# Introduction to innsight

In the last decade, it has been demonstrated in an impressive way how efficiently and successfully neural networks can analyze and understand enormous amounts of data. They can recognize patterns and associations and transfer this knowledge to new data points with remarkable accuracy. Moreover, their flexibility eliminates the feature engineering step that was often necessary before and allows them to work directly with raw data. Nevertheless, these associations and internal findings are hidden somewhere in the black box and it is unclear to the user what the crucial aspects of the prediction are. One way to open the black box is through so-called feature attribution methods. These are local methods that — based on a single data point (image, tabular instance,…) — assign a relevance of a previously defined output class or node to each input variable. In general, only a normal forward pass and a method-specific backward pass are required, making the implementation much faster compared to perturbation- or optimization-based methods like LIME or Occlusion. Figure 1 illustrates the basic approach of the feature attribution methods.

Figure 1: Feature attribution methods

### Why innsight?

Of course, we are not the first to provide several feature attribution methods for neural networks in one package. For example, there are several packages for Python, such as iNNvestigate, captum and zennit. Due to the great and extremely efficient deep learning libraries Keras/TensorFlow and PyTorch, it is only reasonable that these are all Python-exclusive. However, in recent years these libraries have been integrated more and more successfully into the R programming language. We fill this lack of feature attribution methods for neural networks in R with our package innsight.

Figure 2: innsight package

In addition to the availability in R, the package is also outstanding for the following aspects:

Deep-learning-library-agnostic: To be as flexible as possible and available to a range of users, we do not limit ourselves to models from a particular deep learning library, as is the case with all Python variants. Using the `Converter`, each passed model (from keras, torch or neuralnet) is first converted into a list with all relevant information about the model. Then, a torch-model is created from this list, which has the available feature attribution methods pre-implemented for each layer. If our package does not support your favorite library, there is also the option to do the converting step by yourself and pass a list directly.

- No Python dependency: In R, there are currently two major deep learning libraries, namely keras/tensorflow and torch. However, keras/tensorflow, accesses the corresponding Python methods via the package reticulate. We use the fast and efficient torch package for all computations, which runs without Python and directly accesses the C++ variant of PyTorch called LibTorch (see Fig. 2).

- Unified framework: It does not matter which model and method you choose, it is always the same three steps that lead to a visual illustration of the results (see the next section for details):

  model \(\xrightarrow{\text{Step 1}}\) `Converter` \(\xrightarrow{\text{Step 2}}\) method \(\xrightarrow{\text{Step 3}}\) `plot()` or `plot_global()`/`boxplot()`

- Visualization tools: Our package innsight offers several visualization methods for individual or summarized results regardless of whether it is tabular, 1D signal, 2D image data or a mix of these. Additionally, interactive plots can be created based on the plotly package.

## How to use

The following is more of a high-level overview that only explains some of the details of the three steps. In case you are looking for a more detailed overview of all configuration options, we refer you to the vignette “In-depth explanation” (same as `vignette("detailed_overview", package = "innsight")`). The three steps for explaining individual predictions with the provided methods are unified in this package and follow a strict scheme. This will hopefully allow any user a smooth and easy introduction to the possibilities of this package. The steps are:

    # Step 0: Model creation
    model <- ... # this step is left to the user

    # Step 1: Convert the model
    converter <- convert(model)
    converter <- Converter$new(model) # the same but without helper function

    # Step 2: Apply selected method to your data
    result <- run_method(converter, data)
    result <- Method$new(converter, data) # the same but without helper function

    # Step 3: Show and plot the results
    get_result(result) # get the result as an `array`, `data.frame` or `torch_tensor`
    plot(result) # for individual results (local)
    plot_global(result) # for summarized results (global)
    boxplot(result) # alias for `plot_global` for tabular and signal data

### Step 1: Model creation and converting

The innsight package aims to be as flexible as possible and independent of any particular deep learning package in which the passed network was learned or defined. For this reason, there are several ways in this package to pass a neural network to the `Converter` object, but the call is always the same:

    # Using the helper function `convert`
    converter <- convert(model, ...)
    # It simply passes all arguments to the initialization function of
    # the corresponding R6 class, i.e., it is equivalent to
    converter <- Converter$new(model, ...)

