Explain a forecast from time series models with dependence-aware (conditional/observational) Shapley values

Computes dependence-aware Shapley values for observations in explain_idx from the specified model by using the method specified in approach to estimate the conditional expectation. See Aas, et. al (2021) for a thorough introduction to dependence-aware prediction explanation with Shapley values.

explain_forecast(
  model,
  y,
  xreg = NULL,
  train_idx = NULL,
  explain_idx,
  explain_y_lags,
  explain_xreg_lags = explain_y_lags,
  horizon,
  approach,
  phi0,
  max_n_coalitions = NULL,
  iterative = NULL,
  group_lags = TRUE,
  group = NULL,
  n_MC_samples = 1000,
  seed = 1,
  predict_model = NULL,
  get_model_specs = NULL,
  verbose = "basic",
  extra_computation_args = list(),
  iterative_args = list(),
  output_args = list(),
  ...
)

Arguments

model

Model object. Specifies the model whose predictions we want to explain. Run get_supported_models() for a table of which models explain supports natively. Unsupported models can still be explained by passing predict_model and (optionally) get_model_specs, see details for more information.

y

Matrix, data.frame/data.table or a numeric vector. Contains the endogenous variables used to estimate the (conditional) distributions needed to properly estimate the conditional expectations in the Shapley formula including the observations to be explained.

xreg

Matrix, data.frame/data.table or a numeric vector. Contains the exogenous variables used to estimate the (conditional) distributions needed to properly estimate the conditional expectations in the Shapley formula including the observations to be explained. As exogenous variables are used contemporaneously when producing a forecast, this item should contain nrow(y) + horizon rows.

train_idx

Numeric vector. The row indices in data and reg denoting points in time to use when estimating the conditional expectations in the Shapley value formula. If train_idx = NULL (default) all indices not selected to be explained will be used.

explain_idx

Numeric vector. The row indices in data and reg denoting points in time to explain.

explain_y_lags

Numeric vector. Denotes the number of lags that should be used for each variable in y when making a forecast.

explain_xreg_lags

Numeric vector. If xreg != NULL, denotes the number of lags that should be used for each variable in xreg when making a forecast.

horizon

Numeric. The forecast horizon to explain. Passed to the predict_model function.

approach

Character vector of length 1 or one less than the number of features. All elements should, either be "gaussian", "copula", "empirical", "ctree", "vaeac", "categorical", "timeseries", "independence", "regression_separate", or "regression_surrogate". The two regression approaches can not be combined with any other approach. See details for more information.

phi0

Numeric. The prediction value for unseen data, i.e. an estimate of the expected prediction without conditioning on any features. Typically we set this value equal to the mean of the response variable in our training data, but other choices such as the mean of the predictions in the training data are also reasonable.

max_n_coalitions

Integer. The upper limit on the number of unique feature/group coalitions to use in the iterative procedure (if iterative = TRUE). If iterative = FALSE it represents the number of feature/group coalitions to use directly. The quantity refers to the number of unique feature coalitions if group = NULL, and group coalitions if group != NULL. max_n_coalitions = NULL corresponds to max_n_coalitions=2^n_features.

iterative

Logical or NULL If NULL (default), the argument is set to TRUE if there are more than 5 features/groups, and FALSE otherwise. If eventually TRUE, the Shapley values are estimated iteratively in an iterative manner. This provides sufficiently accurate Shapley value estimates faster. First an initial number of coalitions is sampled, then bootsrapping is used to estimate the variance of the Shapley values. A convergence criterion is used to determine if the variances of the Shapley values are sufficently small. If the variances are too high, we estimate the number of required samples to reach convergence, and thereby add more coalitions. The process is repeated until the variances are below the threshold. Specifics related to the iterative process and convergence criterion are set through iterative_args.

group_lags

Logical. If TRUE all lags of each variable are grouped together and explained as a group. If FALSE all lags of each variable are explained individually.

group

List. If NULL regular feature wise Shapley values are computed. If provided, group wise Shapley values are computed. group then has length equal to the number of groups. The list element contains character vectors with the features included in each of the different groups. See Jullum et al. (2021) for more information on group wise Shapley values.

n_MC_samples

Positive integer. For most approaches, it indicates the maximum number of samples to use in the Monte Carlo integration of every conditional expectation. For approach="ctree", n_MC_samples corresponds to the number of samples from the leaf node (see an exception related to the ctree.sample argument setup_approach.ctree()). For approach="empirical", n_MC_samples is the \(K\) parameter in equations (14-15) of Aas et al. (2021), i.e. the maximum number of observations (with largest weights) that is used, see also the empirical.eta argument setup_approach.empirical().

