Getting Started with NNS: Clustering and Regression

Fred Viole

require(NNS)
require(knitr)
require(rgl)
require(data.table)

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.

## $order
## [1] 4
## 
## $dt
##          x         y quadrant prior.quadrant
##   1: -5.00 -125.0000    q4444           q444
##   2: -4.95 -121.2874    q4444           q444
##   3: -4.90 -117.6490    q4444           q444
##   4: -4.85 -114.0841    q4444           q444
##   5: -4.80 -110.5920    q4444           q444
##  ---                                        
## 197:  4.80  110.5920    q1111           q111
## 198:  4.85  114.0841    q1111           q111
## 199:  4.90  117.6490    q1111           q111
## 200:  4.95  121.2874    q1111           q111
## 201:  5.00  125.0000    q1111           q111
## 
## $regression.points
##     quadrant      x          y
##  1:     q111  4.600  98.164000
##  2:     q113  4.150  71.473375
##  3:     q114  3.650  49.448375
##  4:     q131  3.025  27.746813
##  5:     q134  2.700  19.764000
##  6:     q141  2.050   9.076375
##  7:     q143  1.425   2.924813
##  8:     q144  0.650   0.528125
##  9:     q411 -0.600  -0.450000
## 10:     q412 -1.400  -2.786000
## 11:     q414 -2.025  -8.712562
## 12:     q421 -2.650 -18.689125
## 13:     q424 -3.000 -27.090000
## 14:     q441 -3.625 -48.366563
## 15:     q442 -4.125 -70.197187
## 16:     q444 -4.600 -98.164000

X-only Partitioning

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

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 -4.375 -85.585938
## 2:     q112 -3.100 -31.000000
## 3:     q121 -1.850  -7.053125
## 4:     q122 -0.600  -0.450000
## 5:     q211  0.650   0.528125
## 6:     q212  1.900   7.600000
## 7:     q221  3.150  32.484375
## 8:     q222  4.400  86.900000

Clusters Used in Regression

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

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:

## $R2
## [1] 0.9998899
## 
## $SE
## [1] 0.7461974
## 
## $Prediction.Accuracy
## NULL
## 
## $equation
## NULL
## 
## $x.star
## NULL
## 
## $derivative
##     Coefficient X.Lower.Range X.Upper.Range
##  1:    67.09000        -5.000        -4.600
##  2:    58.87750        -4.600        -4.125
##  3:    43.66125        -4.125        -3.625
##  4:    34.04250        -3.625        -3.000
##  5:    24.00250        -3.000        -2.650
##  6:    15.96250        -2.650        -2.025
##  7:     9.48250        -2.025        -1.400
##  8:     2.92000        -1.400        -0.600
##  9:     0.78250        -0.600         0.650
## 10:     3.09250         0.650         1.425
## 11:     9.84250         1.425         2.050
## 12:    16.44250         2.050         2.700
## 13:    24.56250         2.700         3.025
## 14:    34.72250         3.025         3.650
## 15:    44.05000         3.650         4.150
## 16:    59.31250         4.150         4.600
## 17:    67.09000         4.600         5.000
## 
## $Point
## NULL
## 
## $Point.est
## NULL
## 
## $regression.points
##          x           y
##  1: -5.000 -125.000000
##  2: -4.600  -98.164000
##  3: -4.125  -70.197187
##  4: -3.625  -48.366563
##  5: -3.000  -27.090000
##  6: -2.650  -18.689125
##  7: -2.025   -8.712562
##  8: -1.400   -2.786000
##  9: -0.600   -0.450000
## 10:  0.650    0.528125
## 11:  1.425    2.924813
## 12:  2.050    9.076375
## 13:  2.700   19.764000
## 14:  3.025   27.746813
## 15:  3.650   49.448375
## 16:  4.150   71.473375
## 17:  4.600   98.164000
## 18:  5.000  125.000000
## 
## $Fitted
##          y.hat
##   1: -125.0000
##   2: -121.6455
##   3: -118.2910
##   4: -114.9365
##   5: -111.5820
##  ---          
## 197:  111.5820
## 198:  114.9365
## 199:  118.2910
## 200:  121.6455
## 201:  125.0000
## 
## $Fitted.xy
##          x         y     y.hat NNS.ID gradient
##   1: -5.00 -125.0000 -125.0000  q4444    67.09
##   2: -4.95 -121.2874 -121.6455  q4444    67.09
##   3: -4.90 -117.6490 -118.2910  q4444    67.09
##   4: -4.85 -114.0841 -114.9365  q4444    67.09
##   5: -4.80 -110.5920 -111.5820  q4444    67.09
##  ---                                          
## 197:  4.80  110.5920  111.5820  q1111    67.09
## 198:  4.85  114.0841  114.9365  q1111    67.09
## 199:  4.90  117.6490  118.2910  q1111    67.09
## 200:  4.95  121.2874  121.6455  q1111    67.09
## 201:  5.00  125.0000  125.0000  q1111    67.09

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.

## $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
##             y.hat
##     1:   0.000000
##     2:   3.562625
##     3:   7.051000
##     4:  10.465875
##     5:  13.808000
##    ---           
## 40397: -13.808000
## 40398: -10.465875
## 40399:  -7.051000
## 40400:  -3.562625
## 40401:   0.000000
## 
## $Fitted.xy
##         Var1 Var2          y      y.hat      NNS.ID
##     1: -5.00   -5   0.000000   0.000000     201.201
##     2: -4.95   -5   3.562625   3.562625     402.201
##     3: -4.90   -5   7.051000   7.051000     603.201
##     4: -4.85   -5  10.465875  10.465875     804.201
##     5: -4.80   -5  13.808000  13.808000    1005.201
##    ---                                             
## 40397:  4.80    5 -13.808000 -13.808000 39597.40401
## 40398:  4.85    5 -10.465875 -10.465875 39798.40401
## 40399:  4.90    5  -7.051000  -7.051000 39999.40401
## 40400:  4.95    5  -3.562625  -3.562625 40200.40401
## 40401:  5.00    5   0.000000   0.000000 40401.40401

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 $Point.est.

Classification

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

##  [1] 0.9915350 0.9923555 1.0000000 0.9908216 1.0000000 1.0000000 1.0000000
##  [8] 0.9908216 1.0000000 0.9923555

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 $equation.

##        Variable Coefficient
## 1: Sepal.Length   0.7825612
## 2:  Sepal.Width  -0.4266576
## 3: Petal.Length   0.9490347
## 4:  Petal.Width   0.9565473
## 5:  DENOMINATOR   4.0000000

Thus, our model for this regression would be: \[Species = \frac{0.7825612*Sepal.Length -0.4266576*Sepal.Width + 0.9490347*Petal.Length + 0.9565473*Petal.Width}{4} \]

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.

##        Variable Coefficient
## 1: Sepal.Length   0.7825612
## 2:  Sepal.Width   0.0000000
## 3: Petal.Length   0.9490347
## 4:  Petal.Width   0.9565473
## 5:  DENOMINATOR   3.0000000

Thus, our model for this further reduced dimension regression would be: \[Species = \frac{0.7825612*Sepal.Length -0*Sepal.Width + 0.9490347*Petal.Length + 0.9565473*Petal.Width}{3} \]

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

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

References

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