Except for a neuralnet model, no names of inputs or outputs are stored in the given model. If no further arguments are set for the `Converter` instance or `convert()` function, default labels are generated for the input (e.g. `'X1'`, `'X2'`, …) and output names (`'Y1'`, `'Y2'`, … ). In the converter, however, there is the possibility with the optional arguments `input_names` and `output_names` to pass the names, which will then be used in all results and plots created by this object.

#### Usage with torch models

Currently, only models created by `torch::nn_sequential` are accepted. However, the most popular standard layers and activation functions are available (see the detailed vignette for details).

> 📝 Note
> If you want to create an instance of the class `Converter` with a torch model that meets the above conditions, you have to specify the shape of the inputs with the argument `input_dim` because this information is not stored in every given torch model.

Example

    library(torch)
    library(innsight)
    torch_manual_seed(123)

    # Create model
    model <- nn_sequential(
      nn_linear(3, 10),
      nn_relu(),
      nn_linear(10, 2, bias = FALSE),
      nn_softmax(2)
    )
    # Convert the model
    conv_dense <- convert(model, input_dim = c(3))
    # Convert model with input and output names
    conv_dense_with_names <-
      convert(model,
        input_dim = c(3),
        input_names = list(c("Price", "Weight", "Height")),
        output_names = list(c("Buy it!", "Don't buy it!"))
      )

#### Usage with keras models

Models created by `keras_model_sequential` or `keras_model` with the keras package are accepted. Within these functions, the most popular layers and activation functions are accepted (see the in-depth vignette for details).

Example

    library(keras)

    # Create model
    model <- keras_model_sequential()
    model <- model %>%
      layer_conv_2d(4, c(5, 4), input_shape = c(10, 10, 3), activation = "softplus") %>%
      layer_max_pooling_2d(c(2, 2), strides = c(1, 1)) %>%
      layer_conv_2d(6, c(3, 3), activation = "relu", padding = "same") %>%
      layer_max_pooling_2d(c(2, 2)) %>%
      layer_conv_2d(4, c(2, 2), strides = c(2, 1), activation = "relu") %>%
      layer_flatten() %>%
      layer_dense(5, activation = "softmax")

    # Convert the model
    conv_cnn <- convert(model)

#### Usage with neuralnet models

The usage with nets from the package neuralnet is very simple and straightforward, because the package offers much fewer options than torch or keras. The only thing to note is that no custom activation function can be used. However, the package saves the names of the inputs and outputs, which can, of course, be overwritten with the arguments `input_names` and `output_names` when creating the converter object.

Example

    library(neuralnet)
    data(iris)

    # Create model
    model <- neuralnet(Species ~ Petal.Length + Petal.Width, iris,
      linear.output = FALSE
    )

    # Convert model
    conv_dense <- convert(model)

#### Usage with a model as a named list

If you have not trained your net with keras, torch or neuralnet, you can also pass your model as a list, i.e., you write your own wrapper for your library. But you have to consider a few points, which are explained in detail in the in-depth vignette.

Example

    model <- list(
      input_dim = 2,
      input_names = list(c("X1", "Feat2")),
      input_nodes = 1,
      output_nodes = 2,
      layers = list(
        list(
          type = "Dense", weight = matrix(rnorm(10), 5, 2), bias = rnorm(5),
          activation_name = "relu", input_layers = 0, output_layers = 2
        ),
        list(
          type = "Dense", weight = matrix(rnorm(5), 1, 5), bias = rnorm(1),
          activation_name = "sigmoid", input_layers = 1, output_layers = -1
        )
      )
    )

    converter <- convert(model)

After an instance of the `Converter` class has been created, the base `print()` method can be used to output the most important components of the object in summary form:

    converter
    #>
    #> ── Converter (innsight) ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
    #> Fields:
    #>   • input_dim: (*, 2)
    #>   • output_dim: (*, 1)
    #>   • input_names:
    #>       ─ Feature (2): X1, Feat2
    #>   • output_names:
    #>       ─ Output node/Class (1): Y1
    #>   • model_as_list: not included
    #>   • model (class ConvertedModel):
    #>     1. Dense_Layer: input_dim: (*, 2), output_dim: (*, 5)
    #>     2. Dense_Layer: input_dim: (*, 5), output_dim: (*, 1)
    #>
    #> ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────

