PD Bases Importance (Experimental)

Experimental variable importance method based on partial dependence functions. While related to Greenwell et al., our suggestion measures not only main effect strength but also interaction effects. It is very closely related to $H^2_j$, see Details. Use plot() to get a barplot.

pd_importance(object, ...)

# S3 method for default
pd_importance(object, ...)

# S3 method for hstats
pd_importance(
  object,
  normalize = TRUE,
  squared = TRUE,
  sort = TRUE,
  zero = TRUE,
  ...
)

Arguments

object: Object of class "hstats".
...: Currently unused.
normalize: Should statistics be normalized? Default is TRUE.
squared: Should squared statistics be returned? Default is TRUE.
sort: Should results be sorted? Default is TRUE. (Multi-output is sorted by row means.)
zero: Should rows with all 0 be shown? Default is TRUE.

Value

An object of class "hstats_matrix" containing these elements:

M: Matrix of statistics (one column per prediction dimension), or NULL.
SE: Matrix with standard errors of M, or NULL. Multiply with sqrt(m_rep) to get standard deviations instead. Currently, supported only for perm_importance().
m_rep: The number of repetitions behind standard errors SE, or NULL. Currently, supported only for perm_importance().
statistic: Name of the function that generated the statistic.
description: Description of the statistic.

Details

If $x_j$ has no effects, the (centered) prediction function $F$ equals the (centered) partial dependence $F_{\setminus j}$ on all other features $\mathbf{x}_{\setminus j}$, i.e., $$ F(\mathbf{x}) = F_{\setminus j}(\mathbf{x}_{\setminus j}). $$ Therefore, the following measure of variable importance follows: $$ \textrm{PDI}_j = \frac{\frac{1}{n} \sum_{i = 1}^n\big[F(\mathbf{x}_i) - \hat F_{\setminus j}(\mathbf{x}_{i\setminus j})\big]^2}{\frac{1}{n} \sum_{i = 1}^n \big[F(\mathbf{x}_i)\big]^2}. $$ It differs from $H^2_j$ only by not subtracting the main effect of the $j$-th feature in the numerator. It can be read as the proportion of prediction variability unexplained by all other features. As such, it measures variable importance of the $j$-th feature, including its interaction effects (check partial_dep() for all definitions).

Remarks 1 to 4 of h2_overall() also apply here.

Methods (by class)

pd_importance(default): Default method of PD based feature importance.
pd_importance(hstats): PD based feature importance from "hstats" object.

References

Greenwell, Brandon M., Bradley C. Boehmke, and Andrew J. McCarthy. A Simple and Effective Model-Based Variable Importance Measure. Arxiv (2018).

Examples

# MODEL 1: Linear regression
fit <- lm(Sepal.Length ~ . , data = iris)
s <- hstats(fit, X = iris[, -1])
#> 1-way calculations...
#> 
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plot(pd_importance(s))


# MODEL 2: Multi-response linear regression
fit <- lm(as.matrix(iris[, 1:2]) ~ Petal.Length + Petal.Width + Species, data = iris)
s <- hstats(fit, X = iris[, 3:5])
#> 1-way calculations...
#> 
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plot(pd_importance(s))