plot the potato that is most often ranked heaviest (P8), and the potato that is most often ranked lightest (P12).

assess_convergence(model_fit, type = "rho", items = c(8, 12))

Convergence of Cluster Probabilities

We also see that the probability of each cluster, \(\tau_{1}\) and \(\tau_{2}\), fluctuate around 0.5, which is just as expected, since we reverted half of the rankings.

assess_convergence(model_fit, type = "cluster_probs")

We now go on to show clustering in a real data example.

Sushi Data

The BayesMallows package comes with a set of sushi preference data, in which 5000 assessors each have ranked a set of 10 types of sushi (Kamishima (2003)). Here are the first few rows of the dataset.

sushi_rankings

First 6 rows of example dataset `sushi_rankings`.
shrimp	sea eel	tuna	squid	sea urchin	salmon roe	egg	fatty tuna	tuna roll	cucumber roll
2	8	10	3	4	1	5	9	7	6
1	8	6	4	10	9	3	5	7	2
2	8	3	4	6	7	10	1	5	9
4	7	5	6	1	2	8	3	9	10
4	10	7	5	9	3	2	8	1	6
4	6	2	10	7	5	1	9	8	3

With clustering, we can see if there are subsets of assessors with similar preferences. We set nmc = 1000 here, to speed up the vignette building. The argument include_wcd specifies whether to compute within-cluster distance during MCMC. When n_cluster > 1 it defaults to TRUE, otherwise to FALSE. Hence, in order to get the within-cluster distance for the one-cluster case, we set include_wcd = TRUE when calling compute_mallows:

model_fit1 <- compute_mallows(sushi_rankings, nmc = 1000, include_wcd = TRUE)
model_fit2 <- compute_mallows(sushi_rankings, n_clusters = 2, nmc = 1000)

Algorithm Tuning

It is useful to look at the trace plots of \(\alpha_{1}, \dots, \alpha_{C}\).

assess_convergence(model_fit2)

We can also look at the trace plot of the cluster probabilities, \(\tau_{1}\) and \(\tau_{2}\):

assess_convergence(model_fit2, type = "cluster_probs")

It seems like \(\alpha\) is mixing rapidly. Let us set burnin = 400, and compute the CP consensus for the case with two clusters.

model_fit1$burnin <- 400
model_fit2$burnin <- 400

Posterior Distributions

CP Consensus

We can find the CP consensus for each of the two clusters.

cp_consensus_sushi <- compute_cp_consensus(model_fit2)

We can now look at each cluster:

filter(cp_consensus_sushi, cluster == "Cluster 1")

cluster	ranking	item	cumprob
Cluster 1	1	fatty tuna	1
Cluster 1	2	tuna	1
Cluster 1	3	shrimp	1
Cluster 1	4	squid	1
Cluster 1	5	tuna roll	1
Cluster 1	6	sea eel	1
Cluster 1	7	egg	1
Cluster 1	8	salmon roe	1
Cluster 1	9	cucumber roll	1
Cluster 1	10	sea urchin	1

filter(cp_consensus_sushi, cluster == "Cluster 2")

cluster	ranking	item	cumprob
Cluster 2	1	fatty tuna	1
Cluster 2	2	sea urchin	1
Cluster 2	3	salmon roe	1
Cluster 2	4	sea eel	1
Cluster 2	5	tuna	1
Cluster 2	6	shrimp	1
Cluster 2	7	tuna roll	1
Cluster 2	8	squid	1
Cluster 2	9	egg	1
Cluster 2	10	cucumber roll	1

Determining the Number of Clusters

We can also compute an elbow plot, using plot_elbow. Before doing that, we should look at its arguments:

args(plot_elbow)
#> function (..., burnin = NULL) 
#> NULL

plot_elbow requires the model fits for different number of clusters to be provided as the first arguments, either comma separated, or as a list. The other argument is burnin, which must be named. If each model provided has its burnin element set, then this is taken as the default value of burnin.

In order to systematically investigate how many clusters to include, we can use the function compute_mallows_mixtures, which takes a vector that specifies the number of clusters, and calls compute_mallows once for each number.

We only need the within-cluster distances to compute the elbow plot. Thus, we can basically “thin out” all the other parameters, since we do not need them. This reduce the amount of memory used.

n_clusters <- seq(from = 1, to = 10)
nmc <- 2000
models <- compute_mallows_mixtures(n_clusters = n_clusters, 
                                   rankings = sushi_rankings,
                                   nmc = nmc, 
                                   cluster_assignment_thinning = nmc - 1,
                                   rho_thinning = nmc - 1,
                                   aug_thinning = nmc - 1
                                   )

Even though the result of compute_mallows_mixtures is a list of fitted models, we can specify burnin as an element in this case as well, and it is used by plot_elbow.

We can now provide the models list to plot_elbow.

plot_elbow(models, burnin = 1000)

Let us choose 5 clusters. We then compute this model, now without the extreme thinning.

model <- compute_mallows(rankings = sushi_rankings, nmc = nmc, n_clusters = 5)

plot(model, burnin = 1000, type = "cluster_assignment")

We can also the function assign_cluster to get the a posteriori cluster assignment for each assessor.

cluster_assignment <- assign_cluster(model, burnin = 1000, soft = FALSE)

