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 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, noise.reduction = "off", Voronoi = TRUE)}
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.
for(i in 1 : 4){NNS.part(x, y, order = i, type = "XONLY", Voronoi = TRUE)}
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, Voronoi = TRUE) ; NNS.reg(x, y, order = i)}
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, order = 4, noise.reduction = "off")
## $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 67.09Multivariate:
Multivariate regressions return a plot of \(y\) and \(\hat{y}\).
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")
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 $Point.est.
Classification
For a classification problem, we simply set NNS.reg(x, y, type = "CLASS", ...)
NNS.reg(iris[ , 1 : 4], iris[ , 5], point.est = iris[1 : 10, 1 : 4], type = "CLASS", location = "topleft")$Point.est
## [1] 1 1 1 1 1 1 1 1 1 1NNS 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 $equation.
NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft")$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 4.0000000Thus, 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.
NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, location = "topleft")$equation
## Variable Coefficient
## 1: Sepal.Length 0.7825612
## 2: Sepal.Width 0.0000000
## 3: Petal.Length 0.9490347
## 4: Petal.Width 0.9565473
## 5: DENOMINATOR 3.0000000Thus, 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 $Point.est.
NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, point.est = iris[1 : 10, 1 : 4], location = "topleft")$Point.est
## [1] 1 1 1 1 1 1 1 1 1 1