Exploratory Data Analysis

It is good practice to view your data before beginning an analysis, what Tukey (1977) refers to as Exploratory Data Analysis (EDA). To facilitate this, we use ggplot2 figures with the ggplot2::facet_wrap command to create two sets of panel plots, one for categorical variables with boxplots at each level, and one of scatter plots for continuous variables. Each variable is plotted along a selected continuous variable on the X-axis. These figures help to find outliers, missing values and other data anomalies in each variable before getting deep into the analysis. We have also created a separate Shiny app, available at (https://ehrlinger.shinyapps.io/xportEDA), for creating similar figures with an arbitrary data set, to make the EDA process easier for users.

The Boston housing data consists almost entirely of continuous variables, with the exception of the “Charles river” logical variable. A simple EDA visualization to use for this data is a single panel plot of the continuous variables, with observation points colored by the logical variable. Missing values in our continuous variable plots are indicated by the rug marks along the x-axis, of which there are none in this data. We used the Boston housing response variable, the median value of homes (medv), for X variable.

> # Use reshape2::melt to transform the data into long format.
> dta <- melt(Boston, id.vars=c("medv","chas"))
> 
> # plot panels for each covariate colored by the logical chas variable.
> ggplot(dta, aes(x=medv, y=value, color=chas))+
+   geom_point(alpha=.4)+
+   geom_rug(data=dta %>% filter(is.na(value)))+
+   labs(y="", x=st.labs["medv"]) +
+   scale_color_brewer(palette="Set2")+
+   facet_wrap(~variable, scales="free_y", ncol=3)

This figure is loosely related to a pairs scatter plot (Becker, Chambers, and Wilks 1988), but in this case we only examine the relation between the response variable against the remainder. Plotting the data against the response also gives us a “sanity check” when viewing our model results. It’s pretty obvious from this figure that we should find a strong relation between median home values and the lstat and rm variables.

Variable Importance.

Variable importance (VIMP) was originally defined in CART using a measure involving surrogate variables (see Chapter 5 of (Breiman et al. 1984)). The most popular VIMP method uses a prediction error approach involving “noising-up” each variable in turn. VIMP for a variable x_v is the difference between prediction error when x_v is noised up by randomly permuting its values, compared to prediction error under the observed values (Breiman 2001; Liaw and Wiener 2002; Ishwaran 2007; Ishwaran et al. 2008).

Since VIMP is the difference between OOB prediction error before and after permutation, a large VIMP value indicates that misspecification detracts from the variable predictive accuracy in the forest. VIMP close to zero indicates the variable contributes nothing to predictive accuracy, and negative values indicate the predictive accuracy improves when the variable is mispecified. In the later case, we assume noise is more informative than the true variable. As such, we ignore variables with negative and near zero values of VIMP, relying on large positive values to indicate that the predictive power of the forest is dependent on those variables.

The gg_vimp function extracts VIMP measures for each of the variables used to grow the forest. The plot.gg_vimp function shows the variables, in VIMP rank order, from the largest (Lower Status) at the top, to smallest (Charles River) at the bottom. VIMP measures are shown using bars to compare the scale of the error increase under permutation.

> plot(gg_vimp(rfsrc_Boston), lbls=st.labs)

For our random forest, the top two variables (lstat and rm) have the largest VIMP, with a sizable difference to the remaining variables, which mostly have similar VIMP measure. This indicates we should focus attention on these two variables, at least, over the others.

In this example, all VIMP measures are positive, though some are small. When there are both negative and positive VIMP values, the plot.gg_vimp function will color VIMP by the sign of the measure. We use the lbls argument to pass a named vector of meaningful text descriptions to the plot.gg_vimp function, replacing the often terse variable names used by default.

Minimal Depth.

In VIMP, prognostic risk factors are determined by testing the forest prediction under alternative data settings, ranking the most important variables according to their impact on predictive ability of the forest. An alternative method uses inspection of the forest construction to rank variables. Minimal depth assumes that variables with high impact on the prediction are those that most frequently split nodes nearest to the trunks of the trees (i.e. at the root node) where they partition large samples of the population.

