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 (c) 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."
More information, cheat sheets, and worked examples can be found at https://github.com/nredell/forecastML.
forecastML
For reference, below are some resources for learning more about multi-step-ahead forecasting strategies:
In contrast to the recursive or iterated method for producing multi-step-ahead forecasts used in traditional forecasting
methods like ARIMA, direct forecasting involves creating a series of distinct horizon-specific models. Though
several hybrid methods exist for producing multi-step forecasts, the simple direct forecasting method
used in forecastML
lets us avoid the exponentially more difficult problem of having to
"predict the predictors" for forecast horizons beyond 1-step-ahead.
The direct forecasting approach used in forecastML
involves the following steps:
Build a series of horizon-specific short-, medium-, and long-term forecast models.
Assess model generalization performance across a variety of heldout datasets through time.
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).
{width=715px, height=400px}
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 hyperparameters that minimize forecast error over the relevant forecast horizons and re-train.
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 NA
s 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.
create_lagged_df
: Create model training and forecasting datasets with lagged, grouped, dynamic, and static features.
create_windows
: Create time-contiguous validation datasets for model evaluation.
train_model
: Train the user-defined model across forecast horizons and validation datasets.
return_error
: Compute forecast error across forecast horizons and validation datasets.
return_hyper
: Return user-defined model hyperparameters across validation datasets.
combine_forecasts
: Combine multiple horizon-specific forecast models to produce one forecast.
knitr::opts_chunk$set(fig.width = 7.15, fig.height = 4) knitr::opts_knit$set(fig.width = 7.15, fig.height = 4)
In this walkthrough of forecastML
we'll compare the forecast performance of two machine learning
methods, LASSO and Random Forest, across forecast horizons using the Seatbelts dataset from the datasets
package.
Here's a summary of the problem at hand:
DriversKilled
- car drivers killed per month in the UK.DriversKilled
- car drivers killed per month in the UK.kms
- a measure of distance driven.PetrolPrice
- the price of gas.law
- A binary indicator of the presence of a seatbelt law.forecastML
install.packages("forecastML")
library(forecastML) library(dplyr) library(DT) library(ggplot2) library(glmnet) library(randomForest) data("data_seatbelts", package = "forecastML") data <- data_seatbelts 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) data$PetrolPrice <- round(data$PetrolPrice, 3) data <- data[, c("DriversKilled", "kms", "PetrolPrice", "law")] DT::datatable(head(data, 5))
We'll build our models on data_train
and evaluate their out-of-sample performance on data_test
.
data_train <- data[1:(nrow(data) - 12), ] data_test <- data[(nrow(data) - 12 + 1):nrow(data), ]
p <- ggplot(data, aes(x = dates, y = DriversKilled)) p <- p + geom_line() p <- p + geom_vline(xintercept = dates[nrow(data_train)], color = "red", size = 1.1) p <- p + theme_bw() + xlab("Dataset index") p
forecastML::create_lagged_df
We'll create a list of datasets for model training, one for each forecast horizon.
outcome_col <- 1 # The column index of our DriversKilled outcome. horizons <- c(1, 3, 6, 12) # 4 models that forecast 1, 1:3, 1:6, and 1:12 time steps ahead. # A lookback across select time steps in the past. Feature lags 1 through 9, for instance, will be # silently dropped from the 12-step-ahead model. lookback <- c(1:6, 9, 12, 15) # A non-lagged feature that changes through time whose value we either know (e.g., month) or whose # value we would like to forecast. dynamic_features <- "law" data_list <- forecastML::create_lagged_df(data_train, outcome_col = outcome_col, type = "train", horizons = horizons, lookback = lookback, date = dates[1:nrow(data_train)], frequency = date_frequency, dynamic_features = dynamic_features )
Let's view the modeling dataset for a forecast horizon of 6. Notice that "lag
DT::datatable(head(data_list$horizon_6, 10), options = list(scrollX = TRUE))
The plot below illustrates, for a given lagged feature, the number and position (in dataset rows) of lagged features
created for each forecast horizon/model. The lookback
argument in created_lagged_df()
was set to
create lagged features from a minimum of 1 lag to a maximum of 15 lags; however, feature lags that don't support
direct forecasting at a given forecast horizon are silently removed from the modeling dataset.
plot(data_list)
forecastML::create_windows
create_windows()
creates indices for partitioning the training dataset in the outer loop of a nested
cross-validation setup. The validation datasets are created in contiguous blocks of window_length
, as opposed to
randomly selected rows, to mimic forecasting over multi-step-ahead forecast horizons. The skip
,
window_start
, and window_stop
arguments take dataset indices--or dates if a vector of dates is supplied to
create_lagged_df()
--that allow the user to adjust the number and placement of outer loop validation datasets.
windows <- forecastML::create_windows(lagged_df = data_list, window_length = 12, skip = 48, window_start = NULL, window_stop = NULL, include_partial_window = TRUE) windows
Below is a plot of the nested cross-validation outer loop datasets or windows. In our example,
a window_length
of 12 (months) resulted in r nrow(windows)
validation windows.
