Description Usage Arguments Details Value Author(s) References See Also Examples

Computes and plots accumulated local effects (ALE) 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`

).

1 |

`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 ALE plot will be calculated. |

`K` |
A numeric scalar that specifies the number of intervals into which the predictor range is divided when calculating the ALE plot effects. If |

`NA.plot` |
A logical value that is only used if |

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 ALE main effect of *x_j*, which is denoted by *f_j,ALE(x_j)* in Apley (2016). It also plots *f_j,ALE(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 ALE second-order interaction effect of *(x_j1,x_j2)*, which is denoted by *f_{j1,j2},ALE(x_j1,x_j2)* in Apley (2016). It also plots *f_{j1,j2},ALE(x_j1,x_j2)*.

`K` |
The same as the input argument |

`f.values` |
If |

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

Dan Apley

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

See `PDPlot`

for partial dependence plots.

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########################################################################
## 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 ALE main and second-order interaction effects of x1, x2, x3
par(mfrow = c(2,3))
ALE.1=ALEPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=1, K=50, NA.plot = TRUE)
ALE.2=ALEPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=2, K=50, NA.plot = TRUE)
ALE.3=ALEPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=3, K=50, NA.plot = TRUE)
ALE.12=ALEPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=c(1,2), K=20, NA.plot = TRUE)
ALE.13=ALEPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=c(1,3), K=20, NA.plot = TRUE)
ALE.23=ALEPlot(DAT[,2:4], lm.DAT, pred.fun=yhat, J=c(2,3), K=20, NA.plot = TRUE)
## The following manually recreates the same plots produced by the above ALEPlot function calls
par(mfrow = c(2,3))
plot(ALE.1$x.values, ALE.1$f.values, type="l", xlab="x1", ylab="ALE main effect for x1")
plot(ALE.2$x.values, ALE.2$f.values, type="l", xlab="x2", ylab="ALE main effect for x2")
plot(ALE.3$x.values, ALE.3$f.values, type="l", xlab="x3", ylab="ALE main effect for x3")
image(ALE.12$x.values[[1]], ALE.12$x.values[[2]], ALE.12$f.values, xlab = "x1", ylab = "x2")
contour(ALE.12$x.values[[1]], ALE.12$x.values[[2]], ALE.12$f.values, add=TRUE, drawlabels=TRUE)
image(ALE.13$x.values[[1]], ALE.13$x.values[[2]], ALE.13$f.values, xlab = "x1", ylab = "x3")
contour(ALE.13$x.values[[1]], ALE.13$x.values[[2]], ALE.13$f.values, add=TRUE, drawlabels=TRUE)
image(ALE.23$x.values[[1]], ALE.23$x.values[[2]], ALE.23$f.values, xlab = "x2", ylab = "x3")
contour(ALE.23$x.values[[1]], ALE.23$x.values[[2]], ALE.23$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 ALE main and second-order interaction effects of x1, x2, x3
par(mfrow = c(2,3))
ALE.1=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=1, K=50, NA.plot = TRUE)
ALE.2=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=2, K=50, NA.plot = TRUE)
ALE.3=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=3, K=50, NA.plot = TRUE)
ALE.12=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,2), K=20, NA.plot = TRUE)
ALE.13=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,3), K=20, NA.plot = TRUE)
ALE.23=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(2,3), K=20, NA.plot = TRUE)
########################################################################
## 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 ALE main and second-order interaction effects of x1, x2, x3
par(mfrow = c(2,3))
ALE.1=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=1, K=50, NA.plot = TRUE)
ALE.2=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=2, K=50, NA.plot = TRUE)
ALE.3=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=3, K=50, NA.plot = TRUE)
ALE.12=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,2), K=20, NA.plot = TRUE)
ALE.13=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(1,3), K=20, NA.plot = TRUE)
ALE.23=ALEPlot(DAT[,2:4], nnet.DAT, pred.fun=yhat, J=c(2,3), K=20, NA.plot = TRUE)
``` |

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