### Step 2: Apply selected method

The innsight package provides several tools for analyzing black box neural networks based on dense or convolution layers. For the sake of uniform usage, all implemented methods inherit from the `InterpretingMethod` super class (see `?InterpretingMethod` for details) and differ in each case only by method-specific arguments and settings. Therefore, each method has the following initialization structure:

    method <- Method$new(converter, data, # required arguments
      channels_first = TRUE, # optional settings
      output_idx = NULL, # .
      ignore_last_act = TRUE, # .
      ... # other args and method-specific args
    )

However, you can also use the helper functions (e.g., `run_grad()`, `run_deeplift()`, etc.) for initializing a new object:

    method <- run_method(converter, data, # required arguments
      channels_first = TRUE, # optional settings
      output_idx = NULL, # .
      ignore_last_act = TRUE, # .
      ... # other args and method-specific args
    )

The most important arguments are explained below. For a complete and detailed explanation, however, we refer to the R documentation (see `?InterpretingMethod`) or the vignette “In-depth explanation” (`vignette("detailed_overview", package = "innsight")`).

- `converter`: This is the converter object created in the first step.

- `data`: 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 either an `array`, `data.frame`, `torch_tensor` or array-like format of size \(\left(\text{batchsize}, \text{input_dim}\right)\), if e.g., the model has only one input layer, or a `list` of the respective input sizes for each of the input layers.

- `channels_first`: 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 remaining 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`: These indices specify the output nodes or classes for which the method is to be applied. 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 your model has more than one output layer, you can pass the respective output nodes in a list which is described in detail in the R documentation (see `?InterpretingMethod`) or in the in-depth vignette

- `output_label`: These values specify the output nodes for which the method is to be applied and can be used as an alternative to the argument `output_idx`. Only values that were previously passed with the argument `output_names` in the `converter` can be used.

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

The package innsight now offers the following methods for interpreting your model. To use them, simply replace the name `"Method"` with one of the method’s names below. There are also method-specific arguments, but these are explained in detail along with the methods in the R documentation (e.g., `?Gradient` or `?LRP`) or in the in-depth vignette. Let \(x \in \mathbb{R}^p\) the input instance, \(i\) is the feature index of the input and \(c\) the index of the output node or class to be explained:

- `Gradient` : Calculation of the model output Gradients with respect to the model inputs including the attribution method Gradient\(\times\)Input:

  \[ \begin{align} \text{Gradient}(x)_i^c &= \frac{\partial f(x)_c}{\partial x_i}\\ \text{Gradient$\times$Input}(x)_i^c &= x_i \cdot \text{Gradient}(x)_i^c \end{align} \]
  Examples

      # Apply method 'Gradient' for the dense network
      grad_dense <- Gradient$new(conv_dense, iris[-c(1, 2, 5)])

      # You can also use the helper function `run_grad`
      grad_dense <- run_grad(conv_dense, iris[-c(1, 2, 5)])

      # Apply method 'Gradient x Input' for CNN
      x <- torch_randn(c(10, 3, 10, 10))
      grad_cnn <- run_grad(conv_cnn, x, times_input = TRUE)

- `SmoothGrad` : Calculation of the smoothed model output gradients (SmoothGrad) with respect to the model inputs by averaging the gradients over number of inputs with added noise (including SmoothGrad\(\times\)Input): \[ \text{SmoothGrad}(x)_i^c = \mathbb{E}_{\varepsilon \sim \mathcal{N}(0, \sigma^2)}\left[\frac{\partial f(x + \varepsilon)_c}{\partial (x + \varepsilon)_i}\right] \approx \frac{1}{n} \sum_{k=1}^n \frac{\partial f(x + \varepsilon_k)_c}{\partial (x + \varepsilon_k)_i} \] with \(\varepsilon_1, \ldots \varepsilon_n \sim \mathcal{N}(0, \sigma^2)\).