In this nested cross-validation setup, a model is trained with data from r nrow(windows) - 1
windows and
forecast accuracy is assessed on the left-out window. This means that we'll need to train r nrow(windows)
models
for each direct forecast horizon, each potentially selecting different optimal
hyperparameters and having different coefficients--if available--from the inner cross-validation loop.
Assessing the differences between these models is a good way to determine the stability of a given
modeling approach under various time series dynamics.
After model training and exploration, it's entirely possible that a single multi-step-ahead forecast may
use different ML algorithms (e.g., a neural network for shorter horizons and linear regression for
longer horizons) to produce the short- and long-term forecasts.
plot(windows, data_list, show_labels = TRUE)
We'll compare the forecasting performance of two models: (a) a cross-validated LASSO and (b) a non-tuned Random Forest. The following user-defined functions are needed for each model:
create_lagged_df(..., type = "train")
(e.g., my_lagged_df$horizon_h),train_model()
or set with default arguments in the model function.predict()
function.Any data transformations, hyperparameter tuning, or inner loop cross-validation procedures should take
place within this function, with the limitation that it ultimately needs to return()
a model suitable for
the user-defined predict()
function; a list can be returned to capture meta-data and data pre-processing pipelines.
# Example 1 - LASSO # Alternatively, we could define an outcome column identifier argument, say, 'outcome_col = 1' in # this function or just 'outcome_col' and then set the argument as 'outcome_col = 1' in train_model(). model_function <- function(data) { # The 'law' feature is constant during some of our outer-loop validation datasets so we'll # simply drop it so that glmnet converges. constant_features <- which(unlist(lapply(data[, -1], function(x) {!(length(unique(x)) > 1)}))) if (length(constant_features) > 1) { data <- data[, -c(constant_features + 1)] # +1 because we're skipping over the outcome column. } x <- data[, -(1), drop = FALSE] y <- data[, 1, drop = FALSE] x <- as.matrix(x, ncol = ncol(x)) y <- as.matrix(y, ncol = ncol(y)) model <- glmnet::cv.glmnet(x, y, nfolds = 3) return(list("model" = model, "constant_features" = constant_features)) } # Example 2 - Random Forest # Alternatively, we could define an outcome column identifier argument, say, 'outcome_col = 1' in # this function or just 'outcome_col' and then set the argument as 'outcome_col = 1' in train_model(). model_function_2 <- function(data) { outcome_names <- names(data)[1] model_formula <- formula(paste0(outcome_names, "~ .")) model <- randomForest::randomForest(formula = model_formula, data = data, ntree = 200) return(model) }
forecastML::train_model
For each modeling approach, LASSO and Random Forest, a total of N forecast horizons
* N validation windows
models are trained. In this example, that means training r length(data_list) * nrow(windows)
models
for each algorithm.
These models could be trained in parallel on any OS with the very flexible future
package by un-commenting the code below and
setting use_future = TRUE
. To avoid nested parallelization, models are either trained in parallel across forecast horizons or
validation windows, whichever is longer (when equal, the default is parallel across forecast horizons).
#future::plan(future::multiprocess) model_results <- forecastML::train_model(data_list, windows, model_name = "LASSO", model_function, use_future = FALSE) model_results_2 <- forecastML::train_model(data_list, windows, model_name = "RF", model_function_2, use_future = FALSE)
The following user-defined prediction function is needed for each model:
data.frame
of the model features from create_lagged_df(..., type = "train")
.data.frame
of predictions with 1 or 3 columns. A 1-column data.frame will produce point forecasts,
and a 3-column data.frame can be used to return point, lower, and upper forecasts (column names and order do not matter).# Example 1 - LASSO. prediction_function <- function(model, data_features) { if (length(model$constant_features) > 1) { # 'model' was passed as a list. data_features <- data_features[, -c(model$constant_features )] } x <- as.matrix(data_features, ncol = ncol(data_features)) data_pred <- data.frame("y_pred" = predict(model$model, x, s = "lambda.min")) return(data_pred) } # Example 2 - Random Forest. prediction_function_2 <- function(model, data_features) { data_pred <- data.frame("y_pred" = predict(model, data_features)) return(data_pred) }
The predict.forecast_model()
S3 method takes any number of trained models from train_model()
and a
list of user-defined prediction functions. The list of prediction functions should appear in the same order
as the models. Note that the prediction_function
and data
arguments need to be named because the first
function argument is ...
