Getting Started with NNS: Clustering and Regression

Fred Viole

library(NNS)
library(data.table)
require(knitr)
require(rgl)
require(meboot)
require(tdigest)
require(dtw)

Clustering and Regression

Below are some examples demonstrating unsupervised learning with NNS clustering and nonlinear regression using the resulting clusters. As always, for a more thorough description and definition, please view the References.

NNS Partitioning NNS.part()

NNS.part is both a partitional and hierarchical clustering method. NNS iteratively partitions the joint distribution into partial moment quadrants, and then assigns a quadrant identification (1:4) at each partition.

NNS.part returns a data.table of observations along with their final quadrant identification. It also returns the regression points, which are the quadrant means used in NNS.reg.

x = seq(-5, 5, .05); y = x ^ 3

for(i in 1 : 4){NNS.part(x, y, order = i, Voronoi = TRUE, obs.req = 0)}

X-only Partitioning

NNS.part offers a partitioning based on \(x\) values only NNS.part(x, y, type = "XONLY", ...), using the entire bandwidth in its regression point derivation, and shares the same limit condition as partitioning via both \(x\) and \(y\) values.

for(i in 1 : 4){NNS.part(x, y, order = i, type = "XONLY", Voronoi = TRUE)}

Note the partition identifications are limited to 1’s and 2’s (left and right of the partition respectively), not the 4 values per the \(x\) and \(y\) partitioning.

## $order
## [1] 4
## 
## $dt
##          x         y quadrant prior.quadrant
##   1: -5.00 -125.0000    q1111           q111
##   2: -4.95 -121.2874    q1111           q111
##   3: -4.90 -117.6490    q1111           q111
##   4: -4.85 -114.0841    q1111           q111
##   5: -4.80 -110.5920    q1111           q111
##  ---                                        
## 197:  4.80  110.5920    q2222           q222
## 198:  4.85  114.0841    q2222           q222
## 199:  4.90  117.6490    q2222           q222
## 200:  4.95  121.2874    q2222           q222
## 201:  5.00  125.0000    q2222           q222
## 
## $regression.points
##    quadrant          x           y
## 1:     q111 -4.3733596 -79.5965469
## 2:     q112 -3.0984252 -27.8292362
## 3:     q121 -1.8484252  -5.7529911
## 4:     q122 -0.5984252  -0.2162835
## 5:     q211  0.6515748   0.2682589
## 6:     q212  1.9015748   6.2296142
## 7:     q221  3.1515748  29.2102884
## 8:     q222  4.3984252  81.2340157

Clusters Used in Regression

The right column of plots shows the corresponding regression for the order of NNS partitioning.

for(i in 1 : 3){NNS.part(x, y, order = i, obs.req = 0, Voronoi = TRUE) ; NNS.reg(x, y, order = i, ncores = 1)}

NNS Regression NNS.reg()

NNS.reg can fit any \(f(x)\), for both uni- and multivariate cases. NNS.reg returns a self-evident list of values provided below.

Univariate:

NNS.reg(x, y, ncores = 1)

