The *Expected Gradients* method (Erion et al., 2021), also known as
*GradSHAP*, is a local feature attribution technique which extends the
`IntegratedGradient`

method and provides approximate Shapley values. In
contrast to IntegratedGradient, it considers not only a single reference
value \(x'\) but the whole distribution of reference values
\(X' \sim x'\) and averages the IntegratedGradient values over this
distribution. Mathematically, the method can be described as follows:
$$
E_{x'\sim X', \alpha \sim U(0,1)}[(x - x') \times \frac{\partial f(x' + \alpha (x - x'))}{\partial x}]
$$
The distribution of the reference values is specified with the argument
`data_ref`

, of which `n`

samples are taken at random for each instance
during the estimation.

The R6 class can also be initialized using the `run_expgrad`

function
as a helper function so that no prior knowledge of R6 classes is required.

G. Erion et al. (2021) *Improving performance of deep learning models with *
*axiomatic attribution priors and expected gradients.* Nature Machine
Intelligence 3, pp. 620-631.

Other methods:
`ConnectionWeights`

,
`DeepLift`

,
`DeepSHAP`

,
`Gradient`

,
`IntegratedGradient`

,
`LIME`

,
`LRP`

,
`SHAP`

,
`SmoothGrad`

`innsight::InterpretingMethod`

-> `innsight::GradientBased`

-> `ExpectedGradient`

`n`

(

`integer(1)`

)

Number of samples from the distribution of reference values and number of samples for the approximation of the integration path along \(\alpha\) (default: \(50\)).`data_ref`

(

`list`

)

The reference input for the ExpectedGradient method. This value is stored as a list of`torch_tensor`

s of shape*( , dim_in)*for each input layer.

`new()`

Create a new instance of the `ExpectedGradient`

R6 class. When
initialized, the method *Expected Gradient* is applied to the given
data and baseline values and the results are stored in the field `result`

.

```
ExpectedGradient$new(
converter,
data,
data_ref = NULL,
n = 50,
channels_first = TRUE,
output_idx = NULL,
output_label = NULL,
ignore_last_act = TRUE,
verbose = interactive(),
dtype = "float"
)
```

`converter`

(

`Converter`

)

An instance of the`Converter`

class that includes the torch-converted model and some other model-specific attributes. See`Converter`

for details.`data`

(

`array`

,`data.frame`

,`torch_tensor`

or`list`

)

The data to which the method is to be applied. These must have the same format as the input data of the passed model to the converter object. This means eitheran

`array`

,`data.frame`

,`torch_tensor`

or array-like format of size*(batch_size, dim_in)*, if e.g., the model has only one input layer, ora

`list`

with the corresponding input data (according to the upper point) for each of the input layers.

`data_ref`

(

`array`

,`data.frame`

,`torch_tensor`

or`list`

)

The reference inputs for the ExpectedGradient method. This value must have the same format as the input data of the passed model to the converter object. This means eitheran

`array`

,`data.frame`

,`torch_tensor`

or array-like format of size*( , dim_in)*, if e.g., the model has only one input layer, ora

`list`

with the corresponding input data (according to the upper point) for each of the input layers.It is also possible to use the default value

`NULL`

to take only zeros as reference input.

`n`

(

`integer(1)`

)

Number of samples from the distribution of reference values and number of samples for the approximation of the integration path along \(\alpha\) (default: \(50\)).`channels_first`

(

`logical(1)`

)

The channel position of the given data (argument`data`

). If`TRUE`

, the channel axis is placed at the second position between the batch size and the rest of the input axes, e.g.,`c(10,3,32,32)`

for a batch of ten images with three channels and a height and width of 32 pixels. Otherwise (`FALSE`

), the channel axis is at the last position, i.e.,`c(10,32,32,3)`

. If the data has no channel axis, use the default value`TRUE`

.`output_idx`

(

`integer`

,`list`

or`NULL`

)

These indices specify the output nodes for which the method is to be applied. In order to allow models with multiple output layers, there are the following possibilities to select the indices of the output nodes in the individual output layers:An

`integer`

vector of indices: If the model has only one output layer, the values correspond to the indices of the output nodes, e.g.,`c(1,3,4)`

for the first, third and fourth output node. If there are multiple output layers, the indices of the output nodes from the first output layer are considered.A

`list`

of`integer`

vectors of indices: If the method is to be applied to output nodes from different layers, a list can be passed that specifies the desired indices of the output nodes for each output layer. Unwanted output layers have the entry`NULL`

instead of a vector of indices, e.g.,`list(NULL, c(1,3))`

for the first and third output node in the second output layer.`NULL`

(default): The method is applied to all output nodes in the first output layer but is limited to the first ten as the calculations become more computationally expensive for more output nodes.

