In this document the utsf package is described. This package offers a meta engine for applying different regression models for univariate time series forecasting using an autoregressive approach.

Univariate time series forecasting and autoregressive models

An univariate time series forecasting method is one in which the future values of a series are predicted using only information from the series. For example, the future values of a series can be forecast by its mean historical value. An advantage of this type of prediction is that, apart from the series being forecast, there is no need to collect any further information in order to train the model used to predict the future values of the series.

An autoregressive model is a kind of univariate time series forecasting model in which a value of a time series is expressed as a function of some of its past values. That is, an autoregressive model is a regression model in which the independent variables are lagged values (previous values) of the response variable. For example, given a time series with the following historical values: $t = \{1, 3, 6, 7, 9, 11, 16\}$, suppose that we want to develop an autoregressive model in which a target “is explained” by its first, second and fourth past values (in this context, a previous value is also called a lag, so lag 1 is the value immediately preceding a given value in the series). Given this series and lags (1, 2 and 4), the training set would be:

Lag 4	Lag 2	Lag 1	Target
1	6	7	9
3	7	9	11
6	9	11	16

In this model the next future value of the series is predicted as $f(Lag4, Lag2, Lag1)$, where $f$ is the regression function and $Lag4$, $Lag2$ and $Lag1$ are the fourth, second and first lagged values of the next future value. So, the next future value of series $t$ is predicted as $f(7, 11, 16)$, producing a value that will be called $F1$.

Suppose that the forecast horizon (the number of future values to be forecast into the future) is greater than 1. In the case that the regression function only predicts the next future value of the series, a recursive approach can be applied to forecast all the future values of the forecast horizon. Using a recursive approach, the regression function is applied recursively until all horizons are forecast. For instance, following the previous example, suppose that the forecast horizon is 3. As we have explained, to forecast the next future value of the series (horizon 1) the regression function is fed with the vector $[7, 11, 16]$, producing $F1$. To forecast horizon 2 the regression function is fed with the vector $[9, 16, F1]$. The forecast for horizon 1, $F1$, is used as the first lag for horizon 2 because the actual value is unknown. Finally, to predict horizon 3 the regression function is fed with the vector [$11, F1, F2]$. This example of recursive forecast is summarized in the following table:

Horizon	Autoregressive values	Forecast
1	7, 11, 16	F1
2	9, 16, F1	F2
3	11, F1, F2	F3

The recursive approach for forecasting several values into the future is applied in classical statistical models such as ARIMA or exponential smoothing.

The utsf package

The utsf package makes it easy the use of classical regression models for univariate time series forecasting employing the autoregressive approach and the recursive prediction strategy explained in the previous section. All the supported models are applied using an uniform interface: the forecast() function. Let us see an example in which a regression tree model is used to forecast the next future values of a time series:

f <- forecast(AirPassengers, h = 12, lags = 1:12, method = "rt")

In this example, an autoregressive tree model (method = "rt") is trained using the historical values of the AirPassengers time series and a forecast for its 12 next future values (h = 12) is done. The forecast() function returns an S3 object of class utsf with information about the trained model and the forecast. The information about the forecast is included in the component pred as an object of class ts (a time series):

f$pred
#>           Jan      Feb      Mar      Apr      May      Jun      Jul      Aug
#> 1961 460.2980 428.4915 467.0304 496.9833 499.9819 554.7891 627.5849 628.0503
#>           Sep      Oct      Nov      Dec
#> 1961 533.2803 482.4221 448.7926 453.6920

library(ggplot2)
autoplot(f)

