Automatic Time Series Forecasting with Ata Method in R: ATAforecasting Package

Ata method is a new univariate time series forecasting method that provides innovative solutions to issues faced during the initialization and optimization stages of existing methods. The Ata method’s forecasting performance is superior to existing methods in terms of easy implementation and accurate forecasting. It can be applied to non-seasonal or deseasonalized time series, where the deseasonalization can be performed via any preferred decomposition method. The R package ATAforecasting was developed as a comprehensive toolkit for automatic time series forecasting. It focuses on modeling all types of time series components with any preferred Ata methods and handling seasonality patterns by utilizing some popular decomposition techniques. The ATAforecasting package allows researchers to model seasonality with STL, STLplus, TBATS, stR, and TRAMO/SEATS, and power family transformation and analyze the any time series with a simple Ata method and additive, multiplicative, damped trend the Ata methods and level fixed Ata trended methods. It offers functions for researchers and data analysts to model any type of time series data sets without requiring specialization. However, an expert user may use the functions that can model all possible time series behaviors. The package also incorporates types of model specifications and their graphs, uses different accuracy measures that surely increase the Ata method’s performance.

Ali Sabri Taylan (Dokuz Eylul University) , Guckan Yapar (Dokuz Eylul University) , Hanife Taylan Selamlar (Dokuz Eylul University)
2021-12-15

1 Introduction

Ata method (Yapar et al. 2017, 2018, 2019; Yapar 2018; Yilmaz et al. 2019; Cetin and Yavuz 2020) is a new univariate time series forecasting method which provides innovative solutions to issues faced during the initialization and optimization stages of existing methods. ATAforecasting performance is superior to existing methods both in terms of easy implementation and accurate forecasting. It can be applied to non-seasonal or deseasonalized time series, where the deseasonalization can be performed via any preferred decomposition method. This methodology performed extremely well on the M3 and M4-Competition data.

The original exponential smoothing has accomplished well in a wide range of practical researches, and it is well built as a precise and optimal forecasting method. Nonetheless, two essential difficulties are to choose the smoothing constant and starting value in exponential smoothing theory. The Ata method suggests an alternative method for smoothing constant and initial value. The Ata method places more emphasis than the classical method on most recent activities. The forecasting error is compared to the error in forecasts obtained by the original model.

Exponential smoothing (ES) is not the only model. In fact, a family of models. ES models suppose that a time series has four components: seasonality, trend, level, and remainder. Bergmeir et al. (2016) recommended the bootstrap aggregation of ES methods. The bootstrap aggregation employs a Box–Cox transformation afterwards an a seasonal trend decomposition based on LOESS (LOcally Estimated Scatter-plot Smoother) (STL) to segregate the time series sub three part: remainder, seasonal and trend. The remainder is then bootstrapped via a moving block, and a new data is gathered via this bootstrapped residual part. Thereafter, an ensemble of ES models is calculated with the bootstrapped series.

Incorporating other types of model specifications and using different accuracy measures will surely increase the Ata method’s performance. Like other approaches, the method can also benefit from certain transformations and decompositions of other types of more involved combinations, outlier detection, and other more complicated model selection strategies. The fact that these simple selection and combination strategies can perform better than existing methods is fascinating, and this further strengthens the idea that simplicity is indeed a prerequisite for forecasting accuracy.

Several decomposition techniques are used before applying the Ata method for selecting an optimized model. The high performance of these combined models is indicated with an empirical practice. The Ata method is analyzed in contrast practically with the most well-known forecasting techniques based on ES and ARIMA in accordance with its predictive performance on the M3 (Makridakis and Hibon 2000) and M4-Competitions (Makridakis et al. 2018) data set and is illustrated to outperform its contestants.

Over the past few years, the preliminary research on ES (Brown 1959; Pegels 1969; Gardner and McKenzie 1985) expanded to an approach based on a model so that there are 30 potential ES models for various types of trend, seasonality, and errors. The well-known of these are the simple ES, Holt’s linear trend model, and Holt-Winter’s model. Then, Gardner and McKenzie (1985) proposed damped trend model to help deal with overtrending. The reputation and universality of ES can also be attributed to its proven record against more sophisticated techniques (Makridakis et al. 1984; Makridakis and Hibon 2000; Koning et al. 2005). The forecast package (Hyndman et al. 2020) in the programming language R (R Core Team 2016) means that a fully automated software for fitting ETS models is available. These have led to a broadly appropriate ES modelling background, and with the use of latterly developed software packages, these ES models handle seasonality, trend, and other attributes of series without any human intervention (Hyndman et al. 2002, 2008; Hyndman and Athanasopoulos 2019).

