package::forecastML

The purpose of forecastML is to provide a series of functions and visualizations that simplify the process of
multi-step-ahead forecasting with standard machine learning algorithms. It’s a wrapper package aimed at providing maximum flexibility in model-building–choose any machine learning algorithm from any R or Python package–while helping the user quickly assess the (a) accuracy, (b) stability, and © generalizability of grouped (i.e.,
multiple related time series) and ungrouped forecasts produced from potentially high-dimensional modeling datasets.

This package is inspired by Bergmeir, Hyndman, and Koo’s 2018 paper
A note on the validity of cross-validation for evaluating autoregressive time series prediction.
which supports–under certain conditions–forecasting with high-dimensional ML models without having to use methods that are time series specific.

The following quote from Bergmeir et al.'s article nicely sums up the aim of this package:

“When purely (non-linear, nonparametric) autoregressive methods are applied to forecasting problems, as is often the case
(e.g., when using Machine Learning methods), the aforementioned problems of CV are largely
irrelevant, and CV can and should be used without modification, as in the independent case.”

Featured Notebooks

• Forecasting with big data - Spark and H2O

• Forecasting with Python - scikit-learn in parallel

• Forecast reconciliation across planning horizons - coherent weekly ML and monthly ARIMA forecasts

User-contributed notebooks welcome!

Lightning Example

• Requires packageVersion("forecastML") >= v0.9.1

library(glmnet)
library(forecastML)

data("data_seatbelts", package = "forecastML")

data_train <- forecastML::create_lagged_df(data_seatbelts, type = "train", method = "direct",
                                           outcome_col = 1, lookback = 1:15, horizons = 1:12)

windows <- forecastML::create_windows(data_train, window_length = 0)

model_fn <- function(data) {
  x <- as.matrix(data[, -1, drop = FALSE])
  y <- as.matrix(data[, 1, drop = FALSE])
  model <- glmnet::cv.glmnet(x, y)
}

model_results <- forecastML::train_model(data_train, windows, model_name = "LASSO", model_function = model_fn)

predict_fn <- function(model, data) {
  data_pred <- as.data.frame(predict(model, as.matrix(data)))
}

data_fit <- predict(model_results, prediction_function = list(predict_fn), data = data_train)

residuals <- residuals(data_fit)

data_forecast <- forecastML::create_lagged_df(data_seatbelts, type = "forecast", method = "direct",
                                              outcome_col = 1, lookback = 1:15, horizons = 1:12)

data_forecasts <- predict(model_results, prediction_function = list(predict_fn), data = data_forecast)

data_forecasts <- forecastML::combine_forecasts(data_forecasts)

set.seed(224)
data_forecasts <- forecastML::calculate_intervals(data_forecasts, residuals, 
                                                  levels = seq(.5, .95, .05), times = 200)

plot(data_forecasts, data_seatbelts[-(1:160), ], (1:nrow(data_seatbelts))[-(1:160)], interval_alpha = seq(.1, .2, length.out = 10))

README Contents

• Install
• Approach to forecasting
• Vignettes
• Cheat sheets
• FAQ
• Examples
  • Forecasting numeric outcomes
    • Direct forecasting
    • Multi-output forecasting
  • Forecasting factor outcomes (forecasting sequences)

Install

• CRAN

install.packages("forecastML")
library(forecastML)

• Development

remotes::install_github("nredell/forecastML")
library(forecastML)

Approach to Forecasting

Direct forecasting

The direct forecasting approach used in forecastML involves the following steps:

1. Build a series of horizon-specific short-, medium-, and long-term forecast models.

2. Assess model generalization performance across a variety of heldout datasets through time.

3. Select those models that consistently performed the best at each forecast horizon and
combine them to produce a single ensemble forecast.

• Below is a plot of 5 forecast models used to produce a single 12-step-ahead forecast where each color
  represents a distinct horizon-specific ML model. From left to right these models are:

• 1: A feed-forward neural network (purple); 2: An ensemble of ML models;
  3: A boosted tree model; 4: A LASSO regression model; 5: A LASSO regression model (yellow).

• Below is a similar combination of horizon-specific models with a factor outcome and forecasting factor
  probabilities 12 steps ahead.