The training set used to fit the model is built from the historical values of the time series using the autoregressive approach explained in the previous section. The lags parameter of the forecast() function is used to specify the autoregressive lags. In the example: lags = 1:12, so a target is a function of its 12 previous values. Next, we consult the first targets (and their associated features) with which the regression model has been trained:

head(f$targets)  # first targets
#> [1] -11.6666667  -0.9166667  13.4166667   6.6666667  -3.8333333  19.8333333

head(f$features) # and its associated features
#>         Lag12     Lag11     Lag10      Lag9       Lag8       Lag7       Lag6
#> 1 -14.6666667 -8.666667  5.333333  2.333333  -5.666667   8.333333  21.333333
#> 2  -8.9166667  5.083333  2.083333 -5.916667   8.083333  21.083333  21.083333
#> 3   4.4166667  1.416667 -6.583333  7.416667  20.416667  20.416667   8.416667
#> 4   0.6666667 -7.333333  6.666667 19.666667  19.666667   7.666667  -9.333333
#> 5  -7.8333333  6.166667 19.166667 19.166667   7.166667  -9.833333 -24.833333
#> 6   5.8333333 18.833333 18.833333  6.833333 -10.166667 -25.166667 -11.166667
#>         Lag5       Lag4       Lag3       Lag2       Lag1
#> 1  21.333333   9.333333  -7.666667 -22.666667  -8.666667
#> 2   9.083333  -7.916667 -22.916667  -8.916667 -11.916667
#> 3  -8.583333 -23.583333  -9.583333 -12.583333  -1.583333
#> 4 -24.333333 -10.333333 -13.333333  -2.333333  12.666667
#> 5 -10.833333 -13.833333  -2.833333  12.166667   6.166667
#> 6 -14.166667  -3.166667  11.833333   5.833333  -4.166667

Using the example of the previous section:

t <- ts(c(1, 3, 6, 7, 9, 11, 16))
out <- forecast(t, h = 3, lags = c(1, 2, 4), preProcess = NULL)
cbind(out$features, Target = out$targets)
#>   Lag4 Lag2 Lag1 Target
#> 1    1    6    7      9
#> 2    3    7    9     11
#> 3    6    9   11     16

Supported models

The forecast() function provides a common interface to applying an autoregressive approach for time series forecasting using different regression models. These models are implemented in several R packages. Currently, the forecast() function is mainly focused on regression tree models, supporting the following approaches:

k-nearest neighbors: In this case no model is trained and the function FNN::knn.reg() is used, as regression function, to recursively predict the future values of the time series.
Regression trees: The model is trained using the function rpart::rpart() and its associated method rpart::predict.rpart() is applied recursively for the forecasts, i.e., as regression function.
Model trees: The model is trained with the function Cubist::cubist() and its associated method Cubist::predict.cubist() is used for predictions.
Bagging: The model is trained with the function ipred::bagging() and its associated method ipred::predict.regbagg() is used for forecasting.
Random forest: The model is trained with the function ranger::ranger() and its associated method ranger::predict.ranger() is used for predictions.

The S3 object of class utsf returned by the forecast() function contains a component with the trained autoregressive model:

f <- forecast(fdeaths, h = 12, lags = 1:12, method = "rt")
f$model
#> n= 60 
#> 
#> node), split, n, deviance, yval
#>       * denotes terminal node
#> 
#> 1) root 60 1967124.00   -6.465278  
#>   2) Lag12< 73 38  212851.90 -124.414500  
#>     4) Lag6>=-66.45833 30   57355.07 -153.847200  
#>       8) Lag12< -170.7083 10   13293.09 -195.991700 *
#>       9) Lag12>=-170.7083 20   17419.67 -132.775000 *
#>     5) Lag6< -66.45833 8   32051.01  -14.041670 *
#>   3) Lag12>=73 22  312482.00  197.265200  
#>     6) Lag5>=-131.7917 7   24738.12  114.500000 *
#>     7) Lag5< -131.7917 15  217416.40  235.888900 *

In this case, the model is the result of training a regression tree using the function rpart::rpart() with the training set consisting of the features f$features and targets f$targets. Once the model is trained, the rpart::predict.rpart() function is used recursively to forecast the future values of the time series.

Using your own models

One interesting feature of the utsf package is that you can use the forecast() function to apply your own regression models for time series forecasting in an autoregressive way. Thus, your regression models can benefit from the features implemented in the forecast() function, such as preprocessing, parameter tuning, the building of the training set, the implementation of recursive forecasts or the estimation of the forecast accuracy of the model.