The Theta method (Assimakopoulos and Nikolopoulos 2000) was introduced as a new univariate forecasting method which is similar to a simple ES model with drift, and its performance in terms of forecasting accuracy was prominent in M3-Competition. As confirmed once again in Assimakopoulos and Nikolopoulos (2000), it is well known that combining forecasts (Bates and Granger 1969; Clemen 1989) under certain circumstances improves forecasting accuracy (Makridakis et al. 1982; Makridakis and Winkler 1983; Armstrong 1989, 2001). Due to this, the research focuses on transformations, decompositions, rules, and combinations of ES and ARIMA (a few examples are (Clemen 1989; Cleveland et al. 1990; Adya et al. 2000) to improve the forecasting performance rather than suggesting new forecasting methods.

Several other studies that are based on automatic forecasting procedures exist. Particularly for seasonal time series, the forecast package offers the TBATS model (Livera et al. 2011). TBATS uses a parsimonious trigonometric representation of seasonality instead of conventional seasonal indices and also incorporates ARMA errors. In addition, the function also automatically performs Box-Cox transformation of the time series if necessary.

This study introduces ATAforecasting (available from the Comprehensive R Archive Net- work at https://cran.r-project.org/package=ATAforecasting), a software application for R which performs a novel decomposition and power transformation-based approaches to time series forecasting using Ata method without any academic expertise. To sum up, the ATAforecasting package (Taylan et al. 2021a) provides a novel R interface for researchers interested in automatic time series analysis and students and academics who teach courses related to univariate time series analysis topics. There are main 13 functions available in the ATAforecasting package; see Table 1. We are going to describe all of them as we go on to explain the theoretical procedure. The rest of the paper is organized as follows. Section 2 presents a novel forecasting approach using the Ata method, gives an overview of the main estimation methods of the Ata method, and provides some technical details about the ATAforecasting package. Section 3 illustrates M-forecasting Competition dataset examples showing the package’s functionality. Section 4 contains some concluding remarks.

Table 1: A summary of the functions available in the ATAforecasting package.
Function Description
ATA Time series analysis and forecasting using the Ata method.
ATA.Core The core algorithm of the Ata method.
ATA.Forecast Produces forecasts from the output of ATA function.
ATA.Accuracy Computes fitting and forecasting accuracy measures.
ATA.Transform Computes transformed data using power transformation techniques.
ATA.BackTransform Computes back transformed data using power transformation techniques.
ATA.BoxCoxAttr Assigns attributes set for unit root and seasonality tests.
ATA.Seasonality Tests seasonality.
ATA.Decomposition Decomposes a time series into seasonal, trend, and irregular.
ATA.SeasAttr Assigns attributes set for unit root and seasonality tests.
ATA.Plot Specialized plot function of the output of ATA function.
ATA.Print Specialized print screen function of the output of ATA function.
ATA.CI Confidence interval function for the Ata method forecasts.

2 Methodology

The objective of this study is to introduce a new decomposition-based approach to time series forecasting with the Ata method to provide the automation and the optimization of the Ata method which is an innovative and accurate univariate time series analysis method without any expertise of the R program. Specifically, we propose an analytical methodology of time series method with ATAforecasting R package, as it combines several stationarities and seasonality tests, Box–Cox transformations, seasonal decomposition techniques with the Ata method. We merge the various preceding concepts to attain a robust and broadly practicable automatic forecasting algorithm. The methodology involves 2 alternative algorithms with 6 steps as described and summarized below:

First Algorithm in Figure 1a

Second Algorithm in Figure 1b

To summarize, the ATAforecasting procedure is given in Figure 1. As default, initially, the selected power family transformation is implemented, and the series are decomposed into the seasonal part and trend + remainder part, using the selected decomposition technique. Then, the Ata method is applied to the trend + remainder part. The components are added together again, and the selected power family transformation is inverted.

graphic without alt text graphic without alt text
  1. First Algorithm
  1. Second Algorithm
Figure 1: Algorithms of ATAforecasting procedure.