The first few rows of `cluster_assignment`.
assessor	probability	map_cluster
1	0.861	Cluster 5
2	0.635	Cluster 5
3	0.661	Cluster 4
4	0.854	Cluster 1
5	0.601	Cluster 5
6	0.785	Cluster 5
7	0.590	Cluster 1
8	0.654	Cluster 4
9	0.568	Cluster 4
10	0.536	Cluster 5
11	0.655	Cluster 3
12	0.894	Cluster 2
13	0.706	Cluster 2
14	0.839	Cluster 4
15	0.671	Cluster 2
16	0.705	Cluster 2
17	0.696	Cluster 5
18	0.945	Cluster 2
19	0.588	Cluster 1
20	0.797	Cluster 1
21	0.926	Cluster 1
22	0.529	Cluster 2
23	0.728	Cluster 5
24	0.848	Cluster 1
25	0.874	Cluster 4
26	0.753	Cluster 1
27	0.937	Cluster 5
28	0.417	Cluster 1
29	0.601	Cluster 3
30	0.700	Cluster 2
31	0.834	Cluster 3
32	0.453	Cluster 4
33	0.710	Cluster 2
34	0.392	Cluster 3
35	0.761	Cluster 4
36	0.699	Cluster 2
37	0.409	Cluster 2
38	0.500	Cluster 3
39	0.822	Cluster 2
40	0.889	Cluster 4
41	0.939	Cluster 5
42	0.856	Cluster 4
43	0.333	Cluster 1
44	0.651	Cluster 1
45	0.533	Cluster 5
46	0.627	Cluster 4
47	0.883	Cluster 2
48	0.822	Cluster 4
49	0.818	Cluster 5
50	0.741	Cluster 3
51	0.482	Cluster 2
52	0.910	Cluster 5
53	0.594	Cluster 1
54	0.810	Cluster 4
55	0.856	Cluster 2
56	0.475	Cluster 2
57	0.669	Cluster 2
58	0.731	Cluster 5
59	0.466	Cluster 5
60	0.483	Cluster 3
61	0.840	Cluster 2
62	0.628	Cluster 4
63	0.847	Cluster 3
64	0.853	Cluster 4
65	0.949	Cluster 5
66	0.398	Cluster 4
67	0.708	Cluster 2
68	0.860	Cluster 2
69	0.586	Cluster 4
70	0.386	Cluster 2
71	0.450	Cluster 2
72	0.579	Cluster 4
73	0.937	Cluster 4
74	0.604	Cluster 5
75	0.669	Cluster 1
76	0.428	Cluster 2
77	0.676	Cluster 4
78	0.771	Cluster 4
79	0.952	Cluster 4
80	0.531	Cluster 2
81	0.839	Cluster 5
82	0.495	Cluster 3
83	0.757	Cluster 3
84	0.962	Cluster 4
85	0.870	Cluster 3
86	0.378	Cluster 3
87	0.895	Cluster 2
88	0.579	Cluster 2
89	0.451	Cluster 2
90	0.535	Cluster 1
91	0.496	Cluster 1
92	0.758	Cluster 3
93	0.566	Cluster 2
94	0.629	Cluster 3
95	0.886	Cluster 5
96	0.718	Cluster 1
97	0.534	Cluster 1
98	0.572	Cluster 2
99	0.415	Cluster 3
100	0.702	Cluster 4
101	0.953	Cluster 4
102	0.376	Cluster 1
103	0.716	Cluster 2
104	0.819	Cluster 5
105	0.790	Cluster 2
106	0.537	Cluster 1
107	0.757	Cluster 3
108	0.554	Cluster 2
109	0.426	Cluster 3
110	0.895	Cluster 2
111	0.887	Cluster 2
112	0.960	Cluster 4
113	0.480	Cluster 1
114	0.961	Cluster 4
115	0.916	Cluster 4
116	0.730	Cluster 2
117	0.592	Cluster 5
118	0.688	Cluster 1
119	0.572	Cluster 1
120	0.553	Cluster 2
121	0.800	Cluster 1
122	0.955	Cluster 5
123	0.342	Cluster 3
124	0.385	Cluster 3
125	0.543	Cluster 1
126	0.627	Cluster 1
127	0.755	Cluster 1
128	0.551	Cluster 3
129	0.671	Cluster 5
130	0.486	Cluster 2
131	0.708	Cluster 1
132	0.390	Cluster 2
133	0.823	Cluster 4
134	0.899	Cluster 2
135	0.819	Cluster 4
136	0.634	Cluster 3
137	0.880	Cluster 4
138	0.510	Cluster 3
139	0.946	Cluster 5
140	0.604	Cluster 5
141	0.639	Cluster 5
142	0.964	Cluster 5
143	0.883	Cluster 4
144	0.520	Cluster 4
145	0.918	Cluster 4
146	0.859	Cluster 1
147	0.564	Cluster 4
148	0.602	Cluster 5
149	0.323	Cluster 4
150	0.734	Cluster 1
151	0.846	Cluster 2
152	0.905	Cluster 2
153	0.790	Cluster 1
154	0.390	Cluster 3
155	0.590	Cluster 2
156	0.719	Cluster 4
157	0.845	Cluster 1
158	0.830	Cluster 4
159	0.517	Cluster 4
160	0.964	Cluster 2
161	0.546	Cluster 2
162	0.607	Cluster 3
163	0.722	Cluster 2
164	0.762	Cluster 4
165	0.741	Cluster 4
166	0.897	Cluster 4
167	0.937	Cluster 4
168	0.951	Cluster 2
169	0.884	Cluster 2
170	0.910	Cluster 3
171	0.789	Cluster 3
172	0.836	Cluster 2
173	0.744	Cluster 2
174	0.576	Cluster 1
175	0.842	Cluster 1
176	0.503	Cluster 5
177	0.682	Cluster 2
178	0.560	Cluster 4
179	0.428	Cluster 4
180	0.777	Cluster 3
181	0.883	Cluster 2
182	0.298	Cluster 1
183	0.585	Cluster 1
184	0.607	Cluster 4
185	0.794	Cluster 2
186	0.802	Cluster 2
187	0.331	Cluster 1
188	0.866	Cluster 4
189	0.433	Cluster 2
190	0.949	Cluster 5
191	0.519	Cluster 1
192	0.853	Cluster 1
193	0.843	Cluster 4
194	0.499	Cluster 3
195	0.545	Cluster 1
196	0.928	Cluster 4
197	0.880	Cluster 5
198	0.836	Cluster 3
199	0.831	Cluster 2
200	0.740	Cluster 2
201	0.647	Cluster 4
202	0.705	Cluster 4
203	0.816	Cluster 4
204	0.512	Cluster 2
205	0.539	Cluster 3
206	0.811	Cluster 3
207	0.392	Cluster 2
208	0.706	Cluster 2
209	0.768	Cluster 5
210	0.818	Cluster 4
211	0.743	Cluster 3
212	0.389	Cluster 2
213	0.712	Cluster 4
214	0.827	Cluster 2
215	0.943	Cluster 4
216	0.603	Cluster 4
217	0.502	Cluster 4
218	0.761	Cluster 2
219	0.908	Cluster 4
220	0.289	Cluster 4
221	0.748	Cluster 4
222	0.594	Cluster 3
223	0.922	Cluster 4
224	0.735	Cluster 2
225	0.808	Cluster 1
226	0.611	Cluster 3
227	0.650	Cluster 3
228	0.971	Cluster 5
229	0.522	Cluster 1
230	0.902	Cluster 4
231	0.715	Cluster 2
232	0.744	Cluster 3
233	0.944	Cluster 2
234	0.961	Cluster 4
235	0.922	Cluster 4
236	0.906	Cluster 4
237	0.617	Cluster 5
238	0.871	Cluster 1
239	0.922	Cluster 2
240	0.478	Cluster 3
241	0.944	Cluster 4
242	0.846	Cluster 4
243	0.430	Cluster 4
244	0.427	Cluster 4
245	0.526	Cluster 2
246	0.530	Cluster 3
247	0.736	Cluster 4
248	0.727	Cluster 2
249	0.750	Cluster 3
250	0.412	Cluster 2
251	0.752	Cluster 3
252	0.725	Cluster 2
253	0.484	Cluster 2
254	0.833	Cluster 4
255	0.513	Cluster 3
256	0.574	Cluster 1
257	0.455	Cluster 3
258	0.406	Cluster 3
259	0.550	Cluster 2
260	0.329	Cluster 3
261	0.953	Cluster 4
262	0.622	Cluster 2
263	0.629	Cluster 1
264	0.835	Cluster 1
265	0.764	Cluster 1
266	0.711	Cluster 1
267	0.703	Cluster 4
268	0.714	Cluster 2
269	0.749	Cluster 1
270	0.903	Cluster 4
271	0.369	Cluster 2
272	0.780	Cluster 4
273	0.848	Cluster 4
274	0.808	Cluster 4
275	0.862	Cluster 3
276	0.709	Cluster 3
277	0.408	Cluster 1
278	0.403	Cluster 2
279	0.878	Cluster 4
280	0.560	Cluster 1
281	0.446	Cluster 3
282	0.965	Cluster 4
283	0.895	Cluster 3
284	0.704	Cluster 1
285	0.493	Cluster 5
286	0.824	Cluster 2
287	0.881	Cluster 4
288	0.976	Cluster 4
289	0.678	Cluster 4
290	0.833	Cluster 3
291	0.799	Cluster 2
292	0.577	Cluster 4
293	0.598	Cluster 3
294	0.701	Cluster 5
295	0.358	Cluster 3
296	0.878	Cluster 4
297	0.815	Cluster 5
298	0.670	Cluster 5
299	0.558	Cluster 1
300	0.528	Cluster 3
301	0.649	Cluster 5
302	0.919	Cluster 3
303	0.541	Cluster 2
304	0.448	Cluster 4
305	0.355	Cluster 2
306	0.644	Cluster 2
307	0.506	Cluster 2
308	0.946	Cluster 5
309	0.875	Cluster 2
310	0.826	Cluster 4
311	0.400	Cluster 1
312	0.909	Cluster 1
313	0.601	Cluster 1
314	0.760	Cluster 4
315	0.544	Cluster 1
316	0.849	Cluster 4
317	0.898	Cluster 4
318	0.323	Cluster 4
319	0.930	Cluster 2
320	0.531	Cluster 3
321	0.468	Cluster 2
322	0.769	Cluster 4
323	0.824	Cluster 2
324	0.930	Cluster 5
325	0.414	Cluster 4
326	0.920	Cluster 2
327	0.951	Cluster 5
328	0.614	Cluster 4
329	0.882	Cluster 2
330	0.569	Cluster 3
331	0.916	Cluster 3
332	0.718	Cluster 4
333	0.739	Cluster 3
334	0.975	Cluster 4
335	0.391	Cluster 1
336	0.468	Cluster 1
337	0.573	Cluster 3
338	0.610	Cluster 3
339	0.532	Cluster 1
340	0.641	Cluster 5
341	0.692	Cluster 3
342	0.713	Cluster 1
343	0.772	Cluster 1
344	0.877	Cluster 3
345	0.530	Cluster 3
346	0.572	Cluster 1
347	0.755	Cluster 5
348	0.648	Cluster 2
349	0.867	Cluster 4
350	0.523	Cluster 5
351	0.702	Cluster 2
352	0.430	Cluster 2
353	0.721	Cluster 2
354	0.609	Cluster 4
355	0.764	Cluster 1
356	0.798	Cluster 2
357	0.777	Cluster 3
358	0.811	Cluster 4
359	0.743	Cluster 2
360	0.815	Cluster 4
361	0.552	Cluster 1
362	0.763	Cluster 1
363	0.883	Cluster 4
364	0.953	Cluster 5
365	0.855	Cluster 4
366	0.907	Cluster 2
367	0.975	Cluster 4
368	0.702	Cluster 2
369	0.796	Cluster 2
370	0.500	Cluster 3
371	0.400	Cluster 5
372	0.742	Cluster 5
373	0.683	Cluster 3
374	0.532	Cluster 4
375	0.554	Cluster 2
376	0.478	Cluster 2
377	0.517	Cluster 3
378	0.941	Cluster 2
379	0.768	Cluster 1
380	0.389	Cluster 4
381	0.779	Cluster 3
382	0.509	Cluster 3
383	0.734	Cluster 2
384	0.480	Cluster 1
385	0.885	Cluster 5
386	0.936	Cluster 4
387	0.761	Cluster 1
388	0.507	Cluster 4
389	0.930	Cluster 5
390	0.786	Cluster 2
391	0.918	Cluster 5
392	0.803	Cluster 2
393	0.819	Cluster 2
394	0.485	Cluster 3
395	0.647	Cluster 5
396	0.559	Cluster 2
397	0.909	Cluster 4
398	0.935	Cluster 5
399	0.847	Cluster 3
400	0.390	Cluster 2
401	0.364	Cluster 3
402	0.573	Cluster 2
403	0.661	Cluster 4
404	0.870	Cluster 4
405	0.668	Cluster 2
406	0.706	Cluster 2
407	0.312	Cluster 1
408	0.447	Cluster 4
409	0.890	Cluster 5
410	0.597	Cluster 2
411	0.297	Cluster 1
412	0.690	Cluster 1
413	0.334	Cluster 1
414	0.675	Cluster 3
415	0.761	Cluster 1
416	0.933	Cluster 4
417	0.931	Cluster 2
418	0.515	Cluster 1
419	0.811	Cluster 5
420	0.786	Cluster 4
421	0.566	Cluster 1
422	0.548	Cluster 4
423	0.737	Cluster 1
424	0.925	Cluster 4
425	0.806	Cluster 2
426	0.505	Cluster 2
427	0.935	Cluster 3
428	0.579	Cluster 3
429	0.875	Cluster 3
430	0.366	Cluster 3
431	0.554	Cluster 2
432	0.651	Cluster 3
433	0.602	Cluster 4
434	0.903	Cluster 1
435	0.532	Cluster 5
436	0.904	Cluster 2
437	0.893	Cluster 4
438	0.745	Cluster 2
439	0.827	Cluster 3
440	0.804	Cluster 1
441	0.688	Cluster 2
442	0.868	Cluster 1
443	0.828	Cluster 2
444	0.521	Cluster 2
445	0.502	Cluster 3
446	0.890	Cluster 2
447	0.497	Cluster 4
448	0.643	Cluster 2
449	0.725	Cluster 2
450	0.933	Cluster 4
451	0.526	Cluster 5
452	0.803	Cluster 4
453	0.870	Cluster 2
454	0.350	Cluster 1
455	0.721	Cluster 4
456	0.642	Cluster 3
457	0.918	Cluster 2
458	0.502	Cluster 3
459	0.607	Cluster 1
460	0.566	Cluster 1
461	0.518	Cluster 2
462	0.722	Cluster 2
463	0.709	Cluster 2
464	0.299	Cluster 1
465	0.900	Cluster 2
466	0.581	Cluster 1
467	0.717	Cluster 2
468	0.966	Cluster 4
469	0.658	Cluster 5
470	0.695	Cluster 2
471	0.604	Cluster 5
472	0.701	Cluster 2
473	0.927	Cluster 5
474	0.564	Cluster 2
475	0.973	Cluster 5
476	0.823	Cluster 2
477	0.303	Cluster 5
478	0.507	Cluster 2
479	0.615	Cluster 3
480	0.946	Cluster 4
481	0.528	Cluster 2
482	0.964	Cluster 2
483	0.752	Cluster 4
484	0.885	Cluster 2
485	0.926	Cluster 4
486	0.566	Cluster 1
487	0.921	Cluster 5
488	0.925	Cluster 4
489	0.596	Cluster 4
490	0.306	Cluster 1
491	0.630	Cluster 5
492	0.757	Cluster 2
493	0.948	Cluster 4
494	0.636	Cluster 4
495	0.876	Cluster 4
496	0.680	Cluster 2
497	0.765	Cluster 5
498	0.719	Cluster 1
499	0.973	Cluster 5
500	0.676	Cluster 2
501	0.779	Cluster 1
502	0.656	Cluster 2
503	0.390	Cluster 2
504	0.824	Cluster 4
505	0.756	Cluster 1
506	0.925	Cluster 4
507	0.692	Cluster 4
508	0.522	Cluster 4
509	0.962	Cluster 5
510	0.808	Cluster 4
511	0.556	Cluster 3
512	0.714	Cluster 1
513	0.708	Cluster 3
514	0.505	Cluster 2
515	0.797	Cluster 2
516	0.731	Cluster 2
517	0.745	Cluster 3
518	0.914	Cluster 4
519	0.894	Cluster 4
520	0.901	Cluster 2
521	0.489	Cluster 1
522	0.890	Cluster 4
523	0.795	Cluster 4
524	0.709	Cluster 1
525	0.769	Cluster 4
526	0.719	Cluster 1
527	0.872	Cluster 4
528	0.456	Cluster 5
529	0.713	Cluster 5
530	0.592	Cluster 1
531	0.812	Cluster 2
532	0.571	Cluster 1
533	0.587	Cluster 1
534	0.597	Cluster 3
535	0.928	Cluster 1
536	0.389	Cluster 1
537	0.845	Cluster 2
538	0.853	Cluster 3
539	0.822	Cluster 5
540	0.474	Cluster 3
541	0.932	Cluster 4
542	0.675	Cluster 2
543	0.773	Cluster 3
544	0.584	Cluster 1
545	0.653	Cluster 2
546	0.706	Cluster 3
547	0.849	Cluster 2
548	0.574	Cluster 2
549	0.900	Cluster 3
550	0.703	Cluster 2
551	0.540	Cluster 2
552	0.505	Cluster 4
553	0.877	Cluster 3
554	0.524	Cluster 3
555	0.677	Cluster 4
556	0.908	Cluster 2
557	0.463	Cluster 2
558	0.571	Cluster 5
559	0.875	Cluster 4
560	0.486	Cluster 2
561	0.634	Cluster 1
562	0.702	Cluster 2
563	0.945	Cluster 2
564	0.553	Cluster 4
565	0.485	Cluster 5
566	0.772	Cluster 2
567	0.887	Cluster 4
568	0.357	Cluster 3
569	0.650	Cluster 4
570	0.541	Cluster 2
571	0.776	Cluster 2
572	0.909	Cluster 2
573	0.295	Cluster 5
574	0.618	Cluster 3
575	0.688	Cluster 3
576	0.666	Cluster 4
577	0.938	Cluster 4
578	0.655	Cluster 3
579	0.427	Cluster 5
580	0.903	Cluster 2
581	0.832	Cluster 2
582	0.371	Cluster 2
583	0.536	Cluster 2
584	0.570	Cluster 1
585	0.802	Cluster 3
586	0.802	Cluster 3
587	0.829	Cluster 3
588	0.584	Cluster 1
589	0.543	Cluster 4
590	0.855	Cluster 5
591	0.574	Cluster 2
592	0.737	Cluster 4
593	0.575	Cluster 1
594	0.687	Cluster 2
595	0.344	Cluster 2
596	0.777	Cluster 3
597	0.701	Cluster 4
598	0.428	Cluster 2
599	0.912	Cluster 5
600	0.521	Cluster 1
601	0.672	Cluster 2
602	0.899	Cluster 4
603	0.530	Cluster 2
604	0.436	Cluster 1
605	0.967	Cluster 5
606	0.553	Cluster 2
607	0.775	Cluster 2
608	0.804	Cluster 4
609	0.473	Cluster 4
610	0.898	Cluster 2
611	0.936	Cluster 2
612	0.636	Cluster 3
613	0.335	Cluster 4
614	0.959	Cluster 4
615	0.824	Cluster 2
616	0.924	Cluster 3
617	0.682	Cluster 1
618	0.709	Cluster 2
619	0.699	Cluster 1
620	0.622	Cluster 4
621	0.605	Cluster 4
622	0.736	Cluster 1
623	0.520	Cluster 2
624	0.533	Cluster 4
625	0.625	Cluster 2
626	0.558	Cluster 4
627	0.957	Cluster 4
628	0.587	Cluster 4
629	0.502	Cluster 1
630	0.854	Cluster 4
631	0.892	Cluster 5
632	0.474	Cluster 4
633	0.666	Cluster 4
634	0.797	Cluster 2
635	0.872	Cluster 1
636	0.601	Cluster 1
637	0.567	Cluster 4
638	0.501	Cluster 4
639	0.477	Cluster 5
640	0.837	Cluster 2
641	0.932	Cluster 4
642	0.675	Cluster 3
643	0.892	Cluster 2
644	0.892	Cluster 4
645	0.879	Cluster 4
646	0.930	Cluster 2
647	0.476	Cluster 4
648	0.828	Cluster 1
649	0.373	Cluster 3
650	0.940	Cluster 4
651	0.801	Cluster 2
652	0.567	Cluster 1
653	0.528	Cluster 2
654	0.613	Cluster 4
655	0.678	Cluster 3
656	0.710	Cluster 5
657	0.970	Cluster 5
658	0.447	Cluster 2
659	0.653	Cluster 2
660	0.323	Cluster 3
661	0.365	Cluster 2
662	0.464	Cluster 1
663	0.624	Cluster 3
664	0.384	Cluster 2
665	0.899	Cluster 1
666	0.924	Cluster 5
667	0.765	Cluster 2
668	0.792	Cluster 4
669	0.509	Cluster 5
670	0.780	Cluster 3
671	0.563	Cluster 1
672	0.845	Cluster 4
673	0.728	Cluster 5
674	0.362	Cluster 1
675	0.619	Cluster 4
676	0.820	Cluster 4
677	0.530	Cluster 4
678	0.382	Cluster 3
679	0.838	Cluster 4
680	0.932	Cluster 2
681	0.974	Cluster 5
682	0.878	Cluster 4
683	0.684	Cluster 5
684	0.622	Cluster 2
685	0.726	Cluster 1
686	0.661	Cluster 2
687	0.592	Cluster 4
688	0.682	Cluster 1
689	0.606	Cluster 4
690	0.946	Cluster 5
691	0.923	Cluster 4
692	0.825	Cluster 3
693	0.505	Cluster 2
694	0.407	Cluster 3
695	0.817	Cluster 4
696	0.733	Cluster 2
697	0.834	Cluster 4
698	0.759	Cluster 3
699	0.962	Cluster 4
700	0.558	Cluster 2
701	0.930	Cluster 4
702	0.972	Cluster 3
703	0.700	Cluster 2
704	0.945	Cluster 2
705	0.513	Cluster 3
706	0.864	Cluster 4
707	0.540	Cluster 1
708	0.724	Cluster 4
709	0.832	Cluster 2
710	0.764	Cluster 5
711	0.815	Cluster 2
712	0.939	Cluster 5
713	0.478	Cluster 1
714	0.418	Cluster 3
715	0.933	Cluster 1
716	0.951	Cluster 4
717	0.600	Cluster 3
718	0.806	Cluster 3
719	0.568	Cluster 1
720	0.560	Cluster 2
721	0.700	Cluster 2
722	0.890	Cluster 2
723	0.501	Cluster 1
724	0.784	Cluster 4
725	0.567	Cluster 2
726	0.946	Cluster 4
727	0.517	Cluster 1
728	0.949	Cluster 4
729	0.798	Cluster 2
730	0.422	Cluster 2
731	0.494	Cluster 2
732	0.795	Cluster 4
733	0.889	Cluster 4
734	0.944	Cluster 5
735	0.527	Cluster 3
736	0.814	Cluster 2
737	0.706	Cluster 5
738	0.475	Cluster 4
739	0.691	Cluster 1
740	0.664	Cluster 1
741	0.962	Cluster 2
742	0.740	Cluster 2
743	0.405	Cluster 1
744	0.507	Cluster 4
745	0.660	Cluster 2
746	0.824	Cluster 1
747	0.899	Cluster 3
748	0.817	Cluster 2
749	0.530	Cluster 3
750	0.574	Cluster 4
751	0.505	Cluster 4
752	0.727	Cluster 2
753	0.865	Cluster 1
754	0.834	Cluster 5
755	0.708	Cluster 2
756	0.680	Cluster 4
757	0.951	Cluster 5
758	0.950	Cluster 4
759	0.930	Cluster 5
760	0.714	Cluster 1
761	0.442	Cluster 1
762	0.910	Cluster 2
763	0.969	Cluster 4
764	0.702	Cluster 1
765	0.437	Cluster 1
766	0.576	Cluster 1
767	0.799	Cluster 4
768	0.870	Cluster 4
769	0.641	Cluster 5
770	0.847	Cluster 4
771	0.927	Cluster 1
772	0.483	Cluster 2
773	0.873	Cluster 4
774	0.587	Cluster 3
775	0.340	Cluster 1
776	0.634	Cluster 4
777	0.288	Cluster 3
778	0.835	Cluster 2
779	0.395	Cluster 5
780	0.654	Cluster 3
781	0.782	Cluster 5
782	0.684	Cluster 2
783	0.952	Cluster 3
784	0.927	Cluster 4
785	0.780	Cluster 5
786	0.537	Cluster 1
787	0.593	Cluster 5
788	0.399	Cluster 4
789	0.862	Cluster 1
790	0.797	Cluster 1
791	0.327	Cluster 3
792	0.428	Cluster 1
793	0.793	Cluster 2
794	0.781	Cluster 3
795	0.409	Cluster 2
796	0.703	Cluster 4
797	0.658	Cluster 2
798	0.989	Cluster 5
799	0.686	Cluster 2
800	0.647	Cluster 2
801	0.727	Cluster 2
802	0.488	Cluster 3
803	0.411	Cluster 4
804	0.584	Cluster 2
805	0.774	Cluster 1
806	0.892	Cluster 4
807	0.828	Cluster 1
808	0.789	Cluster 5
809	0.539	Cluster 1
810	0.470	Cluster 5
811	0.992	Cluster 5
812	0.543	Cluster 2
813	0.710	Cluster 4
814	0.512	Cluster 3
815	0.807	Cluster 4
816	0.685	Cluster 2
817	0.680	Cluster 4
818	0.794	Cluster 2
819	0.412	Cluster 1
820	0.854	Cluster 5
821	0.800	Cluster 2
822	0.435	Cluster 1
823	0.705	Cluster 5
824	0.711	Cluster 2
825	0.499	Cluster 5
826	0.972	Cluster 5
827	0.562	Cluster 2
828	0.839	Cluster 3
829	0.660	Cluster 2
830	0.934	Cluster 2
831	0.695	Cluster 4
832	0.725	Cluster 3
833	0.673	Cluster 2
834	0.525	Cluster 2
835	0.938	Cluster 4
836	0.481	Cluster 2
837	0.547	Cluster 3
838	0.896	Cluster 4
839	0.488	Cluster 4
840	0.604	Cluster 5
841	0.910	Cluster 5
842	0.716	Cluster 2
843	0.776	Cluster 4
844	0.935	Cluster 4
845	0.773	Cluster 4
846	0.800	Cluster 2
847	0.805	Cluster 4
848	0.499	Cluster 2
849	0.494	Cluster 3
850	0.558	Cluster 2
851	0.356	Cluster 3
852	0.566	Cluster 2
853	0.661	Cluster 3
854	0.797	Cluster 3
855	0.707	Cluster 2
856	0.811	Cluster 2
857	0.584	Cluster 2
858	0.848	Cluster 4
859	0.706	Cluster 4
860	0.894	Cluster 5
861	0.942	Cluster 4
862	0.967	Cluster 5
863	0.578	Cluster 2
864	0.730	Cluster 5
865	0.716	Cluster 2
866	0.921	Cluster 4
867	0.847	Cluster 3
868	0.680	Cluster 5
869	0.556	Cluster 3
870	0.846	Cluster 2
871	0.483	Cluster 5
872	0.851	Cluster 1
873	0.622	Cluster 4
874	0.677	Cluster 1
875	0.474	Cluster 5
876	0.873	Cluster 1
877	0.857	Cluster 3
878	0.838	Cluster 2
879	0.972	Cluster 4
880	0.887	Cluster 2
881	0.918	Cluster 3
882	0.870	Cluster 2
883	0.458	Cluster 4
884	0.947	Cluster 4
885	0.762	Cluster 3
886	0.790	Cluster 4
887	0.957	Cluster 4
888	0.885	Cluster 3
889	0.822	Cluster 4
890	0.876	Cluster 4
891	0.768	Cluster 3
892	0.919	Cluster 4
893	0.940	Cluster 4
894	0.651	Cluster 5
895	0.953	Cluster 3
896	0.593	Cluster 2
897	0.809	Cluster 5
898	0.607	Cluster 2
899	0.671	Cluster 1
900	0.949	Cluster 1
901	0.837	Cluster 4
902	0.470	Cluster 2
903	0.695	Cluster 5
904	0.578	Cluster 1
905	0.797	Cluster 4
906	0.329	Cluster 3
907	0.494	Cluster 1
908	0.925	Cluster 4
909	0.530	Cluster 1
910	0.771	Cluster 3
911	0.571	Cluster 1
912	0.542	Cluster 2
913	0.739	Cluster 2
914	0.894	Cluster 2
915	0.559	Cluster 1
916	0.859	Cluster 2
917	0.927	Cluster 5
918	0.491	Cluster 3
919	0.816	Cluster 2
920	0.940	Cluster 2
921	0.782	Cluster 3
922	0.488	Cluster 2
923	0.735	Cluster 3
924	0.675	Cluster 2
925	0.421	Cluster 3
926	0.684	Cluster 1
927	0.871	Cluster 5
928	0.756	Cluster 1
929	0.748	Cluster 2
930	0.537	Cluster 2
931	0.372	Cluster 2
932	0.851	Cluster 3
933	0.735	Cluster 3
934	0.551	Cluster 3
935	0.620	Cluster 4
936	0.899	Cluster 3
937	0.698	Cluster 2
938	0.458	Cluster 4
939	0.476	Cluster 3
940	0.873	Cluster 2
941	0.602	Cluster 1
942	0.780	Cluster 4
943	0.735	Cluster 2
944	0.570	Cluster 2
945	0.618	Cluster 2
946	0.569	Cluster 2
947	0.581	Cluster 1
948	0.910	Cluster 3
949	0.831	Cluster 4
950	0.737	Cluster 1
951	0.831	Cluster 3
952	0.407	Cluster 3
953	0.759	Cluster 1
954	0.899	Cluster 2
955	0.918	Cluster 5
956	0.826	Cluster 2
957	0.972	Cluster 5
958	0.970	Cluster 5
959	0.584	Cluster 5
960	0.902	Cluster 2
961	0.782	Cluster 3
962	0.548	Cluster 2
963	0.783	Cluster 3
964	0.681	Cluster 1
965	0.654	Cluster 4
966	0.395	Cluster 2
967	0.842	Cluster 3
968	0.561	Cluster 1
969	0.727	Cluster 2
970	0.854	Cluster 4
971	0.924	Cluster 1
972	0.835	Cluster 2
973	0.689	Cluster 1
974	0.515	Cluster 3
975	0.480	Cluster 2
976	0.684	Cluster 2
977	0.791	Cluster 1
978	0.450	Cluster 4
979	0.356	Cluster 1
980	0.849	Cluster 2
981	0.823	Cluster 2
982	0.698	Cluster 4
983	0.497	Cluster 3
984	0.681	Cluster 2
985	0.556	Cluster 2
986	0.747	Cluster 5
987	0.679	Cluster 2
988	0.985	Cluster 5
989	0.701	Cluster 3
990	0.900	Cluster 4
991	0.954	Cluster 4
992	0.917	Cluster 4
993	0.963	Cluster 4
994	0.608	Cluster 4
995	0.602	Cluster 2
996	0.887	Cluster 5
997	0.568	Cluster 2
998	0.557	Cluster 2
999	0.949	Cluster 3
1000	0.567	Cluster 2
1001	0.715	Cluster 4
1002	0.594	Cluster 4
1003	0.846	Cluster 3
1004	0.700	Cluster 1
1005	0.637	Cluster 2
1006	0.931	Cluster 3
1007	0.387	Cluster 3
1008	0.413	Cluster 2
1009	0.790	Cluster 3
1010	0.522	Cluster 2
1011	0.517	Cluster 1
1012	0.947	Cluster 3
1013	0.581	Cluster 1
1014	0.817	Cluster 4
1015	0.439	Cluster 5
1016	0.690	Cluster 5
1017	0.861	Cluster 4
1018	0.566	Cluster 2
1019	0.814	Cluster 2
1020	0.666	Cluster 5
1021	0.703	Cluster 3
1022	0.968	Cluster 4
1023	0.538	Cluster 4
1024	0.826	Cluster 5
1025	0.964	Cluster 4
1026	0.482	Cluster 5
1027	0.242	Cluster 4
1028	0.943	Cluster 2
1029	0.704	Cluster 1
1030	0.518	Cluster 2
1031	0.578	Cluster 2
1032	0.767	Cluster 1
1033	0.839	Cluster 2
1034	0.701	Cluster 1
1035	0.875	Cluster 4
1036	0.962	Cluster 5
1037	0.706	Cluster 1
1038	0.856	Cluster 5
1039	0.612	Cluster 1