## $R2
## [1] 1
## 
## $SE
## [1] 0
## 
## $Prediction.Accuracy
## NULL
## 
## $equation
## NULL
## 
## $x.star
## NULL
## 
## $derivative
##      Coefficient X.Lower.Range X.Upper.Range
##   1:     74.2525         -5.00         -4.95
##   2:     72.7675         -4.95         -4.90
##   3:     71.2975         -4.90         -4.85
##   4:     69.8425         -4.85         -4.80
##   5:     68.4025         -4.80         -4.75
##  ---                                        
## 196:     68.4025          4.75          4.80
## 197:     69.8425          4.80          4.85
## 198:     71.2975          4.85          4.90
## 199:     72.7675          4.90          4.95
## 200:     74.2525          4.95          5.00
## 
## $Point.est
## NULL
## 
## $regression.points
##          x         y
##   1: -5.00 -125.0000
##   2: -4.95 -121.2874
##   3: -4.90 -117.6490
##   4: -4.85 -114.0841
##   5: -4.80 -110.5920
##  ---                
## 197:  4.80  110.5920
## 198:  4.85  114.0841
## 199:  4.90  117.6490
## 200:  4.95  121.2874
## 201:  5.00  125.0000
## 
## $Fitted.xy
##          x         y     y.hat      NNS.ID gradient residuals
##   1: -5.00 -125.0000 -125.0000 q4444444444  74.2525         0
##   2: -4.95 -121.2874 -121.2874 q4444441444  72.7675         0
##   3: -4.90 -117.6490 -117.6490 q4444432222  71.2975         0
##   4: -4.85 -114.0841 -114.0841 q4444414444  69.8425         0
##   5: -4.80 -110.5920 -110.5920 q4444411444  68.4025         0
##  ---                                                         
## 197:  4.80  110.5920  110.5920 q1111142444  69.8425         0
## 198:  4.85  114.0841  114.0841 q1111141444  71.2975         0
## 199:  4.90  117.6490  117.6490 q1111114444  72.7675         0
## 200:  4.95  121.2874  121.2874 q1111112222  74.2525         0
## 201:  5.00  125.0000  125.0000 q1111111444  74.2525         0

Multivariate:

Multivariate regressions return a plot of \(y\) and \(\hat{y}\), as well as the regression points ($RPM) and partitions ($rhs.partitions) for each regressor.

f= function(x, y) x ^ 3 + 3 * y - y ^ 3 - 3 * x
y = x ; z = expand.grid(x, y)
g = f(z[ , 1], z[ , 2])
NNS.reg(z, g, order = "max", ncores = 1)

## $R2
## [1] 1
## 
## $rhs.partitions
##         Var1 Var2
##     1: -5.00   -5
##     2: -4.95   -5
##     3: -4.90   -5
##     4: -4.85   -5
##     5: -4.80   -5
##    ---           
## 40397:  4.80    5
## 40398:  4.85    5
## 40399:  4.90    5
## 40400:  4.95    5
## 40401:  5.00    5
## 
## $RPM
##        Var1  Var2         y.hat
##     1: -4.8 -4.80 -7.105427e-15
##     2: -4.8 -2.55 -8.726063e+01
##     3: -4.8 -2.50 -8.806700e+01
##     4: -4.8 -2.45 -8.883587e+01
##     5: -4.8 -2.40 -8.956800e+01
##    ---                         
## 40397: -2.6 -2.80  3.776000e+00
## 40398: -2.6 -2.75  2.770875e+00
## 40399: -2.6 -2.70  1.807000e+00
## 40400: -2.6 -2.65  8.836250e-01
## 40401: -2.6 -2.60  1.776357e-15
## 
## $Point.est
## NULL
## 
## $Fitted.xy
##         Var1 Var2          y      y.hat      NNS.ID residuals
##     1: -5.00   -5   0.000000   0.000000     201.201         0
##     2: -4.95   -5   3.562625   3.562625     402.201         0
##     3: -4.90   -5   7.051000   7.051000     603.201         0
##     4: -4.85   -5  10.465875  10.465875     804.201         0
##     5: -4.80   -5  13.808000  13.808000    1005.201         0
##    ---                                                       
## 40397:  4.80    5 -13.808000 -13.808000 39597.40401         0
## 40398:  4.85    5 -10.465875 -10.465875 39798.40401         0
## 40399:  4.90    5  -7.051000  -7.051000 39999.40401         0
## 40400:  4.95    5  -3.562625  -3.562625 40200.40401         0
## 40401:  5.00    5   0.000000   0.000000 40401.40401         0

Inter/Extrapolation

NNS.reg can inter- or extrapolate any point of interest. The NNS.reg(x, y, point.est = ...) parameter permits any sized data of similar dimensions to \(x\) and called specifically with NNS.reg(...)$Point.est.

NNS Dimension Reduction Regression

NNS.reg also provides a dimension reduction regression by including a parameter NNS.reg(x, y, dim.red.method = "cor", ...). Reducing all regressors to a single dimension using the returned equation NNS.reg(..., dim.red.method = "cor", ...)$equation.

NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft", ncores = 1)$equation

##        Variable Coefficient
## 1: Sepal.Length   0.7980781
## 2:  Sepal.Width  -0.4402896
## 3: Petal.Length   0.9354305
## 4:  Petal.Width   0.9381792
## 5:  DENOMINATOR   4.0000000

Thus, our model for this regression would be: \[Species = \frac{0.798*Sepal.Length -0.44*Sepal.Width +0.935*Petal.Length +0.938*Petal.Width}{4} \]

Threshold

NNS.reg(x, y, dim.red.method = "cor", threshold = ...) offers a method of reducing regressors further by controlling the absolute value of required correlation.

NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, location = "topleft", ncores = 1)$equation

##        Variable Coefficient
## 1: Sepal.Length   0.7980781
## 2:  Sepal.Width   0.0000000
## 3: Petal.Length   0.9354305
## 4:  Petal.Width   0.9381792
## 5:  DENOMINATOR   3.0000000

Thus, our model for this further reduced dimension regression would be: \[Species = \frac{\: 0.798*Sepal.Length + 0*Sepal.Width +0.935*Petal.Length +0.938*Petal.Width}{3} \]

and the point.est = (...) operates in the same manner as the full regression above, again called with NNS.reg(...)$Point.est.

NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, point.est = iris[1 : 10, 1 : 4], location = "topleft", ncores = 1)$Point.est

##  [1] 1 1 1 1 1 1 1 1 1 1

Classification

For a classification problem, we simply set NNS.reg(x, y, type = "CLASS", ...).

NOTE: Base category of response variable should be 1, not 0 for classification problems.

NNS.reg(iris[ , 1 : 4], iris[ , 5], type = "CLASS", point.est = iris[1 : 10, 1 : 4], location = "topleft", ncores = 1)$Point.est

##  [1] 1 1 1 1 1 1 1 1 1 1

Cross-Validation NNS.stack()

The NNS.stack routine cross-validates for a given objective function the n.best parameter in the multivariate NNS.reg function as well as the threshold parameter in the dimension reduction NNS.reg version. NNS.stack can be used for classification:

NNS.stack(..., type = "CLASS", ...)

or continuous dependent variables:

NNS.stack(..., type = NULL, ...).

Any objective function obj.fn can be called using expression() with the terms predicted and actual.

NNS.stack(IVs.train = iris[ , 1 : 4], 
          DV.train = iris[ , 5], 
          IVs.test = iris[1 : 10, 1 : 4],
          obj.fn = expression( mean(round(predicted) == actual) ),
          objective = "max", type = "CLASS", 
          folds = 1, ncores = 1)
## $OBJfn.reg
## [1] 0.9733333
## 
## $NNS.reg.n.best
## [1] 1
## 
## $probability.threshold
## [1] 0.43875
## 
## $OBJfn.dim.red
## [1] 0.9533333
## 
## $NNS.dim.red.threshold
## [1] 0.43
## 
## $reg
##  [1] 1 1 1 1 1 1 1 1 1 1
## 
## $dim.red
##  [1] 1 1 1 1 1 1 1 1 1 1
## 
## $stack
##  [1] 1 1 1 1 1 1 1 1 1 1

Increasing Dimensions

Given multicollinearity is not an issue for nonparametric regressions as it is for OLS, in the case of an ill-fit univariate model a better option may be to increase the dimensionality of regressors with a copy of itself and cross-validate the number of clusters n.best via:

NNS.stack(IVs.train = cbind(x, x), DV.train = y, method = 1, ...).

set.seed(123)
x <- rnorm(100); y <- rnorm(100)

nns.params <- NNS.stack(IVs.train = cbind(x, x),
                        DV.train = y,
                        method = 1)

NNS.reg(cbind(x, x), y, 
        n.best = nns.params$NNS.reg.n.best,
        point.est = cbind(x, x))

References

If the user is so motivated, detailed arguments further examples are provided within the following: