hstats (version 1.2.2)

h2_overall: Overall Interaction Strength

Description

Friedman and Popescu's statistic of overall interaction strength per feature, see Details. Use plot() to get a barplot.

Usage

h2_overall(object, ...)
# S3 method for default
h2_overall(object, ...)
# S3 method for hstats
h2_overall(
  object,
  normalize = TRUE,
  squared = TRUE,
  sort = TRUE,
  zero = 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.

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.

Methods (by class)

• h2_overall(default): Default method of overall interaction strength.

• h2_overall(hstats): Overall interaction strength from "hstats" object.

Details

The logic of Friedman and Popescu (2008) is as follows: If there are no interactions involving feature \(x_j\), we can decompose the (centered) prediction function \(F\) into the sum of the (centered) partial dependence \(F_j\) on \(x_j\) and the (centered) partial dependence \(F_{\setminus j}\) on all other features \(\mathbf{x}_{\setminus j}\), i.e., $$ F(\mathbf{x}) = F_j(x_j) + F_{\setminus j}(\mathbf{x}_{\setminus j}). $$ Correspondingly, Friedman and Popescu's statistic of overall interaction strength of \(x_j\) is given by $$ H_j^2 = \frac{\frac{1}{n} \sum_{i = 1}^n\big[F(\mathbf{x}_i) - \hat F_j(x_{ij}) - \hat F_{\setminus j}(\mathbf{x}_{i\setminus j}) \big]^2}{\frac{1}{n} \sum_{i = 1}^n\big[F(\mathbf{x}_i)\big]^2} $$ (check partial_dep() for all definitions).

Remarks:

1. Partial dependence functions (and \(F\)) are all centered to (possibly weighted) mean 0.

2. Partial dependence functions (and \(F\)) are evaluated over the data distribution. This is different to partial dependence plots, where one uses a fixed grid.

3. Weighted versions follow by replacing all arithmetic means by corresponding weighted means.

4. Multivariate predictions can be treated in a component-wise manner.

5. Due to (typically undesired) extrapolation effects of partial dependence functions, depending on the model, values above 1 may occur.

6. \(H^2_j = 0\) means there are no interactions associated with \(x_j\). The higher the value, the more prediction variability comes from interactions with \(x_j\).

7. Since the denominator is the same for all features, the values of the test statistics can be compared across features.

References

Friedman, Jerome H., and Bogdan E. Popescu. "Predictive Learning via Rule Ensembles." The Annals of Applied Statistics 2, no. 3 (2008): 916-54.

See Also

hstats(), h2(), h2_pairwise(), h2_threeway()

Examples

# MODEL 1: Linear regression
fit <- lm(Sepal.Length ~ . + Petal.Width:Species, data = iris)
s <- hstats(fit, X = iris[, -1])
h2_overall(s)
plot(h2_overall(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], verbose = FALSE)
plot(h2_overall(s, zero = FALSE))