.
Outer loop nested cross-validation forecasts are returned for each user-defined model, forecast horizon, and validation window.
data_results <- predict(model_results, model_results_2, prediction_function = list(prediction_function, prediction_function_2), data = data_list)
Let's view the models' predictions.
data_results$DriversKilled_pred <- round(data_results$DriversKilled_pred, 0) DT::datatable(head(data_results, 30), options = list(scrollX = TRUE))
Below is a plot of the historical forecasts for each validation window at select forecast horizons.
plot(data_results, type = "prediction", horizons = c(1, 6, 12))
Below is a plot of the historical forecast error for select validation windows at select forecast horizons.
plot(data_results, type = "residual", horizons = c(1, 6, 12))
The plot below is a diagnostic plot to check how forecasts for a target point in time have changed through time by looking at a history of forecasts. In this example, we have 4 direct forecast horizons--1, 3, 6, 12--so each of the colored points represents the origin of the forecast for the black point. In most cases it would be reasonable to expect shorter-horizon forecasts to be more accurate than longer-horizon forecasts.
plot(data_results, type = "forecast_stability", windows = 3)
forecastML::return_error
Let's calculate several common forecast error metrics for our holdout data sets in the training data.
The forecast errors for nested cross-validation are returned at 3 levels of granularity:
data_error <- forecastML::return_error(data_results) # Global error. data_error$error_global[, -1] <- lapply(data_error$error_global[, -1], round, 1) DT::datatable(data_error$error_global, options = list(scrollX = TRUE), )
Below is a plot of error metrics across time for select validation windows and forecast horizons.
plot(data_error, type = "window", facet = ~ horizon, horizons = c(1, 6, 12))
Below is a plot of forecast error metrics by forecast model horizon collapsed across validation windows.
plot(data_error, type = "horizon", facet = ~ horizon, horizons = c(1, 6, 12))
Below is a plot of error metrics collapsed across validation windows and forecast horizons.
plot(data_error, type = "global", facet = ~ horizon)
While it may be reasonable to have distinct models for each forecast horizon or even forecasting model ensembles across horizons, at this point we still have slightly different LASSO and Random Forest models from the outer loop of the nested cross-validation within each horizon-specific model. Here, we'll take a look at the stability of the hyperparameters for the LASSO model to better understand if we can train one model across forecast horizons or if we need additional predictors or modeling strategies to forecast well under various conditions or time series dynamics.
The following user-defined hyperparameter function is needed for each model:
data.frame
.hyper_function <- function(model) { lambda_min <- model$model$lambda.min lambda_1se <- model$model$lambda.1se data_hyper <- data.frame("lambda_min" = lambda_min, "lambda_1se" = lambda_1se) return(data_hyper) }
forecastML::return_hyper
Below are two plots which show (a) univariate hyperparameter variability across the training data and (b) the relationship between each error metric and hyperparameter values.
data_hyper <- forecastML::return_hyper(model_results, hyper_function) plot(data_hyper, data_results, data_error, type = "stability", horizons = c(1, 6, 12)) plot(data_hyper, data_results, data_error, type = "error", c(1, 6, 12))
forecastML::create_lagged_df
To forecast with the direct forecasting method, we need to create another dataset of forward-looking features.
We can do this by running create_lagged_df()
and setting type = "forecast"
.
Below is the forecast dataset for a 6-step-ahead forecast.
The forecast dataset has the following columns:
1:max(horizons)
.type = "train"
, dataset.data_forecast_list <- forecastML::create_lagged_df(data_train, outcome_col = outcome_col, type = "forecast", horizons = horizons, lookback = lookback, date = dates[1:nrow(data_train)], frequency = date_frequency, dynamic_features = dynamic_features ) DT::datatable(head(data_forecast_list$horizon_6), options = list(scrollX = TRUE))
Because we didn't treat law
as a lagged feature, we'll have to fill in its future values when direct
forecasting 1, 3, 6, and 12 steps ahead. In this example, we know that law <- 1
for the next 12 months.
If we did not know the future values of law
we would either have to use a class of models that can predict
with missing features or forecast the value of law
1:12 months ahead.
for (i in seq_along(data_forecast_list)) { data_forecast_list[[i]]$law <- 1 }
Running the predict method, predict.forecast_model()
, on the dataset created
above--with type = "forecast"
--and placing it in the data
argument in predict.forecast_model()
below, returns
a data.frame of forecasts.
An S3 object of class, forecast_results
, is returned. This object will have different plotting and error methods than
the training_results
class from earlier.
data_forecast <- predict(model_results, model_results_2, # ... supports any number of ML models. prediction_function = list(prediction_function, prediction_function_2), data = data_forecast_list) data_forecast$DriversKilled_pred <- round(data_forecast$DriversKilled_pred, 0) DT::datatable(head(data_forecast, 10), options = list(scrollX = TRUE))
Below is a plot of the forecasts vs. the actuals for each model at select forecast horizons.