With a tree, node levels are numbered based on their relative distance to the trunk of the tree (with the root at 0). Minimal depth measures the important risk factors by averaging the depth of the first split for each variable over all trees within the forest. Lower values of this measure indicate variables important in splitting large groups of patients.

The maximal subtree for a variable \(x\) is the largest subtree whose root node splits on \(x\). All parent nodes of \(x\)’s maximal subtree have nodes that split on variables other than \(x\). The largest maximal subtree possible is at the root node. If a variable does not split the root node, it can have more than one maximal subtree, or a maximal subtree may also not exist if there are no splits on the variable. The minimal depth of a variables is a surrogate measure of predictiveness of the variable. The smaller the minimal depth, the more impact the variable has sorting observations, and therefore on the forest prediction.

The gg_minimal_depth function is analogous to the gg_vimp function for minimal depth. Variables are ranked from most important at the top (minimal depth measure), to least at the bottom (maximal minimal depth). The vertical dashed line indicates the minimal depth threshold where smaller minimal depth values indicate higher importance and larger indicate lower importance.

The randomForestSRC::var.select call is again a computationally intensive function, as it traverses the forest finding the maximal subtree within each tree for each variable before averaging the results we use in the gg_minimal_depth call. We again use the cached object strategy here to save computational time. The var.select call is included in the comment of this code block.

> # Load the data, from the call:
> # varsel_Boston <- var.select(rfsrc_Boston)
> data(varsel_Boston)
> 
> # Save the gg_minimal_depth object for later use.
> gg_md <- gg_minimal_depth(varsel_Boston)
> 
> # plot the object
> plot(gg_md, lbls=st.labs)

In general, the selection of variables according to VIMP is to rather arbitrarily examine the values, looking for some point along the ranking where there is a large difference in VIMP measures. The minimal depth threshold method has a more quantitative approach to determine a selection threshold. Given minimal depth is a quantitative property of the forest construction, Ishwaran et al. (2010) also construct an analytic threshold for evidence of variable impact. A simple optimistic threshold rule uses the mean of the minimal depth distribution, classifying variables with minimal depth lower than this threshold as important in forest prediction. The minimal depth plot for our model indicates there are ten variables which have a higher impact (minimal depth below the mean value threshold) than the remaining three.

Since the VIMP and Minimal Depth measures use different criteria, we expect the variable ranking to be somewhat different. We use gg_minimal_vimp function to compare rankings between minimal depth and VIMP. In this call, we plot the stored gg_minimal_depth object (gg_md), which would be equivalent to calling plot.gg_minimal_vimp(varsel_Boston) or plot(gg_minimal_vimp(varsel_Boston)).

> plot.gg_minimal_vimp(gg_md)

The points along the red dashed line indicates where the measures are in agreement. Points above the red dashed line are ranked higher by VIMP than by minimal depth, indicating the variables are sensitive to misspecification. Those below the line have a higher minimal depth ranking, indicating they are better at dividing large portions of the population. The further the points are from the line, the more the discrepancy between measures. The construction of this figure is skewed towards a minimal depth approach, by ranking variables along the y-axis, though points are colored by the sign of VIMP.

In our example, both minimal depth and VIMP indicate the strong relation of lstat and rm variables to the forest prediction, which agrees with our expectation from the EDA done at the beginning of this document. We now turn to investigating how these, and other variables, are related to the predicted response.

Variable Dependence

Variable dependence plots show the predicted response as a function of a covariate of interest, where each observation is represented by a point on the plot. Each predicted point is an individual observations, dependent on the full combination of all other covariates, not only on the covariate of interest. Interpretation of variable dependence plots can only be in general terms, as point predictions are a function of all covariates in that particular observation. However, variable dependence is straight forward to calculate, involving only the getting the predicted response for each observation.

We use the gg_variable function call to extract the training set variables and the predicted OOB response from randomForestSRC::rfsrc and randomForestSRC::predict objects. In the following code block, we will store the gg_variable data object for later use, as all remaining variable dependence plots can be constructed from this (gg_v) object. We will also use the minimal depth selected variables (minimal depth lower than the threshold value) from the previously stored gg_minimal_depth object (gg_md$topvars) to filter the variables of interest.