To apply your own regression model with the forecast() function you have to use the method parameter, providing a function that is able to train your model. This function should return an object with the trained regression model. Also, it must have at least two input parameters:

X: it is a data frame with the features of the training examples. This data frame is built from the time series taking into account the autoregressive lags as explained in a previous section. This is the same object as the features component of the object returned by the forecast() function.
y: a vector with the targets of the training examples. It is built as explained in a previous section. It is the same object as the targets component of the object returned by the forecast() function.

Furthermore, if the function that trains the model (the function provided in the method parameter) returns a model of class model_class, a method with the signature predict.model_class(object, new_value) should be implemented. This method uses your model to predict a new value, that is, it is the regression function associated with the model.

Let us see an example in which the forecast() function is used to forecast a time series using the k-nearest neighbors regression model implemented in the package FNN:

# Function to train the regression model
my_knn_model <- function(X, y, k = 3) {
  structure(list(X = X, y = y, k = k), class = "my_knn")
}

# Function to predict a new example
predict.my_knn <- function(object, new_value) {
  FNN::knn.reg(train = object$X, test = new_value, 
               y = object$y, k = object$k)$pred
}

f <- forecast(AirPassengers, h = 12, lags = 1:12, method = my_knn_model)
print(f$pred)
#>           Jan      Feb      Mar      Apr      May      Jun      Jul      Aug
#> 1961 455.9167 434.3264 480.7703 490.1678 506.1262 568.0534 640.6689 640.8636
#>           Sep      Oct      Nov      Dec
#> 1961 549.5467 495.4255 441.6554 476.7934

autoplot(f)

The new_value parameter of the predict method receives a data frame with the same structure as the X parameter of the function for building the model. The new_value data frame only has one row, with the features of the example to be predicted.

The k-nearest neighbors algorithm is so simple that it can be easily implemented without using functionality of any R package:

# Function to train the regression model
my_knn_model2 <- function(X, y, k = 3) {
  structure(list(X = X, y = y, k = k), class = "my_knn2")
}

# Function to predict a new example
predict.my_knn2 <- function(object, new_value) {
  distances <- sapply(1:nrow(object$X), function(i) sum((object$X[i, ] - new_value)^2))
  k_nearest <- order(distances)[1:object$k]
  mean(object$y[k_nearest])
}

f2 <- forecast(AirPassengers, h = 12, lags = 1:12, method = my_knn_model2)
print(f2$pred)
#>           Jan      Feb      Mar      Apr      May      Jun      Jul      Aug
#> 1961 455.9167 434.3264 480.7703 490.1678 506.1262 568.0534 640.6689 640.8636
#>           Sep      Oct      Nov      Dec
#> 1961 549.5467 495.4255 441.6554 476.7934

Finally, we are going to forecast an artificial time series with a trend by means of a simple linear regression model.

set.seed(7)
t <- 1:15 + rnorm(15, sd = 0.5) # time series
my_lm <- function(X, y) lm(y ~ ., data = data.frame(cbind(X, y = y)))
f <- forecast(t, h = 5, lags = 1, method = my_lm, preProcess = NULL)
library(ggplot2)
autoplot(f)

In this case, we rely on the predict.lm method to predict new values. Let us see, the model:

f$model
#> 
#> Call:
#> lm(formula = y ~ ., data = data.frame(cbind(X, y = y)))
#> 
#> Coefficients:
#> (Intercept)         Lag1  
#>      0.7914       1.0251

The forecast of a future value is computed as $0.7914 + 1.0251Lag1$, where Lag1 is the previous value to the future value being forecast.

