Exploring Learner Predictions with Partial Dependence and Functional ANOVA

Learners use features to make predictions but how those features are used is often not apparent. mlr can estimate the dependence of a learned function on a subset of the feature space using generatePartialDependenceData.

Partial dependence plots reduce the potentially high dimensional function estimated by the learner, and display a marginalized version of this function in a lower dimensional space. For example suppose $\mathbb{E}[Y \ | \ X = x] = f(x)$. With $(x, y)$ pairs drawn independently, a learner may estimate $\hat{f}$, which, if $X$ is high dimensional can be uninterpretable. Suppose we want to approximate the relationship between some column-wise subset of $X$. We partition $X$ into two sets, $X_s$ and $X_c$ such that $X = X_s \cup X_c$, where $X_s$ is a subset of $X$ of interest.

The partial dependence of $f$ on $X_c$ is

We can use the following estimator:

This is described by Friedman (2001) and in Hastie, Tibsharani, and Friedman (2009).

The individual conditional expectation of an observation can also be estimated using the above algorithm absent the averaging, giving $\hat{f}^{(i)}_{x_s}$ as described in Goldstein, Kapelner, Bleich, and Pitkin (2014). This allows the discovery of features of $\hat{f}$ that may be obscured by an aggregated summary of $\hat{f}$.

The partial derivative of the partial dependence function, $\frac{\partial \hat f_{x_s}}{\partial x_s}$, and the individual conditional expectation function, $\frac{\partial \hat{f}^{(i)}_{x_s}}{\partial x_s}$, can also be computed. For regression and survival tasks the partial derivative of a single feature $x_s$ is the gradient of the partial dependence function, and for classification tasks where the learner can output class probabilities the Jacobian. Note that if the learner produces discontinuous partial dependence (e.g., piecewise constant functions such as decision trees, ensembles of decision trees, etc.) the derivative will be 0 (where the function is not changing) or trending towards positive or negative infinity (at the discontinuities where the derivative is undefined). Plotting the partial dependence function of such learners may give the impression that the function is not discontinuous because the prediction grid is not composed of all discontinuous points in the predictor space. This results in a line interpolating that makes the function appear to be piecewise linear (where the derivative would be defined except at the boundaries of each piece).

The partial derivative can be informative regarding the additivity of the learned function in certain features. If $\hat{f}^{(i)}_{x_s}$ is an additive function in a feature $x_s$, then its partial derivative will not depend on any other features ($x_c$) that may have been used by the learner. Variation in the estimated partial derivative indicates that there is a region of interaction between $x_s$ and $x_c$ in $\hat{f}$. Similarly, instead of using the mean to estimate the expected value of the function at different values of $x_s$, instead computing the variance can highlight regions of interaction between $x_s$ and $x_c$.

Again, see Goldstein, Kapelner, Bleich, and Pitkin (2014) for more details and their package ICEbox for the original implementation. The algorithm works for any supervised learner with classification, regression, and survival tasks.

Partial Dependence

Our implementation, following mlr’s visualization pattern, consists of the above mentioned function generatePartialDependenceData and plotPartialDependence. The former generates input (objects of class PartialDependenceData) for the latter.

The first step executed by generatePartialDependenceData is to generate a feature grid for every element of the character vector features passed. The data are given by the input argument, which can be a Task or a data.frame. The feature grid can be generated in several ways. A uniformly spaced grid of length gridsize (default 10) from the empirical minimum to the empirical maximum is created by default, but arguments fmin and fmax may be used to override the empirical default (the lengths of fmin and fmax must match the length of features). Alternatively the feature data can be resampled, either by using a bootstrap or by subsampling.

Results from generatePartialDependenceData can be visualized with plotPartialDependence.

library(mlr)

## Loading required package: ParamHelpers

lrn.classif = makeLearner("classif.ksvm", predict.type = "prob")
fit.classif = train(lrn.classif, iris.task)
pd = generatePartialDependenceData(fit.classif, iris.task, "Petal.Width")
pd

## PartialDependenceData
## Task: iris-example
## Features: Petal.Width
## Target: setosa, versicolor, virginica
## Derivative: FALSE
## Interaction: FALSE
## Individual: FALSE
##    Class Probability Petal.Width
## 1 setosa   0.4872634   0.1000000
## 2 setosa   0.4553176   0.3666667
## 3 setosa   0.3940011   0.6333333
## 4 setosa   0.3107103   0.9000000
## 5 setosa   0.2114968   1.1666667
## 6 setosa   0.1394269   1.4333333
## ... (30 rows, 3 cols)

plotPartialDependence(pd, data = iris)