seed

Positive integer. Specifies the seed before any randomness based code is being run. If NULL no seed is set in the calling environment.

predict_model

Function. The prediction function used when model is not natively supported. (Run get_supported_models() for a list of natively supported models.) The function must have two arguments, model and newdata which specify, respectively, the model and a data.frame/data.table to compute predictions for. The function must give the prediction as a numeric vector. NULL (the default) uses functions specified internally. Can also be used to override the default function for natively supported model classes.

get_model_specs

Function. An optional function for checking model/data consistency when model is not natively supported. (Run get_supported_models() for a list of natively supported models.) The function takes model as argument and provides a list with 3 elements:

labels

Character vector with the names of each feature.

classes

Character vector with the classes of each features.

factor_levels

Character vector with the levels for any categorical features.

If NULL (the default) internal functions are used for natively supported model classes, and the checking is disabled for unsupported model classes. Can also be used to override the default function for natively supported model classes.

verbose

String vector or NULL. Specifies the verbosity (printout detail level) through one or more of strings "basic", "progress", "convergence", "shapley" and "vS_details". "basic" (default) displays basic information about the computation which is being performed. "progress displays information about where in the calculation process the function currently is. #' "convergence" displays information on how close to convergence the Shapley value estimates are (only when iterative = TRUE) . "shapley" displays intermediate Shapley value estimates and standard deviations (only when iterative = TRUE)

• the final estimates. "vS_details" displays information about the v_S estimates. This is most relevant for approach %in% c("regression_separate", "regression_surrogate", "vaeac"). NULL means no printout. Note that any combination of four strings can be used. E.g. verbose = c("basic", "vS_details") will display basic information + details about the v(S)-estimation process.

extra_computation_args

Named list. Specifices extra arguments related to the computation of the Shapley values. See get_extra_comp_args_default() for description of the arguments and their default values.

iterative_args

Named list. Specifices the arguments for the iterative procedure. See get_iterative_args_default() for description of the arguments and their default values.

output_args

Named list. Specifices certain arguments related to the output of the function. See get_output_args_default() for description of the arguments and their default values.

...

Arguments passed on to setup_approach.categorical, setup_approach.copula, setup_approach.ctree, setup_approach.empirical, setup_approach.gaussian, setup_approach.independence, setup_approach.timeseries, setup_approach.vaeac

categorical.joint_prob_dt

Data.table. (Optional) Containing the joint probability distribution for each combination of feature values. NULL means it is estimated from the x_train and x_explain.

categorical.epsilon

Numeric value. (Optional) If categorical.joint_probability_dt is not supplied, probabilities/frequencies are estimated using x_train. If certain observations occur in x_explain and NOT in x_train, then epsilon is used as the proportion of times that these observations occurs in the training data. In theory, this proportion should be zero, but this causes an error later in the Shapley computation.

internal

List. Not used directly, but passed through from explain().

ctree.mincriterion

Numeric scalar or vector. Either a scalar or vector of length equal to the number of features in the model. The value is equal to 1 - \(\alpha\) where \(\alpha\) is the nominal level of the conditional independence tests. If it is a vector, this indicates which value to use when conditioning on various numbers of features. The default value is 0.95.

ctree.minsplit

Numeric scalar. Determines minimum value that the sum of the left and right daughter nodes required for a split. The default value is 20.

ctree.minbucket

Numeric scalar. Determines the minimum sum of weights in a terminal node required for a split The default value is 7.

ctree.sample

Boolean. If TRUE (default), then the method always samples n_MC_samples observations from the leaf nodes (with replacement). If FALSE and the number of observations in the leaf node is less than n_MC_samples, the method will take all observations in the leaf. If FALSE and the number of observations in the leaf node is more than n_MC_samples, the method will sample n_MC_samples observations (with replacement). This means that there will always be sampling in the leaf unless sample = FALSE and the number of obs in the node is less than n_MC_samples.

empirical.type

Character. (default = "fixed_sigma") Should be equal to either "independence","fixed_sigma", "AICc_each_k" "AICc_full". "independence" is deprecated. Use approach = "independence" instead. "fixed_sigma" uses a fixed bandwidth (set through empirical.fixed_sigma) in the kernel density estimation. "AICc_each_k" and "AICc_full" optimize the bandwidth using the AICc criterion, with respectively one bandwidth per coalition size and one bandwidth for all coalition sizes.