Multi-output forecasting

The multi-output forecasting approach used in forecastML involves the following steps:

1. Build a single multi-output model that simultaneously forecasts over both short- and long-term forecast horizons.

2. Assess model generalization performance across a variety of heldout datasets through time.

3. Select the hyperparamters that minimize forecast error over all the relevant forecast horizons and re-train.

Vignettes

The main functions covered in each vignette are shown below as function().

• Detailed forecastML overview vignette.
  create_lagged_df(), create_windows(), train_model(), return_error(), return_hyper(), combine_forecasts()

• Creating custom feature lags for model training. create_lagged_df(lookback_control = ...)

• Direct Forecasting with multiple or grouped time series.
  fill_gaps(),
  create_lagged_df(dates = ..., dynamic_features = ..., groups = ..., static_features = ...), create_windows(), train_model(), combine_forecasts()

• Direct Forecasting with multiple or grouped time series - Sequences.
  fill_gaps(),
  create_lagged_df(dates = ..., dynamic_features = ..., groups = ..., static_features = ...), create_windows(), train_model(), combine_forecasts()

• Customizing the user-defined wrapper functions.
  train() and predict()

• Forecast combinations. combine_forecasts()

Cheat Sheets

1. fill_gaps: Optional if no temporal gaps/missing rows in data collection. Fill gaps in data collection and
   prepare a dataset of evenly-spaced time series for modeling with lagged features. Returns a ‘data.frame’ with
   missing rows added in so that you can either (a) impute, remove, or ignore NAs prior to the forecastML pipeline
   or (b) impute, remove, or ignore them in the user-defined modeling function–depending on the NA handling
   capabilities of the user-specified model.

2. create_lagged_df: Create model training and forecasting datasets with lagged, grouped, dynamic, and static features.

3. create_windows: Create time-contiguous validation datasets for model evaluation.

4. train_model: Train the user-defined model across forecast horizons and validation datasets.

5. return_error: Compute forecast error across forecast horizons and validation datasets.

6. return_hyper: Return user-defined model hyperparameters across validation datasets.

7. combine_forecasts: Combine multiple horizon-specific forecast models to produce one forecast.

FAQ

• Q: Where does forecastML fit in with respect to popular R machine learning packages like mlr3 and caret?

• A: The idea is that forecastML takes care of the tedious parts of forecasting with ML methods: creating training and forecasting datasets with different
  types of features–grouped, static, and dynamic–as well as simplifying validation dataset creation to assess model performance at specific points in time.
  That said, the workflow for packages like mlr3 and caret would mostly occur inside of the user-supplied
  modeling function which is passed into forecastML::train_model(). Refer to the wrapper function customization
  vignette for more details.

• Q: How do I get the model training and forecasting datasets as well as the trained models out of the
  forecastML pipeline?

• A: After running forecastML::create_lagged_df() with either type = "train" or type = "forecast",
  the data.frames can be accessed with my_lagged_df$horizon_h where “h” is an integer marking the
  horizon-specific dataset (e.g., the value(s) passed in horizons = ...). The trained models from
  forecastML::train_model() can be accessed with my_trained_model$horizon_h$window_w$model where “w” is
  the validation window number from forecastML::create_windows().

Examples - Numeric Outcomes with R and Python

Direct forecast in R

Below is an example of how to create 12 horizon-specific ML models to forecast the number of DriversKilled
12 time periods into the future using the Seatbelts dataset. Notice in the last plot that there are multiple forecasts;
these are from the slightly different LASSO models trained in the nested cross-validation. An example of selecting optimal
hyperparameters and retraining to create a single forecast model (i.e., create_windows(..., window_length = 0)) can be found
in the overview vignette.

library(glmnet)
library(forecastML)

# Sampled Seatbelts data from the R package datasets.
data("data_seatbelts", package = "forecastML")

# Example - Training data for 12 horizon-specific models w/ common lags per feature. The data do 
# not have any missing rows or temporal gaps in data collection; if there were gaps, 
# we would need to use fill_gaps() first.
horizons <- 1:12  # 12 models that forecast 1, 1:2, 1:3, ..., and 1:12 time steps ahead.
lookback <- 1:15  # A lookback of 1 to 15 dataset rows (1:15 * 'date frequency' if dates are given).