As noted above, $x_s$ does not have to be unidimensional. If it is not, the interaction flag must be set to TRUE. Then the individual feature grids are combined using the Cartesian product, and the estimator above is applied, producing the partial dependence for every combination of unique feature values. If the interaction flag is FALSE (the default) then by default $x_s$ is assumed unidimensional, and partial dependencies are generated for each feature separately. The resulting output when interaction = FALSE has a column for each feature, and NA where the feature was not used. With one feature and a regression task the output is a line plot, with a point for each point in the corresponding feature’s grid. For classification tasks there is a line for each class (except for binary classification tasks, where the negative class is automatically dropped). The data argument to plotPartialPrediction allows the training data to be input to show the empirical marginal distribution of the data.

pd.lst = generatePartialDependenceData(fit.classif, iris.task, c("Petal.Width", "Petal.Length"))
head(pd.lst$data)

##    Class Probability Petal.Width Petal.Length
## 1 setosa   0.4872634   0.1000000           NA
## 2 setosa   0.4553176   0.3666667           NA
## 3 setosa   0.3940011   0.6333333           NA
## 4 setosa   0.3107103   0.9000000           NA
## 5 setosa   0.2114968   1.1666667           NA
## 6 setosa   0.1394269   1.4333333           NA

tail(pd.lst$data)

##        Class Probability Petal.Width Petal.Length
## 55 virginica   0.2622712          NA     3.622222
## 56 virginica   0.3697957          NA     4.277778
## 57 virginica   0.4854980          NA     4.933333
## 58 virginica   0.6436039          NA     5.588889
## 59 virginica   0.7344045          NA     6.244444
## 60 virginica   0.7195496          NA     6.900000

plotPartialDependence(pd.lst, data = iris)

pd.int = generatePartialDependenceData(fit.classif, iris.task, c("Petal.Width", "Petal.Length"), interaction = TRUE)
pd.int

## PartialDependenceData
## Task: iris-example
## Features: Petal.Width, Petal.Length
## Target: setosa, versicolor, virginica
## Derivative: FALSE
## Interaction: TRUE
## Individual: FALSE
##    Class Probability Petal.Width Petal.Length
## 1 setosa   0.6043750   0.1000000            1
## 2 setosa   0.6023896   0.3666667            1
## 3 setosa   0.5660952   0.6333333            1
## 4 setosa   0.4858965   0.9000000            1
## 5 setosa   0.3759139   1.1666667            1
## 6 setosa   0.2968125   1.4333333            1
## ... (300 rows, 4 cols)

plotPartialDependence(pd.int, facet = "Petal.Length")

When interaction = TRUE, plotPartialDependence can either facet over one feature, showing the conditional relationship between the other feature and $\hat{f}$ in each panel, or a tile plot. The latter is, however, not possible with multiclass classification (an example of a tile plot will be shown later).

At each step in the estimation of $\hat{f}_{x_s}$ a set of predictions of length $N$ is generated. By default the mean prediction is used. For classification where predict.type = "prob" this entails the mean class probabilities. However, other summaries of the predictions may be used. For regression and survival tasks the function used here must either return one number or three, and, if the latter, the numbers must be sorted lowest to highest. For classification tasks the function must return a number for each level of the target feature.

As noted, the fun argument can be a function which returns three numbers (sorted low to high) for a regression task. This allows further exploration of relative feature importance. If a feature is relatively important, the bounds are necessarily tighter because the feature accounts for more of the variance of the predictions, i.e., it is “used” more by the learner. More directly setting fun = var identifies regions of interaction between $x_s$ and $x_c$. This can also be accomplished by computing quantiles. The wider the quantile bounds, the more variation in $\hat{f}$ is due to features other than $x_s$ that is shown in the plot.

lrn.regr = makeLearner("regr.ksvm")
fit.regr = train(lrn.regr, bh.task)

pd.ci = generatePartialDependenceData(fit.regr, bh.task, "lstat",
  fun = function(x) quantile(x, c(.25, .5, .75)))
pd.ci

## PartialDependenceData
## Task: BostonHousing-example
## Features: lstat
## Target: medv
## Derivative: FALSE
## Interaction: FALSE
## Individual: FALSE
##       medv     lstat    lower    upper
## 1 24.92828  1.730000 21.33593 29.56459
## 2 23.76195  5.756667 20.77038 27.37917
## 3 22.37299  9.783333 19.85725 25.00549
## 4 20.70007 13.810000 18.70272 23.63418
## 5 19.57004 17.836667 16.54922 23.02968
## 6 18.95529 21.863333 14.80123 22.28493
## ... (10 rows, 4 cols)

plotPartialDependence(pd.ci)

In addition to bounds based on a summary of the distribution of the conditional expectation of each observation, learners which can estimate the variance of their predictions can also be used. The argument bounds is a numeric vector of length two which is added (so the first number should be negative) to the point prediction to produce a confidence interval for the partial dependence. The default is the .025 and .975 quantiles of the Gaussian distribution.