  Examples

      # Apply method 'SmoothGrad' for the dense network
      smooth_dense <- run_smoothgrad(conv_dense, iris[-c(1, 2, 5)])

      # Apply method 'SmoothGrad x Input' for CNN
      x <- torch_randn(c(10, 3, 10, 10))
      smooth_cnn <- run_smoothgrad(conv_cnn, x, times_input = TRUE)

- `IntegratedGradient` : Calculation of the integrated gradients (Sundararajan et al. (2017)) with respect to a reference input \(\tilde{x}\): \[ \text{IntGrad}(x)_i^c = (x - \tilde{x}) \int_{\alpha = 0}^1 \frac{\partial f(\tilde{x} + \alpha (x - \tilde{x}))}{\partial x} d\alpha. \]

  Examples

      # Apply method 'IntegratedGradient' for the dense network
      intgrad_dense <- run_intgrad(conv_dense, iris[-c(1, 2, 5)])

      # Apply method 'IntegratedGradient' for CNN with the average baseline
      x <- torch_randn(c(10, 3, 10, 10))
      x_ref <- x$mean(1, keepdim = TRUE)
      intgrad_cnn <- run_intgrad(conv_cnn, x, x_ref = x_ref)

- `ExpectedGradient` : Calculation of the integrated gradients (Erion et al., 2021) with respect to a whole reference dataset \(\tilde{X} \sim \tilde{x}\): \[ \text{ExpGrad}(x)_i^c = \mathbb{E}_{\tilde{x}\sim \tilde{X}, \alpha \sim U(0,1)} \left[(x - \tilde{x}) \times \frac{\partial f(\tilde{x} + \alpha (x - \tilde{x}))}{\partial x} \right] \]

  Examples

      # Apply method 'ExpectedGradient' for the dense network
      expgrad_dense <- run_expgrad(conv_dense, iris[-c(1, 2, 5)],
                                   data_ref = iris[-c(1, 2, 5)])

      # Apply method 'ExpectedGradient' for CNN
      x <- torch_randn(c(10, 3, 10, 10))
      data_ref <- torch_randn(c(20, 3, 10, 10))
      expgrad_cnn <- run_expgrad(conv_cnn, x, data_ref = data_ref)

- `LRP` : Back-propagating the model output to the model input neurons to obtain relevance scores for the model prediction which is known as Layer-wise Relevance Propagation: \[ f(x)_c \approx \sum_{i=1}^d R_i \] with \(R_i\) relevance score for input neuron \(i\).

  Examples

      # Apply method 'LRP' for the dense network
      lrp_dense <- run_lrp(conv_dense, iris[-c(1, 2, 5)])

      # Apply method 'LRP' for CNN with alpha-beta-rule
      x <- torch_randn(c(10, 10, 10, 3))
      lrp_cnn <- run_lrp(conv_cnn, x,
        rule_name = "alpha_beta", rule_param = 1,
        channels_first = FALSE
      )

- `DeepLift` : Calculation of a decomposition of the model output with respect to the model inputs and a reference input which is known as Deep Learning Important Features (DeepLift): \[ \Delta y_c = f(x)_c - f(x_\text{ref})_c = \sum_{i=1}^d C_i \] with \(C_i\) contribution score for input neuron \(i\) to the difference-from-reference model output \(\Delta y_c\).

  Examples

      # Define reference value
      x_ref <- array(colMeans(iris[-c(1, 2, 5)]), dim = c(1, 2))
      # Apply method 'DeepLift' for the dense network
      deeplift_dense <- run_deeplift(conv_dense, iris[-c(1, 2, 5)], x_ref = x_ref)

      # Apply method 'DeepLift' for CNN (default is a zero baseline)
      x <- torch_randn(c(10, 3, 10, 10))
      deeplift_cnn <- run_deeplift(conv_cnn, x)

- `ConnectionWeights` : This is a naive and old approach by calculating the product of all weights from an input to an output neuron and then adding them up (see Connection Weights).

  Examples

      # Apply global method 'ConnectionWeights' for a dense network
      connectweights_dense <- run_cw(conv_dense)

      # Apply local method 'ConnectionWeights' for a CNN
      # Note: This variant requires input data
      x <- torch_randn(c(10, 3, 10, 10))
      connectweights_cnn <- run_cw(conv_cnn, x, times_input = TRUE)

- Additionally, the method `DeepSHAP` and the model-agnostic methods `LIME` and `SHAP` are implemented (by the functions `run_deepshap()`, `run_lime()` and `run_shap()`). For details, we refer to our vignette “In-depth explanation”.