The plot.gg_variable function call operates in the gg_variable object. We pass it the list of variables of interest (xvar) and request a single panel (panel=TRUE) to display the figures. By default, the plot.gg_variable function returns a list of ggplot objects, one figure for each variable named in xvar argument. The next three arguments are passed to internal ggplot plotting routines. The se and span arguments are used to modify the internal call to ggplot2::geom_smooth for fitting smooth lines to the data. The alpha argument lightens the coloring points in the ggplot2::geom_point call, making it easier to see point over plotting. We also demonstrate modification of the plot labels using the ggplot2::labs function.

> # Create the variable dependence object from the random forest
> gg_v <- gg_variable(rfsrc_Boston)
> 
> # We want the top minimal depth variables only,
> # plotted in minimal depth rank order. 
> xvar <- gg_md$topvars
> 
> # plot the variable list in a single panel plot
> plot(gg_v, xvar=xvar, panel=TRUE, 
+      se=.95, span=1.2, alpha=.4)+
+   labs(y=st.labs["medv"], x="")

This figure looks very similar to the EDA figure, although with transposed axis as we plot the response variable on the y-axis. The closer the panels match, the better the RF prediction. The panels are sorted to match the order of variables in the xvar argument and include a smooth loess line (Cleveland 1981; Cleveland and Devlin 1988), with 95% shaded confidence band, to indicates the trend of the prediction dependence over the covariate values.

There is not a convenient method to panel scatter plots and boxplots together, so we recommend creating panel plots for each variable type separately. The Boston housing data does contain a single categorical variable, the Charles river logical variable. Variable dependence plots for categorical variables are constructed using boxplots to show the distribution of the predictions within each category. Although the Charles river variable has the lowest importance scores in both VIMP and minimal depth measures, we include the variable dependence plot as an example of categorical variable dependence.

> plot(gg_v, xvar="chas", points=FALSE,
+      se=FALSE, notch=TRUE, alpha=.4)+
+   labs(y=st.labs["medv"])+
+   coord_cartesian(ylim=c(5,49))

The figure shows that most housing tracts do not border the Charles river (chas=FALSE), and comparing the distributions of the predicted median housing values indicates no significant difference in home values. This reinforces the findings in both VIMP and Minimal depth, the Charles river variable has very little impact on the forest prediction.

Partial Dependence.

Partial variable dependence plots are a risk adjusted alternative to variable dependence. Partial plots are generated by integrating out the effects of all variables beside the covariate of interest. Partial dependence data are constructed by selecting points evenly spaced along the distribution of the \(X\) variable of interest. For each value (\(X = x\)), we calculate the average RF prediction over all other covariates in \(X\) by \[ \tilde{f}(x) = \frac{1}{n} \sum_{i = 1}^n \hat{f}(x, x_{i, o}), \] where \(\hat{f}\) is the predicted response from the random forest and \(x_{i, o}\) is the value for all other covariates other than \(X = x\) for the observation \(i\) (Friedman 2000). Essentially, we average a set of predictions for each observation in the training set at the value of \(X=x\). We repeating the process for a sequence of \(X=x\) values to generate the estimated points to create a partial dependence plot.

Partial plots are another computationally intensive analysis, especially when there are a large number of observations. We again turn to our data caching strategy here. The default parameters for the randomForestSRC::plot.variable function generate partial dependence estimates at npts=25 points along the variable of interest. For each point of interest, the plot.variable function averages n response predictions. This is repeated for each of the variables of interest and the results are returned for later analysis.

> # Load the data, from the call:
> # partial_Boston <- plot.variable(rfsrc_Boston, 
> #                                 xvar=gg_md$topvars, 
> #                                 partial=TRUE, sorted=FALSE, 
> #                                 show.plots = FALSE )
> data(partial_Boston)
> 
> # generate a list of gg_partial objects, one per xvar.
> gg_p <- gg_partial(partial_Boston)
> 
> # plot the variable list in a single panel plot
> plot(gg_p, xvar=xvar, panel=TRUE, se=FALSE) +
+   labs(y=st.labs["medv"], x="")

We again order the panels by minimal depth ranking. We see again how the lstat and rm variables are strongly related to the median value response, making the partial dependence of the remaining variables look flat. We also see strong nonlinearity of these two variables. The lstat variable looks rather quadratic, while the rm shape is more complex.