`output_label`

(

`character`

,`factor`

,`list`

or`NULL`

)

These values specify the output nodes for which the method is to be applied. Only values that were previously passed with the argument`output_names`

in the`converter`

can be used. In order to allow models with multiple output layers, there are the following possibilities to select the names of the output nodes in the individual output layers:A

`character`

vector or`factor`

of labels: If the model has only one output layer, the values correspond to the labels of the output nodes named in the passed`Converter`

object, e.g.,`c("a", "c", "d")`

for the first, third and fourth output node if the output names are`c("a", "b", "c", "d")`

. If there are multiple output layers, the names of the output nodes from the first output layer are considered.A

`list`

of`charactor`

/`factor`

vectors of labels: If the method is to be applied to output nodes from different layers, a list can be passed that specifies the desired labels of the output nodes for each output layer. Unwanted output layers have the entry`NULL`

instead of a vector of labels, e.g.,`list(NULL, c("a", "c"))`

for the first and third output node in the second output layer.`NULL`

(default): The method is applied to all output nodes in the first output layer but is limited to the first ten as the calculations become more computationally expensive for more output nodes.

`ignore_last_act`

(

`logical(1)`

)

Set this logical value to include the last activation functions for each output layer, or not (default:`TRUE`

). In practice, the last activation (especially for softmax activation) is often omitted.`verbose`

(

`logical(1)`

)

This logical argument determines whether a progress bar is displayed for the calculation of the method or not. The default value is the output of the primitive R function`interactive()`

.`dtype`

(

`character(1)`

)

The data type for the calculations. Use either`'float'`

for torch_float or`'double'`

for torch_double.

```
#----------------------- Example 1: Torch ----------------------------------
library(torch)
# Create nn_sequential model and data
model <- nn_sequential(
nn_linear(5, 12),
nn_relu(),
nn_linear(12, 2),
nn_softmax(dim = 2)
)
data <- torch_randn(25, 5)
ref <- torch_randn(1, 5)
# Create Converter
converter <- convert(model, input_dim = c(5))
# Apply method IntegratedGradient
int_grad <- IntegratedGradient$new(converter, data, x_ref = ref)
# You can also use the helper function `run_intgrad` for initializing
# an R6 IntegratedGradient object
int_grad <- run_intgrad(converter, data, x_ref = ref)
# Print the result as a torch tensor for first two data points
get_result(int_grad, "torch.tensor")[1:2]
#> torch_tensor
#> (1,.,.) =
#> -0.0181 -0.0321
#> 0.1131 -0.1167
#> 0.0834 0.3268
#> -0.0247 -0.0560
#> 0.0043 -0.1922
#>
#> (2,.,.) =
#> 0.1135 -0.1107
#> 0.0307 -0.0320
#> 0.0236 0.0393
#> 0.0039 0.0124
#> -0.0020 0.1557
#> [ CPUFloatType{2,5,2} ]
# Plot the result for both classes
plot(int_grad, output_idx = 1:2)
# Plot the boxplot of all datapoints and for both classes
boxplot(int_grad, output_idx = 1:2)
# ------------------------- Example 2: Neuralnet ---------------------------
if (require("neuralnet")) {
library(neuralnet)
data(iris)
# Train a neural network
nn <- neuralnet((Species == "setosa") ~ Petal.Length + Petal.Width,
iris,
linear.output = FALSE,
hidden = c(3, 2), act.fct = "tanh", rep = 1
)
# Convert the model
converter <- convert(nn)
# Apply IntegratedGradient with a reference input of the feature means
x_ref <- matrix(colMeans(iris[, c(3, 4)]), nrow = 1)
int_grad <- run_intgrad(converter, iris[, c(3, 4)], x_ref = x_ref)
# Get the result as a dataframe and show first 5 rows
get_result(int_grad, type = "data.frame")[1:5, ]
# Plot the result for the first datapoint in the data
plot(int_grad, data_idx = 1)
# Plot the result as boxplots
boxplot(int_grad)
}
# ------------------------- Example 3: Keras -------------------------------
if (require("keras") & keras::is_keras_available()) {
library(keras)
# Make sure keras is installed properly
is_keras_available()
data <- array(rnorm(10 * 32 * 32 * 3), dim = c(10, 32, 32, 3))
model <- keras_model_sequential()
model %>%
layer_conv_2d(
input_shape = c(32, 32, 3), kernel_size = 8, filters = 8,
activation = "softplus", padding = "valid") %>%
layer_conv_2d(
kernel_size = 8, filters = 4, activation = "tanh",
padding = "same") %>%
layer_conv_2d(
kernel_size = 4, filters = 2, activation = "relu",
padding = "valid") %>%
layer_flatten() %>%
layer_dense(units = 64, activation = "relu") %>%
layer_dense(units = 2, activation = "softmax")
# Convert the model
converter <- convert(model)
# Apply the IntegratedGradient method with a zero baseline and n = 20
# iteration steps
int_grad <- run_intgrad(converter, data,
channels_first = FALSE,
n = 20
)
# Plot the result for the first image and both classes
plot(int_grad, output_idx = 1:2)
# Plot the pixel-wise median of the results
plot_global(int_grad, output_idx = 1)
}
#------------------------- Plotly plots ------------------------------------
if (require("plotly")) {
# You can also create an interactive plot with plotly.
# This is a suggested package, so make sure that it is installed
library(plotly)
boxplot(int_grad, as_plotly = TRUE)
}
#> Warning: The `boxplot()` function is only intended for tabular or signal data. It is
#> called `plot_global()` instead.
```