mid.breakdown() calculates the contribution of each component function of a fitted MID model to a single prediction.
It breaks down the total prediction into the effects of the intercept, main effects, and interactions.
Usage
mid.breakdown(
object,
data = NULL,
row = NULL,
sort = TRUE,
format = c("%s", "%s, %s")
)Arguments
- object
a "mid" object.
- data
a data frame containing one or more observations for which to calculate the MID breakdown. If not provided, data is automatically extracted based on the function call.
- row
an optional numeric value or character string specifying the row of
datato be used for the breakdown. IfNULL, and thedatacontains two or more observations, only the first observation is used.- sort
logical. If
TRUE, the output data frame is sorted by the absolute contribution of each effect.- format
a character vector of length two to be used as a format string for
sprintf()to display the values of main effects and interactions, respectively.
Value
mid.breakdown() returns an object of class "mid.breakdown". This is a list with the following components:
- breakdown
a data frame containing the breakdown of the prediction.
- data
the data frame containing the predictor variable values used for the prediction.
- intercept
the intercept of the MID model.
- prediction
the predicted value from the MID model.
Details
mid.breakdown() is a method for local interpretability.
For a given observation, it provides a clear answer to the question, "How much did each component of the MID model contribute to the final prediction?"
The function calculates the value of each term in the MID model's additive structure for the specified observation. The total prediction is the sum of these individual contributions. The prediction, denoted \(\mathcal{F}(\mathbf{x})\), is decomposed as: $$\mathcal{F}(\mathbf{x}) = f_\phi + \sum_{j} f_{j}(x_j) + \sum_{j<k} f_{jk}(x_j, x_k)$$
The output data frame itemizes the numerical value of each main effect (\(f_{j}(x_j)\)) and interaction effect (\(f_{jk}(x_j, x_k)\)), along with the intercept (\(f_\phi\)). This makes the prediction transparent and easy to understand.
Examples
data(airquality, package = "datasets")
mid <- interpret(Ozone ~ .^2, data = airquality, lambda = 1)
#> 'model' not passed: response variable in 'data' is used
# Calculate the breakdown for the first observation in the data
mbd <- mid.breakdown(mid, data = airquality, row = 1)
print(mbd)
#>
#> MID Breakdown of a Prediction
#>
#> Intercept: 42.099
#>
#> Prediction: 39.739
#>
#> Breakdown of Effects:
#> term value mid order
#> 1 Temp 67 -1.5043e+01 1
#> 2 Day 1 4.9770e+00 1
#> 3 Month 5 3.7575e+00 1
#> 4 Wind:Month 7.4, 5 2.0404e+00 2
#> 5 Temp:Day 67, 1 1.3885e+00 2
#> 6 Solar.R:Month 190, 5 1.3124e+00 2
#> 7 Wind 7.4 1.0227e+00 1
#> 8 Solar.R 190 -9.4228e-01 1
#> 9 Wind:Temp 7.4, 67 -6.2914e-01 2
#> 10 Solar.R:Temp 190, 67 -5.6533e-01 2
#> 11 Month:Day 5, 1 5.0500e-01 2
#> 12 Solar.R:Day 190, 1 -2.4755e-01 2
#> 13 Wind:Day 7.4, 1 4.2771e-02 2
#> 14 Temp:Month 67, 5 2.0660e-02 2
#> 15 Solar.R:Wind 190, 7.4 -3.4803e-04 2
# Calculate the breakdown for the third observation in the data
mbd <- mid.breakdown(mid, data = airquality, row = 3)
print(mbd)
#>
#> MID Breakdown of a Prediction
#>
#> Intercept: 42.099
#>
#> Prediction: 9.9693
#>
#> Breakdown of Effects:
#> term value mid order
#> 1 Temp 74 -16.686548 1
#> 2 Day 3 -11.078821 1
#> 3 Wind 12.6 -9.263680 1
#> 4 Month 5 3.757459 1
#> 5 Solar.R:Month 149, 5 0.702906 2
#> 6 Wind:Day 12.6, 3 0.501892 2
#> 7 Wind:Month 12.6, 5 -0.461498 2
#> 8 Solar.R:Wind 149, 12.6 0.321963 2
#> 9 Month:Day 5, 3 0.180867 2
#> 10 Temp:Day 74, 3 0.120489 2
#> 11 Solar.R:Day 149, 3 -0.069100 2
#> 12 Temp:Month 74, 5 -0.068488 2
#> 13 Solar.R 149 -0.034390 1
#> 14 Wind:Temp 12.6, 74 -0.030288 2
#> 15 Solar.R:Temp 149, 74 -0.022524 2