We could stop here, indicating that the RF analysis has found these ten variables to be important in predicting the median home values. That strongest associations to home values where there is a decrease with rising lstat variable and an increase when rm\(>6\). However, we may also be interested in investigating how variables these work together to help random forest prediction.

Partial dependence coplots

By characterizing conditional plots as stratified variable dependence plots, the next logical step would be to generate an analogous conditional partial dependence plot. The process is similar to variable dependence coplots, first determine conditional group membership, then calculate the partial dependence estimates on each subgroup using the randomForestSRC::plot.variable function with a the subset argument for each grouped interval. The ggRandomForests::gg_partial_coplot function is a wrapper for generating a conditional partial dependence data object. Given a random forest (randomForestSRC::rfsrc object) and a groups vector for conditioning the training data set observations, gg_partial_coplot calls the randomForestSRC::plot.variable function for a set of training set observations conditional on groups membership. The function returns a gg_partial_coplot object, a sub class of the gg_partial object, which can be plotted with the plot.gg_partial function.

The following code block will generate the data object for creating partial dependence coplot of the predicted median home value as a function of lstat conditional on membership within the 6 groups of rm “intervals” that we examined in the previous section.

> partial_coplot_Boston <- gg_partial_coplot(rfsrc_Boston, xvar="lstat", 
+                                            groups=rm_grp,
+                                            show.plots=FALSE)

Since the gg_partial_coplot makes a call to randomForestSRC::plot.variable for each group (6) in the conditioning set, we again resort to the data caching strategy, and load the stored result data from the ggRandomForests package. We modify the legend label to indicate we’re working with groups of the “Room” variable, and use the palette="Set1" Color Brewer color palette to choose a nice color theme for displaying the six curves.

> # Load the stored partial coplot data.
> data(partial_coplot_Boston)
> 
> # Partial coplot
> plot(partial_coplot_Boston, se=FALSE)+
+   labs(x=st.labs["lstat"], y=st.labs["medv"], 
+        color="Room", shape="Room")+
+   scale_color_brewer(palette="Set1")

Unlike variable dependence coplots, we do not need to use a panel format for partial dependence coplots because we are looking risk adjusted estimates (points) instead of population estimates. The figure has a loess curve through the point estimates conditional on the rm interval groupings. The figure again indicates that larger homes (rm from 6.87 and up, shown in yellow) have a higher median value then the others. In neighborhoods with higher lstat percentage, the Median values decrease with rm until it stabilizes from the intervals between 5.73 and 6.47, then decreases again for values smaller than 5.73. In lower lstat neighborhoods, the effect of smaller rm is not as noticeable.

We can view the partial coplot curves as slices along a surface viewed into the page, either along increasing or decreasing rm values. This is made more difficult by our choice to select groups of similar population size, as the curves are not evenly spaced along the rm variable. We return to this problem in the next section.

We also construct the complement view, for partial dependence coplot of the predicted median home value as a function of rm conditional on membership within the 6 groups of lstat “intervals”, and cache the following gg_partial_coplot data call.

> partial_coplot_Boston2 <- gg_partial_coplot(rfsrc_Boston, xvar="rm", 
+                                             groups=lstat_grp,
+                                             show.plots=FALSE)

We plot these results with the following plot.gg_variable call:

> # Load the stored partial coplot data.
> data(partial_coplot_Boston2)
> 
> # Partial coplot
> plot(partial_coplot_Boston2, se=FALSE)+
+   labs(x=st.labs["rm"], y=st.labs["medv"], 
+        color="Lower Status", shape="Lower Status")+
+   scale_color_brewer(palette="Set1")

This figure indicates that the median home value does not change much until the rm increases above 6.5, then flattens again above 8, regardless of the lstat value. This agrees well with the rm partial plot shown earlier. Again, care must be taken in interpreting the even spacing of these curves along the percentage of lstat groupings, as again, we chose these groups to have similar sized populations, not to be evenly spaced along the lstat variable.