empirical.eta

Numeric scalar. Needs to be 0 < eta <= 1. The default value is 0.95. Represents the minimum proportion of the total empirical weight that data samples should use. If e.g. eta = .8 we will choose the K samples with the largest weight so that the sum of the weights accounts for 80\ eta is the \(\eta\) parameter in equation (15) of Aas et al. (2021).

empirical.fixed_sigma

Positive numeric scalar. The default value is 0.1. Represents the kernel bandwidth in the distance computation used when conditioning on all different coalitions. Only used when empirical.type = "fixed_sigma"

empirical.n_samples_aicc

Positive integer. Number of samples to consider in AICc optimization. The default value is 1000. Only used for empirical.type is either "AICc_each_k" or "AICc_full".

empirical.eval_max_aicc

Positive integer. Maximum number of iterations when optimizing the AICc. The default value is 20. Only used for empirical.type is either "AICc_each_k" or "AICc_full".

empirical.start_aicc

Numeric. Start value of the sigma parameter when optimizing the AICc. The default value is 0.1. Only used for empirical.type is either "AICc_each_k" or "AICc_full".

empirical.cov_mat

Numeric matrix. (Optional) The covariance matrix of the data generating distribution used to define the Mahalanobis distance. NULL means it is estimated from x_train.

gaussian.mu

Numeric vector. (Optional) Containing the mean of the data generating distribution. NULL means it is estimated from the x_train.

gaussian.cov_mat

Numeric matrix. (Optional) Containing the covariance matrix of the data generating distribution. NULL means it is estimated from the x_train.

timeseries.fixed_sigma

Positive numeric scalar. Represents the kernel bandwidth in the distance computation. The default value is 2.

timeseries.bounds

Numeric vector of length two. Specifies the lower and upper bounds of the timeseries. The default is c(NULL, NULL), i.e. no bounds. If one or both of these bounds are not NULL, we restrict the sampled time series to be between these bounds. This is useful if the underlying time series are scaled between 0 and 1, for example.

vaeac.depth

Positive integer (default is 3). The number of hidden layers in the neural networks of the masked encoder, full encoder, and decoder.

vaeac.width

Positive integer (default is 32). The number of neurons in each hidden layer in the neural networks of the masked encoder, full encoder, and decoder.

vaeac.latent_dim

Positive integer (default is 8). The number of dimensions in the latent space.

vaeac.lr

Positive numeric (default is 0.001). The learning rate used in the torch::optim_adam() optimizer.

vaeac.activation_function

An torch::nn_module() representing an activation function such as, e.g., torch::nn_relu() (default), torch::nn_leaky_relu(), torch::nn_selu(), or torch::nn_sigmoid().

vaeac.n_vaeacs_initialize

Positive integer (default is 4). The number of different vaeac models to initiate in the start. Pick the best performing one after vaeac.extra_parameters$epochs_initiation_phase epochs (default is 2) and continue training that one.

vaeac.epochs

Positive integer (default is 100). The number of epochs to train the final vaeac model. This includes vaeac.extra_parameters$epochs_initiation_phase, where the default is 2.

vaeac.extra_parameters

Named list with extra parameters to the vaeac approach. See vaeac_get_extra_para_default() for description of possible additional parameters and their default values.

Value

Object of class c("shapr", "list"). Contains the following items:

shapley_values_est

data.table with the estimated Shapley values with explained observation in the rows and features along the columns. The column none is the prediction not devoted to any of the features (given by the argument phi0)

shapley_values_sd

data.table with the standard deviation of the Shapley values reflecting the uncertainty. Note that this only reflects the coalition sampling part of the kernelSHAP procedure, and is therefore by definition 0 when all coalitions is used. Only present when extra_computation_args$compute_sd=TRUE, which is the default when iterative = TRUE

internal

List with the different parameters, data, functions and other output used internally.

pred_explain

Numeric vector with the predictions for the explained observations

MSEv

List with the values of the MSEv evaluation criterion for the approach. See the MSEv evaluation section in the general usage for details.

timing

List containing timing information for the different parts of the computation. init_time and end_time gives the time stamps for the start and end of the computation. total_time_secs gives the total time in seconds for the complete execution of explain(). main_timing_secs gives the time in seconds for the main computations. iter_timing_secs gives for each iteration of the iterative estimation, the time spent on the different parts iterative estimation routine.