> 📝 Notes

> - By default, the last activation function is not taken into account for all data-based methods. Because often, this is a sigmoid/logistic or softmax function, which has increasingly smaller gradients with a growing distance from 0, which leads to the so-called saturation problem. But if you still want to consider the last activation function, use the argument `ignore_last_act = FALSE`.

> - For data with channels, it is impossible to determine exactly on which axis the channels are located. Internally, all data and the converted model are in the data format “channels first”, i.e., directly after the batch dimension \(\left(\text{batchsize}, \text{channels}, \text{input_dim}\right)\). In case you want to pass data with “channels last” (e.g., for MNIST-data \(\left(\text{batchsize}, 28,28,3\right)\)), you have to indicate that with argument `channels_first` in the applied method.

> - It can happen with very large and deep neural networks that the calculation for all outputs requires the entire memory and takes a very long time. But often, the results are needed only for certain output nodes. By default, only the results for the first 10 outputs are calculated, which can be adjusted individually with the argument `output_idx` by passing the relevant output indices.

Similar to the instances of the `Converter` class, the default `print()` function for R6 classes was also overridden for each method object, so that all important contents of the corresponding method are displayed:

    smooth_cnn
    #>
    #> ── Method SmoothGrad (innsight) ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
    #> Fields (method-specific):
    #>   • times_input: TRUE (→ SmoothGrad x Input method)
    #>   • n: 50
    #>   • noise_level: 0.1
    #> Fields (other):
    #>   • output_idx: 1, 2, 3, 4, 5 (→ corresponding labels: 'Y1', 'Y2', 'Y3', 'Y4', 'Y5')
    #>   • ignore_last_act: TRUE
    #>   • channels_first: TRUE
    #>   • dtype: 'float'
    #>
    #> ── Result (result) ──
    #>
    #>   ─ Shape: (10, 3, 10, 10, 5)
    #>   ─ Range: min: -0.20168, median: 9.60946e-05, max: 0.357108
    #>   ─ Number of NaN values: 0
    #>
    #> ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────

### Step 3: Show and plot the results

Once a method object has been created, the results can be returned as an `array`, `data.frame`, or `torch_tensor`, and can be further processed as desired. In addition, for each of the three sizes of the inputs (tabular, 1D signals or 2D images) suitable plot and boxplot functions based on ggplot2 are implemented. Due to the complexity of higher dimensional inputs, these plots and boxplots can also be displayed as an interactive plotly plots by using the argument `as_plotly`.

#### Get results

Each instance of the interpretability methods has the class method `get_result()`, which is used to return the results. You can choose between the data formats `array`, `data.frame` or `torch_tensor` by passing the name as an character for argument `type`. This method is also implemented as a S3 method. For a deeper view in this method look this section in the in-depth vignette.

    # Get the result with the class method
    method$get_result(type = "array")

    # or use the S3 function
    get_result(method, type = "array")

Examples:
`array` (default)

    # Get result (make sure 'grad_dense' is defined!)
    result_array <- grad_dense$get_result()

    # or with the S3 method
    result_array <- get_result(grad_dense)

    # Show the result for data point 1 and 71
    result_array[c(1, 71), , ]
    #> , , setosa
    #>
    #>      Petal.Length  Petal.Width
    #> [1,]  -0.09324544   -0.2008985
    #> [2,] -97.20391083 -209.4270325
    #>
    #> , , versicolor
    #>
    #>       Petal.Length  Petal.Width
    #> [1,] -0.0005318918 -0.001145968
    #> [2,] -0.5544717908 -1.194616318
    #>
    #> , , virginica
    #>
    #>      Petal.Length Petal.Width
    #> [1,]  0.004687082  0.01009838
    #> [2,]  4.886058807 10.52707481