Partial coplot surfaces

Visualizing two dimensional projections of three dimensional data is difficult, though there are tools available to make the data more understandable. To make the interplay of lower status and average room size a bit more understandable, we will generate a contour plot of the median home values. We could generate this figure with the data we already have, but the resolution would be a bit strange. To generate the plot of lstat conditional on rm groupings, we would end up with contours over a grid of lstat=\(25 \times\) rm=\(6\), for the alternative rm conditional on lstat groups, we’d have the transpose grid of lstat=\(6 \times\) rm=\(25\).

Since we are already using the data caching strategy, we will generate another gg_partial_coplot data set with increased resolution in both the lstat and rm dimensions. For this exercise, we will create 50 rm groups and generate the partial plot data at npts=50 points along the lstat dimension for each group within the plot.variable call. This code block generates the 50 rm groups, each containing about 9 observations.

> # Find the quantile points to create 50 interval groups
> rm_pts <- quantile_cuts(rfsrc_Boston$xvar$rm, groups=50)
> 
> # generate the grouping intervals.
> rm_grp <- cut(rfsrc_Boston$xvar$rm, breaks=rm_pts)

We use the following data call to generate the gg_partial_coplot data object. This took about 15 minutes to run on a quad core Mac Air.

> # Generate the gg_partial_coplot data object
> system.time(partial_coplot_Boston_surf <- gg_partial_coplot(rfsrc_Boston, xvar="lstat", 
+                                                  groups=rm_grp, npts=50,
+                                                  show.plots=FALSE))
> 
> # user  system elapsed 
> # 848.266  79.705 934.584 

The cached gg_partial_coplot data object is included as a data set in the ggRandomForests package. We load the data, attach numeric values for the rm groups, and generate the figure.

> # Load the stored partial coplot data.
> data(partial_coplot_Boston_surf)
> 
> # Instead of groups, we want the raw rm point values,
> # To make the dimensions match, we need to repeat the values
> # for each of the 50 points in the lstat direction
> rm.tmp <- do.call(c,lapply(rm_pts[-1], 
+                  function(grp){rep(grp, 50)}))
> 
> # attach the data to the gg_partial_coplot
> partial_coplot_Boston_surf$rm <- rm.tmp
> 
> # ggplot2 contour plot of x, y and z data.
> ggplot(partial_coplot_Boston_surf, aes(x=lstat, y=rm, z=yhat))+
+   stat_contour(aes(colour = ..level..), binwidth = 1)+
+   labs(x=st.labs["lstat"], y=st.labs["rm"], 
+        color="Median Home Values")+
+   scale_colour_gradientn(colours=topo.colors(10))

The contours are generated over the raw gg_partial estimation points, not smooth curves as shown in the partial plot and coplot figures. We can also generate a surface with this data using the plot3D package and the plot3D::surf3D function. Viewed in 3D, a surface can help to better understand what the contour lines mean.

> # Modify the figure margins to make the figure larger
> par(mai = c(0,0,0,0))
> 
> # Transform the gg_partial_coplot object into a list of three named matrices
> # for surface plotting with plot3D::surf3D
> srf <- surface_matrix(partial_coplot_Boston_surf, c("lstat", "rm", "yhat"))
> 
> # Generate the figure.
> surf3D(x=srf$x, y=srf$y, z=srf$z, col=topo.colors(10),
+        colkey=FALSE, border = "black", bty="b2", 
+        shade = 0.5, expand = 0.5, 
+        lighting = TRUE, lphi = -50,
+        xlab="Lower Status", ylab="Average Rooms", zlab="Median Value"
+        )

These figures reinforce the previous findings, where lower home values are associated with higher lstat percentage, and higher values are associated with larger rm. The difference in this figure is we can see how the predicted values change as we move around the map of lstat and rm combinations.