Setting the parameters of the regression models

Normally, a regression model can be adjusted using different parameters. By default, the models supported by the forecast() function are set using some specific parameters, usually the default values of the functions used to train the models (these functions are listed in a previous section). However, the user can set the parameters used to train the regression models with the param argument of the forecast() function. The param argument must be a list with the names and values of the parameters to be set. Let us see an example:

# A bagging model set with default parameters
f <- forecast(AirPassengers, h = 12, lags = 1:12, method = "bagging")
length(f$model$mtrees) # number of regression trees (25 by default)
#> [1] 25

# A bagging model set with 3 regression tress
f <- forecast(AirPassengers, h = 12, 
              lags = 1:12, 
              method = "bagging", 
              param = list(nbagg = 3)
)
length(f$model$mtrees) # number of regression trees
#> [1] 3

In the previous example, two bagging models (using regression trees) are trained with the forecast() function. In the first model the number of trees is 25, the default value of the function ipred::ipredbagg() used to train the model. In the second model the number of trees is set to 3. Of course, in order to set some specific parameters the user must consult the arguments of the function used internally by the forecast() function to train the model. In the example, ipred::ipredbagg().

In the following example the user sets the parameters of a regression model implemented by himself/herself:

# Function to train the model
my_knn_model <- function(X, y, k = 3) {
  structure(list(X = X, y = y, k = k), class = "my_knn")
}

# Regression function for object of class my_knn
predict.my_knn <- function(object, new_value) {
  FNN::knn.reg(train = object$X, test = new_value, 
               y = object$y, k = object$k)$pred
}

# The model is trained with default parameters (k = 3)
f <- forecast(AirPassengers, h = 12, lags = 1:12,  method = my_knn_model)
print(f$model$k)
#> [1] 3

# The model is trained with k = 5
f <- forecast(AirPassengers, h = 12, 
              method = my_knn_model, param = list(k = 5))
print(f$model$k)
#> [1] 5

Estimating forecast accuracy

This section explains how to estimate the forecast accuracy of a regression model predicting a time series with the utsf package. Let us see an example:

f <- forecast(UKgas, h = 4, lags = 1:4, method = "knn", efa = "fixed")
f$efa 
#>       MAE      MAPE     sMAPE      RMSE 
#> 49.304102  7.226569  7.499436 53.862046

To assess the forecast accuracy of a regression model you can use the forecast() function to specify the regression task, using the efa parameter to choose how the forecast accuracy is estimated. In this case a k- nearest neighbors model is used (model = "knn") with the autoregressive lags 1 to 4 and an estimation of its forecast accuracy on the UKgas time series for a forecast horizon of 4 (h = 4) is obtained using a fixed origin strategy. The result of this estimation can be found in the efa component of the object returned by the forecast() function. It is a vector with estimates of forecast errors according to different forecast accuracy measures. For instance, in the example, the estimated mean absolute error (MAE) for horizon 4 is 49.3 approximately. Currently, the following forecasting accuracy measures are computed:

MAE: mean absolute error
MAPE: mean absolute percentage error
sMAPE: symmetric MAPE
RMSE: root mean squared error

Next, we describe how the forecasting accuracy measures are computed for a forecasting horizon $h$ ($y_t$ and $\hat{y}_t$ are the actual future value and its forecast for horizon $t$ respectively):

\[ MAE = \frac{1}{h}\sum_{t=1}^{h} |y_t-\hat{y}_t| \]

\[ MAPE = \frac{1}{h}\sum_{t=1}^{h} 100\frac{|y_t-\hat{y}_t|}{y_t} \] \[ sMAPE = \frac{1}{h}\sum_{t=1}^{h} 200\frac{\left|y_{t}-\hat{y}_{t}\right|}{|y_t|+|\hat{y}_t|} \]

\[ RMSE = \sqrt{\frac{1}{h}\sum_{t=1}^{h} (y_t-\hat{y}_t)^2} \]

The efa parameter can be:

"fixed": In that case a fixed origin evaluation is followed to estimate the forecast error. The test set is formed by the last h observations of the time series and the training set is formed by its previous values.
"rolling": The rolling origin strategy is applied. First, rolling origin uses a test set formed by the last h historical values and a training set consisting of the previous historical values. This procedure is repeated $h -1$ times, moving the origin of the test set one value ahead each time. This way, h one step ahead predictions can be assessed, $h - 1$ two steps ahead predictions, and so on. This is a great contrast with the fixed origin evaluation that, for a validation set of size h, is only able to assess one prediction for each one of the h horizons. However, rolling origin evaluation is more computationally intensive than fixed origin evaluation.