Setting the data_actual = ...
and actual_indices = ...
arguments plots a background dataset (gray line in the plots below).
It's clear from the plots that our Random Forest model is producing less accurate forecasts and is more sensitive to the data on which it was trained.
plot(data_forecast, data_actual = data[-(1:150), ], # Actuals from the training and test data sets. actual_indices = dates[-(1:150)], horizons = c(1, 6, 12))
forecastML::return_error
Finally, we'll look at our out-of-sample forecast error by forecast horizon for our two models by setting data_test = data_test
.
If the first argument of return_error()
is an object of class forecast_results
and
the data_test
argument is a data.frame like data_test from our beginning train-test split, a data.frame
of forecast error metrics with the following columns is returned:
train_model()
.Below are 3 forecast error plots at various levels of aggregation.
data_error <- forecastML::return_error(data_forecast, data_test = data_test, test_indices = dates[(nrow(data_train) + 1):length(dates)]) plot(data_error, facet = ~ horizon, type = "window") plot(data_error, facet = ~ horizon, type = "horizon") plot(data_error, facet = ~ horizon, type = "global")
Because our LASSO model is both more stable and accurate, we'll re-train this model across
the entire training dataset to get our final r length(horizons)
models--1 for each forecast horizon.
Note that for a real-world forecasting problem this is when we would do additional model tuning
to improve forecast accuracy across validation windows as well as narrow the hyperparameter search
in the user-specified modeling functions.
data_list <- forecastML::create_lagged_df(data_train, outcome_col = outcome_col, type = "train", horizons = horizons, lookback = lookback, date = dates[1:nrow(data_train)], frequency = date_frequency, dynamic_features = dynamic_features )
To create a dataset without nested cross-validation, set window_length = 0
in create_windows()
.
windows <- forecastML::create_windows(data_list, window_length = 0) plot(windows, data_list, show_labels = TRUE)
Without nested cross-validation and holdout windows, the prediction plot is essentially a plot of model fit.
model_results <- forecastML::train_model(data_list, windows, model_name = "LASSO", model_function) data_results <- predict(model_results, prediction_function = list(prediction_function), data = data_list) DT::datatable(head(data_results, 10), options = list(scrollX = TRUE)) plot(data_results, type = "prediction", horizons = c(1, 6, 12))
forecastML::return_error
Below is a the training error collapsed across our 4 direct forecast horizons/models.
data_error <- forecastML::return_error(data_results) data_error$error_global[, -1] <- lapply(data_error$error_global[, -1], round, 1) DT::datatable(head(data_error$error_global), options = list(scrollX = TRUE))
data_forecast_list <- forecastML::create_lagged_df(data_train, outcome_col = outcome_col, type = "forecast", horizons = horizons, lookback = lookback, date = dates[1:nrow(data_train)], frequency = date_frequency, dynamic_features = dynamic_features ) for (i in seq_along(data_forecast_list)) { data_forecast_list[[i]]$law <- 1 } data_forecast <- predict(model_results, prediction_function = list(prediction_function), data = data_forecast_list) plot(data_forecast, data_actual = data[-(1:150), ], actual_indices = dates[-(1:150)], horizons = c(1, 6, 12))
forecastML::return_error
data_error <- forecastML::return_error(data_forecast, data_test = data_test, test_indices = dates[(nrow(data_train) + 1):nrow(data)]) plot(data_error, type = "horizon", facet = ~ horizon)
forecastML::combine_forecasts
The final step in the forecastML
framework is to combine multiple direct-horizon forecast
models with combine_forecasts()
to produce a single h-step-ahead forecast.
The default approach, type = 'horizon'
, is to combine forecasts across models such that short-term
models produce the shorter-term forecasts and long-term models produce the longer-term forecasts.
This implies that, for our 12-month-ahead forecast,
data_combined <- forecastML::combine_forecasts(data_forecast) # Plot a background dataset of actuals using the most recent data. data_actual <- data[dates >= as.Date("1980-01-01"), ] actual_indices <- dates[dates >= as.Date("1980-01-01")] # Plot all final forecasts plus historical data. plot(data_combined, data_actual = data_actual, actual_indices = actual_indices) # Error by forecast horizon. DT::datatable(return_error(data_combined, data_actual, actual_indices)$error_by_horizon, options = list(scrollX = TRUE)) # Error aggregated across forecast horizons. DT::datatable(return_error(data_combined, data_actual, actual_indices)$error_global, options = list(scrollX = TRUE))
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