#------------------------------------------------------------------------------
# Create a dataset of lagged features for modeling.
data_train <- forecastML::create_lagged_df(data_seatbelts, type = "train",
                                           outcome_col = 1, lookback = lookback,
                                           horizon = horizons)

#------------------------------------------------------------------------------
# Create validation datasets for outer-loop nested cross-validation.
windows <- forecastML::create_windows(data_train, window_length = 12)

#------------------------------------------------------------------------------
# User-define model - LASSO
# A user-defined wrapper function for model training that takes the following
# arguments: (1) a horizon-specific data.frame made with create_lagged_df(..., type = "train")
# (e.g., my_lagged_df$horizon_h) and, optionally, (2) any number of additional named arguments
# which can also be passed in '...' in train_model(). The function returns a model object suitable for 
# the user-defined predict function. The returned model may also be a list that holds meta-data such 
# as hyperparameter settings.

model_function <- function(data, my_outcome_col) {  # my_outcome_col = 1 could be defined here.

  x <- data[, -(my_outcome_col), drop = FALSE]
  y <- data[, my_outcome_col, drop = FALSE]
  x <- as.matrix(x, ncol = ncol(x))
  y <- as.matrix(y, ncol = ncol(y))

  model <- glmnet::cv.glmnet(x, y)
  return(model)  # This model is the first argument in the user-defined predict() function below.
}

#------------------------------------------------------------------------------
# Train a model across forecast horizons and validation datasets.
# my_outcome_col = 1 is passed in ... but could have been defined in the user-defined model function.
model_results <- forecastML::train_model(data_train,
                                         windows = windows,
                                         model_name = "LASSO", 
                                         model_function = model_function,
                                         my_outcome_col = 1,  # ...
                                         use_future = FALSE)

#------------------------------------------------------------------------------
# User-defined prediction function - LASSO
# The predict() wrapper function takes 2 positional arguments. First,
# the returned model from the user-defined modeling function (model_function() above).
# Second, a data.frame of model features. If predicting on validation data, expect the input data to be 
# passed in the same format as returned by create_lagged_df(type = 'train') but with the outcome column 
# removed. If forecasting, expect the input data to be in the same format as returned by 
# create_lagged_df(type = 'forecast') but with the 'index' and 'horizon' columns removed. The function 
# can return a 1- or 3-column data.frame with either (a) point
# forecasts or (b) point forecasts plus lower and upper forecast bounds (column order and names do not matter).

prediction_function <- function(model, data_features) {

  x <- as.matrix(data_features, ncol = ncol(data_features))
  data_pred <- data.frame("y_pred" = predict(model, x, s = "lambda.min"),  # 1 column is required.
                          "y_pred_lower" = predict(model, x, s = "lambda.min") - 50,  # optional.
                          "y_pred_upper" = predict(model, x, s = "lambda.min") + 50)  # optional.
  return(data_pred)
}

# Predict on the validation datasets.
data_valid <- predict(model_results, prediction_function = list(prediction_function), data = data_train)

#------------------------------------------------------------------------------
# Plot forecasts for each validation dataset.
plot(data_valid, horizons = c(1, 6, 12))

#------------------------------------------------------------------------------
# Forecast.

# Forward-looking forecast data.frame.
data_forecast <- forecastML::create_lagged_df(data_seatbelts, type = "forecast",
                                              outcome_col = 1, lookback = lookback, horizons = horizons)

# Forecasts.
data_forecasts <- predict(model_results, prediction_function = list(prediction_function), data = data_forecast)

# We'll plot a background dataset of actuals as well.
plot(data_forecasts,
     data_actual = data_seatbelts[-(1:150), ], 
     actual_indices = as.numeric(row.names(data_seatbelts[-(1:150), ])), 
     horizons = c(1, 6, 12), windows = c(5, 10, 15))

---

Direct forecast in R & Python

Now we’ll look at an example similar to above. The main difference is that our user-defined modeling
and prediction functions are now written in Python. Thanks to the reticulate
R package, entire ML workflows already written in Python can be imported into forecastML with the
simple addition of 2 lines of R code.

• The reticulate::source_python() function will run a .py file and import any objects into your R environment. As we’ll
  see below, we’ll only be importing library calls and functions to keep our R environment clean.

library(forecastML)
library(reticulate)  # Move Python objects in and out of R. See the reticulate package for setup info.

reticulate::source_python("modeling_script.py")  # Run a Python file and import objects into R.

• Below is a simple, slightly different forecastML setup for the seatbelt forecasting problem from the
  previous example.

data("data_seatbelts", package = "forecastML")

horizons <- c(1, 12)  # 2 models that forecast 1 and 1:12 time steps ahead.