Details

This function explains a forecast of length horizon. The argument train_idx is analogous to x_train in explain(), however, it just contains the time indices of where in the data the forecast should start for each training sample. In the same way explain_idx defines the time index (indices) which will precede a forecast to be explained.

As any autoregressive forecast model will require a set of lags to make a forecast at an arbitrary point in time, explain_y_lags and explain_xreg_lags define how many lags are required to "refit" the model at any given time index. This allows the different approaches to work in the same way they do for time-invariant models.

See the forecasting section of the general usages for further details.

References

• Aas, K., Jullum, M., & Løland, A. (2021). Explaining individual predictions when features are dependent: More accurate approximations to Shapley values. Artificial Intelligence, 298, 103502

• Frye, C., Rowat, C., & Feige, I. (2020). Asymmetric Shapley values: incorporating causal knowledge into model-agnostic explainability. Advances in neural information processing systems, 33, 1229-1239

• Heskes, T., Sijben, E., Bucur, I. G., & Claassen, T. (2020). Causal shapley values: Exploiting causal knowledge to explain individual predictions of complex models. Advances in neural information processing systems, 33, 4778-4789

• Jullum, M., Redelmeier, A. & Aas, K. (2021). Efficient and simple prediction explanations with groupShapley: A practical perspective. Italian Workshop on Explainable Artificial Intelligence 2021.

• Redelmeier, A., Jullum, M., & Aas, K. (2020). Explaining predictive models with mixed features using Shapley values and conditional inference trees. In Machine Learning and Knowledge Extraction: International Cross-Domain Conference, CD-MAKE 2020, Dublin, Ireland, August 25–28, 2020, Proceedings 4 (pp. 117-137). Springer International Publishing.

• Sellereite N., & Jullum, M. (2019). shapr: An R-package for explaining machine learning models with dependence-aware Shapley values. Journal of Open Source Software, 5(46), 2027

• Olsen, L. H., Glad, I. K., Jullum, M., & Aas, K. (2022). Using Shapley values and variational autoencoders to explain predictive models with dependent mixed features. Journal of machine learning research, 23(213), 1-51

• Olsen, L. H. B., Glad, I. K., Jullum, M., & Aas, K. (2024). A comparative study of methods for estimating model-agnostic Shapley value explanations. Data Mining and Knowledge Discovery, 1-48

• Olsen, L. H. B., & Jullum, M. (2024). Improving the Sampling Strategy in KernelSHAP. arXiv e-prints, arXiv-2410

Author

Jon Lachmann, Martin Jullum

Examples

# Load example data
data("airquality")
data <- data.table::as.data.table(airquality)

# Fit an AR(2) model.
model_ar_temp <- ar(data$Temp, order = 2)

# Calculate the zero prediction values for a three step forecast.
p0_ar <- rep(mean(data$Temp), 3)

# Empirical approach, explaining forecasts starting at T = 152 and T = 153.
explain_forecast(
  model = model_ar_temp,
  y = data[, "Temp"],
  train_idx = 2:151,
  explain_idx = 152:153,
  explain_y_lags = 2,
  horizon = 3,
  approach = "empirical",
  phi0 = p0_ar,
  group_lags = FALSE
)
#> Note: Feature names extracted from the model contains NA.
#> Consistency checks between model and data is therefore disabled.
#> Success with message:
#> max_n_coalitions is NULL or larger than or 2^n_features = 4, 
#> and is therefore set to 2^n_features = 4.
#> 
#> ── Starting `shapr::explain()` at 2024-12-23 10:48:22 ──────────────────────────
#> • Model class: <ar>
#> • Approach: empirical
#> • Iterative estimation: FALSE
#> • Number of feature-wise Shapley values: 2
#> • Number of observations to explain: 2
#> • Computations (temporary) saved at: /tmp/RtmpdiQJC0/shapr_obj_2a4d53c0ba51.rds
#> 
#> ── Main computation started ──
#> 
#> ℹ Using 4 of 4 coalitions. 
#>    explain_idx horizon  none  Temp.1  Temp.2
#>          <int>   <int> <num>   <num>   <num>
#> 1:         152       1 77.88 -0.3972 -1.3912
#> 2:         153       1 77.88 -6.6177 -0.1835
#> 3:         152       2 77.88 -0.3285 -1.2034
#> 4:         153       2 77.88 -6.0208 -0.3371
#> 5:         152       3 77.88 -0.2915 -1.0552
#> 6:         153       3 77.88 -5.2122 -0.2553