fit.se = train(makeLearner("regr.randomForest", predict.type = "se"), bh.task)
pd.se = generatePartialDependenceData(fit.se, bh.task, c("lstat", "crim"))
head(pd.se$data)

##       medv     lstat crim    lower    upper
## 1 31.16866  1.730000   NA 27.33761 34.99971
## 2 25.98765  5.756667   NA 23.53881 28.43648
## 3 23.49603  9.783333   NA 21.33348 25.65858
## 4 22.04595 13.810000   NA 20.29177 23.80014
## 5 20.40087 17.836667   NA 18.69259 22.10916
## 6 19.75266 21.863333   NA 17.96500 21.54032

tail(pd.se$data)

##        medv lstat     crim    lower    upper
## 15 21.80198    NA 39.54849 19.50150 24.10246
## 16 21.77178    NA 49.43403 19.46862 24.07494
## 17 21.75443    NA 59.31957 19.45200 24.05685
## 18 21.74859    NA 69.20512 19.43800 24.05917
## 19 21.74845    NA 79.09066 19.43743 24.05946
## 20 21.74862    NA 88.97620 19.43756 24.05969

plotPartialDependence(pd.se)

As previously mentioned if the aggregation function is not used, i.e., it is the identity, then the conditional expectation of $\hat{f}^{(i)}_{x_s}$ is estimated. If individual = TRUE then generatePartialDependenceData returns $N$ partial dependence estimates made at each point in the prediction grid constructed from the features.

pd.ind.regr = generatePartialDependenceData(fit.regr, bh.task, "lstat", individual = TRUE)
pd.ind.regr

## PartialDependenceData
## Task: BostonHousing-example
## Features: lstat
## Target: medv
## Derivative: FALSE
## Interaction: FALSE
## Individual: TRUE
## Predictions centered: FALSE
##       medv     lstat idx
## 1 27.34360  1.730000   1
## 2 26.21121  5.756667   1
## 3 24.91534  9.783333   1
## 4 23.63205 13.810000   1
## 5 22.47994 17.836667   1
## 6 21.51669 21.863333   1
## ... (5060 rows, 3 cols)

plotPartialDependence(pd.ind.regr)

The resulting output, particularly the element data in the returned object, has an additional column idx which gives the index of the observation to which the row pertains.

For classification tasks this index references both the class and the observation index.

pd.ind.classif = generatePartialDependenceData(fit.classif, iris.task, "Petal.Length", individual = TRUE)
pd.ind.classif

## PartialDependenceData
## Task: iris-example
## Features: Petal.Length
## Target: setosa, versicolor, virginica
## Derivative: FALSE
## Interaction: FALSE
## Individual: TRUE
## Predictions centered: FALSE
##    Class Probability Petal.Length      idx
## 1 setosa   0.9824128            1 1.setosa
## 2 setosa   0.9717450            1 2.setosa
## 3 setosa   0.9830395            1 3.setosa
## 4 setosa   0.9810668            1 4.setosa
## 5 setosa   0.9820302            1 5.setosa
## 6 setosa   0.9733090            1 6.setosa
## ... (4500 rows, 4 cols)

plotPartialDependence(pd.ind.classif)

The plots, at least in these forms, are difficult to interpet. Individual estimates of partial dependence can also be centered by predictions made at all $N$ observations for a particular point in the prediction grid created by the features. This is controlled by the argument center which is a list of the same length as the length of the features argument and contains the values of the features desired.

pd.ind.classif = generatePartialDependenceData(fit.classif, iris.task, "Petal.Length", individual = TRUE, center = list("Petal.Length" = min(iris$Petal.Length)))
plotPartialDependence(pd.ind.classif)

Partial derivatives can also be computed for individual partial dependence estimates and aggregate partial dependence. This is restricted to a single feature at a time. The derivatives of individual partial dependence estimates can be useful in finding regions of interaction between the feature for which the derivative is estimated and the features excluded. Applied to the aggregated partial dependence function they are not very informative, but when applied to the individual conditional expectations, they can be used to find regions of interaction.

pd.regr.der.ind = generatePartialDependenceData(fit.regr, bh.task, "lstat", derivative = TRUE, individual = TRUE)
head(pd.regr.der.ind$data)

##         medv     lstat idx
## 1 -0.2446031  1.730000   1
## 2 -0.3096243  5.756667   1
## 3 -0.3265486  9.783333   1
## 4 -0.3059314 13.810000   1
## 5 -0.2640377 17.836667   1
## 6 -0.2136498 21.863333   1

plotPartialDependence(pd.regr.der.ind)

pd.classif.der.ind = generatePartialDependenceData(fit.classif, iris.task, "Petal.Width", derivative = TRUE, individual = TRUE)
head(pd.classif.der.ind$data)

##    Class Probability Petal.Width      idx
## 1 setosa  0.04088195         0.1 1.setosa
## 2 setosa  0.03010277         0.1 2.setosa
## 3 setosa  0.02441696         0.1 3.setosa
## 4 setosa  0.02231513         0.1 4.setosa
## 5 setosa  0.04178867         0.1 5.setosa
## 6 setosa  0.05093621         0.1 6.setosa

plotPartialDependence(pd.classif.der.ind)

This suggests that Petal.Width interacts with some other feature in the neighborhood of $(1.5, 2)$ for classes “virginica” and “versicolor”.