Power transformation family for ATAforecasting package

Traditional statistical procedures often assume that the data is homoscedastic and normally distributed. Despite its apparent restrictions, the logarithmic transformation has been used mostly when data violates these assumptions. The purpose of a particular transformation for better fitting is additivity, convergence to normality, stationarity, linearity, reduction of skewness, stabilizing variance. These purposes, which may even be inconsistent, are quite significant as it is just under such assumptions that particular statistical methods are relevant. Generally, logarithmic transformations almost stabilize the variance for time series consisting of large values. Some of the problems that arise when implementing a specific transformation are argued in different settings by Sakia (1992), Staniswalis et al. (1993), Quiroz et al. (1996), Yeo and Johnson (2000), Chen et al. (2002), Mu and He (2007), Horowitz (2009) and Meintanis and Stupfler (2015). There are many proposed methods of transformation and a large amount of research in the literature. Sakia (1992) provided a detailed and extensive review of the Box–Cox ((Box and Cox 1964)) and some alternative versions. Different methodology recommended for choosing the appropriate value of transformation parameters based on maximizing the likelihood function ((Box and Cox 1964)) or alternatively, Kullback-Leibler information-based method (Hernandez and Johnson 1980), robust adaptive method (Carroll 1980) and a method based on Kendall’s rank correlation, (Han 1987).

A chiefly used algorithm of the Box–Cox family is the logarithm transformation, which is convenient for multiplicative process data. Moreover, the asymptotic variance of a time series can be stabilized by the log-transformation. A shift parameter was additionally proposed to apply the log transformations more responsive and handy. The parameterizations of the shift parameter depend on knowledge of the data e.g., data range, data distribution, so user intervention is usually required. ATAforecasting package automates the selection of shift parameter, which is an important contribution of automatic times series forecasting.

Selected transformation functions included in the ATAforecasting R package provide the applicability of different types of transformation techniques for the data to which the Ata method will be applied. The ATA.Transform function works with many different types of inputs. Many power transformation methods are available to stabilize linearity and variance. In this package, power transformation family is consist of "Box–Cox", "Sqrt", "Reciprocal", "Log", "NegLog", "Modulus", "Bickel–Doksum", "Manly", "Dual", "Yeo–Johnson", "GPower", "GLog". If the transformation process needs a shift parameter, ATA.Transform will calculate the required shift parameter automatically.

The ATA.BoxCoxAttr function

Since the main ATA function and ATA.Transform are designed by some attributes of Box–Cox power transformation family, we provide the user with the function ATA.BoxCoxAttr.

The R function ATA.BoxCoxAttr can be utilized with the following code,

    ATA.BoxCoxAttr(bcMethod = "loglik", bcLower = 0
    , bcUpper = 1,  bcBiasAdj = FALSE),

and makes use of four parameters. These are

The ATA.Transform function

The main function of power transformations, the ATA.Transform, can be called with

    ATA.Transform(X, tMethod = "Box_Cox", tLambda
    , tShift = 0, bcMethod = "loglik", bcLower = 0, bcUpper = 1)

and it makes use of seven parameters and returns three outputs. The inputs are

The outputs are

To apply this algorithm to an example in the tsibbledata package "aus_retail", monthly retail turnover (in million AUD) in Australian states from April 1982 to December 2018, we use the following commands.

    library(tsibbledata)
    library(lubridate)
    library(dplyr)
    library(tsbox)
    library(ATAforecasting)
    train_data <- aus_retail %>% filter(State == "New South Wales"
     , Industry == "Department stores"
     , `Series ID`== "A3349790V")
    train_data <- tsbox::ts_ts(train_data)
    bc_attr_set <- ATA.BoxCoxAttr(bcMethod = "loglik", bcLower = 0, bcUpper = 1)
    fit_bc <- ATA(train_data, seasonal.type = "M", model.type = "A"
    , seasonal.test = TRUE, seasonal.model = "decomp", plot.out = TRUE
    , transform.method = "Box_Cox", transform.order = "before"
    , transform.attr = bc_attr_set, negative.forecast = FALSE)

Seasonality for ATAforecasting package

Seasonality is a well-known phenomenon observed in many economic time series. Seasonal decomposition, which is the first stage of a time series modeling, is also the first stage of the Ata method. The performance of the Ata method has been improved after the seasonal decomposition.

Specifically, our proposed methodology to identify seasonality in time series is as follows. After or before implementing a Box—Cox transformation (if necessary) to the data, the data is decomposed into remainder, seasonal, and trend components. The trend and remainder components are then forecasted via the Ata method, the seasonal component is added back in, and the Box—Cox transformation is inverted. Then, point forecasts are calculated using each of the different models, and/or the resulting forecasts are able to be combined.

