PDPlot {ALEPlot} | R Documentation |

## Partial Dependence (PD) Plots

### Description

Computes and plots partial dependence (PD) plots for a fitted supervised learning model. The effects can be either a main effect for an individual predictor (`length(J) = 1`

) or a second-order interaction effect for a pair of predictors (`length(J) = 2`

).

### Usage

```
PDPlot(X, X.model, pred.fun, J, K)
```

### Arguments

`X` |
The data frame of predictor variables to which the supervised learning model was fit. The names of the predictor variables must be the same as when the model was fit. The response variable should not be included in |

`X.model` |
The fitted supervised learning model object (e.g., a tree, random forest, neural network, etc.), typically an object to which a built-in |

`pred.fun` |
A user-supplied function that will be used to predict the response for |

`J` |
A numeric scalar or two-length vector of indices of the predictors for which the PD plot will be calculated. |

`K` |
A numeric scalar that represents the number of discrete points at which the PD plot will be calculated. If |

### Details

This function calculates and plots the partial dependence (PD) plots first introduced in Friedman (2001). See the Apley (2016) reference paper listed below for details. For `J = j`

(i.e., if the index for a single predictor `x_j`

is specified), the function calculates and returns the PD main effect of `x_j`

, which is denoted by `f_{j,PD}(x_j)`

in Apley (2016). It also plots `f_{j,PD}(x_j)`

. For `J = c(j1,j2)`

(i.e., if the indices for a pair of predictors `(x_{j1},x_{j2})`

are specified), the function calculates and returns the PD second-order interaction effect of `(x_{j1},x_{j2})`

, which is denoted by `f_{{j1,j2},PD}(x_{j1},x_{j2})`

in Apley (2016). It also plots `f_{{j1,j2},PD}(x_{j1},x_{j2})`

.

### Value

`f.values` |
If |

`x.values` |
For numeric predictors, if |

### Author(s)

Dan Apley

### References

Friedman, J. H., (2001), "Greedy function approximation: A gradient boosting machine," Annals of Statistics, 29(5), pp. 1189-1232.

Apley, D. W. (2016), "Visualizing the Effects of Predictor Variables in Black Box Supervised Learning Models," submitted for publication.

### See Also

See `ALEPlot`

for partial dependence plots.

### Examples

```
########################################################################
## A transparent example in which the supervised learning model is a linear regression \code{lm},
## but we will pretend it is black-box
########################################################################
## Generate some data and fit a \code{lm} supervised learning model
N=500
x1 <- runif(N, min=0, max=1)
x2 <- runif(N, min=0, max=1)
x3 <- runif(N, min=0, max=1)
y = x1 + 2*x2^2 + rnorm(N, 0, 0.1)
DAT = data.frame(y, x1, x2, x3)
lm.DAT = lm(y ~ .^2 + I(x1^2) + I(x2^2) + I(x3^2), DAT)
## Define the predictive function (easy in this case, since \code{lm} has
## a built-in predict function that suffices)
yhat <- function(X.model, newdata) as.numeric(predict(X.model, newdata))
## Calculate and plot the PD main effects and second-order interaction effects of x1, x2, x3
par(mfrow = c(2,3))
PD.1=PDPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=1, K=50)
PD.2=PDPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=2, K=50)
PD.3=PDPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=3, K=50)
PD.12=PDPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=c(1,2), K=30)
PD.13=PDPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=c(1,3), K=30)
PD.23=PDPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=c(2,3), K=30)
## The following manually recreates the same plots produced by the above PDPlot function calls
par(mfrow = c(2,3))
plot(PD.1$x.values, PD.1$f.values, type="l", xlab="x1", ylab="PD main effect for x1")
plot(PD.2$x.values, PD.2$f.values, type="l", xlab="x2", ylab="PD main effect for x2")
plot(PD.3$x.values, PD.3$f.values, type="l", xlab="x3", ylab="PD main effect for x3")
image(PD.12$x.values[[1]], PD.12$x.values[[2]], PD.12$f.values, xlab = "x1", ylab = "x2")
contour(PD.12$x.values[[1]], PD.12$x.values[[2]], PD.12$f.values, add=TRUE, drawlabels=TRUE)
image(PD.13$x.values[[1]], PD.13$x.values[[2]], PD.13$f.values, xlab = "x1", ylab = "x3")
contour(PD.13$x.values[[1]], PD.13$x.values[[2]], PD.13$f.values, add=TRUE, drawlabels=TRUE)
image(PD.23$x.values[[1]], PD.23$x.values[[2]], PD.23$f.values, xlab = "x2", ylab = "x3")
contour(PD.23$x.values[[1]], PD.23$x.values[[2]], PD.23$f.values, add=TRUE, drawlabels=TRUE)
########################################################################
## A larger example in which the supervised learning model is a neural network (\code{nnet})
########################################################################
## Generate some data and fit a \code{nnet} supervised learning model
library(nnet)
N=5000
x1 <- runif(N, min=0, max=1)
x2 <- runif(N, min=0, max=1)
x3 <- runif(N, min=0, max=1)
y = x1 + 2*x2^2 +(x1-0.5)*(x3-0.5) + rnorm(N, 0, 0.1)
DAT = data.frame(y, x1, x2, x3)
nnet.DAT<-nnet(y~., data=DAT, linout=TRUE, skip=FALSE, size=10, decay=0.01,
maxit=1000, trace=FALSE)
## Define the predictive function
yhat <- function(X.model, newdata) as.numeric(predict(X.model, newdata, type="raw"))
## Calculate and plot the PD main and second-order interaction effects of x1, x2, x3
par(mfrow = c(2,3))
PD.1=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=1, K=50)
PD.2=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=2, K=50)
PD.3=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=3, K=50)
PD.12=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,2), K=20)
PD.13=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,3), K=20)
PD.23=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(2,3), K=20)
########################################################################
## A binary classification example in which the supervised learning model is
## a neural network (\code{nnet}), and the log-odds of the predicted class
## probability is the function to be plotted
########################################################################
## Generate some data and fit a \code{nnet} supervised learning model
library(nnet)
N=5000
x1 <- runif(N, min=0, max=1)
x2 <- runif(N, min=0, max=1)
x3 <- runif(N, min=0, max=1)
z = -3.21 + 2.81*x1 + 5.62*x2^2 + 2.81*(x1-0.5)*(x3-0.5) #true log-odds
p = exp(z)/(1+exp(z))
u = runif(N)
y = u < p
DAT = data.frame(y, x1, x2, x3)
nnet.DAT<-nnet(y~., data=DAT, linout=FALSE, skip=FALSE, size=10, decay=0.05,
maxit=1000, trace=FALSE)
## Define the ALE function to be the log-odds of the predicted probability that y = TRUE
yhat <- function(X.model, newdata) {
p.hat = as.numeric(predict(X.model, newdata, type="raw"))
log(p.hat/(1-p.hat))
}
## Calculate and plot the PD main and second-order interaction effects of x1, x2, x3
par(mfrow = c(2,3))
PD.1=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=1, K=50)
PD.2=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=2, K=50)
PD.3=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=3, K=50)
PD.12=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,2), K=20)
PD.13=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,3), K=20)
PD.23=PDPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(2,3), K=20)
```

*ALEPlot*version 1.1 Index]