1040	0.515	Cluster 3
1041	0.547	Cluster 2
1042	0.695	Cluster 2
1043	0.744	Cluster 5
1044	0.463	Cluster 5
1045	0.686	Cluster 4
1046	0.443	Cluster 3
1047	0.529	Cluster 4
1048	0.417	Cluster 3
1049	0.564	Cluster 1
1050	0.538	Cluster 2
1051	0.652	Cluster 2
1052	0.799	Cluster 4
1053	0.891	Cluster 2
1054	0.880	Cluster 5
1055	0.852	Cluster 1
1056	0.985	Cluster 5
1057	0.921	Cluster 3
1058	0.486	Cluster 1
1059	0.779	Cluster 5
1060	0.530	Cluster 5
1061	0.694	Cluster 2
1062	0.534	Cluster 4
1063	0.652	Cluster 4
1064	0.919	Cluster 4
1065	0.819	Cluster 2
1066	0.896	Cluster 4
1067	0.879	Cluster 4
1068	0.556	Cluster 2
1069	0.520	Cluster 3
1070	0.580	Cluster 3
1071	0.625	Cluster 1
1072	0.953	Cluster 2
1073	0.909	Cluster 4
1074	0.694	Cluster 4
1075	0.640	Cluster 2
1076	0.613	Cluster 2
1077	0.907	Cluster 3
1078	0.545	Cluster 1
1079	0.541	Cluster 1
1080	0.756	Cluster 2
1081	0.587	Cluster 4
1082	0.774	Cluster 3
1083	0.827	Cluster 2
1084	0.891	Cluster 4
1085	0.409	Cluster 3
1086	0.398	Cluster 4
1087	0.499	Cluster 3
1088	0.970	Cluster 2
1089	0.517	Cluster 2
1090	0.774	Cluster 3
1091	0.709	Cluster 1
1092	0.422	Cluster 5
1093	0.506	Cluster 1
1094	0.855	Cluster 4
1095	0.495	Cluster 2
1096	0.649	Cluster 5
1097	0.535	Cluster 4
1098	0.571	Cluster 1
1099	0.395	Cluster 3
1100	0.954	Cluster 5
1101	0.653	Cluster 2
1102	0.862	Cluster 1
1103	0.733	Cluster 2
1104	0.583	Cluster 3
1105	0.909	Cluster 2
1106	0.986	Cluster 5
1107	0.450	Cluster 4
1108	0.492	Cluster 2
1109	0.500	Cluster 4
1110	0.921	Cluster 4
1111	0.804	Cluster 5
1112	0.515	Cluster 3
1113	0.797	Cluster 4
1114	0.608	Cluster 1
1115	0.444	Cluster 2
1116	0.464	Cluster 4
1117	0.982	Cluster 5
1118	0.713	Cluster 4
1119	0.543	Cluster 2
1120	0.510	Cluster 2
1121	0.706	Cluster 1
1122	0.805	Cluster 4
1123	0.996	Cluster 5
1124	0.744	Cluster 3
1125	0.458	Cluster 1
1126	0.933	Cluster 4
1127	0.500	Cluster 2
1128	0.788	Cluster 3
1129	0.506	Cluster 2
1130	0.493	Cluster 3
1131	0.940	Cluster 5
1132	0.583	Cluster 2
1133	0.476	Cluster 1
1134	0.945	Cluster 2
1135	0.565	Cluster 3
1136	0.733	Cluster 4
1137	0.730	Cluster 4
1138	0.377	Cluster 2
1139	0.540	Cluster 1
1140	0.845	Cluster 2
1141	0.382	Cluster 2
1142	0.841	Cluster 2
1143	0.583	Cluster 2
1144	0.686	Cluster 2
1145	0.453	Cluster 2
1146	0.367	Cluster 2
1147	0.793	Cluster 4
1148	0.834	Cluster 2
1149	0.902	Cluster 5
1150	0.878	Cluster 2
1151	0.802	Cluster 4
1152	0.839	Cluster 1
1153	0.880	Cluster 3
1154	0.793	Cluster 3
1155	0.852	Cluster 2
1156	0.738	Cluster 4
1157	0.488	Cluster 2
1158	0.719	Cluster 3
1159	0.483	Cluster 3
1160	0.775	Cluster 3
1161	0.802	Cluster 2
1162	0.901	Cluster 2
1163	0.672	Cluster 4
1164	0.727	Cluster 2
1165	0.898	Cluster 5
1166	0.714	Cluster 2
1167	0.653	Cluster 4
1168	0.820	Cluster 3
1169	0.829	Cluster 4
1170	0.447	Cluster 3
1171	0.385	Cluster 4
1172	0.728	Cluster 4
1173	0.828	Cluster 2
1174	0.917	Cluster 2
1175	0.684	Cluster 5
1176	0.464	Cluster 5
1177	0.685	Cluster 2
1178	0.443	Cluster 5
1179	0.885	Cluster 3
1180	0.495	Cluster 5
1181	0.686	Cluster 2
1182	0.537	Cluster 2
1183	0.511	Cluster 1
1184	0.634	Cluster 4
1185	0.907	Cluster 4
1186	0.700	Cluster 2
1187	0.859	Cluster 4
1188	0.883	Cluster 5
1189	0.960	Cluster 2
1190	0.484	Cluster 2
1191	0.966	Cluster 4
1192	0.587	Cluster 4
1193	0.833	Cluster 1
1194	0.565	Cluster 2
1195	0.731	Cluster 2
1196	0.792	Cluster 1
1197	0.462	Cluster 3
1198	0.678	Cluster 1
1199	0.888	Cluster 3
1200	0.711	Cluster 2
1201	0.900	Cluster 3
1202	0.500	Cluster 4
1203	0.652	Cluster 1
1204	0.442	Cluster 2
1205	0.628	Cluster 1
1206	0.786	Cluster 5
1207	0.697	Cluster 2
1208	0.410	Cluster 2
1209	0.920	Cluster 4
1210	0.637	Cluster 1
1211	0.867	Cluster 4
1212	0.624	Cluster 5
1213	0.796	Cluster 4
1214	0.837	Cluster 2
1215	0.719	Cluster 2
1216	0.483	Cluster 2
1217	0.535	Cluster 1
1218	0.847	Cluster 3
1219	0.937	Cluster 4
1220	0.808	Cluster 5
1221	0.867	Cluster 1
1222	0.530	Cluster 4
1223	0.810	Cluster 2
1224	0.724	Cluster 1
1225	0.533	Cluster 5
1226	0.870	Cluster 1
1227	0.863	Cluster 5
1228	0.921	Cluster 4
1229	0.734	Cluster 2
1230	0.511	Cluster 1
1231	0.692	Cluster 2
1232	0.415	Cluster 1
1233	0.867	Cluster 5
1234	0.922	Cluster 1
1235	0.704	Cluster 5
1236	0.659	Cluster 3
1237	0.623	Cluster 2
1238	0.748	Cluster 1
1239	0.861	Cluster 4
1240	0.547	Cluster 1
1241	0.430	Cluster 2
1242	0.868	Cluster 4
1243	0.728	Cluster 2
1244	0.602	Cluster 1
1245	0.527	Cluster 4
1246	0.774	Cluster 2
1247	0.414	Cluster 3
1248	0.894	Cluster 2
1249	0.498	Cluster 2
1250	0.936	Cluster 4
1251	0.727	Cluster 3
1252	0.813	Cluster 2
1253	0.872	Cluster 1
1254	0.546	Cluster 4
1255	0.690	Cluster 2
1256	0.946	Cluster 5
1257	0.925	Cluster 4
1258	0.444	Cluster 1
1259	0.353	Cluster 2
1260	0.489	Cluster 3
1261	0.711	Cluster 3
1262	0.318	Cluster 5
1263	0.741	Cluster 1
1264	0.533	Cluster 2
1265	0.391	Cluster 1
1266	0.667	Cluster 2
1267	0.547	Cluster 3
1268	0.703	Cluster 2
1269	0.440	Cluster 3
1270	0.506	Cluster 1
1271	0.779	Cluster 3
1272	0.577	Cluster 2
1273	0.914	Cluster 5
1274	0.450	Cluster 5
1275	0.808	Cluster 2
1276	0.509	Cluster 3
1277	0.714	Cluster 4
1278	0.720	Cluster 2
1279	0.972	Cluster 2
1280	0.815	Cluster 4
1281	0.937	Cluster 3
1282	0.548	Cluster 2
1283	0.526	Cluster 1
1284	0.387	Cluster 3
1285	0.565	Cluster 2
1286	0.881	Cluster 4
1287	0.410	Cluster 3
1288	0.561	Cluster 1
1289	0.602	Cluster 2
1290	0.794	Cluster 4
1291	0.757	Cluster 1
1292	0.818	Cluster 2
1293	0.903	Cluster 2
1294	0.924	Cluster 5
1295	0.807	Cluster 4
1296	0.859	Cluster 1
1297	0.836	Cluster 4
1298	0.417	Cluster 4
1299	0.933	Cluster 1
1300	0.744	Cluster 3
1301	0.820	Cluster 2
1302	0.643	Cluster 3
1303	0.838	Cluster 1
1304	0.599	Cluster 2
1305	0.522	Cluster 2
1306	0.672	Cluster 4
1307	0.482	Cluster 1
1308	0.419	Cluster 5
1309	0.527	Cluster 1
1310	0.860	Cluster 4
1311	0.454	Cluster 1
1312	0.503	Cluster 2
1313	0.826	Cluster 5
1314	0.850	Cluster 2
1315	0.890	Cluster 3
1316	0.703	Cluster 2
1317	0.829	Cluster 2
1318	0.388	Cluster 2
1319	0.658	Cluster 2
1320	0.862	Cluster 3
1321	0.955	Cluster 4
1322	0.689	Cluster 3
1323	0.793	Cluster 2
1324	0.901	Cluster 4
1325	0.908	Cluster 1
1326	0.930	Cluster 5
1327	0.614	Cluster 4
1328	0.817	Cluster 2
1329	0.341	Cluster 1
1330	0.732	Cluster 2
1331	0.551	Cluster 4
1332	0.890	Cluster 2
1333	0.593	Cluster 1
1334	0.916	Cluster 4
1335	0.575	Cluster 2
1336	0.748	Cluster 1
1337	0.938	Cluster 4
1338	0.721	Cluster 2
1339	0.790	Cluster 3
1340	0.315	Cluster 1
1341	0.908	Cluster 3
1342	0.564	Cluster 4
1343	0.698	Cluster 1
1344	0.384	Cluster 1
1345	0.513	Cluster 1
1346	0.898	Cluster 2
1347	0.934	Cluster 5
1348	0.321	Cluster 5
1349	0.579	Cluster 1
1350	0.838	Cluster 5
1351	0.800	Cluster 3
1352	0.379	Cluster 4
1353	0.747	Cluster 3
1354	0.529	Cluster 2
1355	0.712	Cluster 5
1356	0.946	Cluster 2
1357	0.632	Cluster 1
1358	0.947	Cluster 4
1359	0.429	Cluster 5
1360	0.522	Cluster 3
1361	0.806	Cluster 2
1362	0.830	Cluster 5
1363	0.451	Cluster 2
1364	0.750	Cluster 2
1365	0.839	Cluster 1
1366	0.678	Cluster 4
1367	0.851	Cluster 4
1368	0.715	Cluster 2
1369	0.922	Cluster 4
1370	0.657	Cluster 3
1371	0.623	Cluster 3
1372	0.727	Cluster 3
1373	0.708	Cluster 2
1374	0.579	Cluster 1
1375	0.466	Cluster 3
1376	0.866	Cluster 1
1377	0.364	Cluster 3
1378	0.748	Cluster 1
1379	0.956	Cluster 5
1380	0.735	Cluster 1
1381	0.833	Cluster 5
1382	0.399	Cluster 3
1383	0.804	Cluster 2
1384	0.424	Cluster 4
1385	0.571	Cluster 1
1386	0.963	Cluster 3
1387	0.761	Cluster 1
1388	0.754	Cluster 1
1389	0.779	Cluster 2
1390	0.598	Cluster 2
1391	0.712	Cluster 2
1392	0.837	Cluster 3
1393	0.832	Cluster 2
1394	0.471	Cluster 1
1395	0.939	Cluster 4
1396	0.841	Cluster 1
1397	0.900	Cluster 1
1398	0.390	Cluster 3
1399	0.591	Cluster 2
1400	0.748	Cluster 4
1401	0.463	Cluster 1
1402	0.779	Cluster 1
1403	0.801	Cluster 5
1404	0.852	Cluster 3
1405	0.738	Cluster 1
1406	0.515	Cluster 5
1407	0.378	Cluster 2
1408	0.541	Cluster 2
1409	0.710	Cluster 5
1410	0.795	Cluster 5
1411	0.882	Cluster 4
1412	0.353	Cluster 3
1413	0.344	Cluster 1
1414	0.836	Cluster 2
1415	0.464	Cluster 4
1416	0.518	Cluster 2
1417	0.386	Cluster 1
1418	0.546	Cluster 5
1419	0.728	Cluster 2
1420	0.618	Cluster 1
1421	0.895	Cluster 3
1422	0.879	Cluster 2
1423	0.421	Cluster 2
1424	0.400	Cluster 3
1425	0.509	Cluster 4
1426	0.591	Cluster 5
1427	0.933	Cluster 4
1428	0.831	Cluster 2
1429	0.942	Cluster 4
1430	0.891	Cluster 3
1431	0.970	Cluster 2
1432	0.376	Cluster 4
1433	0.338	Cluster 3
1434	0.418	Cluster 3
1435	0.734	Cluster 2
1436	0.466	Cluster 2
1437	0.600	Cluster 3
1438	0.831	Cluster 2
1439	0.441	Cluster 2
1440	0.572	Cluster 5
1441	0.923	Cluster 2
1442	0.814	Cluster 5
1443	0.780	Cluster 4
1444	0.725	Cluster 5
1445	0.543	Cluster 1
1446	0.548	Cluster 2
1447	0.632	Cluster 3
1448	0.672	Cluster 5
1449	0.367	Cluster 5
1450	0.521	Cluster 1
1451	0.590	Cluster 2
1452	0.553	Cluster 4
1453	0.894	Cluster 4
1454	0.823	Cluster 3
1455	0.616	Cluster 1
1456	0.562	Cluster 1
1457	0.868	Cluster 4
1458	0.961	Cluster 4
1459	0.959	Cluster 4
1460	0.410	Cluster 2
1461	0.879	Cluster 5
1462	0.725	Cluster 2
1463	0.890	Cluster 5
1464	0.508	Cluster 1
1465	0.902	Cluster 5
1466	0.563	Cluster 1
1467	0.334	Cluster 2
1468	0.742	Cluster 5
1469	0.726	Cluster 4
1470	0.848	Cluster 5
1471	0.567	Cluster 2
1472	0.606	Cluster 1
1473	0.716	Cluster 3
1474	0.583	Cluster 1
1475	0.328	Cluster 3
1476	0.936	Cluster 3
1477	0.526	Cluster 3
1478	0.401	Cluster 2
1479	0.677	Cluster 1
1480	0.603	Cluster 2
1481	0.741	Cluster 4
1482	0.934	Cluster 4
1483	0.602	Cluster 1
1484	0.718	Cluster 1
1485	0.852	Cluster 4
1486	0.736	Cluster 4
1487	0.351	Cluster 5
1488	0.958	Cluster 1
1489	0.899	Cluster 1
1490	0.476	Cluster 4
1491	0.662	Cluster 2
1492	0.852	Cluster 2
1493	0.472	Cluster 5
1494	0.400	Cluster 1
1495	0.465	Cluster 3
1496	0.747	Cluster 1
1497	0.896	Cluster 3
1498	0.853	Cluster 3
1499	0.766	Cluster 1
1500	0.665	Cluster 4
1501	0.627	Cluster 5
1502	0.502	Cluster 2
1503	0.495	Cluster 4
1504	0.616	Cluster 2
1505	0.454	Cluster 5
1506	0.541	Cluster 4
1507	0.619	Cluster 4
1508	0.720	Cluster 4
1509	0.761	Cluster 5
1510	0.821	Cluster 2
1511	0.253	Cluster 3
1512	0.579	Cluster 3
1513	0.748	Cluster 2
1514	0.589	Cluster 1
1515	0.890	Cluster 2
1516	0.630	Cluster 3
1517	0.378	Cluster 2
1518	0.474	Cluster 5
1519	0.610	Cluster 3
1520	0.273	Cluster 3
1521	0.666	Cluster 4
1522	0.767	Cluster 2
1523	0.622	Cluster 1
1524	0.380	Cluster 2
1525	0.665	Cluster 3
1526	0.749	Cluster 3
1527	0.767	Cluster 1
1528	0.952	Cluster 2
1529	0.869	Cluster 4
1530	0.986	Cluster 5
1531	0.466	Cluster 2
1532	0.815	Cluster 2
1533	0.634	Cluster 1
1534	0.806	Cluster 4
1535	0.360	Cluster 1
1536	0.485	Cluster 2
1537	0.841	Cluster 1
1538	0.691	Cluster 3
1539	0.617	Cluster 2
1540	0.794	Cluster 4
1541	0.885	Cluster 2
1542	0.726	Cluster 4
1543	0.861	Cluster 4
1544	0.510	Cluster 1
1545	0.839	Cluster 4
1546	0.890	Cluster 4
1547	0.767	Cluster 5
1548	0.909	Cluster 3
1549	0.816	Cluster 4
1550	0.752	Cluster 5
1551	0.584	Cluster 3
1552	0.767	Cluster 4
1553	0.869	Cluster 4
1554	0.889	Cluster 2
1555	0.656	Cluster 3
1556	0.695	Cluster 2
1557	0.962	Cluster 4
1558	0.660	Cluster 3
1559	0.846	Cluster 1
1560	0.820	Cluster 2
1561	0.502	Cluster 3
1562	0.421	Cluster 2
1563	0.946	Cluster 2
1564	0.933	Cluster 4
1565	0.770	Cluster 4
1566	0.538	Cluster 1
1567	0.886	Cluster 2
1568	0.734	Cluster 2
1569	0.957	Cluster 4
1570	0.503	Cluster 1
1571	0.750	Cluster 5
1572	0.794	Cluster 2
1573	0.670	Cluster 5
1574	0.926	Cluster 4
1575	0.381	Cluster 3
1576	0.427	Cluster 5
1577	0.606	Cluster 4
1578	0.607	Cluster 5
1579	0.933	Cluster 4
1580	0.532	Cluster 1
1581	0.875	Cluster 4
1582	0.749	Cluster 5
1583	0.560	Cluster 1
1584	0.773	Cluster 3
1585	0.477	Cluster 5
1586	0.906	Cluster 1
1587	0.787	Cluster 2
1588	0.434	Cluster 4
1589	0.920	Cluster 4
1590	0.439	Cluster 1
1591	0.485	Cluster 3
1592	0.616	Cluster 1
1593	0.723	Cluster 2
1594	0.514	Cluster 4
1595	0.826	Cluster 2
1596	0.886	Cluster 1
1597	0.509	Cluster 4
1598	0.704	Cluster 2
1599	0.906	Cluster 1
1600	0.591	Cluster 4
1601	0.993	Cluster 5
1602	0.899	Cluster 5
1603	0.901	Cluster 4
1604	0.832	Cluster 2
1605	0.625	Cluster 3
1606	0.700	Cluster 2
1607	0.445	Cluster 1
1608	0.724	Cluster 4
1609	0.609	Cluster 1
1610	0.753	Cluster 1
1611	0.748	Cluster 1
1612	0.823	Cluster 2
1613	0.658	Cluster 4
1614	0.438	Cluster 2
1615	0.708	Cluster 1
1616	0.949	Cluster 4
1617	0.394	Cluster 3
1618	0.557	Cluster 2
1619	0.510	Cluster 2
1620	0.785	Cluster 3
1621	0.835	Cluster 1
1622	0.859	Cluster 4
1623	0.499	Cluster 5
1624	0.905	Cluster 4
1625	0.738	Cluster 2
1626	0.598	Cluster 1
1627	0.496	Cluster 4
1628	0.451	Cluster 4
1629	0.665	Cluster 4
1630	0.583	Cluster 1
1631	0.959	Cluster 3
1632	0.833	Cluster 5
1633	0.526	Cluster 2
1634	0.597	Cluster 2
1635	0.712	Cluster 5
1636	0.638	Cluster 1
1637	0.840	Cluster 1
1638	0.718	Cluster 2
1639	0.640	Cluster 4
1640	0.409	Cluster 4
1641	0.651	Cluster 3
1642	0.834	Cluster 5
1643	0.629	Cluster 4
1644	0.952	Cluster 5
1645	0.711	Cluster 5
1646	0.546	Cluster 2
1647	0.634	Cluster 3
1648	0.548	Cluster 3
1649	0.994	Cluster 5
1650	0.765	Cluster 2
1651	0.536	Cluster 1
1652	0.433	Cluster 1
1653	0.809	Cluster 5
1654	0.666	Cluster 2
1655	0.386	Cluster 3
1656	0.912	Cluster 4
1657	0.833	Cluster 1
1658	0.408	Cluster 3
1659	0.543	Cluster 5
1660	0.650	Cluster 2
1661	0.536	Cluster 3
1662	0.708	Cluster 2
1663	0.655	Cluster 3
1664	0.587	Cluster 2
1665	0.712	Cluster 4
1666	0.980	Cluster 4
1667	0.806	Cluster 1
1668	0.570	Cluster 1
1669	0.678	Cluster 1
1670	0.935	Cluster 4
1671	0.414	Cluster 3
1672	0.947	Cluster 2
1673	0.867	Cluster 4
1674	0.954	Cluster 4
1675	0.405	Cluster 3
1676	0.651	Cluster 4
1677	0.573	Cluster 1
1678	0.538	Cluster 1
1679	0.415	Cluster 3
1680	0.497	Cluster 3
1681	0.328	Cluster 3
1682	0.602	Cluster 5
1683	0.853	Cluster 3
1684	0.632	Cluster 1
1685	0.732	Cluster 1
1686	0.400	Cluster 3
1687	0.385	Cluster 2
1688	0.814	Cluster 2
1689	0.790	Cluster 5
1690	0.821	Cluster 2
1691	0.457	Cluster 3
1692	0.941	Cluster 4
1693	0.413	Cluster 5
1694	0.659	Cluster 1
1695	0.467	Cluster 3
1696	0.895	Cluster 3
1697	0.693	Cluster 5
1698	0.863	Cluster 3
1699	0.520	Cluster 1
1700	0.494	Cluster 3
1701	0.561	Cluster 1
1702	0.646	Cluster 1
1703	0.781	Cluster 3
1704	0.578	Cluster 2
1705	0.482	Cluster 3
1706	0.563	Cluster 1
1707	0.577	Cluster 4
1708	0.901	Cluster 4
1709	0.977	Cluster 4
1710	0.635	Cluster 5
1711	0.629	Cluster 2
1712	0.602	Cluster 1
1713	0.864	Cluster 2
1714	0.728	Cluster 2
1715	0.814	Cluster 1
1716	0.407	Cluster 2
1717	0.566	Cluster 4
1718	0.481	Cluster 5
1719	0.984	Cluster 4
1720	0.889	Cluster 1
1721	0.974	Cluster 5
1722	0.470	Cluster 1
1723	0.825	Cluster 2
1724	0.554	Cluster 2
1725	0.905	Cluster 2
1726	0.757	Cluster 4
1727	0.447	Cluster 3
1728	0.441	Cluster 3
1729	0.897	Cluster 2
1730	0.885	Cluster 3
1731	0.705	Cluster 2
1732	0.452	Cluster 3
1733	0.907	Cluster 2
1734	0.557	Cluster 2
1735	0.420	Cluster 3
1736	0.425	Cluster 5
1737	0.775	Cluster 5
1738	0.665	Cluster 4
1739	0.771	Cluster 3
1740	0.649	Cluster 3
1741	0.439	Cluster 2
1742	0.744	Cluster 1
1743	0.583	Cluster 1
1744	0.957	Cluster 2
1745	0.610	Cluster 1
1746	0.916	Cluster 5
1747	0.562	Cluster 4
1748	0.559	Cluster 1
1749	0.233	Cluster 3
1750	0.543	Cluster 2
1751	0.567	Cluster 5
1752	0.862	Cluster 4
1753	0.533	Cluster 3
1754	0.895	Cluster 1
1755	0.795	Cluster 4
1756	0.674	Cluster 3
1757	0.889	Cluster 4
1758	0.913	Cluster 3
1759	0.825	Cluster 3
1760	0.769	Cluster 2
1761	0.700	Cluster 5
1762	0.450	Cluster 1
1763	0.416	Cluster 1
1764	0.962	Cluster 4
1765	0.663	Cluster 2
1766	0.641	Cluster 4
1767	0.543	Cluster 1
1768	0.618	Cluster 2
1769	0.488	Cluster 5
1770	0.826	Cluster 2
1771	0.482	Cluster 1
1772	0.674	Cluster 4
1773	0.557	Cluster 2
1774	0.899	Cluster 3
1775	0.706	Cluster 4
1776	0.819	Cluster 2
1777	0.619	Cluster 3
1778	0.614	Cluster 1
1779	0.708	Cluster 2
1780	0.481	Cluster 1
1781	0.954	Cluster 2
1782	0.718	Cluster 2
1783	0.934	Cluster 4
1784	0.600	Cluster 2
1785	0.552	Cluster 2
1786	0.753	Cluster 2
1787	0.588	Cluster 2
1788	0.686	Cluster 2
1789	0.486	Cluster 1
1790	0.455	Cluster 3
1791	0.553	Cluster 5
1792	0.732	Cluster 1
1793	0.801	Cluster 2
1794	0.707	Cluster 2
1795	0.649	Cluster 3
1796	0.569	Cluster 2
1797	0.665	Cluster 3
1798	0.598	Cluster 4
1799	0.818	Cluster 3
1800	0.919	Cluster 3
1801	0.883	Cluster 2
1802	0.703	Cluster 1
1803	0.571	Cluster 5
1804	0.698	Cluster 2
1805	0.410	Cluster 4
1806	0.854	Cluster 4
1807	0.840	Cluster 1
1808	0.820	Cluster 2
1809	0.972	Cluster 2
1810	0.602	Cluster 4
1811	0.554	Cluster 3
1812	0.503	Cluster 2
1813	0.940	Cluster 5
1814	0.545	Cluster 3
1815	0.827	Cluster 2
1816	0.930	Cluster 4
1817	0.280	Cluster 3
1818	0.662	Cluster 4
1819	0.351	Cluster 2
1820	0.711	Cluster 4
1821	0.868	Cluster 3
1822	0.795	Cluster 4
1823	0.786	Cluster 3
1824	0.744	Cluster 4
1825	0.885	Cluster 4
1826	0.712	Cluster 4
1827	0.754	Cluster 4
1828	0.465	Cluster 1
1829	0.439	Cluster 1
1830	0.480	Cluster 4
1831	0.953	Cluster 5
1832	0.849	Cluster 1
1833	0.845	Cluster 2
1834	0.634	Cluster 2
1835	0.723	Cluster 4
1836	0.711	Cluster 2
1837	0.556	Cluster 2
1838	0.712	Cluster 3
1839	0.901	Cluster 2
1840	0.374	Cluster 4
1841	0.620	Cluster 3
1842	0.591	Cluster 4
1843	0.878	Cluster 1
1844	0.803	Cluster 2
1845	0.523	Cluster 1
1846	0.646	Cluster 5
1847	0.806	Cluster 4
1848	0.835	Cluster 5
1849	0.402	Cluster 4
1850	0.841	Cluster 3
1851	0.986	Cluster 5
1852	0.364	Cluster 2
1853	0.529	Cluster 5
1854	0.922	Cluster 2
1855	0.528	Cluster 5
1856	0.637	Cluster 4
1857	0.701	Cluster 3
1858	0.931	Cluster 4
1859	0.594	Cluster 2
1860	0.758	Cluster 2
1861	0.623	Cluster 4
1862	0.858	Cluster 1
1863	0.816	Cluster 3
1864	0.804	Cluster 2
1865	0.935	Cluster 4
1866	0.565	Cluster 2
1867	0.820	Cluster 2
1868	0.594	Cluster 4
1869	0.666	Cluster 3
1870	0.485	Cluster 3
1871	0.342	Cluster 4
1872	0.823	Cluster 2
1873	0.510	Cluster 5
1874	0.880	Cluster 5
1875	0.941	Cluster 4
1876	0.446	Cluster 4
1877	0.589	Cluster 2
1878	0.488	Cluster 1
1879	0.729	Cluster 1
1880	0.601	Cluster 5
1881	0.509	Cluster 1
1882	0.782	Cluster 1
1883	0.655	Cluster 4
1884	0.562	Cluster 3
1885	0.575	Cluster 5
1886	0.827	Cluster 2
1887	0.896	Cluster 4
1888	0.482	Cluster 1
1889	0.894	Cluster 5
1890	0.825	Cluster 4
1891	0.833	Cluster 2
1892	0.494	Cluster 1
1893	0.472	Cluster 2
1894	0.696	Cluster 4
1895	0.532	Cluster 2
1896	0.707	Cluster 5
1897	0.719	Cluster 3
1898	0.613	Cluster 2
1899	0.476	Cluster 3
1900	0.363	Cluster 4
1901	0.674	Cluster 2
1902	0.508	Cluster 3
1903	0.736	Cluster 5
1904	0.512	Cluster 1
1905	0.726	Cluster 3
1906	0.695	Cluster 2
1907	0.819	Cluster 2
1908	0.457	Cluster 4
1909	0.846	Cluster 3
1910	0.872	Cluster 5
1911	0.879	Cluster 4
1912	0.729	Cluster 3
1913	0.889	Cluster 4
1914	0.863	Cluster 2
1915	0.629	Cluster 2
1916	0.826	Cluster 4
1917	0.325	Cluster 1
1918	0.888	Cluster 4
1919	0.524	Cluster 1
1920	0.892	Cluster 4
1921	0.657	Cluster 2
1922	0.536	Cluster 3
1923	0.414	Cluster 2
1924	0.825	Cluster 2
1925	0.976	Cluster 5
1926	0.772	Cluster 1
1927	0.463	Cluster 3
1928	0.591	Cluster 2
1929	0.964	Cluster 2
1930	0.513	Cluster 2
1931	0.971	Cluster 4
1932	0.693	Cluster 1
1933	0.825	Cluster 4
1934	0.709	Cluster 4
1935	0.638	Cluster 2
1936	0.588	Cluster 2
1937	0.800	Cluster 4
1938	0.971	Cluster 2
1939	0.599	Cluster 2
1940	0.912	Cluster 4
1941	0.831	Cluster 3
1942	0.894	Cluster 1
1943	0.494	Cluster 4
1944	0.801	Cluster 3
1945	0.833	Cluster 2
1946	0.363	Cluster 4
1947	0.652	Cluster 2
1948	0.819	Cluster 2
1949	0.492	Cluster 1
1950	0.814	Cluster 4
1951	0.541	Cluster 1
1952	0.621	Cluster 4
1953	0.406	Cluster 5
1954	0.694	Cluster 4
1955	0.612	Cluster 3
1956	0.810	Cluster 4
1957	0.831	Cluster 1
1958	0.901	Cluster 2
1959	0.756	Cluster 4
1960	0.938	Cluster 5
1961	0.738	Cluster 4
1962	0.570	Cluster 3
1963	0.423	Cluster 2
1964	0.912	Cluster 3
1965	0.791	Cluster 4
1966	0.638	Cluster 5
1967	0.360	Cluster 3
1968	0.785	Cluster 4
1969	0.960	Cluster 4
1970	0.594	Cluster 5
1971	0.685	Cluster 2
1972	0.826	Cluster 2
1973	0.974	Cluster 5
1974	0.569	Cluster 2
1975	0.823	Cluster 2
1976	0.517	Cluster 5
1977	0.813	Cluster 1
1978	0.808	Cluster 2
1979	0.867	Cluster 5
1980	0.916	Cluster 4
1981	0.436	Cluster 3
1982	0.690	Cluster 2
1983	0.406	Cluster 1
1984	0.637	Cluster 3
1985	0.750	Cluster 2
1986	0.807	Cluster 4
1987	0.697	Cluster 1
1988	0.954	Cluster 5
1989	0.964	Cluster 4
1990	0.461	Cluster 1
1991	0.583	Cluster 5
1992	0.928	Cluster 5
1993	0.733	Cluster 1
1994	0.700	Cluster 2
1995	0.845	Cluster 4
1996	0.525	Cluster 3
1997	0.480	Cluster 2
1998	0.922	Cluster 1
1999	0.662	Cluster 5
2000	0.819	Cluster 1
2001	0.896	Cluster 5
2002	0.381	Cluster 3
2003	0.868	Cluster 2
2004	0.614	Cluster 1
2005	0.924	Cluster 4
2006	0.817	Cluster 2
2007	0.348	Cluster 3
2008	0.382	Cluster 3
2009	0.991	Cluster 5
2010	0.478	Cluster 4
2011	0.648	Cluster 5
2012	0.932	Cluster 4
2013	0.556	Cluster 4
2014	0.782	Cluster 1
2015	0.719	Cluster 5
2016	0.717	Cluster 5
2017	0.551	Cluster 3
2018	0.630	Cluster 4
2019	0.557	Cluster 4
2020	0.540	Cluster 1
2021	0.392	Cluster 4
2022	0.721	Cluster 4
2023	0.966	Cluster 4
2024	0.421	Cluster 3
2025	0.701	Cluster 2
2026	0.645	Cluster 2
2027	0.698	Cluster 2
2028	0.454	Cluster 4
2029	0.817	Cluster 2
