predict.krls {KRLS}

Predict method for Kernel-based Regularized Least Squares (KRLS) Model Fits

Description

Predicted values and standard errors based on krls model object.

Usage

 ## S3 method for class 'krls'
predict(object, newdata, se.fit = FALSE , ...)

Arguments

object

Fitted KRLS model, i.e. an object of class krls

newdata

A data frame or matrix with variables values at which to predict the outcome. Number and order of columns in newdata have to match the corresponding predictors used in the fitted krls model given in object.

se.fit

logical flag if standard errors should be computed for pointwise predictions.

...

additional arguments affecting the predictions produced.

Details

Function produces predicted values, obtained by evaluating the fitted krls function with the newdata (ie. the test points). The prediction at a new test point x_i is based on

f(x_i)= \sum_j=1^n c_j K_{x_j}(x_i)

where K is the kernel matrix and thus K_{x_j} is a vector whose j-th entry is K(x_j,x_i) (e.g. the distance between the test point x_i and the training point x_j). The training points are passed to the function with the krls fit in object.

When data are missing in newdata during prediction, the value of each k(x_i,x_j) is computed by using an adjusted Euclidean distance in the kernel definition. Assume x is D-dimensional but a given pair of observations x_i and x_j have only D' < D non-missing dimensions in common. The adjusted Euclidean distance computes the sum of squared differences over the D' non-missing dimensions, rescales this sum by D/D', and takes the square root. The result corresponds to an assumption that conditional on the observed data, the missing values would not have contributed new information predictive of the outcome.

Value

fit

M by 1 vector of fitted values for M test points.

se.fit

M by 1 vector of standard errors for the fitted values for M test points (NULL unless se.fit=TRUE is specified).

vcov.fit

M by M variance-covariance matrix for the fitted values for M test points (NULL unless se.fit=TRUE is specified).

newdata

M by D data matrix of of M test points with D predictors.

newdataK

M by N data matrix for pairwise Gauss Kernel distances between M test points and N training points from krls model fit in object.

Author(s)

Jens Hainmueller (Stanford) and Chad Hazlett (MIT)

See Also

krls

Examples

# make up data
X <- seq(-3,3,.1)
Y <- sin(X) + rnorm(length(X),.1)

# fit krls
krlsout <- krls(y=Y,X=X)

# get in-sample prediction 
predin <- predict(krlsout,newdata=X,se.fit=TRUE)

# get out-of-sample prediction
X2 <- runif(5)
predout <- predict(krlsout,newdata=X2,se.fit=TRUE)

# plot true function and predictions
plot(y=sin(X),x=X,type="l",col="red",ylim=c(-1.8,1.8),lwd=2,ylab="f(X)")
points(y=predin$fit,x=X,col="blue",pch=19)
arrows(y1=predin$fit+2*predin$se.fit,
       y0=predin$fit-2*predin$se.fit,
       x1=X,x0=X,col="blue",length=0)
       
points(y=predout$fit,x=X2,col="green",pch=17)
arrows(y1=predout$fit+2*predout $se.fit,
       y0=predout$fit-2*predout $se.fit,
       x1=X2,x0=X2,col="green",length=0)

legend("bottomright",
       legend=c("true f(x)=sin(X)",
                "KRLS fitted in-sample",
                "KRLS fitted out-of-sample"),
       lty=c(1,NA,NA),pch=c(NA,19,17),
       lwd=c(2,NA,NA),
       col=c("red","blue","green"),
       cex=.8)