knitr::opts_chunk$set(echo = TRUE) library(NNS) library(data.table) data.table::setDTthreads(2L) options(mc.cores = 1) Sys.setenv("OMP_THREAD_LIMIT" = 2)
library(NNS) library(data.table) require(knitr) require(rgl)
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.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)}
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.
NNS.part(x,y,order = 4, type = "XONLY")
The right column of plots shows the corresponding regression (plus endpoints and central point) for the order of NNS
partitioning.
```r,results='hide'}
for(i in 1 : 3){NNS.part(x, y, order = i, obs.req = 0, Voronoi = TRUE, type = "XONLY") ; 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: ```r NNS.reg(x, y, ncores = 1)
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", plot = FALSE, ncores = 1)
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(...)$Point.est
.
NNS.reg
also provides a dimension reduction regression by including a parameter NNS.reg(x, y, dim.red.method = "cor", ...)
. Reducing all regressors to a single dimension using the returned equation NNS.reg(..., dim.red.method = "cor", ...)$equation
.
NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft", ncores = 1)$equation
a = NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft", ncores = 1, plot = FALSE)$equation
Thus, our model for this regression would be:
$$Species = \frac{r round(a$Coefficient[1],3)
Sepal.Length r round(a$Coefficient[2],3)
Sepal.Width +r round(a$Coefficient[3],3)
Petal.Length +r round(a$Coefficient[4],3)
Petal.Width}{4} $$
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)$equation
a = NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, location = "topleft", ncores = 1, plot = FALSE)$equation
Thus, our model for this further reduced dimension regression would be:
$$Species = \frac{\: r round(a$Coefficient[1],3)
Sepal.Length + r round(a$Coefficient[2],3)
Sepal.Width +r round(a$Coefficient[3],3)
Petal.Length +r round(a$Coefficient[4],3)
Petal.Width}{3} $$
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
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
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
, even from external packages such as Metrics
.
NNS.stack(..., obj.fn = expression(Metrics::mape(actual, predicted)), objective = "min")
.
NNS.stack(IVs.train = iris[ , 1 : 4], DV.train = iris[ , 5], IVs.test = iris[1 : 10, 1 : 4], dim.red.method = "cor", obj.fn = expression( mean(round(predicted) == actual) ), objective = "max", type = "CLASS", folds = 1, ncores = 1)
Folds Remaining = 0 Current NNS.reg(... , threshold = 0.935 ) MAX Iterations Remaining = 2 Current NNS.reg(... , threshold = 0.795 ) MAX Iterations Remaining = 1 Current NNS.reg(... , threshold = 0.44 ) MAX Iterations Remaining = 0 Current NNS.reg(... , n.best = 1 ) MAX Iterations Remaining = 12 Current NNS.reg(... , n.best = 2 ) MAX Iterations Remaining = 11 Current NNS.reg(... , n.best = 3 ) MAX Iterations Remaining = 10 Current NNS.reg(... , n.best = 4 ) MAX Iterations Remaining = 9 $OBJfn.reg [1] 1 $NNS.reg.n.best [1] 4 $probability.threshold [1] 0.43875 $OBJfn.dim.red [1] 0.9666667 $NNS.dim.red.threshold [1] 0.935 $reg [1] 1 1 1 1 1 1 1 1 1 1 $reg.pred.int NULL $dim.red [1] 1 1 1 1 1 1 1 1 1 1 $dim.red.pred.int NULL $stack [1] 1 1 1 1 1 1 1 1 1 1 $pred.int NULL
Given multicollinearity is not an issue for nonparametric regressions as it is for OLS, in the case of an ill-fit univariate model a better option may be to increase the dimensionality of regressors with a copy of itself and cross-validate the number of clusters n.best
via:
NNS.stack(IVs.train = cbind(x, x), DV.train = y, method = 1, ...)
.
set.seed(123) x = rnorm(100); y = rnorm(100) nns.params = NNS.stack(IVs.train = cbind(x, x), DV.train = y, method = 1, ncores = 1)
set.seed(123) x = rnorm(100); y = rnorm(100) nns.params = list() nns.params$NNS.reg.n.best = 100
NNS.reg(cbind(x, x), y, n.best = nns.params$NNS.reg.n.best, point.est = cbind(x, x), residual.plot = TRUE, ncores = 1, confidence.interval = .95)
If the user is so motivated, detailed arguments further examples are provided within the following:
Sys.setenv("OMP_THREAD_LIMIT" = "")
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