2030	0.737	Cluster 2
2031	0.972	Cluster 4
2032	0.551	Cluster 4
2033	0.502	Cluster 2
2034	0.822	Cluster 4
2035	0.457	Cluster 4
2036	0.812	Cluster 2
2037	0.545	Cluster 3
2038	0.774	Cluster 4
2039	0.870	Cluster 2
2040	0.673	Cluster 1
2041	0.796	Cluster 3
2042	0.756	Cluster 5
2043	0.741	Cluster 4
2044	0.774	Cluster 1
2045	0.712	Cluster 2
2046	0.969	Cluster 4
2047	0.444	Cluster 2
2048	0.827	Cluster 3
2049	0.505	Cluster 4
2050	0.480	Cluster 2
2051	0.365	Cluster 2
2052	0.526	Cluster 4
2053	0.798	Cluster 1
2054	0.547	Cluster 4
2055	0.900	Cluster 5
2056	0.570	Cluster 2
2057	0.866	Cluster 1
2058	0.613	Cluster 2
2059	0.528	Cluster 2
2060	0.920	Cluster 4
2061	0.774	Cluster 5
2062	0.586	Cluster 4
2063	0.686	Cluster 2
2064	0.974	Cluster 4
2065	0.470	Cluster 1
2066	0.544	Cluster 5
2067	0.755	Cluster 3
2068	0.838	Cluster 4
2069	0.389	Cluster 2
2070	0.792	Cluster 1
2071	0.817	Cluster 2
2072	0.433	Cluster 5
2073	0.809	Cluster 2
2074	0.590	Cluster 5
2075	0.760	Cluster 4
2076	0.589	Cluster 1
2077	0.571	Cluster 1
2078	0.730	Cluster 2
2079	0.622	Cluster 1
2080	0.777	Cluster 2
2081	0.831	Cluster 2
2082	0.531	Cluster 2
2083	0.469	Cluster 5
2084	0.966	Cluster 4
2085	0.914	Cluster 1
2086	0.819	Cluster 4
2087	0.564	Cluster 2
2088	0.775	Cluster 2
2089	0.354	Cluster 1
2090	0.396	Cluster 5
2091	0.577	Cluster 4
2092	0.907	Cluster 3
2093	0.796	Cluster 2
2094	0.909	Cluster 1
2095	0.882	Cluster 4
2096	0.497	Cluster 5
2097	0.601	Cluster 4
2098	0.668	Cluster 1
2099	0.710	Cluster 4
2100	0.703	Cluster 2
2101	0.532	Cluster 1
2102	0.491	Cluster 3
2103	0.866	Cluster 5
2104	0.386	Cluster 3
2105	0.755	Cluster 4
2106	0.502	Cluster 4
2107	0.711	Cluster 1
2108	0.763	Cluster 2
2109	0.496	Cluster 5
2110	0.683	Cluster 4
2111	0.647	Cluster 5
2112	0.408	Cluster 5
2113	0.720	Cluster 2
2114	0.971	Cluster 4
2115	0.625	Cluster 1
2116	0.578	Cluster 4
2117	0.819	Cluster 2
2118	0.683	Cluster 2
2119	0.679	Cluster 2
2120	0.812	Cluster 2
2121	0.611	Cluster 2
2122	0.709	Cluster 2
2123	0.899	Cluster 2
2124	0.978	Cluster 4
2125	0.591	Cluster 3
2126	0.548	Cluster 2
2127	0.841	Cluster 5
2128	0.519	Cluster 2
2129	0.527	Cluster 5
2130	0.816	Cluster 2
2131	0.903	Cluster 2
2132	0.909	Cluster 4
2133	0.678	Cluster 2
2134	0.721	Cluster 5
2135	0.744	Cluster 5
2136	0.812	Cluster 2
2137	0.541	Cluster 3
2138	0.583	Cluster 2
2139	0.642	Cluster 1
2140	0.604	Cluster 1
2141	0.495	Cluster 4
2142	0.692	Cluster 5
2143	0.931	Cluster 5
2144	0.560	Cluster 2
2145	0.990	Cluster 5
2146	0.985	Cluster 5
2147	0.742	Cluster 2
2148	0.575	Cluster 1
2149	0.561	Cluster 5
2150	0.610	Cluster 2
2151	0.576	Cluster 2
2152	0.868	Cluster 1
2153	0.729	Cluster 4
2154	0.916	Cluster 4
2155	0.998	Cluster 5
2156	0.663	Cluster 4
2157	0.439	Cluster 4
2158	0.488	Cluster 1
2159	0.598	Cluster 4
2160	0.578	Cluster 3
2161	0.796	Cluster 3
2162	0.553	Cluster 2
2163	0.673	Cluster 2
2164	0.906	Cluster 4
2165	0.884	Cluster 4
2166	0.338	Cluster 1
2167	0.970	Cluster 4
2168	0.739	Cluster 1
2169	0.806	Cluster 3
2170	0.643	Cluster 1
2171	0.612	Cluster 3
2172	0.930	Cluster 5
2173	0.450	Cluster 1
2174	0.795	Cluster 4
2175	0.747	Cluster 2
2176	0.967	Cluster 5
2177	0.964	Cluster 5
2178	0.630	Cluster 3
2179	0.711	Cluster 2
2180	0.440	Cluster 2
2181	0.635	Cluster 4
2182	0.879	Cluster 2
2183	0.945	Cluster 2
2184	0.612	Cluster 3
2185	0.896	Cluster 4
2186	0.530	Cluster 2
2187	0.966	Cluster 4
2188	0.786	Cluster 1
2189	0.762	Cluster 4
2190	0.944	Cluster 2
2191	0.589	Cluster 1
2192	0.476	Cluster 2
2193	0.544	Cluster 1
2194	0.485	Cluster 2
2195	0.560	Cluster 3
2196	0.827	Cluster 2
2197	0.824	Cluster 5
2198	0.766	Cluster 5
2199	0.573	Cluster 2
2200	0.631	Cluster 2
2201	0.917	Cluster 1
2202	0.686	Cluster 3
2203	0.944	Cluster 4
2204	0.498	Cluster 4
2205	0.794	Cluster 2
2206	0.872	Cluster 4
2207	0.585	Cluster 2
2208	0.955	Cluster 4
2209	0.963	Cluster 5
2210	0.570	Cluster 5
2211	0.728	Cluster 1
2212	0.613	Cluster 4
2213	0.633	Cluster 4
2214	0.597	Cluster 1
2215	0.948	Cluster 2
2216	0.880	Cluster 2
2217	0.649	Cluster 1
2218	0.466	Cluster 2
2219	0.949	Cluster 4
2220	0.882	Cluster 2
2221	0.664	Cluster 3
2222	0.841	Cluster 4
2223	0.454	Cluster 2
2224	0.575	Cluster 1
2225	0.542	Cluster 3
2226	0.360	Cluster 1
2227	0.892	Cluster 4
2228	0.884	Cluster 1
2229	0.811	Cluster 3
2230	0.728	Cluster 4
2231	0.680	Cluster 4
2232	0.942	Cluster 5
2233	0.833	Cluster 3
2234	0.619	Cluster 1
2235	0.344	Cluster 5
2236	0.745	Cluster 2
2237	0.868	Cluster 1
2238	0.309	Cluster 1
2239	0.970	Cluster 1
2240	0.575	Cluster 2
2241	0.461	Cluster 3
2242	0.836	Cluster 1
2243	0.739	Cluster 1
2244	0.413	Cluster 5
2245	0.664	Cluster 1
2246	0.963	Cluster 4
2247	0.808	Cluster 4
2248	0.494	Cluster 3
2249	0.533	Cluster 3
2250	0.864	Cluster 4
2251	0.492	Cluster 5
2252	0.904	Cluster 5
2253	0.782	Cluster 3
2254	0.364	Cluster 4
2255	0.849	Cluster 1
2256	0.384	Cluster 2
2257	0.522	Cluster 2
2258	0.618	Cluster 5
2259	0.537	Cluster 2
2260	0.594	Cluster 1
2261	0.467	Cluster 1
2262	0.891	Cluster 4
2263	0.671	Cluster 2
2264	0.510	Cluster 1
2265	0.523	Cluster 1
2266	0.696	Cluster 2
2267	0.888	Cluster 4
2268	0.855	Cluster 4
2269	0.891	Cluster 2
2270	0.379	Cluster 2
2271	0.636	Cluster 5
2272	0.887	Cluster 4
2273	0.588	Cluster 2
2274	0.652	Cluster 4
2275	0.929	Cluster 5
2276	0.682	Cluster 2
2277	0.854	Cluster 4
2278	0.974	Cluster 5
2279	0.678	Cluster 2
2280	0.549	Cluster 1
2281	0.741	Cluster 4
2282	0.424	Cluster 2
2283	0.659	Cluster 2
2284	0.738	Cluster 1
2285	0.885	Cluster 4
2286	0.817	Cluster 5
2287	0.968	Cluster 4
2288	0.897	Cluster 4
2289	0.886	Cluster 3
2290	0.639	Cluster 5
2291	0.833	Cluster 1
2292	0.902	Cluster 2
2293	0.851	Cluster 1
2294	0.707	Cluster 3
2295	0.678	Cluster 1
2296	0.574	Cluster 4
2297	0.961	Cluster 4
2298	0.536	Cluster 4
2299	0.689	Cluster 2
2300	0.541	Cluster 2
2301	0.538	Cluster 2
2302	0.472	Cluster 2
2303	0.851	Cluster 2
2304	0.372	Cluster 3
2305	0.588	Cluster 2
2306	0.823	Cluster 1
2307	0.914	Cluster 1
2308	0.691	Cluster 3
2309	0.905	Cluster 2
2310	0.548	Cluster 2
2311	0.869	Cluster 1
2312	0.533	Cluster 4
2313	0.504	Cluster 2
2314	0.822	Cluster 2
2315	0.719	Cluster 4
2316	0.698	Cluster 2
2317	0.334	Cluster 2
2318	0.481	Cluster 2
2319	0.821	Cluster 2
2320	0.552	Cluster 4
2321	0.467	Cluster 4
2322	0.824	Cluster 4
2323	0.560	Cluster 1
2324	0.457	Cluster 2
2325	0.742	Cluster 3
2326	0.763	Cluster 4
2327	0.961	Cluster 4
2328	0.733	Cluster 2
2329	0.598	Cluster 4
2330	0.780	Cluster 2
2331	0.743	Cluster 4
2332	0.768	Cluster 1
2333	0.518	Cluster 1
2334	0.545	Cluster 2
2335	0.750	Cluster 4
2336	0.587	Cluster 1
2337	0.973	Cluster 2
2338	0.869	Cluster 4
2339	0.958	Cluster 4
2340	0.884	Cluster 2
2341	0.500	Cluster 2
2342	0.863	Cluster 4
2343	0.438	Cluster 4
2344	0.397	Cluster 2
2345	0.937	Cluster 4
2346	0.645	Cluster 2
2347	0.631	Cluster 3
2348	0.626	Cluster 2
2349	0.743	Cluster 4
2350	0.713	Cluster 2
2351	0.515	Cluster 2
2352	0.412	Cluster 3
2353	0.575	Cluster 2
2354	0.461	Cluster 1
2355	0.702	Cluster 2
2356	0.500	Cluster 2
2357	0.391	Cluster 2
2358	0.538	Cluster 4
2359	0.696	Cluster 2
2360	0.908	Cluster 2
2361	0.790	Cluster 2
2362	0.601	Cluster 2
2363	0.335	Cluster 3
2364	0.550	Cluster 1
2365	0.842	Cluster 2
2366	0.401	Cluster 1
2367	0.539	Cluster 4
2368	0.943	Cluster 4
2369	0.757	Cluster 1
2370	0.923	Cluster 4
2371	0.820	Cluster 4
2372	0.814	Cluster 2
2373	0.561	Cluster 1
2374	0.778	Cluster 4
2375	0.791	Cluster 5
2376	0.915	Cluster 4
2377	0.578	Cluster 2
2378	0.590	Cluster 1
2379	0.745	Cluster 2
2380	0.579	Cluster 3
2381	0.389	Cluster 2
2382	0.532	Cluster 2
2383	0.713	Cluster 4
2384	0.397	Cluster 3
2385	0.816	Cluster 2
2386	0.595	Cluster 1
2387	0.901	Cluster 4
2388	0.881	Cluster 4
2389	0.741	Cluster 2
2390	0.581	Cluster 1
2391	0.774	Cluster 1
2392	0.558	Cluster 4
2393	0.472	Cluster 2
2394	0.751	Cluster 2
2395	0.433	Cluster 1
2396	0.642	Cluster 5
2397	0.426	Cluster 5
2398	0.707	Cluster 2
2399	0.490	Cluster 2
2400	0.831	Cluster 5
2401	0.550	Cluster 1
2402	0.930	Cluster 2
2403	0.602	Cluster 1
2404	0.983	Cluster 5
2405	0.527	Cluster 2
2406	0.942	Cluster 4
2407	0.849	Cluster 5
2408	0.624	Cluster 1
2409	0.639	Cluster 4
2410	0.462	Cluster 2
2411	0.806	Cluster 2
2412	0.851	Cluster 4
2413	0.512	Cluster 1
2414	0.567	Cluster 4
2415	0.401	Cluster 1
2416	0.716	Cluster 2
2417	0.692	Cluster 4
2418	0.852	Cluster 3
2419	0.809	Cluster 5
2420	0.882	Cluster 2
2421	0.513	Cluster 1
2422	0.544	Cluster 2
2423	0.497	Cluster 2
2424	0.825	Cluster 2
2425	0.562	Cluster 2
2426	0.817	Cluster 2
2427	0.819	Cluster 2
2428	0.757	Cluster 2
2429	0.363	Cluster 2
2430	0.514	Cluster 3
2431	0.525	Cluster 2
2432	0.665	Cluster 2
2433	0.518	Cluster 1
2434	0.397	Cluster 3
2435	0.835	Cluster 2
2436	0.546	Cluster 1
2437	0.419	Cluster 3
2438	0.582	Cluster 1
2439	0.920	Cluster 5
2440	0.602	Cluster 1
2441	0.869	Cluster 4
2442	0.959	Cluster 5
2443	0.949	Cluster 4
2444	0.450	Cluster 1
2445	0.478	Cluster 2
2446	0.738	Cluster 1
2447	0.928	Cluster 4
2448	0.843	Cluster 1
2449	0.849	Cluster 3
2450	0.677	Cluster 1
2451	0.594	Cluster 1
2452	0.710	Cluster 4
2453	0.718	Cluster 1
2454	0.381	Cluster 1
2455	0.839	Cluster 4
2456	0.678	Cluster 2
2457	0.897	Cluster 4
2458	0.729	Cluster 2
2459	0.890	Cluster 3
2460	0.683	Cluster 3
2461	0.834	Cluster 2
2462	0.932	Cluster 5
2463	0.941	Cluster 5
2464	0.309	Cluster 5
2465	0.437	Cluster 3
2466	0.778	Cluster 2
2467	0.705	Cluster 4
2468	0.830	Cluster 2
2469	0.557	Cluster 2
2470	0.736	Cluster 2
2471	0.889	Cluster 5
2472	0.583	Cluster 4
2473	0.883	Cluster 4
2474	0.840	Cluster 2
2475	0.515	Cluster 2
2476	0.665	Cluster 4
2477	0.828	Cluster 2
2478	0.967	Cluster 4
2479	0.540	Cluster 4
2480	0.438	Cluster 3
2481	0.501	Cluster 1
2482	0.846	Cluster 2
2483	0.798	Cluster 1
2484	0.698	Cluster 2
2485	0.603	Cluster 3
2486	0.696	Cluster 2
2487	0.793	Cluster 4
2488	0.947	Cluster 4
2489	0.491	Cluster 3
2490	0.890	Cluster 4
2491	0.370	Cluster 1
2492	0.825	Cluster 2
2493	0.394	Cluster 2
2494	0.921	Cluster 4
2495	0.942	Cluster 5
2496	0.703	Cluster 3
2497	0.757	Cluster 3
2498	0.942	Cluster 4
2499	0.616	Cluster 4
2500	0.652	Cluster 4
2501	0.897	Cluster 5
2502	0.590	Cluster 1
2503	0.835	Cluster 1
2504	0.763	Cluster 4
2505	0.864	Cluster 4
2506	0.642	Cluster 1
2507	0.919	Cluster 4
2508	0.753	Cluster 1
2509	0.780	Cluster 1
2510	0.700	Cluster 2
2511	0.622	Cluster 1
2512	0.478	Cluster 1
2513	0.436	Cluster 4
2514	0.646	Cluster 4
2515	0.894	Cluster 2
2516	0.831	Cluster 4
2517	0.521	Cluster 2
2518	0.413	Cluster 3
2519	0.637	Cluster 4
2520	0.569	Cluster 1
2521	0.496	Cluster 5
2522	0.734	Cluster 3
2523	0.509	Cluster 1
2524	0.530	Cluster 3
2525	0.812	Cluster 1
2526	0.802	Cluster 3
2527	0.801	Cluster 4
2528	0.862	Cluster 3
2529	0.885	Cluster 4
2530	0.938	Cluster 2
2531	0.941	Cluster 4
2532	0.432	Cluster 5
2533	0.953	Cluster 2
2534	0.922	Cluster 4
2535	0.769	Cluster 3
2536	0.919	Cluster 5
2537	0.628	Cluster 4
2538	0.873	Cluster 4
2539	0.712	Cluster 2
2540	0.719	Cluster 2
2541	0.923	Cluster 5
2542	0.684	Cluster 3
2543	0.839	Cluster 1
2544	0.514	Cluster 5
2545	0.815	Cluster 5
2546	0.809	Cluster 4
2547	0.506	Cluster 5
2548	0.695	Cluster 5
2549	0.421	Cluster 2
2550	0.570	Cluster 4
2551	0.824	Cluster 2
2552	0.715	Cluster 2
2553	0.651	Cluster 3
2554	0.446	Cluster 2
2555	0.340	Cluster 4
2556	0.598	Cluster 2
2557	0.736	Cluster 1
2558	0.455	Cluster 2
2559	0.922	Cluster 4
2560	0.762	Cluster 4
2561	0.808	Cluster 2
2562	0.795	Cluster 2
2563	0.620	Cluster 2
2564	0.508	Cluster 4
2565	0.420	Cluster 2
2566	0.872	Cluster 4
2567	0.750	Cluster 1
2568	0.924	Cluster 1
2569	0.943	Cluster 1
2570	0.845	Cluster 1
2571	0.822	Cluster 4
2572	0.967	Cluster 4
2573	0.386	Cluster 2
2574	0.907	Cluster 4
2575	0.836	Cluster 2
2576	0.574	Cluster 4
2577	0.831	Cluster 3
2578	0.955	Cluster 4
2579	0.918	Cluster 1
2580	0.557	Cluster 2
2581	0.941	Cluster 2
2582	0.569	Cluster 2
2583	0.362	Cluster 2
2584	0.468	Cluster 1
2585	0.868	Cluster 1
2586	0.782	Cluster 5
2587	0.566	Cluster 1
2588	0.598	Cluster 4
2589	0.962	Cluster 5
2590	0.771	Cluster 2
2591	0.436	Cluster 5
2592	0.877	Cluster 4
2593	0.741	Cluster 1
2594	0.955	Cluster 5
2595	0.353	Cluster 5
2596	0.814	Cluster 4
2597	0.680	Cluster 2
2598	0.547	Cluster 4
2599	0.552	Cluster 2
2600	0.429	Cluster 3
2601	0.314	Cluster 1
2602	0.930	Cluster 5
2603	0.806	Cluster 4
2604	0.742	Cluster 4
2605	0.928	Cluster 5
2606	0.838	Cluster 2
2607	0.629	Cluster 2
2608	0.565	Cluster 3
2609	0.915	Cluster 4
2610	0.569	Cluster 3
2611	0.790	Cluster 5
2612	0.310	Cluster 2
2613	0.433	Cluster 5
2614	0.540	Cluster 3
2615	0.904	Cluster 4
2616	0.619	Cluster 3
2617	0.529	Cluster 4
2618	0.528	Cluster 2
2619	0.470	Cluster 1
2620	0.849	Cluster 4
2621	0.651	Cluster 1
2622	0.588	Cluster 4
2623	0.946	Cluster 3
2624	0.579	Cluster 2
2625	0.430	Cluster 2
2626	0.432	Cluster 2
2627	0.984	Cluster 5
2628	0.427	Cluster 1
2629	0.543	Cluster 2
2630	0.428	Cluster 4
2631	0.980	Cluster 4
2632	0.597	Cluster 2
2633	0.523	Cluster 3
2634	0.890	Cluster 2
2635	0.688	Cluster 2
2636	0.556	Cluster 5
2637	0.451	Cluster 1
2638	0.533	Cluster 2
2639	0.729	Cluster 2
2640	0.538	Cluster 2
2641	0.515	Cluster 1
2642	0.436	Cluster 1
2643	0.790	Cluster 2
2644	0.573	Cluster 1
2645	0.415	Cluster 4
2646	0.743	Cluster 2
2647	0.659	Cluster 2
2648	0.425	Cluster 2
2649	0.945	Cluster 4
2650	0.637	Cluster 5
2651	0.312	Cluster 1
2652	0.385	Cluster 3
2653	0.900	Cluster 2
2654	0.723	Cluster 2
2655	0.373	Cluster 4
2656	0.553	Cluster 2
2657	0.956	Cluster 4
2658	0.532	Cluster 3
2659	0.648	Cluster 3
2660	0.850	Cluster 1
2661	0.800	Cluster 5
2662	0.697	Cluster 5
2663	0.806	Cluster 4
2664	0.654	Cluster 2
2665	0.729	Cluster 2
2666	0.977	Cluster 4
2667	0.874	Cluster 4
2668	0.569	Cluster 2
2669	0.477	Cluster 4
2670	0.343	Cluster 2
2671	0.614	Cluster 1
2672	0.833	Cluster 3
2673	0.348	Cluster 1
2674	0.769	Cluster 2
2675	0.841	Cluster 1
2676	0.317	Cluster 4
2677	0.612	Cluster 4
2678	0.630	Cluster 1
2679	0.571	Cluster 5
2680	0.473	Cluster 5
2681	0.659	Cluster 4
2682	0.550	Cluster 3
2683	0.877	Cluster 5
2684	0.934	Cluster 4
2685	0.692	Cluster 3
2686	0.420	Cluster 3
2687	0.826	Cluster 4
2688	0.854	Cluster 5
2689	0.388	Cluster 2
2690	0.461	Cluster 3
2691	0.560	Cluster 1
2692	0.670	Cluster 2
2693	0.425	Cluster 2
2694	0.381	Cluster 4
2695	0.858	Cluster 4
2696	0.736	Cluster 3
2697	0.796	Cluster 2
2698	0.947	Cluster 2
2699	0.844	Cluster 3
2700	0.494	Cluster 5
2701	0.916	Cluster 4
2702	0.970	Cluster 4
2703	0.473	Cluster 4
2704	0.766	Cluster 4
2705	0.910	Cluster 2
2706	0.785	Cluster 3
2707	0.944	Cluster 5
2708	0.581	Cluster 1
2709	0.428	Cluster 3
2710	0.448	Cluster 1
2711	0.904	Cluster 2
2712	0.554	Cluster 2
2713	0.823	Cluster 1
2714	0.613	Cluster 1
2715	0.343	Cluster 2
2716	0.521	Cluster 5
2717	0.282	Cluster 3
2718	0.641	Cluster 5
2719	0.663	Cluster 2
2720	0.469	Cluster 4
2721	0.588	Cluster 4
2722	0.807	Cluster 2
2723	0.609	Cluster 1
2724	0.477	Cluster 2
2725	0.573	Cluster 1
2726	0.763	Cluster 2
2727	0.340	Cluster 1
2728	0.885	Cluster 2
2729	0.603	Cluster 1
2730	0.418	Cluster 3
2731	0.693	Cluster 2
2732	0.802	Cluster 5
2733	0.479	Cluster 3
2734	0.823	Cluster 4
2735	0.648	Cluster 2
2736	0.549	Cluster 3
2737	0.337	Cluster 2
2738	0.345	Cluster 3
2739	0.880	Cluster 4
2740	0.703	Cluster 4
2741	0.599	Cluster 1
2742	0.709	Cluster 5
2743	0.921	Cluster 5
2744	0.429	Cluster 3
2745	0.639	Cluster 3
2746	0.922	Cluster 5
2747	0.831	Cluster 4
2748	0.735	Cluster 1
2749	0.450	Cluster 2
2750	0.534	Cluster 4
2751	0.902	Cluster 1
2752	0.802	Cluster 2
2753	0.834	Cluster 1
2754	0.554	Cluster 4
2755	0.565	Cluster 2
2756	0.729	Cluster 2
2757	0.695	Cluster 2
2758	0.822	Cluster 1
2759	0.781	Cluster 4
2760	0.814	Cluster 2
2761	0.477	Cluster 2
2762	0.464	Cluster 3
2763	0.793	Cluster 2
2764	0.479	Cluster 2
2765	0.339	Cluster 2
2766	0.793	Cluster 2
2767	0.928	Cluster 4
2768	0.819	Cluster 4
2769	0.440	Cluster 3
2770	0.640	Cluster 2
2771	0.878	Cluster 4
2772	0.365	Cluster 5
2773	0.706	Cluster 2
2774	0.804	Cluster 4
2775	0.831	Cluster 4
2776	0.907	Cluster 2
2777	0.479	Cluster 4
2778	0.461	Cluster 4
2779	0.459	Cluster 2
2780	0.813	Cluster 4
2781	0.901	Cluster 4
2782	0.832	Cluster 4
2783	0.829	Cluster 4
2784	0.916	Cluster 3
2785	0.517	Cluster 1
2786	0.400	Cluster 4
2787	0.721	Cluster 2
2788	0.751	Cluster 1
2789	0.614	Cluster 2
2790	0.924	Cluster 4
2791	0.860	Cluster 3
2792	0.917	Cluster 3
2793	0.960	Cluster 2
2794	0.835	Cluster 1
2795	0.675	Cluster 5
2796	0.480	Cluster 2
2797	0.789	Cluster 2
2798	0.898	Cluster 5
2799	0.522	Cluster 5
2800	0.302	Cluster 1
2801	0.393	Cluster 3
2802	0.892	Cluster 4
2803	0.835	Cluster 4
2804	0.710	Cluster 5
2805	0.668	Cluster 3
2806	0.967	Cluster 4
2807	0.454	Cluster 3
2808	0.902	Cluster 2
2809	0.710	Cluster 2
2810	0.906	Cluster 5
2811	0.756	Cluster 4
2812	0.837	Cluster 1
2813	0.958	Cluster 4
2814	0.270	Cluster 3
2815	0.555	Cluster 2
2816	0.460	Cluster 1
2817	0.395	Cluster 3
2818	0.540	Cluster 2
2819	0.654	Cluster 5
2820	0.522	Cluster 1
2821	0.592	Cluster 2
2822	0.852	Cluster 4
2823	0.679	Cluster 2
2824	0.559	Cluster 1
2825	0.753	Cluster 5
2826	0.927	Cluster 4
2827	0.650	Cluster 1
2828	0.758	Cluster 1
2829	0.449	Cluster 1
2830	0.952	Cluster 5
2831	0.399	Cluster 3
2832	0.848	Cluster 4
2833	0.535	Cluster 3
2834	0.587	Cluster 3
2835	0.555	Cluster 2
2836	0.893	Cluster 2
2837	0.519	Cluster 3
2838	0.860	Cluster 1
2839	0.648	Cluster 2
2840	0.676	Cluster 4
2841	0.505	Cluster 1
2842	0.437	Cluster 3
2843	0.358	Cluster 3
2844	0.393	Cluster 5
2845	0.662	Cluster 4
2846	0.524	Cluster 1
2847	0.973	Cluster 2
2848	0.728	Cluster 2
2849	0.899	Cluster 4
2850	0.915	Cluster 2
2851	0.484	Cluster 1
2852	0.964	Cluster 4
2853	0.910	Cluster 3
2854	0.801	Cluster 4
2855	0.640	Cluster 1
2856	0.529	Cluster 1
2857	0.893	Cluster 5
2858	0.899	Cluster 4
2859	0.725	Cluster 2
2860	0.863	Cluster 2
2861	0.572	Cluster 1
2862	0.831	Cluster 2
2863	0.573	Cluster 2
2864	0.899	Cluster 4
2865	0.788	Cluster 5
2866	0.558	Cluster 1
2867	0.561	Cluster 5
2868	0.821	Cluster 2
2869	0.927	Cluster 3
2870	0.585	Cluster 1
2871	0.635	Cluster 3
2872	0.353	Cluster 2
2873	0.730	Cluster 2
2874	0.586	Cluster 1
2875	0.518	Cluster 3
2876	0.645	Cluster 4
2877	0.941	Cluster 1
2878	0.671	Cluster 4
2879	0.685	Cluster 2
2880	0.888	Cluster 4
2881	0.869	Cluster 2
2882	0.443	Cluster 2
2883	0.908	Cluster 2
2884	0.444	Cluster 3
2885	0.630	Cluster 2
2886	0.420	Cluster 1
2887	0.966	Cluster 4
2888	0.406	Cluster 3
2889	0.679	Cluster 2
2890	0.673	Cluster 2
2891	0.444	Cluster 3
2892	0.609	Cluster 3
2893	0.960	Cluster 4
2894	0.457	Cluster 3
2895	0.479	Cluster 2
2896	0.603	Cluster 1
2897	0.660	Cluster 2
2898	0.547	Cluster 2
2899	0.669	Cluster 1
2900	0.718	Cluster 1
2901	0.545	Cluster 3
2902	0.711	Cluster 2
2903	0.680	Cluster 3
2904	0.778	Cluster 1
2905	0.793	Cluster 4
2906	0.632	Cluster 4
2907	0.644	Cluster 1
2908	0.612	Cluster 1
2909	0.607	Cluster 2
2910	0.855	Cluster 4
2911	0.740	Cluster 2
2912	0.568	Cluster 1
2913	0.403	Cluster 2
2914	0.518	Cluster 2
2915	0.773	Cluster 1
2916	0.946	Cluster 4
2917	0.945	Cluster 3
2918	0.647	Cluster 2
2919	0.896	Cluster 1
2920	0.763	Cluster 5
2921	0.608	Cluster 2
2922	0.948	Cluster 5
2923	0.684	Cluster 4
2924	0.897	Cluster 2
2925	0.829	Cluster 2
2926	0.534	Cluster 2
2927	0.763	Cluster 4
2928	0.880	Cluster 4
2929	0.916	Cluster 3
2930	0.790	Cluster 1
2931	0.936	Cluster 4
2932	0.912	Cluster 5
2933	0.776	Cluster 1
2934	0.546	Cluster 5
2935	0.565	Cluster 2
2936	0.792	Cluster 4
2937	0.688	Cluster 4
2938	0.736	Cluster 5
2939	0.540	Cluster 4
2940	0.275	Cluster 1
2941	0.950	Cluster 5
2942	0.435	Cluster 5
2943	0.309	Cluster 1
2944	0.855	Cluster 4
2945	0.430	Cluster 4
2946	0.469	Cluster 3
2947	0.455	Cluster 1
2948	0.663	Cluster 1
2949	0.431	Cluster 5
2950	0.850	Cluster 4
2951	0.606	Cluster 2
2952	0.764	Cluster 3
2953	0.501	Cluster 3
2954	0.613	Cluster 2
2955	0.777	Cluster 3
2956	0.756	Cluster 5
2957	0.860	Cluster 4
2958	0.924	Cluster 4
2959	0.579	Cluster 4
2960	0.606	Cluster 1
2961	0.466	Cluster 5
2962	0.723	Cluster 2
2963	0.494	Cluster 2
2964	0.827	Cluster 3
2965	0.699	Cluster 2
2966	0.817	Cluster 2
2967	0.625	Cluster 3
2968	0.451	Cluster 3
2969	0.920	Cluster 1
2970	0.573	Cluster 4
2971	0.644	Cluster 1
2972	0.892	Cluster 4
2973	0.448	Cluster 2
2974	0.539	Cluster 2
2975	0.800	Cluster 4
2976	0.795	Cluster 2
2977	0.713	Cluster 4
2978	0.574	Cluster 2
2979	0.571	Cluster 2
2980	0.925	Cluster 4
2981	0.868	Cluster 3
2982	0.809	Cluster 2
2983	0.553	Cluster 1
2984	0.587	Cluster 2
2985	0.531	Cluster 2
2986	0.548	Cluster 5
2987	0.945	Cluster 2
2988	0.625	Cluster 3
2989	0.394	Cluster 3
2990	0.986	Cluster 4
2991	0.725	Cluster 4
2992	0.717	Cluster 2
2993	0.251	Cluster 4
2994	0.508	Cluster 1
2995	0.521	Cluster 3
2996	0.677	Cluster 4
2997	0.666	Cluster 5
2998	0.942	Cluster 1
2999	0.766	Cluster 4
3000	0.937	Cluster 4
3001	0.613	Cluster 4
3002	0.575	Cluster 1
3003	0.608	Cluster 1
3004	0.836	Cluster 1
3005	0.588	Cluster 1
3006	0.848	Cluster 2
3007	0.901	Cluster 4
3008	0.518	Cluster 4
3009	0.846	Cluster 4
3010	0.731	Cluster 1
3011	0.815	Cluster 5
3012	0.910	Cluster 4
3013	0.758	Cluster 2
3014	0.892	Cluster 1
3015	0.863	Cluster 2
3016	0.711	Cluster 5
3017	0.666	Cluster 4
3018	0.700	Cluster 2
3019	0.869	Cluster 4
3020	0.924	Cluster 2