# A lookback across select time steps in the past. Feature lags 1 through 9 will be silently dropped from the 12-step-ahead model.
lookback <- c(1, 3, 6, 9, 12, 15)

date_frequency <- "1 month"  # Time step frequency.

# The date indices, which don't come with the stock dataset, should not be included in the modeling data.frame.
dates <- seq(as.Date("1969-01-01"), as.Date("1984-12-01"), by = date_frequency)

# Create a dataset of features for modeling.
data_train <- forecastML::create_lagged_df(data_seatbelts, type = "train", outcome_col = 1,
                                           lookback = lookback, horizon = horizons,
                                           dates = dates, frequency = date_frequency)

# Create 2 custom validation datasets for outer-loop nested cross-validation. The purpose of
# the multiple validation windows is to assess expected forecast accuracy for specific
# time periods while supporting an investigation of the hyperparameter stability for
# models trained on different time periods. Validation windows can overlap.
window_start <- c(as.Date("1983-01-01"), as.Date("1984-01-01"))
window_stop <- c(as.Date("1983-12-01"), as.Date("1984-12-01"))

windows <- forecastML::create_windows(data_train, window_start = window_start, window_stop = window_stop)

modeling_script.py

• Let’s look at the content of our Python modeling file that we source()'d above. The Python wrapper function inputs
  and returns for py_model_function() and py_prediction_function() are the same as their R counterparts. Just
  be sure to expect and return pandas DataFrames as conversion from numpy arrays has not been tested.

import pandas as pd
from sklearn import linear_model
from sklearn.preprocessing import StandardScaler

# User-defined model.
# A user-defined wrapper function for model training that takes the following
# arguments: (1) a horizon-specific pandas DataFrame made with create_lagged_df(..., type = "train")
# (e.g., my_lagged_df$horizon_h)
def py_model_function(data):
  
  X = data.iloc[:, 1:]
  y = data.iloc[:, 0]
  
  scaler = StandardScaler()
  X = scaler.fit_transform(X)
  
  model_lasso = linear_model.Lasso(alpha = 0.1)
  
  model_lasso.fit(X = X, y = y)
  
  return({'model': model_lasso, 'scaler': scaler})

# User-defined prediction function.
# The predict() wrapper function takes 2 positional arguments. First,
# the returned model from the user-defined modeling function (py_model_function() above).
# Second, a pandas DataFrame of model features. For numeric outcomes, the function 
# can return a 1- or 3-column pandas DataFrame with either (a) point
# forecasts or (b) point forecasts plus lower and upper forecast bounds (column order and names do not matter).
def py_prediction_function(model_list, data_x):
  
  data_x = model_list['scaler'].transform(data_x)
  
  data_pred = pd.DataFrame({'y_pred': model_list['model'].predict(data_x)})
  
  return(data_pred)

• Train and predict on historical validation data with the imported Python wrapper functions.

# Train a model across forecast horizons and validation datasets.
model_results <- forecastML::train_model(data_train,
                                         windows = windows,
                                         model_name = "LASSO",
                                         model_function = py_model_function,
                                         use_future = FALSE)

# Predict on the validation datasets.
data_valid <- predict(model_results, prediction_function = list(py_prediction_function), data = data_train)

# Plot forecasts for each validation dataset.
plot(data_valid, horizons = c(1, 12))

• Forecast with the same imported Python wrapper functions. The final wrapper functions may eventually have
  fixed hyperparameters or complicated model ensembles based on repeated model training and investigation.

# Forward-looking forecast data.frame.
data_forecast <- forecastML::create_lagged_df(data_seatbelts, type = "forecast", outcome_col = 1,
                                              lookback = lookback, horizon = horizons,
                                              dates = dates, frequency = date_frequency)

# Forecasts.
data_forecasts <- predict(model_results, prediction_function = list(py_prediction_function),
                          data = data_forecast)

# We'll plot a background dataset of actuals as well.
plot(data_forecasts, data_actual = data_seatbelts[-(1:150), ], 
     actual_indices = dates[-(1:150)], horizons = c(1, 12))

---

Multi-output forecast in R

• This is the same seatbelt dataset example except now, instead of 1 model for each
  forecast horizon, we’ll build 1 multi-output neural network model that forecasts 12
  steps into the future.

