This function calculates the "area of applicability" of a model, as introduced by Meyer and Pebesma (2021). While the initial paper introducing this method focused on spatial models, there is nothing inherently spatial about the method; it can be used with any type of data (and, because it does not care about the spatial arrangement of your data, can be used with 2D or 3D spatial data, and with geographic or projected CRS).

## Usage

ww_area_of_applicability(x, ...)

# S3 method for data.frame
ww_area_of_applicability(x, testing = NULL, importance, ..., na_rm = FALSE)

# S3 method for matrix
ww_area_of_applicability(x, testing = NULL, importance, ..., na_rm = FALSE)

# S3 method for formula
ww_area_of_applicability(
x,
data,
testing = NULL,
importance,
...,
na_rm = FALSE
)

# S3 method for recipe
ww_area_of_applicability(
x,
data,
testing = NULL,
importance,
...,
na_rm = FALSE
)

# S3 method for rset
ww_area_of_applicability(x, y = NULL, importance, ..., na_rm = FALSE)

## Arguments

x

Either a data frame, matrix, formula (specifying predictor terms on the right-hand side), recipe (from recipes::recipe(), or rset object, produced by resampling functions from rsample or spatialsample.

If x is a recipe, it should be the same one used to pre-process the data used in your model. If the recipe used to build the area of applicability doesn't match the one used to build the model, the returned area of applicability won't be correct.

...

Not currently used.

testing

A data frame or matrix containing the data used to validate your model. This should be the same data as used to calculate all model accuracy metrics.

If this argument is NULL, then this function will use the training data (from x or data) to calculate within-sample distances. This may result in the area of applicability threshold being set too high, with the result that too many points are classed as "inside" the area of applicability.

importance

Either:

• A data.frame with two columns: term, containing the names of each variable in the training and testing data, and estimate, containing the (raw or scaled) feature importance for each variable.

• An object of class vi with at least two columns, Variable and Importance.

All variables in the training data (x or data, depending on the context) must have a matching importance estimate, and all terms with importance estimates must be in the training data.

na_rm

A logical of length 1, indicating whether observations (in both training and testing) with NA values in predictors should be removed. Only predictor variables are considered, and this value has no impact on predictions (where NA values produce NA predictions). If na_rm = FALSE and NA values are found, this function returns an error.

data

The data frame representing your "training" data, when using the formula or recipe methods.

y

Optional: a recipe (from recipes::recipe()) or formula.

If y is a recipe, it should be the same one used to pre-process the data used in your model. If the recipe used to build the area of applicability doesn't match the one used to build the model, the returned area of applicability won't be correct.

## Value

A ww_area_of_applicability object, which can be used with predict() to calculate the distance of new data to the original training data, and determine if new data is within a model's area of applicability.

## Details

Predictions made on points "inside" the area of applicability should be as accurate as predictions made on the data provided to testing. That means that generally testing should be your final hold-out set so that predictions on points inside the area of applicability are accurately described by your reported model metrics. When passing an rset object to x, predictions made on points "inside" the area of applicability instead should be as accurate as predictions made on the assessment sets during cross-validation.

This method assumes your model was fit using dummy variables in the place of any non-numeric predictor, and that you have one importance score per dummy variable. Having non-numeric predictors will cause this function to fail.

## Differences from CAST

This implementation differs from Meyer and Pebesma (2021) (and therefore from CAST) when using cross-validated data in order to minimize data leakage. Namely, in order to calculate the dissimilarity index $$DI_{k}$$, CAST:

1. Rescales all data used for cross validation at once, lumping assessment folds in with analysis data.

2. Calculates a single $$\bar{d}$$ as the mean distance between all points in the rescaled data set, including between points in the same assessment fold.

3. For each point $$k$$ that's used in an assessment fold, calculates $$d_{k}$$ as the minimum distance between $$k$$ and any point in its corresponding analysis fold.

4. Calculates $$DI_{k}$$ by dividing $$d_{k}$$ by $$\bar{d}$$ (which was partially calculated as the distance between $$k$$ and the rest of the rescaled data).

Because assessment data is used to calculate constants for rescaling analysis data and $$\bar{d}$$, the assessment data may appear too "similar" to the analysis data when calculating $$DI_{k}$$. As such, waywiser treats each fold in an rset independently:

1. Each analysis set is rescaled independently.

2. Separate $$\bar{d}$$ are calculated for each fold, as the mean distance between all points in the analysis set for that fold.

3. Identically to CAST, $$d_{k}$$ is the minimum distance between a point $$k$$ in the assessment fold and any point in the corresponding analysis fold.

4. $$DI_{k}$$ is then found by dividing $$d_{k}$$ by $$\bar{d}$$, which was calculated independently from k.

Predictions are made using the full training data set, rescaled once (in the same way as CAST), and the mean $$\bar{d}$$ across folds, under the assumption that the "final" model in use will be retrained using the entire data set.

In practice, this means waywiser produces very slightly higher $$\bar{d}$$ values than CAST and a slightly higher area of applicability threshold than CAST when using rset objects.

## References

H. Meyer and E. Pebesma. 2021. "Predicting into unknown space? Estimating the area of applicability of spatial prediction models," Methods in Ecology and Evolution 12(9), pp 1620 - 1633, doi: 10.1111/2041-210X.13650.

Other area of applicability functions: predict.ww_area_of_applicability()

## Examples

train <- vip::gen_friedman(1000, seed = 101) # ?vip::gen_friedman
test <- train[701:1000, ]
train <- train[1:700, ]
pp <- stats::ppr(y ~ ., data = train, nterms = 11)
importance <- vip::vi_permute(
pp,
target = "y",
metric = "rsquared",
pred_wrapper = predict
)

aoa <- ww_area_of_applicability(y ~ ., train, test, importance = importance)
predict(aoa, test)
#> # A tibble: 300 × 2
#>        di aoa
#>     <dbl> <lgl>
#>  1 0.175  TRUE
#>  2 0.191  TRUE
#>  3 0.190  TRUE
#>  4 0.0802 TRUE
#>  5 0.141  TRUE
#>  6 0.287  TRUE
#>  7 0.178  TRUE
#>  8 0.205  TRUE
#>  9 0.134  TRUE
#> 10 0.177  TRUE
#> # ℹ 290 more rows

# Equivalent methods for calculating AOA:
ww_area_of_applicability(train[2:11], test[2:11], importance)
#> # Predictors:
#>    10
#> Area-of-applicability threshold:
#>    0.3080491
ww_area_of_applicability(
as.matrix(train[2:11]),
as.matrix(test[2:11]),
importance
)
#> # Predictors:
#>    10
#> Area-of-applicability threshold:
#>    0.3080491