3021	0.937	Cluster 4
3022	0.956	Cluster 4
3023	0.849	Cluster 1
3024	0.720	Cluster 2
3025	0.639	Cluster 3
3026	0.910	Cluster 2
3027	0.904	Cluster 4
3028	0.655	Cluster 4
3029	0.784	Cluster 1
3030	0.865	Cluster 4
3031	0.916	Cluster 2
3032	0.603	Cluster 2
3033	0.854	Cluster 3
3034	0.678	Cluster 2
3035	0.606	Cluster 5
3036	0.586	Cluster 5
3037	0.714	Cluster 2
3038	0.458	Cluster 3
3039	0.406	Cluster 4
3040	0.411	Cluster 2
3041	0.846	Cluster 5
3042	0.957	Cluster 4
3043	0.812	Cluster 5
3044	0.948	Cluster 1
3045	0.651	Cluster 1
3046	0.527	Cluster 4
3047	0.812	Cluster 4
3048	0.801	Cluster 2
3049	0.376	Cluster 5
3050	0.750	Cluster 2
3051	0.623	Cluster 2
3052	0.824	Cluster 2
3053	0.520	Cluster 2
3054	0.782	Cluster 2
3055	0.727	Cluster 4
3056	0.659	Cluster 3
3057	0.550	Cluster 4
3058	0.834	Cluster 4
3059	0.450	Cluster 2
3060	0.618	Cluster 1
3061	0.528	Cluster 1
3062	0.369	Cluster 4
3063	0.754	Cluster 4
3064	0.388	Cluster 5
3065	0.828	Cluster 2
3066	0.533	Cluster 5
3067	0.546	Cluster 2
3068	0.314	Cluster 5
3069	0.618	Cluster 1
3070	0.571	Cluster 2
3071	0.636	Cluster 1
3072	0.979	Cluster 4
3073	0.588	Cluster 5
3074	0.286	Cluster 4
3075	0.943	Cluster 4
3076	0.648	Cluster 2
3077	0.548	Cluster 2
3078	0.911	Cluster 4
3079	0.731	Cluster 4
3080	0.523	Cluster 5
3081	0.380	Cluster 3
3082	0.480	Cluster 1
3083	0.527	Cluster 2
3084	0.827	Cluster 5
3085	0.808	Cluster 5
3086	0.744	Cluster 4
3087	0.503	Cluster 3
3088	0.700	Cluster 2
3089	0.587	Cluster 2
3090	0.551	Cluster 2
3091	0.733	Cluster 2
3092	0.466	Cluster 2
3093	0.486	Cluster 4
3094	0.586	Cluster 3
3095	0.753	Cluster 3
3096	0.874	Cluster 2
3097	0.805	Cluster 4
3098	0.620	Cluster 4
3099	0.840	Cluster 2
3100	0.935	Cluster 4
3101	0.566	Cluster 4
3102	0.895	Cluster 4
3103	0.962	Cluster 2
3104	0.809	Cluster 4
3105	0.815	Cluster 3
3106	0.710	Cluster 2
3107	0.828	Cluster 1
3108	0.608	Cluster 3
3109	0.779	Cluster 3
3110	0.781	Cluster 1
3111	0.649	Cluster 2
3112	0.929	Cluster 2
3113	0.590	Cluster 4
3114	0.968	Cluster 4
3115	0.891	Cluster 2
3116	0.960	Cluster 5
3117	0.770	Cluster 3
3118	0.591	Cluster 2
3119	0.719	Cluster 1
3120	0.418	Cluster 2
3121	0.645	Cluster 2
3122	0.720	Cluster 5
3123	0.308	Cluster 1
3124	0.805	Cluster 2
3125	0.756	Cluster 3
3126	0.446	Cluster 4
3127	0.863	Cluster 5
3128	0.930	Cluster 5
3129	0.525	Cluster 2
3130	0.929	Cluster 4
3131	0.713	Cluster 4
3132	0.730	Cluster 4
3133	0.469	Cluster 3
3134	0.551	Cluster 5
3135	0.601	Cluster 2
3136	0.912	Cluster 1
3137	0.715	Cluster 2
3138	0.823	Cluster 4
3139	0.762	Cluster 1
3140	0.709	Cluster 2
3141	0.896	Cluster 2
3142	0.860	Cluster 4
3143	0.447	Cluster 3
3144	0.957	Cluster 4
3145	0.488	Cluster 2
3146	0.684	Cluster 5
3147	0.831	Cluster 2
3148	0.601	Cluster 4
3149	0.439	Cluster 3
3150	0.933	Cluster 5
3151	0.517	Cluster 4
3152	0.297	Cluster 3
3153	0.872	Cluster 2
3154	0.871	Cluster 4
3155	0.656	Cluster 2
3156	0.741	Cluster 1
3157	0.595	Cluster 1
3158	0.457	Cluster 2
3159	0.827	Cluster 5
3160	0.821	Cluster 4
3161	0.787	Cluster 5
3162	0.577	Cluster 1
3163	0.807	Cluster 2
3164	0.252	Cluster 3
3165	0.618	Cluster 2
3166	0.676	Cluster 4
3167	0.808	Cluster 2
3168	0.895	Cluster 5
3169	0.574	Cluster 1
3170	0.488	Cluster 3
3171	0.857	Cluster 1
3172	0.526	Cluster 2
3173	0.645	Cluster 4
3174	0.794	Cluster 5
3175	0.374	Cluster 2
3176	0.630	Cluster 4
3177	0.964	Cluster 2
3178	0.832	Cluster 2
3179	0.373	Cluster 4
3180	0.726	Cluster 3
3181	0.792	Cluster 1
3182	0.709	Cluster 4
3183	0.932	Cluster 1
3184	0.977	Cluster 2
3185	0.527	Cluster 4
3186	0.768	Cluster 4
3187	0.639	Cluster 2
3188	0.809	Cluster 3
3189	0.939	Cluster 2
3190	0.930	Cluster 4
3191	0.754	Cluster 1
3192	0.487	Cluster 1
3193	0.809	Cluster 1
3194	0.405	Cluster 1
3195	0.751	Cluster 1
3196	0.968	Cluster 2
3197	0.431	Cluster 4
3198	0.754	Cluster 4
3199	0.518	Cluster 2
3200	0.476	Cluster 1
3201	0.472	Cluster 5
3202	0.840	Cluster 4
3203	0.839	Cluster 2
3204	0.553	Cluster 2
3205	0.666	Cluster 5
3206	0.596	Cluster 1
3207	0.775	Cluster 1
3208	0.670	Cluster 2
3209	0.481	Cluster 4
3210	0.609	Cluster 5
3211	0.288	Cluster 3
3212	0.624	Cluster 3
3213	0.589	Cluster 5
3214	0.837	Cluster 3
3215	0.730	Cluster 4
3216	0.697	Cluster 3
3217	0.507	Cluster 5
3218	0.639	Cluster 1
3219	0.583	Cluster 3
3220	0.931	Cluster 4
3221	0.356	Cluster 2
3222	0.551	Cluster 2
3223	0.759	Cluster 2
3224	0.714	Cluster 2
3225	0.809	Cluster 2
3226	0.833	Cluster 1
3227	0.614	Cluster 2
3228	0.715	Cluster 2
3229	0.458	Cluster 4
3230	0.645	Cluster 1
3231	0.430	Cluster 3
3232	0.716	Cluster 2
3233	0.449	Cluster 1
3234	0.895	Cluster 4
3235	0.934	Cluster 4
3236	0.803	Cluster 1
3237	0.558	Cluster 4
3238	0.732	Cluster 5
3239	0.953	Cluster 4
3240	0.722	Cluster 2
3241	0.734	Cluster 4
3242	0.621	Cluster 3
3243	0.868	Cluster 1
3244	0.405	Cluster 3
3245	0.561	Cluster 2
3246	0.868	Cluster 1
3247	0.443	Cluster 3
3248	0.768	Cluster 5
3249	0.896	Cluster 5
3250	0.371	Cluster 1
3251	0.821	Cluster 1
3252	0.527	Cluster 2
3253	0.835	Cluster 1
3254	0.950	Cluster 4
3255	0.874	Cluster 2
3256	0.849	Cluster 4
3257	0.515	Cluster 2
3258	0.465	Cluster 1
3259	0.941	Cluster 4
3260	0.838	Cluster 3
3261	0.383	Cluster 1
3262	0.980	Cluster 5
3263	0.497	Cluster 4
3264	0.916	Cluster 4
3265	0.984	Cluster 5
3266	0.840	Cluster 2
3267	0.772	Cluster 2
3268	0.641	Cluster 4
3269	0.901	Cluster 3
3270	0.830	Cluster 5
3271	0.575	Cluster 1
3272	0.454	Cluster 3
3273	0.707	Cluster 2
3274	0.556	Cluster 2
3275	0.916	Cluster 3
3276	0.336	Cluster 3
3277	0.392	Cluster 2
3278	0.883	Cluster 2
3279	0.576	Cluster 2
3280	0.492	Cluster 2
3281	0.897	Cluster 2
3282	0.852	Cluster 1
3283	0.765	Cluster 2
3284	0.436	Cluster 3
3285	0.836	Cluster 2
3286	0.553	Cluster 5
3287	0.637	Cluster 1
3288	0.500	Cluster 3
3289	0.561	Cluster 4
3290	0.860	Cluster 3
3291	0.685	Cluster 4
3292	0.659	Cluster 3
3293	0.824	Cluster 2
3294	0.652	Cluster 5
3295	0.658	Cluster 1
3296	0.460	Cluster 2
3297	0.632	Cluster 2
3298	0.644	Cluster 4
3299	0.899	Cluster 1
3300	0.416	Cluster 2
3301	0.658	Cluster 1
3302	0.437	Cluster 4
3303	0.872	Cluster 4
3304	0.331	Cluster 2
3305	0.499	Cluster 1
3306	0.694	Cluster 2
3307	0.963	Cluster 4
3308	0.947	Cluster 2
3309	0.571	Cluster 5
3310	0.934	Cluster 3
3311	0.739	Cluster 2
3312	0.668	Cluster 3
3313	0.504	Cluster 3
3314	0.621	Cluster 5
3315	0.317	Cluster 2
3316	0.843	Cluster 4
3317	0.887	Cluster 2
3318	0.581	Cluster 2
3319	0.883	Cluster 4
3320	0.496	Cluster 1
3321	0.589	Cluster 3
3322	0.748	Cluster 4
3323	0.530	Cluster 1
3324	0.629	Cluster 4
3325	0.692	Cluster 2
3326	0.881	Cluster 5
3327	0.569	Cluster 1
3328	0.637	Cluster 3
3329	0.785	Cluster 4
3330	0.482	Cluster 1
3331	0.853	Cluster 4
3332	0.554	Cluster 2
3333	0.801	Cluster 2
3334	0.422	Cluster 5
3335	0.698	Cluster 2
3336	0.356	Cluster 2
3337	0.486	Cluster 5
3338	0.618	Cluster 3
3339	0.450	Cluster 4
3340	0.849	Cluster 2
3341	0.543	Cluster 2
3342	0.744	Cluster 2
3343	0.806	Cluster 4
3344	0.325	Cluster 2
3345	0.688	Cluster 2
3346	0.965	Cluster 5
3347	0.488	Cluster 2
3348	0.551	Cluster 1
3349	0.948	Cluster 2
3350	0.475	Cluster 2
3351	0.950	Cluster 4
3352	0.423	Cluster 2
3353	0.481	Cluster 5
3354	0.837	Cluster 4
3355	0.550	Cluster 1
3356	0.926	Cluster 4
3357	0.715	Cluster 2
3358	0.553	Cluster 1
3359	0.945	Cluster 3
3360	0.801	Cluster 2
3361	0.482	Cluster 4
3362	0.895	Cluster 2
3363	0.318	Cluster 2
3364	0.444	Cluster 1
3365	0.711	Cluster 2
3366	0.863	Cluster 1
3367	0.826	Cluster 2
3368	0.837	Cluster 2
3369	0.381	Cluster 1
3370	0.599	Cluster 4
3371	0.935	Cluster 4
3372	0.619	Cluster 2
3373	0.846	Cluster 1
3374	0.849	Cluster 1
3375	0.396	Cluster 4
3376	0.940	Cluster 2
3377	0.963	Cluster 1
3378	0.778	Cluster 3
3379	0.611	Cluster 2
3380	0.937	Cluster 2
3381	0.713	Cluster 1
3382	0.950	Cluster 3
3383	0.588	Cluster 5
3384	0.943	Cluster 5
3385	0.944	Cluster 5
3386	0.498	Cluster 1
3387	0.785	Cluster 2
3388	0.965	Cluster 4
3389	0.734	Cluster 4
3390	0.702	Cluster 3
3391	0.702	Cluster 2
3392	0.815	Cluster 3
3393	0.491	Cluster 3
3394	0.783	Cluster 4
3395	0.972	Cluster 2
3396	0.531	Cluster 1
3397	0.883	Cluster 5
3398	0.445	Cluster 4
3399	0.746	Cluster 4
3400	0.812	Cluster 4
3401	0.548	Cluster 1
3402	0.694	Cluster 4
3403	0.935	Cluster 2
3404	0.431	Cluster 2
3405	0.687	Cluster 2
3406	0.561	Cluster 3
3407	0.498	Cluster 4
3408	0.495	Cluster 1
3409	0.540	Cluster 2
3410	0.681	Cluster 5
3411	0.669	Cluster 5
3412	0.722	Cluster 5
3413	0.687	Cluster 3
3414	0.645	Cluster 2
3415	0.697	Cluster 2
3416	0.542	Cluster 2
3417	0.664	Cluster 3
3418	0.891	Cluster 2
3419	0.661	Cluster 3
3420	0.732	Cluster 4
3421	0.749	Cluster 5
3422	0.319	Cluster 5
3423	0.566	Cluster 1
3424	0.742	Cluster 4
3425	0.509	Cluster 2
3426	0.884	Cluster 3
3427	0.825	Cluster 4
3428	0.927	Cluster 4
3429	0.656	Cluster 3
3430	0.904	Cluster 4
3431	0.652	Cluster 3
3432	0.681	Cluster 1
3433	0.473	Cluster 2
3434	0.940	Cluster 2
3435	0.531	Cluster 5
3436	0.892	Cluster 1
3437	0.652	Cluster 3
3438	0.337	Cluster 3
3439	0.814	Cluster 2
3440	0.659	Cluster 1
3441	0.437	Cluster 2
3442	0.384	Cluster 1
3443	0.743	Cluster 1
3444	0.761	Cluster 2
3445	0.719	Cluster 2
3446	0.730	Cluster 2
3447	0.834	Cluster 5
3448	0.879	Cluster 1
3449	0.941	Cluster 4
3450	0.964	Cluster 2
3451	0.691	Cluster 4
3452	0.459	Cluster 3
3453	0.675	Cluster 3
3454	0.567	Cluster 4
3455	0.929	Cluster 4
3456	0.891	Cluster 4
3457	0.823	Cluster 2
3458	0.893	Cluster 2
3459	0.974	Cluster 5
3460	0.820	Cluster 5
3461	0.932	Cluster 4
3462	0.576	Cluster 5
3463	0.578	Cluster 5
3464	0.818	Cluster 2
3465	0.534	Cluster 2
3466	0.779	Cluster 3
3467	0.862	Cluster 2
3468	0.418	Cluster 4
3469	0.633	Cluster 4
3470	0.577	Cluster 1
3471	0.484	Cluster 4
3472	0.743	Cluster 4
3473	0.831	Cluster 3
3474	0.760	Cluster 2
3475	0.881	Cluster 4
3476	0.905	Cluster 4
3477	0.793	Cluster 3
3478	0.608	Cluster 4
3479	0.953	Cluster 5
3480	0.787	Cluster 5
3481	0.731	Cluster 5
3482	0.484	Cluster 3
3483	0.946	Cluster 4
3484	0.367	Cluster 5
3485	0.867	Cluster 4
3486	0.571	Cluster 1
3487	0.722	Cluster 3
3488	0.968	Cluster 5
3489	0.499	Cluster 4
3490	0.383	Cluster 3
3491	0.727	Cluster 1
3492	0.315	Cluster 3
3493	0.563	Cluster 2
3494	0.727	Cluster 2
3495	0.510	Cluster 2
3496	0.529	Cluster 1
3497	0.452	Cluster 3
3498	0.645	Cluster 2
3499	0.724	Cluster 4
3500	0.844	Cluster 2
3501	0.587	Cluster 1
3502	0.508	Cluster 4
3503	0.667	Cluster 1
3504	0.734	Cluster 1
3505	0.590	Cluster 2
3506	0.698	Cluster 1
3507	0.874	Cluster 4
3508	0.823	Cluster 5
3509	0.480	Cluster 3
3510	0.547	Cluster 1
3511	0.806	Cluster 4
3512	0.384	Cluster 1
3513	0.543	Cluster 1
3514	0.967	Cluster 5
3515	0.961	Cluster 4
3516	0.299	Cluster 3
3517	0.787	Cluster 5
3518	0.348	Cluster 3
3519	0.551	Cluster 5
3520	0.792	Cluster 2
3521	0.393	Cluster 3
3522	0.577	Cluster 2
3523	0.903	Cluster 2
3524	0.850	Cluster 1
3525	0.689	Cluster 4
3526	0.283	Cluster 3
3527	0.684	Cluster 2
3528	0.647	Cluster 1
3529	0.609	Cluster 2
3530	0.432	Cluster 4
3531	0.628	Cluster 1
3532	0.422	Cluster 4
3533	0.754	Cluster 3
3534	0.782	Cluster 1
3535	0.909	Cluster 5
3536	0.657	Cluster 2
3537	0.690	Cluster 3
3538	0.371	Cluster 1
3539	0.580	Cluster 1
3540	0.830	Cluster 4
3541	0.917	Cluster 4
3542	0.524	Cluster 4
3543	0.472	Cluster 5
3544	0.506	Cluster 5
3545	0.398	Cluster 5
3546	0.519	Cluster 2
3547	0.808	Cluster 1
3548	0.652	Cluster 3
3549	0.595	Cluster 1
3550	0.662	Cluster 4
3551	0.852	Cluster 3
3552	0.500	Cluster 1
3553	0.747	Cluster 2
3554	0.696	Cluster 3
3555	0.794	Cluster 4
3556	0.863	Cluster 3
3557	0.686	Cluster 2
3558	0.621	Cluster 1
3559	0.360	Cluster 3
3560	0.812	Cluster 5
3561	0.744	Cluster 5
3562	0.681	Cluster 2
3563	0.716	Cluster 1
3564	0.724	Cluster 1
3565	0.565	Cluster 2
3566	0.398	Cluster 4
3567	0.535	Cluster 4
3568	0.929	Cluster 4
3569	0.815	Cluster 4
3570	0.558	Cluster 1
3571	0.419	Cluster 5
3572	0.568	Cluster 1
3573	0.932	Cluster 3
3574	0.401	Cluster 1
3575	0.341	Cluster 1
3576	0.508	Cluster 5
3577	0.767	Cluster 2
3578	0.808	Cluster 3
3579	0.626	Cluster 1
3580	0.867	Cluster 5
3581	0.399	Cluster 2
3582	0.713	Cluster 5
3583	0.604	Cluster 4
3584	0.926	Cluster 4
3585	0.950	Cluster 4
3586	0.500	Cluster 2
3587	0.976	Cluster 5
3588	0.498	Cluster 5
3589	0.938	Cluster 5
3590	0.569	Cluster 3
3591	0.847	Cluster 4
3592	0.569	Cluster 2
3593	0.449	Cluster 1
3594	0.766	Cluster 4
3595	0.574	Cluster 5
3596	0.496	Cluster 3
3597	0.905	Cluster 5
3598	0.709	Cluster 1
3599	0.839	Cluster 2
3600	0.492	Cluster 3
3601	0.732	Cluster 2
3602	0.304	Cluster 3
3603	0.684	Cluster 2
3604	0.509	Cluster 1
3605	0.699	Cluster 1
3606	0.699	Cluster 2
3607	0.909	Cluster 3
3608	0.391	Cluster 5
3609	0.605	Cluster 4
3610	0.447	Cluster 1
3611	0.604	Cluster 3
3612	0.887	Cluster 1
3613	0.519	Cluster 4
3614	0.517	Cluster 4
3615	0.904	Cluster 4
3616	0.549	Cluster 3
3617	0.353	Cluster 3
3618	0.682	Cluster 1
3619	0.646	Cluster 4
3620	0.918	Cluster 3
3621	0.676	Cluster 3
3622	0.734	Cluster 4
3623	0.761	Cluster 4
3624	0.758	Cluster 4
3625	0.706	Cluster 1
3626	0.857	Cluster 4
3627	0.487	Cluster 1
3628	0.774	Cluster 5
3629	0.587	Cluster 2
3630	0.813	Cluster 3
3631	0.643	Cluster 5
3632	0.735	Cluster 3
3633	0.856	Cluster 4
3634	0.698	Cluster 5
3635	0.707	Cluster 2
3636	0.555	Cluster 1
3637	0.471	Cluster 2
3638	0.988	Cluster 5
3639	0.611	Cluster 5
3640	0.545	Cluster 1
3641	0.969	Cluster 5
3642	0.651	Cluster 3
3643	0.886	Cluster 4
3644	0.541	Cluster 3
3645	0.541	Cluster 1
3646	0.556	Cluster 1
3647	0.810	Cluster 2
3648	0.938	Cluster 3
3649	0.413	Cluster 5
3650	0.368	Cluster 1
3651	0.661	Cluster 1
3652	0.592	Cluster 2
3653	0.711	Cluster 2
3654	0.880	Cluster 4
3655	0.610	Cluster 1
3656	0.424	Cluster 2
3657	0.744	Cluster 1
3658	0.867	Cluster 2
3659	0.483	Cluster 2
3660	0.803	Cluster 2
3661	0.627	Cluster 2
3662	0.696	Cluster 3
3663	0.892	Cluster 4
3664	0.655	Cluster 4
3665	0.493	Cluster 2
3666	0.562	Cluster 3
3667	0.523	Cluster 3
3668	0.788	Cluster 1
3669	0.582	Cluster 4
3670	0.801	Cluster 4
3671	0.482	Cluster 3
3672	0.556	Cluster 1
3673	0.554	Cluster 1
3674	0.681	Cluster 3
3675	0.336	Cluster 1
3676	0.819	Cluster 5
3677	0.765	Cluster 1
3678	0.798	Cluster 4
3679	0.816	Cluster 5
3680	0.737	Cluster 2
3681	0.763	Cluster 3
3682	0.715	Cluster 2
3683	0.926	Cluster 4
3684	0.455	Cluster 2
3685	0.692	Cluster 2
3686	0.704	Cluster 2
3687	0.833	Cluster 2
3688	0.904	Cluster 2
3689	0.698	Cluster 4
3690	0.974	Cluster 2
3691	0.818	Cluster 2
3692	0.770	Cluster 5
3693	0.436	Cluster 1
3694	0.815	Cluster 5
3695	0.910	Cluster 2
3696	0.812	Cluster 1
3697	0.846	Cluster 3
3698	0.872	Cluster 4
3699	0.728	Cluster 5
3700	0.588	Cluster 4
3701	0.679	Cluster 5
3702	0.542	Cluster 3
3703	0.339	Cluster 1
3704	0.807	Cluster 4
3705	0.880	Cluster 3
3706	0.369	Cluster 1
3707	0.535	Cluster 2
3708	0.447	Cluster 4
3709	0.436	Cluster 1
3710	0.556	Cluster 4
3711	0.545	Cluster 4
3712	0.828	Cluster 2
3713	0.383	Cluster 2
3714	0.595	Cluster 3
3715	0.536	Cluster 4
3716	0.933	Cluster 2
3717	0.814	Cluster 3
3718	0.597	Cluster 2
3719	0.903	Cluster 5
3720	0.756	Cluster 2
3721	0.680	Cluster 2
3722	0.827	Cluster 2
3723	0.631	Cluster 2
3724	0.487	Cluster 5
3725	0.891	Cluster 2
3726	0.945	Cluster 2
3727	0.545	Cluster 3
3728	0.539	Cluster 5
3729	0.667	Cluster 1
3730	0.478	Cluster 1
3731	0.445	Cluster 2
3732	0.942	Cluster 2
3733	0.750	Cluster 2
3734	0.505	Cluster 1
3735	0.739	Cluster 1
3736	0.558	Cluster 1
3737	0.642	Cluster 2
3738	0.460	Cluster 5
3739	0.532	Cluster 2
3740	0.683	Cluster 2
3741	0.427	Cluster 3
3742	0.636	Cluster 3
3743	0.853	Cluster 4
3744	0.702	Cluster 1
3745	0.907	Cluster 2
3746	0.485	Cluster 3
3747	0.601	Cluster 1
3748	0.875	Cluster 5
3749	0.634	Cluster 3
3750	0.619	Cluster 1
3751	0.502	Cluster 1
3752	0.843	Cluster 2
3753	0.643	Cluster 3
3754	0.803	Cluster 2
3755	0.768	Cluster 1
3756	0.786	Cluster 4
3757	0.705	Cluster 2
3758	0.480	Cluster 2
3759	0.964	Cluster 3
3760	0.520	Cluster 2
3761	0.946	Cluster 5
3762	0.851	Cluster 3
3763	0.741	Cluster 2
3764	0.746	Cluster 5
3765	0.301	Cluster 3
3766	0.675	Cluster 5
3767	0.971	Cluster 5
3768	0.753	Cluster 2
3769	0.445	Cluster 2
3770	0.333	Cluster 1
3771	0.346	Cluster 1
3772	0.800	Cluster 2
3773	0.725	Cluster 1
3774	0.431	Cluster 5
3775	0.307	Cluster 2
3776	0.548	Cluster 1
3777	0.413	Cluster 5
3778	0.400	Cluster 2
3779	0.594	Cluster 4
3780	0.560	Cluster 2
3781	0.442	Cluster 4
3782	0.533	Cluster 3
3783	0.717	Cluster 1
3784	0.788	Cluster 4
3785	0.639	Cluster 1
3786	0.632	Cluster 5
3787	0.518	Cluster 1
3788	0.576	Cluster 4
3789	0.979	Cluster 4
3790	0.419	Cluster 4
3791	0.857	Cluster 3
3792	0.479	Cluster 2
3793	0.557	Cluster 5
3794	0.628	Cluster 4
3795	0.682	Cluster 4
3796	0.535	Cluster 5
3797	0.524	Cluster 2
3798	0.394	Cluster 1
3799	0.374	Cluster 4
3800	0.565	Cluster 2
3801	0.569	Cluster 2
3802	0.886	Cluster 4
3803	0.804	Cluster 3
3804	0.733	Cluster 4
3805	0.743	Cluster 2
3806	0.502	Cluster 1
3807	0.528	Cluster 1
3808	0.512	Cluster 4
3809	0.400	Cluster 3
3810	0.957	Cluster 4
3811	0.852	Cluster 2
3812	0.975	Cluster 4
3813	0.522	Cluster 2
3814	0.955	Cluster 4
3815	0.517	Cluster 5
3816	0.860	Cluster 3
3817	0.807	Cluster 2
3818	0.968	Cluster 4
3819	0.358	Cluster 2
3820	0.823	Cluster 2
3821	0.814	Cluster 4
3822	0.816	Cluster 2
3823	0.370	Cluster 4
3824	0.703	Cluster 1
3825	0.336	Cluster 1
3826	0.551	Cluster 2
3827	0.947	Cluster 2
3828	0.422	Cluster 4
3829	0.815	Cluster 2
3830	0.887	Cluster 2
3831	0.357	Cluster 3
3832	0.794	Cluster 4
3833	0.922	Cluster 3
3834	0.542	Cluster 2
3835	0.977	Cluster 5
3836	0.891	Cluster 4
3837	0.829	Cluster 2
3838	0.380	Cluster 3
3839	0.392	Cluster 1
3840	0.905	Cluster 4
3841	0.895	Cluster 2
3842	0.913	Cluster 2
3843	0.376	Cluster 2
3844	0.808	Cluster 1
3845	0.890	Cluster 4
3846	0.566	Cluster 2
3847	0.439	Cluster 1
3848	0.868	Cluster 4
3849	0.471	Cluster 3
3850	0.944	Cluster 1
3851	0.595	Cluster 2
3852	0.933	Cluster 2
3853	0.920	Cluster 1
3854	0.649	Cluster 1
3855	0.385	Cluster 1
3856	0.755	Cluster 2
3857	0.692	Cluster 2
3858	0.948	Cluster 5
3859	0.666	Cluster 1
3860	0.565	Cluster 2
3861	0.915	Cluster 4
3862	0.524	Cluster 3
3863	0.643	Cluster 1
3864	0.669	Cluster 3
3865	0.731	Cluster 4
3866	0.948	Cluster 5
3867	0.995	Cluster 4
3868	0.650	Cluster 3
3869	0.775	Cluster 1
3870	0.678	Cluster 5
3871	0.919	Cluster 4
3872	0.974	Cluster 2
3873	0.420	Cluster 3
3874	0.708	Cluster 4
3875	0.722	Cluster 4
3876	0.521	Cluster 2
3877	0.718	Cluster 4
3878	0.575	Cluster 4
3879	0.768	Cluster 3
3880	0.844	Cluster 2
3881	0.493	Cluster 1
3882	0.589	Cluster 2
3883	0.933	Cluster 5
3884	0.588	Cluster 4
3885	0.594	Cluster 4
3886	0.578	Cluster 1
3887	0.772	Cluster 1
3888	0.947	Cluster 1
3889	0.595	Cluster 3
3890	0.863	Cluster 4
3891	0.471	Cluster 1
3892	0.758	Cluster 3
3893	0.673	Cluster 3
3894	0.445	Cluster 4
3895	0.756	Cluster 4
3896	0.792	Cluster 5
3897	0.634	Cluster 1
3898	0.306	Cluster 1
3899	0.883	Cluster 2
3900	0.965	Cluster 5
3901	0.730	Cluster 2
3902	0.650	Cluster 2
3903	0.596	Cluster 1
3904	0.509	Cluster 1
3905	0.547	Cluster 2
3906	0.592	Cluster 3
3907	0.388	Cluster 2
3908	0.606	Cluster 3
3909	0.970	Cluster 5
3910	0.584	Cluster 4
3911	0.786	Cluster 2
3912	0.386	Cluster 1
3913	0.831	Cluster 4
3914	0.699	Cluster 2
3915	0.542	Cluster 3
3916	0.264	Cluster 2
3917	0.567	Cluster 2
3918	0.611	Cluster 1
3919	0.598	Cluster 5
3920	0.591	Cluster 2
3921	0.894	Cluster 5
3922	0.870	Cluster 3
3923	0.568	Cluster 2
3924	0.702	Cluster 2
3925	0.878	Cluster 4
3926	0.340	Cluster 1
3927	0.850	Cluster 2
3928	0.510	Cluster 2
3929	0.827	Cluster 2
3930	0.376	Cluster 4
3931	0.940	Cluster 4
3932	0.533	Cluster 1
3933	0.819	Cluster 4
3934	0.805	Cluster 4
3935	0.837	Cluster 3
3936	0.518	Cluster 3
3937	0.908	Cluster 4
3938	0.929	Cluster 4
3939	0.428	Cluster 3
3940	0.522	Cluster 4
3941	0.528	Cluster 1
3942	0.966	Cluster 1
3943	0.989	Cluster 5
3944	0.542	Cluster 3
3945	0.718	Cluster 2
3946	0.640	Cluster 1
3947	0.450	Cluster 1
3948	0.537	Cluster 1
3949	0.469	Cluster 3
3950	0.799	Cluster 2
3951	0.403	Cluster 4
3952	0.576	Cluster 1
3953	0.878	Cluster 5
3954	0.395	Cluster 2
3955	0.851	Cluster 1
3956	0.532	Cluster 5
3957	0.521	Cluster 1
3958	0.545	Cluster 2
3959	0.828	Cluster 4
3960	0.605	Cluster 4
3961	0.336	Cluster 2
3962	0.557	Cluster 2
3963	0.752	Cluster 2
3964	0.614	Cluster 4
3965	0.544	Cluster 1
3966	0.552	Cluster 2
3967	0.841	Cluster 5
3968	0.795	Cluster 4
3969	0.568	Cluster 2
3970	0.648	Cluster 2
3971	0.972	Cluster 5
3972	0.847	Cluster 4
3973	0.672	Cluster 1
3974	0.533	Cluster 5
3975	0.856	Cluster 5
3976	0.416	Cluster 5
3977	0.449	Cluster 2
3978	0.960	Cluster 1
3979	0.713	Cluster 2
3980	0.712	Cluster 2
3981	0.695	Cluster 3
3982	0.475	Cluster 5
3983	0.645	Cluster 5
3984	0.447	Cluster 1
3985	0.526	Cluster 4
3986	0.704	Cluster 2
3987	0.891	Cluster 2
3988	0.627	Cluster 1
3989	0.733	Cluster 3
3990	0.498	Cluster 1
3991	0.804	Cluster 4
3992	0.742	Cluster 3
3993	0.884	Cluster 2
3994	0.512	Cluster 2
3995	0.820	Cluster 4
3996	0.639	Cluster 2
3997	0.530	Cluster 4
3998	0.545	Cluster 2
3999	0.912	Cluster 4
4000	0.844	Cluster 2
4001	0.323	Cluster 1
4002	0.861	Cluster 4
4003	0.434	Cluster 4
4004	0.385	Cluster 1
4005	0.600	Cluster 2
4006	0.532	Cluster 1
4007	0.825	Cluster 2
4008	0.857	Cluster 4
4009	0.966	Cluster 5
4010	0.749	Cluster 1
