Predictor based print

Summary method allows you to check details about transformation of specific variable.

Standard usage print(xspliner_object, variable_name)

Quantitative variable

When predictor is the quantitative variable its transition is based on GAM model. For this case print uses standard print.gam method.

print(model_xs, "Petal.Length")

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## yhat ~ s(Petal.Length, bs = "cr")
## 
## Estimated degrees of freedom:
## 8.82  total = 9.82 
## 
## GCV score: 0.001747755

Qualitative variable

In case of qualitative predictor, standard print.factorMerger method is used.

print(model_xs, "Species")

## Family: gaussian Factor Merger.
## 
## Factor levels were merged in the following order:
## 
##      groupA       groupB                     model   pvalVsFull   pvalVsPrevious
## ---  -----------  --------------------  ----------  -----------  ---------------
## 0                                        -262.1572            1                1
## 1    versicolor   virginica              -278.4864            0                0
## 2    setosa       versicolorvirginica    -354.2443            0                0

Extracting effect

Each transition is built on top of the black box response data. For example the default response for quantitative variables is PDP - for qualitative ones ICE.

In order to extract the effect use transition method with type parameter equals to data

transition(model_xs, predictor = "Petal.Length", type = "data") %>% 
  head

##   Petal.Length      yhat
## 1     1.000000 0.7103410
## 2     1.059596 0.7102940
## 3     1.119192 0.7102940
## 4     1.178788 0.7139648
## 5     1.238384 0.7152855
## 6     1.297980 0.7179491

transition(model_xs, predictor = "Species", type = "data") %>% 
  head

##      Species      yhat yhat.id
## 1     setosa 0.2433265       1
## 2 versicolor 0.6857316       1
## 3  virginica 0.8473016       1
## 4     setosa 0.2157995       2
## 5 versicolor 0.6870610       2
## 6  virginica 0.8488110       2

Extracting transition model

After we built transition basing on continuity of variable specific model is created. In case of quantitative predictor we build GAM model in order to get spline approximation of effect. In case of qualitative predictor we build factorMerger object and get optimal factor division on that.

To extract the model, use transition method with type = "base":

transition(model_xs, predictor = "Petal.Length", type = "base")

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## yhat ~ s(Petal.Length, bs = "cr")
## 
## Estimated degrees of freedom:
## 8.82  total = 9.82 
## 
## GCV score: 0.001747755

transition(model_xs, predictor = "Species", type = "base")

## Family: gaussian Factor Merger.
## 
## Factor levels were merged in the following order:
## 
##      groupA       groupB                     model   pvalVsFull   pvalVsPrevious
## ---  -----------  --------------------  ----------  -----------  ---------------
## 0                                        -262.1572            1                1
## 1    versicolor   virginica              -278.4864            0                0
## 2    setosa       versicolorvirginica    -354.2443            0                0

Extracting transition function

The final result of building transition is transformation function, that is used in the final GLM model estimation.

To extract the function just use transition method with type = "function".

petal_length_xs <- transition(model_xs, predictor = "Petal.Length", type = "function")
x <- seq(1, 7, length.out = 50)
plot(x, petal_length_xs(x))

species_xf <- transition(model_xs, predictor = "Species", type = "function")
species_xf(c("setosa", "versicolor", "virginica"))

## [1] setosa              versicolorvirginica versicolorvirginica
## Levels: setosa versicolorvirginica

GLM summary

Standard summary method is just wrapper for summary::glm. In order to use this just type:

summary(model_xs)

## 
## Call:
## stats::glm(formula = Petal.Width ~ Sepal.Length + xs(Petal.Length) + 
##     xf(Species), family = family, data = data)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -0.68325  -0.07925  -0.02485   0.10265   0.50723  
## 
## Coefficients:
##                                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    -1.99718    0.14136 -14.128  < 2e-16 ***
## Sepal.Length                   -0.01340    0.03652  -0.367    0.714    
## xs(Petal.Length)                3.21878    0.22377  14.384  < 2e-16 ***
## xf(Species)versicolorvirginica -0.73479    0.12926  -5.685  6.9e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for gaussian family taken to be 0.03553466)
## 
##     Null deviance: 86.5699  on 149  degrees of freedom
## Residual deviance:  5.1881  on 146  degrees of freedom
## AIC: -68.96
## 
## Number of Fisher Scoring iterations: 2

Predictor based summary

Summary method allows you to check details about transformation of specific variable.

Standard usage summary(xspliner_object, variable_name)

Quantitative variable

When predictor is quantitative variable its transition is based on GAM model. For this case summary displays summary of that model.

summary(model_xs, "Petal.Length")

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## yhat ~ s(Petal.Length, bs = "cr")
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.20307    0.00397     303   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                   edf Ref.df     F p-value    
## s(Petal.Length) 8.816   8.99 637.1  <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.983   Deviance explained = 98.5%
## GCV = 0.0017478  Scale est. = 0.0015762  n = 100

Qualitative variable

In case of qualitative predictor, the method displays data.frame storing information how factors were merged during the transition.

summary(model_xs, "Species")

##         orig                pred
## 1     setosa              setosa
## 2 versicolor versicolorvirginica
## 3  virginica versicolorvirginica

Surrogate vs Black Box comparison

Providing model parameter instead of predictor, the summary displays a few statistics that compares original model with surrogate one. All statistics definitions are included in summary.xspline documentation.

Here we show one example for classification model.

For this example we use ISLR::Default data and build svm model as black box. The model aims to predict default variable, indicating whether the customer defaulted on their debt.

library(xspliner)
library(e1071)
set.seed(1)
data <- ISLR::Default
default.svm <- svm(default ~ ., data = data, probability = TRUE)
default.xs <- xspline(default ~ student + xs(balance) + xs(income), model = default.svm)

In order to check the summary, we need to specify prediction functions for each model. In this case predictions are probabilities of success:

prob_svm <- function(object, newdata) attr(predict(object, newdata = newdata, probability = TRUE), "probabilities")[, 2]
prob_xs <- function(object, newdata) predict(object, newdata = newdata, type = "response")

Almost each summary statistic compares models basing on some data.

In this case we’re going to compare models on training data providing:

• newdata parameter as training data
• model parameter with black box model
• prediction_funs as a list of prediction functions (for surrogate and original model respectively)

summary(default.xs, model = default.svm, newdata = data, prediction_funs = list(prob_xs, prob_svm))

## Models comparison 
##   1 - Max prediction normed-diff:  0.5268109 
##   R^2:  0.9185403

## Setting levels: control = No, case = Yes
## Setting levels: control = No, case = Yes

##   1 - Max ROC diff:  0.8712113 
##   1 - Mean ROC diff:  0.9506292

Another set of statistics is generated for prediction functions that return response levels.

response_svm <- function(object, newdata) predict(object, newdata = newdata)
response_xs <- function(object, newdata) {
  y_levels <- levels(newdata[[environment(object)$response]])
  factor(y_levels[(predict.glm(object, newdata = newdata, type = "link") > 0) + 1], levels = y_levels)
}

And similarly to previous example:

summary(default.xs, model = default.svm, newdata = data, prediction_funs = list(response_xs, response_svm))

## Models comparison 
##   Mean predictions similarity:  0.9966 
##   ACC Black Box:  0.9719 
##   ACC Surrogate:  0.9729