Functional ANOVA

Hooker (2004) proposed the decomposition of a learned function $\hat{f}$ as a sum of lower dimensional functions where $p$ is the number of features. generateFunctionalANOVAData estimates the individual $g$ functions using partial dependence. When functions depend only on one feature, they are equivalent to partial dependence, but a $g$ function which depends on more than one feature is the “effect” of only those features: lower dimensional “effects” are removed.

Here $u$ is a subset of ${1, \ldots, p}$. When $|v| = 1$ $g_v$ can be directly computed by computing the bivariate partial dependence of $\hat{f}$ on $x_u$ and then subtracting off the univariate partial dependences of the features contained in $v$.

Although this decomposition is generalizable to classification it is currently only available for regression tasks.

lrn.regr = makeLearner("regr.ksvm")
fit.regr = train(lrn.regr, bh.task)

fa = generateFunctionalANOVAData(fit.regr, bh.task, "lstat", depth = 1, fun = median)
fa

## FunctionalANOVAData
## Task: BostonHousing-example
## Features: lstat
## Target: medv
## 
## 
##   effect     medv     lstat
## 1  lstat 24.88912  1.730000
## 2  lstat 23.75931  5.756667
## 3  lstat 22.35706  9.783333
## 4  lstat 20.67813 13.810000
## 5  lstat 19.60456 17.836667
## 6  lstat 19.01695 21.863333
## ... (10 rows, 3 cols)

pd.regr = generatePartialDependenceData(fit.regr, bh.task, "lstat", fun = median)
pd.regr

## PartialDependenceData
## Task: BostonHousing-example
## Features: lstat
## Target: medv
## Derivative: FALSE
## Interaction: FALSE
## Individual: FALSE
##       medv     lstat
## 1 24.88912  1.730000
## 2 23.75931  5.756667
## 3 22.35706  9.783333
## 4 20.67813 13.810000
## 5 19.60456 17.836667
## 6 19.01695 21.863333
## ... (10 rows, 2 cols)

The depth argument is similar to the interaction argument in generatePartialDependenceData but instead of specifying whether all of joint “effect” of all the features is computed, it determines whether “effects” of all subsets of the features given the specified depth are computed. So, for example, with $p$ features and depth 1, the univariate partial dependence is returned. If, instead, depth = 2, then all possible bivariate functional ANOVA effects are returned. This is done by computing the univariate partial dependence for each feature and subtracting it from the bivariate partial dependence for each possible pair.

fa.bv = generateFunctionalANOVAData(fit.regr, bh.task, c("crim", "lstat", "age"), depth = 2)
fa.bv

## FunctionalANOVAData
## Task: BostonHousing-example
## Features: crim, lstat, age
## Target: medv
## 
## 
##       effect      medv      crim lstat age
## 1 crim:lstat -22.68839  0.006320  1.73  NA
## 2 crim:lstat -23.22034  9.891862  1.73  NA
## 3 crim:lstat -24.78243 19.777404  1.73  NA
## 4 crim:lstat -26.42732 29.662947  1.73  NA
## 5 crim:lstat -27.57391 39.548489  1.73  NA
## 6 crim:lstat -28.27667 49.434031  1.73  NA
## ... (300 rows, 5 cols)

names(table(fa.bv$data$effect)) ## interaction effects estimated

## [1] "crim:age"   "crim:lstat" "lstat:age"

Plotting univariate and bivariate functional ANOVA components works the same as for partial dependence.

fa.bv = generateFunctionalANOVAData(fit.regr, bh.task, c("crim", "lstat"), depth = 2)
plotPartialDependence(fa.bv, geom = "tile", data = getTaskData(bh.task))

When overplotting the training data on the plot it is easy to see that much of the variation of the effect is due to extrapolation. Although it hasn’t been implemented yet, weighting the functional ANOVA appropriately can ensure that the estimated effects do not depend (or depend less) on regions of the feature space which are sparse.

I also plan on implementing the faster estimation algorith for expanding the functionality of the functional ANOVA function include faster computation using the algorithm in Hooker (2007) and weighting (in order to avoid excessive reliance on points of extrapolation) using outlier detection or joint density estimation.