4011	0.755	Cluster 1
4012	0.394	Cluster 1
4013	0.788	Cluster 4
4014	0.627	Cluster 3
4015	0.529	Cluster 3
4016	0.924	Cluster 4
4017	0.805	Cluster 1
4018	0.913	Cluster 2
4019	0.926	Cluster 1
4020	0.441	Cluster 4
4021	0.910	Cluster 1
4022	0.728	Cluster 2
4023	0.791	Cluster 2
4024	0.504	Cluster 5
4025	0.576	Cluster 2
4026	0.482	Cluster 2
4027	0.469	Cluster 3
4028	0.304	Cluster 3
4029	0.945	Cluster 5
4030	0.901	Cluster 2
4031	0.603	Cluster 1
4032	0.846	Cluster 4
4033	0.548	Cluster 2
4034	0.648	Cluster 2
4035	0.756	Cluster 3
4036	0.851	Cluster 2
4037	0.694	Cluster 1
4038	0.987	Cluster 5
4039	0.348	Cluster 3
4040	0.926	Cluster 5
4041	0.617	Cluster 4
4042	0.611	Cluster 5
4043	0.590	Cluster 4
4044	0.885	Cluster 2
4045	0.482	Cluster 5
4046	0.644	Cluster 4
4047	0.419	Cluster 1
4048	0.574	Cluster 1
4049	0.706	Cluster 4
4050	0.387	Cluster 4
4051	0.825	Cluster 4
4052	0.587	Cluster 2
4053	0.444	Cluster 2
4054	0.551	Cluster 3
4055	0.604	Cluster 4
4056	0.653	Cluster 2
4057	0.405	Cluster 1
4058	0.822	Cluster 2
4059	0.387	Cluster 1
4060	0.841	Cluster 3
4061	0.634	Cluster 1
4062	0.920	Cluster 4
4063	0.807	Cluster 5
4064	0.661	Cluster 5
4065	0.934	Cluster 4
4066	0.882	Cluster 4
4067	0.708	Cluster 1
4068	0.396	Cluster 1
4069	0.674	Cluster 2
4070	0.607	Cluster 2
4071	0.625	Cluster 4
4072	0.511	Cluster 4
4073	0.631	Cluster 3
4074	0.931	Cluster 4
4075	0.289	Cluster 4
4076	0.585	Cluster 1
4077	0.512	Cluster 3
4078	0.693	Cluster 3
4079	0.503	Cluster 3
4080	0.684	Cluster 4
4081	0.930	Cluster 4
4082	0.985	Cluster 5
4083	0.554	Cluster 3
4084	0.621	Cluster 5
4085	0.558	Cluster 1
4086	0.777	Cluster 4
4087	0.956	Cluster 3
4088	0.494	Cluster 2
4089	0.578	Cluster 1
4090	0.826	Cluster 3
4091	0.825	Cluster 1
4092	0.886	Cluster 4
4093	0.926	Cluster 4
4094	0.401	Cluster 3
4095	0.475	Cluster 4
4096	0.502	Cluster 3
4097	0.572	Cluster 4
4098	0.552	Cluster 2
4099	0.724	Cluster 1
4100	0.628	Cluster 4
4101	0.429	Cluster 2
4102	0.571	Cluster 2
4103	0.804	Cluster 1
4104	0.963	Cluster 4
4105	0.937	Cluster 5
4106	0.913	Cluster 1
4107	0.407	Cluster 1
4108	0.526	Cluster 5
4109	0.532	Cluster 3
4110	0.839	Cluster 1
4111	0.471	Cluster 1
4112	0.527	Cluster 2
4113	0.951	Cluster 4
4114	0.558	Cluster 5
4115	0.810	Cluster 2
4116	0.567	Cluster 2
4117	0.571	Cluster 4
4118	0.406	Cluster 3
4119	0.924	Cluster 4
4120	0.933	Cluster 4
4121	0.716	Cluster 1
4122	0.688	Cluster 2
4123	0.573	Cluster 4
4124	0.831	Cluster 3
4125	0.544	Cluster 2
4126	0.928	Cluster 2
4127	0.883	Cluster 2
4128	0.531	Cluster 3
4129	0.822	Cluster 3
4130	0.567	Cluster 4
4131	0.818	Cluster 2
4132	0.827	Cluster 2
4133	0.924	Cluster 4
4134	0.731	Cluster 3
4135	0.726	Cluster 3
4136	0.780	Cluster 4
4137	0.476	Cluster 5
4138	0.771	Cluster 1
4139	0.521	Cluster 3
4140	0.479	Cluster 5
4141	0.689	Cluster 5
4142	0.569	Cluster 1
4143	0.555	Cluster 5
4144	0.413	Cluster 3
4145	0.614	Cluster 2
4146	0.562	Cluster 2
4147	0.801	Cluster 2
4148	0.489	Cluster 4
4149	0.737	Cluster 4
4150	0.528	Cluster 3
4151	0.860	Cluster 4
4152	0.861	Cluster 1
4153	0.909	Cluster 2
4154	0.782	Cluster 3
4155	0.576	Cluster 2
4156	0.599	Cluster 2
4157	0.741	Cluster 4
4158	0.704	Cluster 1
4159	0.474	Cluster 3
4160	0.766	Cluster 1
4161	0.773	Cluster 3
4162	0.367	Cluster 5
4163	0.619	Cluster 1
4164	0.543	Cluster 1
4165	0.508	Cluster 2
4166	0.975	Cluster 5
4167	0.651	Cluster 3
4168	0.691	Cluster 4
4169	0.583	Cluster 4
4170	0.805	Cluster 4
4171	0.411	Cluster 3
4172	0.636	Cluster 2
4173	0.928	Cluster 4
4174	0.561	Cluster 2
4175	0.509	Cluster 2
4176	0.923	Cluster 4
4177	0.592	Cluster 2
4178	0.645	Cluster 2
4179	0.792	Cluster 1
4180	0.599	Cluster 1
4181	0.510	Cluster 4
4182	0.740	Cluster 2
4183	0.885	Cluster 4
4184	0.666	Cluster 4
4185	0.872	Cluster 4
4186	0.522	Cluster 5
4187	0.827	Cluster 2
4188	0.824	Cluster 2
4189	0.571	Cluster 5
4190	0.789	Cluster 1
4191	0.927	Cluster 1
4192	0.936	Cluster 4
4193	0.937	Cluster 4
4194	0.652	Cluster 1
4195	0.836	Cluster 1
4196	0.931	Cluster 5
4197	0.906	Cluster 4
4198	0.493	Cluster 2
4199	0.676	Cluster 3
4200	0.462	Cluster 2
4201	0.974	Cluster 2
4202	0.772	Cluster 4
4203	0.385	Cluster 4
4204	0.803	Cluster 2
4205	0.796	Cluster 5
4206	0.789	Cluster 2
4207	0.452	Cluster 2
4208	0.511	Cluster 5
4209	0.709	Cluster 2
4210	0.901	Cluster 5
4211	0.492	Cluster 5
4212	0.471	Cluster 1
4213	0.655	Cluster 2
4214	0.606	Cluster 1
4215	0.860	Cluster 4
4216	0.848	Cluster 1
4217	0.420	Cluster 2
4218	0.471	Cluster 5
4219	0.904	Cluster 4
4220	0.743	Cluster 4
4221	0.590	Cluster 2
4222	0.938	Cluster 2
4223	0.798	Cluster 4
4224	0.430	Cluster 4
4225	0.790	Cluster 2
4226	0.710	Cluster 4
4227	0.730	Cluster 1
4228	0.851	Cluster 4
4229	0.438	Cluster 5
4230	0.563	Cluster 2
4231	0.713	Cluster 4
4232	0.915	Cluster 3
4233	0.518	Cluster 3
4234	0.933	Cluster 3
4235	0.821	Cluster 4
4236	0.802	Cluster 5
4237	0.847	Cluster 5
4238	0.632	Cluster 1
4239	0.695	Cluster 5
4240	0.950	Cluster 5
4241	0.435	Cluster 4
4242	0.537	Cluster 4
4243	0.755	Cluster 5
4244	0.544	Cluster 3
4245	0.807	Cluster 4
4246	0.883	Cluster 2
4247	0.459	Cluster 3
4248	0.724	Cluster 1
4249	0.748	Cluster 2
4250	0.464	Cluster 1
4251	0.852	Cluster 3
4252	0.361	Cluster 3
4253	0.420	Cluster 3
4254	0.496	Cluster 2
4255	0.878	Cluster 2
4256	0.467	Cluster 4
4257	0.601	Cluster 2
4258	0.274	Cluster 3
4259	0.417	Cluster 5
4260	0.649	Cluster 3
4261	0.951	Cluster 4
4262	0.550	Cluster 2
4263	0.651	Cluster 4
4264	0.474	Cluster 4
4265	0.947	Cluster 4
4266	0.839	Cluster 3
4267	0.788	Cluster 3
4268	0.757	Cluster 3
4269	0.651	Cluster 4
4270	0.551	Cluster 2
4271	0.878	Cluster 4
4272	0.815	Cluster 2
4273	0.400	Cluster 2
4274	0.785	Cluster 1
4275	0.538	Cluster 4
4276	0.717	Cluster 2
4277	0.498	Cluster 5
4278	0.840	Cluster 5
4279	0.802	Cluster 1
4280	0.726	Cluster 1
4281	0.963	Cluster 4
4282	0.556	Cluster 1
4283	0.665	Cluster 5
4284	0.875	Cluster 2
4285	0.584	Cluster 1
4286	0.761	Cluster 4
4287	0.687	Cluster 2
4288	0.708	Cluster 1
4289	0.402	Cluster 1
4290	0.770	Cluster 4
4291	0.439	Cluster 3
4292	0.798	Cluster 2
4293	0.422	Cluster 2
4294	0.834	Cluster 4
4295	0.963	Cluster 4
4296	0.327	Cluster 4
4297	0.751	Cluster 1
4298	0.563	Cluster 2
4299	0.882	Cluster 4
4300	0.629	Cluster 2
4301	0.851	Cluster 1
4302	0.477	Cluster 2
4303	0.462	Cluster 1
4304	0.457	Cluster 3
4305	0.883	Cluster 2
4306	0.441	Cluster 3
4307	0.726	Cluster 2
4308	0.461	Cluster 3
4309	0.812	Cluster 4
4310	0.394	Cluster 1
4311	0.509	Cluster 3
4312	0.572	Cluster 2
4313	0.921	Cluster 4
4314	0.406	Cluster 2
4315	0.972	Cluster 4
4316	0.543	Cluster 1
4317	0.913	Cluster 4
4318	0.426	Cluster 1
4319	0.563	Cluster 2
4320	0.420	Cluster 2
4321	0.389	Cluster 4
4322	0.711	Cluster 1
4323	0.879	Cluster 3
4324	0.564	Cluster 1
4325	0.888	Cluster 4
4326	0.742	Cluster 2
4327	0.726	Cluster 1
4328	0.617	Cluster 3
4329	0.717	Cluster 3
4330	0.670	Cluster 4
4331	0.940	Cluster 2
4332	0.821	Cluster 3
4333	0.923	Cluster 4
4334	0.543	Cluster 2
4335	0.494	Cluster 4
4336	0.902	Cluster 2
4337	0.436	Cluster 1
4338	0.616	Cluster 1
4339	0.622	Cluster 4
4340	0.773	Cluster 1
4341	0.530	Cluster 2
4342	0.388	Cluster 2
4343	0.966	Cluster 5
4344	0.535	Cluster 4
4345	0.780	Cluster 5
4346	0.929	Cluster 2
4347	0.938	Cluster 4
4348	0.486	Cluster 4
4349	0.873	Cluster 5
4350	0.939	Cluster 4
4351	0.677	Cluster 2
4352	0.985	Cluster 5
4353	0.838	Cluster 3
4354	0.292	Cluster 4
4355	0.605	Cluster 2
4356	0.823	Cluster 2
4357	0.378	Cluster 3
4358	0.708	Cluster 2
4359	0.867	Cluster 3
4360	0.535	Cluster 1
4361	0.382	Cluster 2
4362	0.919	Cluster 5
4363	0.751	Cluster 2
4364	0.884	Cluster 4
4365	0.830	Cluster 5
4366	0.470	Cluster 2
4367	0.531	Cluster 5
4368	0.524	Cluster 4
4369	0.469	Cluster 2
4370	0.779	Cluster 4
4371	0.892	Cluster 5
4372	0.673	Cluster 2
4373	0.947	Cluster 5
4374	0.877	Cluster 1
4375	0.908	Cluster 3
4376	0.611	Cluster 2
4377	0.726	Cluster 1
4378	0.838	Cluster 4
4379	0.956	Cluster 4
4380	0.424	Cluster 3
4381	0.720	Cluster 1
4382	0.915	Cluster 3
4383	0.894	Cluster 2
4384	0.564	Cluster 2
4385	0.935	Cluster 4
4386	0.585	Cluster 3
4387	0.830	Cluster 2
4388	0.489	Cluster 1
4389	0.714	Cluster 3
4390	0.740	Cluster 5
4391	0.898	Cluster 4
4392	0.606	Cluster 1
4393	0.282	Cluster 5
4394	0.493	Cluster 3
4395	0.925	Cluster 4
4396	0.494	Cluster 5
4397	0.670	Cluster 3
4398	0.483	Cluster 2
4399	0.907	Cluster 2
4400	0.525	Cluster 4
4401	0.870	Cluster 3
4402	0.892	Cluster 4
4403	0.395	Cluster 1
4404	0.560	Cluster 5
4405	0.841	Cluster 4
4406	0.772	Cluster 1
4407	0.881	Cluster 2
4408	0.356	Cluster 1
4409	0.858	Cluster 1
4410	0.958	Cluster 4
4411	0.498	Cluster 1
4412	0.495	Cluster 1
4413	0.475	Cluster 2
4414	0.502	Cluster 3
4415	0.944	Cluster 5
4416	0.950	Cluster 2
4417	0.354	Cluster 2
4418	0.789	Cluster 4
4419	0.522	Cluster 4
4420	0.884	Cluster 4
4421	0.727	Cluster 1
4422	0.938	Cluster 2
4423	0.468	Cluster 1
4424	0.960	Cluster 4
4425	0.966	Cluster 4
4426	0.590	Cluster 1
4427	0.947	Cluster 3
4428	0.781	Cluster 3
4429	0.537	Cluster 2
4430	0.625	Cluster 4
4431	0.556	Cluster 2
4432	0.646	Cluster 4
4433	0.357	Cluster 3
4434	0.729	Cluster 2
4435	0.389	Cluster 3
4436	0.533	Cluster 2
4437	0.542	Cluster 2
4438	0.358	Cluster 4
4439	0.528	Cluster 4
4440	0.749	Cluster 5
4441	0.485	Cluster 2
4442	0.943	Cluster 4
4443	0.948	Cluster 3
4444	0.359	Cluster 5
4445	0.505	Cluster 1
4446	0.915	Cluster 2
4447	0.501	Cluster 5
4448	0.580	Cluster 3
4449	0.728	Cluster 5
4450	0.791	Cluster 4
4451	0.722	Cluster 3
4452	0.963	Cluster 1
4453	0.584	Cluster 1
4454	0.697	Cluster 1
4455	0.440	Cluster 4
4456	0.668	Cluster 4
4457	0.909	Cluster 2
4458	0.874	Cluster 1
4459	0.684	Cluster 1
4460	0.447	Cluster 5
4461	0.769	Cluster 4
4462	0.827	Cluster 2
4463	0.593	Cluster 1
4464	0.651	Cluster 5
4465	0.578	Cluster 4
4466	0.814	Cluster 4
4467	0.899	Cluster 4
4468	0.791	Cluster 4
4469	0.612	Cluster 2
4470	0.506	Cluster 4
4471	0.935	Cluster 3
4472	0.603	Cluster 3
4473	0.815	Cluster 2
4474	0.869	Cluster 1
4475	0.913	Cluster 4
4476	0.653	Cluster 4
4477	0.935	Cluster 2
4478	0.627	Cluster 5
4479	0.378	Cluster 4
4480	0.939	Cluster 4
4481	0.593	Cluster 3
4482	0.717	Cluster 1
4483	0.721	Cluster 4
4484	0.843	Cluster 2
4485	0.832	Cluster 2
4486	0.742	Cluster 1
4487	0.641	Cluster 4
4488	0.603	Cluster 1
4489	0.546	Cluster 1
4490	0.580	Cluster 4
4491	0.758	Cluster 2
4492	0.671	Cluster 2
4493	0.837	Cluster 1
4494	0.810	Cluster 2
4495	0.831	Cluster 4
4496	0.675	Cluster 4
4497	0.758	Cluster 2
4498	0.876	Cluster 4
4499	0.379	Cluster 2
4500	0.722	Cluster 2
4501	0.471	Cluster 1
4502	0.923	Cluster 4
4503	0.811	Cluster 3
4504	0.941	Cluster 2
4505	0.648	Cluster 3
4506	0.441	Cluster 3
4507	0.830	Cluster 4
4508	0.466	Cluster 5
4509	0.402	Cluster 4
4510	0.595	Cluster 2
4511	0.804	Cluster 2
4512	0.452	Cluster 2
4513	0.884	Cluster 4
4514	0.393	Cluster 1
4515	0.674	Cluster 4
4516	0.744	Cluster 2
4517	0.494	Cluster 3
4518	0.828	Cluster 2
4519	0.793	Cluster 2
4520	0.845	Cluster 2
4521	0.897	Cluster 5
4522	0.507	Cluster 2
4523	0.484	Cluster 2
4524	0.480	Cluster 5
4525	0.601	Cluster 2
4526	0.560	Cluster 2
4527	0.598	Cluster 2
4528	0.975	Cluster 5
4529	0.820	Cluster 2
4530	0.942	Cluster 5
4531	0.603	Cluster 1
4532	0.615	Cluster 3
4533	0.699	Cluster 4
4534	0.873	Cluster 2
4535	0.819	Cluster 3
4536	0.859	Cluster 2
4537	0.539	Cluster 4
4538	0.978	Cluster 5
4539	0.811	Cluster 4
4540	0.419	Cluster 3
4541	0.738	Cluster 5
4542	0.604	Cluster 4
4543	0.633	Cluster 3
4544	0.434	Cluster 2
4545	0.946	Cluster 2
4546	0.608	Cluster 1
4547	0.553	Cluster 2
4548	0.546	Cluster 1
4549	0.967	Cluster 4
4550	0.974	Cluster 5
4551	0.487	Cluster 1
4552	0.790	Cluster 5
4553	0.858	Cluster 2
4554	0.845	Cluster 3
4555	0.591	Cluster 1
4556	0.564	Cluster 2
4557	0.803	Cluster 4
4558	0.477	Cluster 1
4559	0.842	Cluster 2
4560	0.798	Cluster 2
4561	0.514	Cluster 2
4562	0.912	Cluster 4
4563	0.507	Cluster 2
4564	0.690	Cluster 2
4565	0.487	Cluster 4
4566	0.842	Cluster 3
4567	0.456	Cluster 3
4568	0.434	Cluster 4
4569	0.836	Cluster 2
4570	0.444	Cluster 3
4571	0.583	Cluster 2
4572	0.562	Cluster 2
4573	0.857	Cluster 4
4574	0.504	Cluster 4
4575	0.587	Cluster 1
4576	0.988	Cluster 5
4577	0.799	Cluster 3
4578	0.603	Cluster 1
4579	0.836	Cluster 2
4580	0.850	Cluster 4
4581	0.914	Cluster 3
4582	0.637	Cluster 1
4583	0.596	Cluster 2
4584	0.912	Cluster 4
4585	0.725	Cluster 1
4586	0.395	Cluster 2
4587	0.650	Cluster 2
4588	0.550	Cluster 2
4589	0.545	Cluster 4
4590	0.473	Cluster 3
4591	0.974	Cluster 2
4592	0.763	Cluster 4
4593	0.801	Cluster 5
4594	0.877	Cluster 2
4595	0.502	Cluster 4
4596	0.603	Cluster 2
4597	0.421	Cluster 1
4598	0.966	Cluster 1
4599	0.791	Cluster 4
4600	0.419	Cluster 4
4601	0.752	Cluster 1
4602	0.872	Cluster 4
4603	0.518	Cluster 2
4604	0.383	Cluster 1
4605	0.508	Cluster 4
4606	0.568	Cluster 3
4607	0.633	Cluster 1
4608	0.938	Cluster 5
4609	0.608	Cluster 2
4610	0.821	Cluster 5
4611	0.639	Cluster 1
4612	0.851	Cluster 2
4613	0.419	Cluster 4
4614	0.506	Cluster 4
4615	0.950	Cluster 4
4616	0.944	Cluster 5
4617	0.540	Cluster 4
4618	0.435	Cluster 5
4619	0.809	Cluster 2
4620	0.642	Cluster 4
4621	0.675	Cluster 2
4622	0.930	Cluster 5
4623	0.619	Cluster 1
4624	0.506	Cluster 2
4625	0.754	Cluster 4
4626	0.352	Cluster 3
4627	0.445	Cluster 1
4628	0.581	Cluster 2
4629	0.776	Cluster 3
4630	0.684	Cluster 2
4631	0.841	Cluster 4
4632	0.497	Cluster 2
4633	0.916	Cluster 4
4634	0.633	Cluster 1
4635	0.677	Cluster 3
4636	0.828	Cluster 2
4637	0.349	Cluster 1
4638	0.561	Cluster 2
4639	0.739	Cluster 4
4640	0.881	Cluster 2
4641	0.830	Cluster 2
4642	0.563	Cluster 2
4643	0.989	Cluster 4
4644	0.955	Cluster 4
4645	0.493	Cluster 4
4646	0.553	Cluster 2
4647	0.843	Cluster 4
4648	0.688	Cluster 2
4649	0.433	Cluster 2
4650	0.339	Cluster 1
4651	0.495	Cluster 1
4652	0.353	Cluster 3
4653	0.899	Cluster 5
4654	0.565	Cluster 2
4655	0.932	Cluster 5
4656	0.286	Cluster 4
4657	0.676	Cluster 2
4658	0.832	Cluster 2
4659	0.554	Cluster 2
4660	0.603	Cluster 4
4661	0.575	Cluster 4
4662	0.746	Cluster 4
4663	0.801	Cluster 4
4664	0.776	Cluster 4
4665	0.532	Cluster 3
4666	0.963	Cluster 4
4667	0.685	Cluster 2
4668	0.388	Cluster 4
4669	0.857	Cluster 3
4670	0.844	Cluster 5
4671	0.514	Cluster 2
4672	0.939	Cluster 2
4673	0.831	Cluster 2
4674	0.648	Cluster 3
4675	0.948	Cluster 2
4676	0.898	Cluster 2
4677	0.553	Cluster 4
4678	0.690	Cluster 4
4679	0.760	Cluster 3
4680	0.701	Cluster 3
4681	0.643	Cluster 5
4682	0.624	Cluster 3
4683	0.642	Cluster 4
4684	0.452	Cluster 1
4685	0.902	Cluster 2
4686	0.767	Cluster 1
4687	0.761	Cluster 3
4688	0.618	Cluster 5
4689	0.850	Cluster 4
4690	0.861	Cluster 4
4691	0.921	Cluster 4
4692	0.557	Cluster 2
4693	0.634	Cluster 3
4694	0.584	Cluster 2
4695	0.536	Cluster 4
4696	0.616	Cluster 2
4697	0.529	Cluster 2
4698	0.407	Cluster 3
4699	0.428	Cluster 2
4700	0.409	Cluster 5
4701	0.759	Cluster 1
4702	0.871	Cluster 5
4703	0.834	Cluster 4
4704	0.457	Cluster 2
4705	0.633	Cluster 5
4706	0.588	Cluster 5
4707	0.940	Cluster 4
4708	0.719	Cluster 4
4709	0.559	Cluster 3
4710	0.991	Cluster 5
4711	0.542	Cluster 2
4712	0.842	Cluster 1
4713	0.459	Cluster 5
4714	0.957	Cluster 5
4715	0.530	Cluster 1
4716	0.319	Cluster 4
4717	0.478	Cluster 3
4718	0.409	Cluster 5
4719	0.972	Cluster 5
4720	0.495	Cluster 5
4721	0.763	Cluster 2
4722	0.599	Cluster 4
4723	0.549	Cluster 2
4724	0.508	Cluster 4
4725	0.584	Cluster 4
4726	0.486	Cluster 3
4727	0.422	Cluster 3
4728	0.702	Cluster 5
4729	0.823	Cluster 2
4730	0.953	Cluster 5
4731	0.339	Cluster 1
4732	0.397	Cluster 2
4733	0.848	Cluster 1
4734	0.604	Cluster 1
4735	0.718	Cluster 2
4736	0.365	Cluster 1
4737	0.832	Cluster 2
4738	0.975	Cluster 4
4739	0.796	Cluster 2
4740	0.614	Cluster 1
4741	0.851	Cluster 3
4742	0.800	Cluster 2
4743	0.645	Cluster 2
4744	0.351	Cluster 2
4745	0.925	Cluster 2
4746	0.790	Cluster 5
4747	0.438	Cluster 2
4748	0.830	Cluster 2
4749	0.851	Cluster 2
4750	0.540	Cluster 2
4751	0.386	Cluster 4
4752	0.878	Cluster 3
4753	0.765	Cluster 3
4754	0.352	Cluster 4
4755	0.558	Cluster 1
4756	0.426	Cluster 1
4757	0.926	Cluster 4
4758	0.676	Cluster 1
4759	0.465	Cluster 2
4760	0.677	Cluster 1
4761	0.452	Cluster 1
4762	0.798	Cluster 4
4763	0.519	Cluster 1
4764	0.689	Cluster 2
4765	0.947	Cluster 2
4766	0.401	Cluster 2
4767	0.624	Cluster 1
4768	0.423	Cluster 3
4769	0.796	Cluster 2
4770	0.455	Cluster 2
4771	0.464	Cluster 2
4772	0.683	Cluster 2
4773	0.696	Cluster 2
4774	0.462	Cluster 5
4775	0.722	Cluster 1
4776	0.661	Cluster 3
4777	0.912	Cluster 5
4778	0.693	Cluster 3
4779	0.662	Cluster 2
4780	0.642	Cluster 5
4781	0.829	Cluster 2
4782	0.694	Cluster 4
4783	0.778	Cluster 3
4784	0.722	Cluster 2
4785	0.960	Cluster 1
4786	0.756	Cluster 2
4787	0.792	Cluster 2
4788	0.951	Cluster 4
4789	0.818	Cluster 4
4790	0.717	Cluster 1
4791	0.783	Cluster 2
4792	0.477	Cluster 1
4793	0.683	Cluster 2
4794	0.712	Cluster 2
4795	0.544	Cluster 2
4796	0.372	Cluster 1
4797	0.803	Cluster 1
4798	0.543	Cluster 5
4799	0.653	Cluster 2
4800	0.731	Cluster 2
4801	0.475	Cluster 5
4802	0.811	Cluster 3
4803	0.868	Cluster 4
4804	0.770	Cluster 4
4805	0.957	Cluster 4
4806	0.681	Cluster 4
4807	0.406	Cluster 1
4808	0.702	Cluster 2
4809	0.862	Cluster 1
4810	0.404	Cluster 3
4811	0.791	Cluster 4
4812	0.384	Cluster 3
4813	0.812	Cluster 3
4814	0.929	Cluster 3
4815	0.822	Cluster 2
4816	0.484	Cluster 5
4817	0.957	Cluster 4
4818	0.874	Cluster 5
4819	0.425	Cluster 4
4820	0.515	Cluster 2
4821	0.912	Cluster 4
4822	0.494	Cluster 3
4823	0.740	Cluster 3
4824	0.886	Cluster 2
4825	0.710	Cluster 2
4826	0.531	Cluster 1
4827	0.570	Cluster 4
4828	0.975	Cluster 5
4829	0.595	Cluster 2
4830	0.436	Cluster 3
4831	0.520	Cluster 5
4832	0.556	Cluster 2
4833	0.949	Cluster 4
4834	0.690	Cluster 2
4835	0.693	Cluster 2
4836	0.719	Cluster 2
4837	0.406	Cluster 3
4838	0.638	Cluster 1
4839	0.416	Cluster 1
4840	0.915	Cluster 4
4841	0.950	Cluster 3
4842	0.916	Cluster 4
4843	0.954	Cluster 4
4844	0.538	Cluster 5
4845	0.846	Cluster 4
4846	0.716	Cluster 4
4847	0.528	Cluster 2
4848	0.629	Cluster 2
4849	0.878	Cluster 2
4850	0.531	Cluster 2
4851	0.470	Cluster 4
4852	0.558	Cluster 1
4853	0.820	Cluster 2
4854	0.918	Cluster 5
4855	0.722	Cluster 2
4856	0.889	Cluster 4
4857	0.933	Cluster 4
4858	0.868	Cluster 1
4859	0.907	Cluster 3
4860	0.669	Cluster 2
4861	0.779	Cluster 5
4862	0.651	Cluster 3
4863	0.951	Cluster 4
4864	0.695	Cluster 5
4865	0.527	Cluster 2
4866	0.609	Cluster 2
4867	0.524	Cluster 2
4868	0.683	Cluster 1
4869	0.888	Cluster 4
4870	0.956	Cluster 5
4871	0.931	Cluster 4
4872	0.816	Cluster 1
4873	0.736	Cluster 2
4874	0.280	Cluster 4
4875	0.333	Cluster 5
4876	0.607	Cluster 1
4877	0.417	Cluster 2
4878	0.872	Cluster 3
4879	0.939	Cluster 4
4880	0.744	Cluster 3
4881	0.730	Cluster 3
4882	0.842	Cluster 2
4883	0.701	Cluster 2
4884	0.639	Cluster 3
4885	0.928	Cluster 5
4886	0.682	Cluster 4
4887	0.416	Cluster 4
4888	0.341	Cluster 3
4889	0.467	Cluster 4
4890	0.471	Cluster 4
4891	0.576	Cluster 3
4892	0.969	Cluster 5
4893	0.551	Cluster 2
4894	0.412	Cluster 4
4895	0.473	Cluster 5
4896	0.622	Cluster 1
4897	0.440	Cluster 4
4898	0.837	Cluster 3
4899	0.571	Cluster 1
4900	0.562	Cluster 3
4901	0.448	Cluster 3
4902	0.849	Cluster 5
4903	0.749	Cluster 4
4904	0.935	Cluster 1
4905	0.596	Cluster 1
4906	0.343	Cluster 3
4907	0.530	Cluster 2
4908	0.928	Cluster 2
4909	0.444	Cluster 5
4910	0.678	Cluster 2
4911	0.811	Cluster 3
4912	0.582	Cluster 5
4913	0.549	Cluster 3
4914	0.913	Cluster 1
4915	0.561	Cluster 1
4916	0.954	Cluster 4
4917	0.364	Cluster 5
4918	0.327	Cluster 3
4919	0.812	Cluster 1
4920	0.407	Cluster 3
4921	0.693	Cluster 5
4922	0.584	Cluster 2
4923	0.468	Cluster 3
4924	0.641	Cluster 1
4925	0.472	Cluster 5
4926	0.535	Cluster 2
4927	0.760	Cluster 1
4928	0.701	Cluster 2
4929	0.858	Cluster 3
4930	0.497	Cluster 5
4931	0.675	Cluster 4
4932	0.961	Cluster 2
4933	0.835	Cluster 4
4934	0.563	Cluster 1
4935	0.552	Cluster 2
4936	0.831	Cluster 5
4937	0.906	Cluster 4
4938	0.580	Cluster 1
4939	0.537	Cluster 3
4940	0.532	Cluster 4
4941	0.588	Cluster 2
4942	0.468	Cluster 2
4943	0.852	Cluster 5
4944	0.803	Cluster 2
4945	0.983	Cluster 4
4946	0.893	Cluster 4
4947	0.551	Cluster 1
4948	0.736	Cluster 1
4949	0.712	Cluster 2
4950	0.620	Cluster 1
4951	0.710	Cluster 2
4952	0.715	Cluster 2
4953	0.421	Cluster 2
4954	0.926	Cluster 4
4955	0.802	Cluster 2
4956	0.802	Cluster 1
4957	0.895	Cluster 2
4958	0.561	Cluster 1
4959	0.454	Cluster 2
4960	0.511	Cluster 2
4961	0.624	Cluster 5
4962	0.948	Cluster 2
4963	0.782	Cluster 3
4964	0.539	Cluster 2
4965	0.626	Cluster 4
4966	0.412	Cluster 1
4967	0.738	Cluster 2
4968	0.949	Cluster 4
4969	0.865	Cluster 5
4970	0.820	Cluster 5
4971	0.815	Cluster 2
4972	0.894	Cluster 4
4973	0.571	Cluster 2
4974	0.717	Cluster 1
4975	0.867	Cluster 2
4976	0.908	Cluster 1
4977	0.379	Cluster 2
4978	0.748	Cluster 3
4979	0.714	Cluster 2
4980	0.548	Cluster 5
4981	0.875	Cluster 5
4982	0.714	Cluster 4
4983	0.792	Cluster 2
4984	0.860	Cluster 4
4985	0.950	Cluster 3
4986	0.888	Cluster 2
4987	0.977	Cluster 4
4988	0.736	Cluster 1
4989	0.668	Cluster 4
4990	0.834	Cluster 4
4991	0.812	Cluster 2
4992	0.866	Cluster 1
4993	0.575	Cluster 5
4994	0.801	Cluster 5
4995	0.712	Cluster 5
4996	0.715	Cluster 2
4997	0.541	Cluster 4
4998	0.750	Cluster 2
4999	0.940	Cluster 4
5000	0.371	Cluster 5