• Given that this is a small dataset, the multi-output approach would require a decent
  amount of tuning to produce accurate results. An alternative would be to forecast, say,
  horizons 6 through 12 if longer term forecasts were of interest to reduce the number of
  parameters; the output neurons do not have to start at a horizon of 1 or even be contiguous.

library(forecastML)
library(keras)  # Using the TensorFlow 2.0 backend.

data("data_seatbelts", package = "forecastML")

data_seatbelts[] <- lapply(data_seatbelts, function(x) {
  (x - mean(x, na.rm = TRUE)) / sd(x, na.rm = TRUE)
})

date_frequency <- "1 month"
dates <- seq(as.Date("1969-01-01"), as.Date("1984-12-01"), by = date_frequency)

data_train <- forecastML::create_lagged_df(data_seatbelts, type = "train", method = "multi_output",
                                           outcome_col = 1, lookback = 1:15, horizons = 1:12,
                                           dates = dates, frequency = date_frequency,
                                           dynamic_features = "law")

# 'window_length = 0' creates 1 historical training dataset with no external validation datasets. 
# Set it to, say, 24 to see the model and forecast stability when trained across different slices 
# of historical data.
windows <- forecastML::create_windows(data_train, window_length = 0)

#------------------------------------------------------------------------------
# 'data_y' consists of 1 column for each forecast horizon--here, 12.
model_fun <- function(data, horizons) {  # 'horizons' is passed in train_model().

  data_x <- apply(as.matrix(data[, -(1:length(horizons))]), 2, function(x){ifelse(is.na(x), 0, x)})
  data_y <- apply(as.matrix(data[, 1:length(horizons)]), 2, function(x){ifelse(is.na(x), 0, x)})

  layers_x_input <- keras::layer_input(shape = ncol(data_x))

  layers_x_output <- layers_x_input %>%
    keras::layer_dense(ncol(data_x), activation = "relu") %>%
    keras::layer_dense(ncol(data_x), activation = "relu") %>%
    keras::layer_dense(length(horizons))

  model <- keras::keras_model(inputs = layers_x_input, outputs = layers_x_output) %>%
    keras::compile(optimizer = 'adam', loss = 'mean_absolute_error')

  early_stopping <- callback_early_stopping(monitor = 'val_loss', patience = 2)

  tensorflow::tf$random$set_seed(224)

  model_results <- model %>%
    keras::fit(x = list(as.matrix(data_x)), y = list(as.matrix(data_y)),
               validation_split = 0.2, callbacks = c(early_stopping), verbose = FALSE)

  return(list("model" = model, "model_results" = model_results))
}
#------------------------------------------------------------------------------
# The predict() wrapper function will return a data.frame with a number of columns 
# equaling the number of forecast horizons.
prediction_fun <- function(model, data_features) {

  data_features[] <- lapply(data_features, function(x){ifelse(is.na(x), 0, x)})
  data_features <- list(as.matrix(data_features, ncol = ncol(data_features)))

  data_pred <- data.frame(predict(model$model, data_features))
  names(data_pred) <- paste0("y_pred_", 1:ncol(data_pred))

  return(data_pred)
}
#------------------------------------------------------------------------------

model_results <- forecastML::train_model(data_train, windows, model_name = "Multi-Output NN",
                                         model_function = model_fun,
                                         horizons = 1:12)

data_valid <- predict(model_results, prediction_function = list(prediction_fun), data = data_train)

# We'll plot select forecast horizons to reduce visual clutter.
plot(data_valid, facet = ~ model, horizons = c(1, 3, 6, 12))

• Forecast combinations from combine_forecasts() aren’t necessary as we’ve trained only 1 model.

data_forecast <- forecastML::create_lagged_df(data_seatbelts, type = "forecast", method = "multi_output",
                                              outcome_col = 1, lookback = 1:15, horizons = 1:12,
                                              dates = dates, frequency = date_frequency,
                                              dynamic_features = "law")

data_forecasts <- predict(model_results, prediction_function = list(prediction_fun), data = data_forecast)

plot(data_forecasts, facet = NULL, data_actual = data_seatbelts[-(1:100), ], actual_indices = dates[-(1:100)])

Examples - Factor Outcomes with R and Python

R

• This example is similar to the numeric outcome examples with the exception that the outcome has been
  factorized to illustrate how factors or sequences are forecasted.

data("data_seatbelts", package = "forecastML")

# Create an artifical factor outcome for illustration' sake.
data_seatbelts$DriversKilled <- cut(data_seatbelts$DriversKilled, 3)

horizons <- c(1, 12)  # 2 models that forecast 1 and 1:12 time steps ahead.