Parameter tuning

Another useful feature of the utsf package is parameter tuning. The forecast() function allows to estimate the forecast accuracy of a model using different combinations of parameters. Furthermore, the best combination of parameters is used to train the model with all the historical values of the series and forecast the future values of the series. Let us see an example:

f <- forecast(UKgas, h = 4, lags = 1:4, method = "knn", 
              tuneGrid = expand.grid(k = 1:7), efa = "fixed")
f$tuneGrid 
#>   k      MAE     MAPE    sMAPE     RMSE
#> 1 1 26.39307 3.781212 3.725138 34.51027
#> 2 2 49.04570 7.209276 7.447296 52.11319
#> 3 3 49.30410 7.226569 7.499436 53.86205
#> 4 4 46.82356 6.604359 6.825456 53.25469
#> 5 5 53.05555 7.267617 7.538089 62.05870
#> 6 6 55.16359 7.104669 7.347932 66.33756
#> 7 7 48.01931 5.719260 5.906827 62.99989

In this example, the tuneGrid parameter is used to specify (using a data frame) the set of parameters to assess. The forecast accuracy of the model using the different combinations of parameters is estimated as explained in the previous section using the last observations of the time series as validation set. The efa parameter is used to specify whether fixed or rolling origin evaluation is applied. The tuneGrid component of the object returned by the forecast() function contains the result of the estimation. In this case, the k-nearest neighbors method using $k=1$ obtains the best results for all the forecast accuracy measures. The best combination of parameters according to RMSE is used to forecast the time series:

f$param
#> $k
#> [1] 1

f$pred 
#>           Qtr1      Qtr2      Qtr3      Qtr4
#> 1987 1217.9250  661.4063  388.1828  817.3785

Let us plot the values of $k$ against their estimated accuracy using RMSE:

plot(f$tuneGrid$k, f$tuneGrid$RMSE, type = "o", pch = 19, xlab = "k (number of nearest neighbors)", ylab = "RMSE", main = "Estimated accuracy")

Preprocessings and transformations

Sometimes, the forecast accuracy of a model can be improved by applying some transformations and/or preprocessings to the time series being forecast. Currently, the utsf package is focused on preprocessings related to forecasting time series with a trend. This is due to the fact that, at present, the models incorporated in the forecast() function (such as regression trees and k-nearest neighbors) predict averages of the targets in the training set (i.e., an average of historical values of the series). In a trended series these averages are probably outside the range of the future values of the series. Let us see an example in which a trended series (the yearly revenue passenger miles flown by commercial airlines in the United States) is forecast using a random forest model:

f <- forecast(airmiles, h = 4, lags = 1:4, method = "rf", preProcess = NULL)
autoplot(f)

In this case, no preprocessing is applied (preProcess = NULL). It can be observed how the forecast does not capture the trending behavior of the series because regression tree models predict averaged values of the training targets.

In the next subsections we explain how to deal with trended series using three different transformations.

Differencing

A common way of dealing with trended series is to transform the original series taking first differences (computing the differences between consecutive observations). Then, the model is trained with the differenced series and the forecast are back-tranformed. Let us see an example:

f <- forecast(airmiles, h = 4, lags = 1:4, method = "rf", preProcess = list(differences(1)))
autoplot(f)

In this case, the preProcess parameter has been used to specify first-order differences. The preProcess parameter consists of a list of preprocessings or transformations. Currently, only one preprocessing can be specified. The differences are specified using the differences() function. This function takes one parameter with the order of first differences (normally 1). It is also possible to specify the value -1, in which case the order of differences is estimated using the ndiffs() function from the forecast package (in that case the chosen order of first differences could be 0, that is, no differences are applied).