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	1	2	6	18	7	4	14	12	13
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	7	16	8	5	12	9	13
A9	8	16	18	19	12	13	6	20	5	3	7	1	4	2	17	10	9	15	14	11
A10	7	17	19	18	9	15	5	20	3	10	11	1	2	6	16	8	4	13	12	14
A11	12	16	19	15	13	18	7	20	3	5	11	1	2	6	17	10	4	14	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	13	10	11

	P7	P9	P10	P12	P13	P14	P16	P17
A1	4	3	5	1	2	NA	NA	NA
A2	NA	4	3	1	2	NA	NA	5
A3	NA	NA	3	2	1	4	5	NA
A4	5	3	4	1	2	NA	NA	NA
A5	2	3	NA	1	4	5	NA	NA
A6	NA	3	NA	1	2	4	NA	5
A7	5	3	NA	1	2	NA	NA	4
A8	NA	4	3	1	2	NA	NA	5
A9	NA	5	3	1	4	2	NA	NA
A10	5	3	NA	1	2	NA	NA	4
A11	NA	3	5	1	2	NA	NA	4
A12	NA	3	4	1	2	NA	NA	5

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	NA	19	NA	6	16	4	20	3	5	12	1	2	9	NA	8	7	14	13	11
A2	10	NA	19	17	11	NA	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	NA	12
A5	12	17	19	NA	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	NA	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	NA	2	6	18	7	4	14	12	NA
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	NA	16	8	5	12	9	13
A9	8	NA	18	NA	12	13	6	20	5	3	NA	1	4	NA	17	10	9	15	14	11
A10	7	17	19	18	NA	15	NA	20	3	10	11	1	2	6	16	8	NA	13	12	14
A11	NA	16	19	15	NA	18	7	20	3	NA	11	NA	2	6	17	10	4	NA	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	NA	10	NA