# A lookback across select time steps in the past. Feature lag 1 will be silently dropped from the 12-step-ahead model.
lookback <- c(1, 12, 18)

date_frequency <- "1 month"  # Time step frequency.

# The date indices, which don't come with the stock dataset, should not be included in the modeling data.frame.
dates <- seq(as.Date("1969-01-01"), as.Date("1984-12-01"), by = date_frequency)

# Create a dataset of features for modeling.
data_train <- forecastML::create_lagged_df(data_seatbelts, type = "train", outcome_col = 1,
                                           lookback = lookback, horizon = horizons,
                                           dates = dates, frequency = date_frequency)

# We won't use nested cross-validation; rather, we'll train a model over the entire training dataset.
windows <- forecastML::create_windows(data_train, window_length = 0)

# This is the model-training dataset.
plot(windows, data_train)

• Model training and historical fit.

model_function <- function(data, my_outcome_col) {  # my_outcome_col = 1 could be defined here.
  
  outcome_names <- names(data)[1]
  model_formula <- formula(paste0(outcome_names,  "~ ."))
  
  set.seed(224)
  model <- randomForest::randomForest(formula = model_formula, data = data, ntree = 3)
  return(model)  # This model is the first argument in the user-defined predict() function below.
}

#------------------------------------------------------------------------------
# Train a model across forecast horizons and validation datasets.
# my_outcome_col = 1 is passed in ... but could have been defined in the user-defined model function.
model_results <- forecastML::train_model(data_train,
                                         windows = windows,
                                         model_name = "RF", 
                                         model_function = model_function,
                                         my_outcome_col = 1,  # ...
                                         use_future = FALSE)

#------------------------------------------------------------------------------
# User-defined prediction function.
#
# The predict() wrapper function takes 2 positional arguments. First,
# the returned model from the user-defined modeling function (model_function() above).
# Second, a data.frame of model features. If predicting on validation data, expect the input data to be 
# passed in the same format as returned by create_lagged_df(type = 'train') but with the outcome column 
# removed. If forecasting, expect the input data to be in the same format as returned by 
# create_lagged_df(type = 'forecast') but with the 'index' and 'horizon' columns removed.
# 
# For factor outcomes, the function can return either (a) a 1-column data.frame with factor level 
# predictions or (b) an L-column data.frame of predicted class probabilities where 'L' equals the 
# number of levels in the outcome; the order of the return()'d columns should match the order of the 
# outcome factor levels from left to right which is the default behavior of most predict() functions.

# Predict/forecast a single factor level.
prediction_function_level <- function(model, data_features) {
  
  data_pred <- data.frame("y_pred" = predict(model, data_features, type = "response"))
  
  return(data_pred)
}

# Predict/forecast outcome class probabilities.
prediction_function_prob <- function(model, data_features) {
  
  data_pred <- data.frame("y_pred" = predict(model, data_features, type = "prob"))
  
  return(data_pred)
}

# Predict on the validation datasets.
data_valid_level <- predict(model_results, 
                            prediction_function = list(prediction_function_level), 
                            data = data_train)
data_valid_prob <- predict(model_results, 
                           prediction_function = list(prediction_function_prob), 
                           data = data_train)

• Predict historical factor levels.

• With window_length = 0 these are essentially plots of model fit.

plot(data_valid_level, horizons = c(1, 12))

• Predict historical class probabilities.

plot(data_valid_prob, horizons = c(1, 12))

• Forecast

# Forward-looking forecast data.frame.
data_forecast <- forecastML::create_lagged_df(data_seatbelts, type = "forecast",
                                              outcome_col = 1, lookback = lookback, horizons = horizons)

# Forecasts.
data_forecasts_level <- predict(model_results,
                                prediction_function = list(prediction_function_level),
                                data = data_forecast)

data_forecasts_prob <- predict(model_results,
                                prediction_function = list(prediction_function_prob),
                                data = data_forecast)

• Forecast factor levels

plot(data_forecasts_level)

• Forecast class probabilities

plot(data_forecasts_prob)