`data.frame`

    # Get result as data.frame (make sure 'lrp_cnn' is defined!)
    result_data.frame <- lrp_cnn$get_result("data.frame")

    # or with the S3 method
    result_data.frame <- get_result(lrp_cnn, "data.frame")

    # Show the first 5 rows
    head(result_data.frame, 5)
    #>     data model_input model_output feature feature_2 channel output_node         value       pred decomp_sum decomp_goal input_dimension
    #> 1 data_1     Input_1     Output_1      H1        W1      C1          Y1 -1.124579e-05 0.20785421 -0.0651063 -0.06510648               3
    #> 2 data_2     Input_1     Output_1      H1        W1      C1          Y1  0.000000e+00 0.19935113 -0.1034352 -0.10343549               3
    #> 3 data_3     Input_1     Output_1      H1        W1      C1          Y1  0.000000e+00 0.07403765 -0.4485819 -0.44858283               3
    #> 4 data_4     Input_1     Output_1      H1        W1      C1          Y1 -1.354794e-07 0.12916298 -0.5135351 -0.51353657               3
    #> 5 data_5     Input_1     Output_1      H1        W1      C1          Y1  0.000000e+00 0.17241161 -0.2322727 -0.23227328               3

`torch_tensor`

    # Get result (make sure 'deeplift_dense' is defined!)
    result_torch <- deeplift_dense$get_result("torch_tensor")

    # or with the S3 method
    result_torch <- get_result(deeplift_dense, "torch_tensor")

    # Show for datapoint 1 and 71 the result
    result_torch[c(1, 71), , ]
    #> torch_tensor
    #> (1,.,.) =
    #>   6.1404  0.0350 -0.3087
    #>   5.6068  0.0320 -0.2818
    #>
    #> (2,.,.) =
    #>  -47.5426  -0.2712   2.3898
    #>  -59.0470  -0.3368   2.9681
    #> [ CPUFloatType{2,2,3} ]

#### Plot results

The package innsight also provides methods for visualizing the results. By default a ggplot2-plot is created, but it can also be rendered as an interactive plotly plot with the `as_plotly` argument. You can use the argument `output_idx` to select the indices of the output nodes for the plot. In addition, if the results have channels, the `aggr_channels` argument can be used to determine how the channels are aggregated. All arguments are explained in detail in the R documentation (see `?InterpretingMethod`) or here for `plot()` and here for `plot_global()`.

    # Create a plot for single data points
    plot(method,
      data_idx = 1, # the data point to be plotted
      output_idx = NULL, # the indices of the output nodes/classes to be plotted
      output_label = NULL, # the class labels to be plotted
      aggr_channels = "sum",
      as_plotly = FALSE, # create an interactive plot
      ... # other arguments
    )

    # Create a plot with summarized results
    plot_global(method,
      output_idx = NULL, # the indices of the output nodes/classes to be plotted
      output_label = NULL, # the class labels to be plotted
      ref_data_idx = NULL, # the index of an reference data point to be plotted
      aggr_channels = "sum",
      as_plotly = FALSE, # create an interactive plot
      ... # other arguments
    )

    # Alias for `plot_global` for tabular and signal data
    boxplot(...)

> 📝 Note
> The argument `output_idx` can be either a vector of indices or a list of vectors of indices but must be a subset of the indices for which the results were calculated, i.e., a subset of the argument `output_idx` passed to the respective method previously. By default (`NULL`), the smallest index of all computed output nodes and output layers is used.

Examples:
`plot()` function (ggplot2)

    # Plot the result of the first data point (default) for the output classes '1', '2' and '3'
    plot(smooth_dense, output_idx = 1:3)

    # You can plot several data points at once
    plot(smooth_dense, data_idx = c(1, 144), output_idx = 1:3)

    # Plot result for the first data point and first and fourth output classes
    # and aggregate the channels by taking the Euclidean norm
    plot(lrp_cnn, aggr_channels = "norm", output_idx = c(1, 4))

`plot()` function (plotly)

    # Create a plotly plot for the first output
    plot(lrp_cnn, aggr_channels = "norm", output_idx = c(1), as_plotly = TRUE)

`plot_global()` function (ggplot2)

    # Create boxplot for the first two output classes
    plot_global(smooth_dense, output_idx = 1:2)

    # Use no preprocess function (default: abs) and plot a reference data point
    plot_global(smooth_dense,
      output_idx = 1:3, preprocess_FUN = identity,
      ref_data_idx = c(55)
    )

`plot_global()` function (plotly)

    # You can do the same with the plotly-based plots
    plot_global(smooth_dense,
      output_idx = 1:3, preprocess_FUN = identity,
      ref_data_idx = c(55), as_plotly = TRUE
    )