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
7	15	3	14	13	2	10	12	6	4	1	5	8	11	9
5	15	14	6	2	13	12	11	1	7	4	10	9	8	3
3	9	7	15	14	2	6	13	1	8	12	11	10	5	4
5	15	2	14	11	8	9	10	1	4	3	13	7	12	6
9	15	7	5	6	1	14	13	4	3	2	12	11	8	10
9	13	5	12	15	3	7	11	1	6	4	14	8	10	2

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	12	5	2	4	11	6	16	1	18	13	10	20	19	15	3	14	17	7	9	8
A8	7	3	1	2	10	6	15	4	17	18	11	20	19	14	5	13	16	9	12	8
A9	13	5	3	2	9	8	15	1	16	18	14	20	17	19	4	11	12	6	7	10
A10	14	4	2	3	12	6	16	1	18	11	10	20	19	15	5	13	17	8	9	7
A11	9	5	2	6	8	3	14	1	18	16	10	20	19	15	4	11	17	7	13	12
A12	7	6	2	5	9	3	13	1	18	17	12	20	19	14	4	15	16	8	11	10

shrimp	sea eel	tuna	squid	sea urchin	salmon roe	egg	fatty tuna	tuna roll	cucumber roll
2	8	10	3	4	1	5	9	7	6
1	8	6	4	10	9	3	5	7	2
2	8	3	4	6	7	10	1	5	9
4	7	5	6	1	2	8	3	9	10
4	10	7	5	9	3	2	8	1	6
4	6	2	10	7	5	1	9	8	3

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	1	2	6	18	7	4	14	12	13
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	7	16	8	5	12	9	13
A9	8	16	18	19	12	13	6	20	5	3	7	1	4	2	17	10	9	15	14	11
A10	7	17	19	18	9	15	5	20	3	10	11	1	2	6	16	8	4	13	12	14
A11	12	16	19	15	13	18	7	20	3	5	11	1	2	6	17	10	4	14	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	13	10	11

	P7	P9	P10	P12	P13	P14	P16	P17
A1	4	3	5	1	2	NA	NA	NA
A2	NA	4	3	1	2	NA	NA	5
A3	NA	NA	3	2	1	4	5	NA
A4	5	3	4	1	2	NA	NA	NA
A5	2	3	NA	1	4	5	NA	NA
A6	NA	3	NA	1	2	4	NA	5
A7	5	3	NA	1	2	NA	NA	4
A8	NA	4	3	1	2	NA	NA	5
A9	NA	5	3	1	4	2	NA	NA
A10	5	3	NA	1	2	NA	NA	4
A11	NA	3	5	1	2	NA	NA	4
A12	NA	3	4	1	2	NA	NA	5

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	NA	19	NA	6	16	4	20	3	5	12	1	2	9	NA	8	7	14	13	11
A2	10	NA	19	17	11	NA	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	NA	12
A5	12	17	19	NA	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	NA	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	NA	2	6	18	7	4	14	12	NA
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	NA	16	8	5	12	9	13
A9	8	NA	18	NA	12	13	6	20	5	3	NA	1	4	NA	17	10	9	15	14	11
A10	7	17	19	18	NA	15	NA	20	3	10	11	1	2	6	16	8	NA	13	12	14
A11	NA	16	19	15	NA	18	7	20	3	NA	11	NA	2	6	17	10	4	NA	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	NA	10	NA

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
7	15	3	14	13	2	10	12	6	4	1	5	8	11	9
5	15	14	6	2	13	12	11	1	7	4	10	9	8	3
3	9	7	15	14	2	6	13	1	8	12	11	10	5	4
5	15	2	14	11	8	9	10	1	4	3	13	7	12	6
9	15	7	5	6	1	14	13	4	3	2	12	11	8	10
9	13	5	12	15	3	7	11	1	6	4	14	8	10	2

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	12	5	2	4	11	6	16	1	18	13	10	20	19	15	3	14	17	7	9	8
A8	7	3	1	2	10	6	15	4	17	18	11	20	19	14	5	13	16	9	12	8
A9	13	5	3	2	9	8	15	1	16	18	14	20	17	19	4	11	12	6	7	10
A10	14	4	2	3	12	6	16	1	18	11	10	20	19	15	5	13	17	8	9	7
A11	9	5	2	6	8	3	14	1	18	16	10	20	19	15	4	11	17	7	13	12
A12	7	6	2	5	9	3	13	1	18	17	12	20	19	14	4	15	16	8	11	10

shrimp	sea eel	tuna	squid	sea urchin	salmon roe	egg	fatty tuna	tuna roll	cucumber roll
2	8	10	3	4	1	5	9	7	6
1	8	6	4	10	9	3	5	7	2
2	8	3	4	6	7	10	1	5	9
4	7	5	6	1	2	8	3	9	10
4	10	7	5	9	3	2	8	1	6
4	6	2	10	7	5	1	9	8	3

BayesMallows: An R Package for Probabilistic Preference Learning with the Mallows Rank Model

Øystein Sørensen

2018-09-30

Mixture of Mallows Models

Overview of Package

Functions

Datasets

Mallows’ Rank Model

Prior Distributions

Metropolis-Hastings Algorithm

Partial Rankings

Pairwise Comparisons

Mixtures of Mallows Models

BayesMallows User Guide

Completely Ranked Data

Algorithm Tuning

Posterior Distributions

Scale Parameter \(\alpha\)

Latent Ranks \(\rho\)

Jumping over \(\alpha\)

Thinning

Varying the Distance Metric

Validation of Input

Top-\(k\) Rankings

Encoding of Missing Ranks

Metropolis-Hastings Algorithm with Missing Ranks

Algorithm Tuning

Posterior Distributions

Other Distance Measures

Ranks Missing at Random

Algorithm Tuning

Posterior Distributions

Pairwise Preferences

Introduction

Data Model

Transitive Closure

Initial Ranking

Beach Preferences

Transitive Closures

Initial Ranking

Algorithm Tuning

Convergence of Rtilde

Assessor 1

Assessor 2

Posterior Distributions

CP Consensus

MAP Consensus

Posterior Intervals

Posterior Probability of Being Ranked Top-\(k\)

Clustering of Assessors

Introduction

Algorithm Tuning

Convergence of \(\alpha\)

Convergence of \(\rho\)

Convergence of Cluster Probabilities

Sushi Data

Algorithm Tuning

Posterior Distributions

CP Consensus

Determining the Number of Clusters

References

`BayesMallows`: An R Package for Probabilistic Preference Learning with the Mallows Rank Model

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	1	2	6	18	7	4	14	12	13
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	7	16	8	5	12	9	13
A9	8	16	18	19	12	13	6	20	5	3	7	1	4	2	17	10	9	15	14	11
A10	7	17	19	18	9	15	5	20	3	10	11	1	2	6	16	8	4	13	12	14
A11	12	16	19	15	13	18	7	20	3	5	11	1	2	6	17	10	4	14	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	13	10	11

	P7	P9	P10	P12	P13	P14	P16	P17
A1	4	3	5	1	2	NA	NA	NA
A2	NA	4	3	1	2	NA	NA	5
A3	NA	NA	3	2	1	4	5	NA
A4	5	3	4	1	2	NA	NA	NA
A5	2	3	NA	1	4	5	NA	NA
A6	NA	3	NA	1	2	4	NA	5
A7	5	3	NA	1	2	NA	NA	4
A8	NA	4	3	1	2	NA	NA	5
A9	NA	5	3	1	4	2	NA	NA
A10	5	3	NA	1	2	NA	NA	4
A11	NA	3	5	1	2	NA	NA	4
A12	NA	3	4	1	2	NA	NA	5

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	NA	19	NA	6	16	4	20	3	5	12	1	2	9	NA	8	7	14	13	11
A2	10	NA	19	17	11	NA	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	NA	12
A5	12	17	19	NA	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	NA	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	9	16	19	17	10	15	5	20	3	8	11	NA	2	6	18	7	4	14	12	NA
A8	14	18	20	19	11	15	6	17	4	3	10	1	2	NA	16	8	5	12	9	13
A9	8	NA	18	NA	12	13	6	20	5	3	NA	1	4	NA	17	10	9	15	14	11
A10	7	17	19	18	NA	15	NA	20	3	10	11	1	2	6	16	8	NA	13	12	14
A11	NA	16	19	15	NA	18	7	20	3	NA	11	NA	2	6	17	10	4	NA	8	9
A12	14	15	19	16	12	18	8	20	3	4	9	1	2	7	17	6	5	NA	10	NA

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
7	15	3	14	13	2	10	12	6	4	1	5	8	11	9
5	15	14	6	2	13	12	11	1	7	4	10	9	8	3
3	9	7	15	14	2	6	13	1	8	12	11	10	5	4
5	15	2	14	11	8	9	10	1	4	3	13	7	12	6
9	15	7	5	6	1	14	13	4	3	2	12	11	8	10
9	13	5	12	15	3	7	11	1	6	4	14	8	10	2

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
A1	10	18	19	15	6	16	4	20	3	5	12	1	2	9	17	8	7	14	13	11
A2	10	18	19	17	11	15	6	20	4	3	13	1	2	7	16	8	5	12	9	14
A3	12	15	18	16	13	11	7	20	6	3	8	2	1	4	19	5	9	14	10	17
A4	9	17	19	16	10	15	5	20	3	4	8	1	2	7	18	11	6	13	14	12
A5	12	17	19	15	7	16	2	20	3	9	13	1	4	5	18	11	6	8	10	14
A6	10	15	19	16	8	18	6	20	3	7	11	1	2	4	17	9	5	13	12	14
A7	12	5	2	4	11	6	16	1	18	13	10	20	19	15	3	14	17	7	9	8
A8	7	3	1	2	10	6	15	4	17	18	11	20	19	14	5	13	16	9	12	8
A9	13	5	3	2	9	8	15	1	16	18	14	20	17	19	4	11	12	6	7	10
A10	14	4	2	3	12	6	16	1	18	11	10	20	19	15	5	13	17	8	9	7
A11	9	5	2	6	8	3	14	1	18	16	10	20	19	15	4	11	17	7	13	12
A12	7	6	2	5	9	3	13	1	18	17	12	20	19	14	4	15	16	8	11	10

shrimp	sea eel	tuna	squid	sea urchin	salmon roe	egg	fatty tuna	tuna roll	cucumber roll
2	8	10	3	4	1	5	9	7	6
1	8	6	4	10	9	3	5	7	2
2	8	3	4	6	7	10	1	5	9
4	7	5	6	1	2	8	3	9	10
4	10	7	5	9	3	2	8	1	6
4	6	2	10	7	5	1	9	8	3