# The order of first differences is estimated using the ndiffs function
f <- forecast(airmiles, h = 4, lags = 1:4, method = "rf", preProcess = list(differences(-1)))
#> Registered S3 method overwritten by 'quantmod':
#>   method            from
#>   as.zoo.data.frame zoo

f$differences
#> Order of first differences (selected automatically): 2

The additive transformation

This transformation has been used to deal with trending series in other packages, such as tsknn and tsfgrnn. The additive transformation works transforming the training examples as follows:

A vector of features is transformed by subtracting the mean of the vector.
The target associated with a vector of features is transformed by subtracting from it the mean of its associated vector of features.
To back transform a prediction, the mean of the input vector is added to it.

It is easy to check how the training examples are modified by the additive transformation using the API of the package. For example, let us see the training examples of a model with no transformation:

timeS <- ts(c(1, 3, 7, 9, 10, 12))
f <- forecast(timeS, h = 1, lags = 1:2, preProcess = NULL)
cbind(f$features, Targets = f$targets)
#>   Lag2 Lag1 Targets
#> 1    1    3       7
#> 2    3    7       9
#> 3    7    9      10
#> 4    9   10      12

Now, let us see the effect of the additive transformation in the training examples:

timeS <- ts(c(1, 3, 7, 9, 10, 12))
f <- forecast(timeS, h = 1, lags = 1:2, preProcess = list("additive"))
cbind(f$features, Targets = f$targets)
#>   Lag2 Lag1 Targets
#> 1 -1.0  1.0     5.0
#> 2 -2.0  2.0     4.0
#> 3 -1.0  1.0     2.0
#> 4 -0.5  0.5     2.5

Finally, we forecast the airmiles series using the additive tranformation and a random forest model:

f <- forecast(airmiles, h = 4, lags = 1:4, method = "rf", preProcess = list("additive"))
autoplot(f)

The multiplicative transformation

The multiplicative transformation is similar to the additive transformation, but it is intended for series with an exponential trend (the additive transformation is suited for series with a linear trend). In the multiplicative transformation a training example is transformed this way:

A vector of features is transformed by dividing it by its mean.
The target associated with a vector of features is transformed by dividing it by the mean of its associated vector of features.
To back transform a prediction, the prediction is multiplied by the mean of the input vector.

Let us see an example of an artificial time series with a multiplicative trend and its forecast using the additive and the multiplicative transformation:

t <- ts(10 * 1.05^(1:20))
f_m <- forecast(t, h = 4, lags = 1:3, method = "rf", preProcess = list("multiplicative"))
f_a <- forecast(t, h = 4, lags = 1:3, method = "rf", preProcess = list("additive"))
library(vctsfr)
plot_predictions(t, predictions = list(Multiplicative = f_m$pred, Additive = f_a$pred))

In this case, the forecast with the multiplicative transformation captures perfectly the exponential trend.

Default parameters

The forecast() function only has two compulsory parameters: the time series being forecast and the forecast horizon. Next, we discuss the default values for the rest of its parameters:

lags: The lags parameter is an integer vector (in increasing order) with the autoregressive lags. If frequency(ts) == f where ts is the time series being forecast and $f > 1$ then the lags used as autoregressive features are 1:f. For example, the lags for quarterly data are 1:4 and for monthly data 1:12. This is useful to capture a possible seasonal behavior of the series. If frequency(ts) == 1, then:
- The lags with significant autocorrelation in the partial autocorrelation function (stats::pacf()) are selected.
- If no lag has a significant autocorrelation, then lags 1:5 are chosen.
- If only one lag has significant autocorrelation and the additive or multiplicative transformation is applied, then lags 1:5 are chosen. This is done because it does not make sense to use these transformations with only one autoregressive lag.
method: By default, the k-nearest neighbors algorithm is applied.
param: By default, the model is trained using some sensible parameters, normally the default values of the function used to train the model.
preProcess: The additive transformation is applied. This transformation have proved its effectiveness to forecast trending series. In general, its application seems beneficial on any kind of series.
efa: No estimation of forecast accuracy is done.
tuneGrid: No estimation of forecast accuracy is done.