In previous studies, the classical decomposition method is much used after the seasonality test. With this package, stl, stlplus, tbats, and stR decomposition techniques are also available choices by the ATAforecasting package, which can be chosen with only one or multiple.

Seasonality for ATAforecasting package enables estimating all of the below components and specifications. The main functions of seasonality in the package are the following

Three seasonality diagnostics methods are able to be applied in the package.

The ATA.SeasAttr function

This function is a class of seasonality tests using corrgram.test from ATAforecasting package, ndiffs and nsdiffs functions from forecast package. Also, ndiffs and nsdiffs functions have been modified according to different unit root testing packages. Please review manual and vignette documents of the latest forecast package. According to forecast package, ndiffs and nsdiffs functions estimate the number of differences requisite to ensure stationary of a given time series.

ndiffs employs unit root tests to define required number of differences for time series to be ensured trend stationary. nsdiffs employs seasonal unit root tests to define required number of seasonal differences for time series to be ensured trend stationary.

The ATA.SeasAttr function works with many different types of inputs. The inputs are below.

Unit root tests

Unit root tests for stationarity have compatibility in almost every practical time series analysis. Choosing which unit root procedure to employ is an issue of active interest. In this study, we implement the three widely used unit root tests. In accordance with past research, the selected unit root tests occasionally disagree in choosing the convenient order of integration for a given data. The following literature shows the basic features of unit root tests. Users who demand details should consult the original resources and standard references (see, for example, (Davidson and MacKinnon 1993; Hamilton 1994; Hayashi 2000)).

In the ATAforecasting package, the following unit roots methods are able to be applied.

The ATA.SeasAttr function for unit root tests

Since the main ATA function and ATA.Seasonality are designed by some attributes of unit root tests, we provide the user with the function ATA.SeasAttr.

For the main function ATA, the attributes of unit root test can be accessed with

    ATA.SeasAttr(uroot.pkg = "tseries", uroot.test = "kpss"
     , uroot.type = "trend", uroot.alpha = 0.05)

The following code uses the unit root test approach to search trend component before the seasonality test of the data in the context of the Ata method.

    seas_attr_set <- ATA.SeasAttr(suroot.test = "correlogram"
    , corrgram.tcrit = 1.28, uroot.pkg = "tseries"
    , uroot.test = "kpss", uroot.type = "trend"
    , uroot.alpha = 0.05)
    
    fit_seas <- ATA(train_data, model.type = "A", seasonal.type = "M"
    , seasonal.test = TRUE, seasonal.model = "tbats", plot.out = TRUE
    , seasonal.test.attr = seas_attr_set, negative.forecast = FALSE)

Seasonality tests

There are numerous studies relating seasonality and unit root. One of these studies uses autocorrelogram. Autocorrelation function and partial autocorrelation function are useful qualitative tools to estimate the existence of autocorrelation at individual lags. The Ljung-Box Q-test is a more quantitative method to test autocorrelation at multiple lags jointly. Other techniques generally use unit root tests. Hylleberg et al. (1990) improved unit root tests in linear time series regarding seasonality and studied with different models, including different combinations of seasonal, trend, remainder, and constant parts. Their purpose is to improve a testing process that will determine what class of seasonality is accountable for the seasonality in a time series process. There exist more studies for testing seasonal unit roots, such as (Ljung and Box 1978), (Dickey et al. 1984), (Osborn et al. 1988), and (Wang et al. 2006).

In the ATAforecasting package, the following methods are able to applied.

The ATA.SeasAttr function for seasonal unit root test

Since the main ATA function and ATA.Seasonality are designed by some attributes of seasonality tests, we provide the user with the function ATA.SeasAttr.

For the main function ATA, the attributes of seasonality test can be accessed with

    seas_attr_set <- ATA.SeasAttr(suroot.test = "correlogram"
      , corrgram.tcrit = 1.28)
      
    seas_attr_set <- ATA.SeasAttr(suroot.test = "ocsb", suroot.alpha = 0.05)

An example of the seasonality test’s call is the following

    seas_attr_set <- ATA.SeasAttr(suroot.test = "ocsb", suroot.alpha = 0.05
    , uroot.pkg = "tseries", uroot.test = "adf", uroot.type = "trend"
    , uroot.alpha = 0.05)
    
    fit_seas <- ATA(train_data, model.type = "A", seasonal.type = "M"
    , seasonal.test = TRUE, seasonal.model = "stl", plot.out = TRUE
    , seasonal.test.attr = seas_attr_set, negative.forecast = FALSE)

The ATA.Seasonality function for Seasonality Tests

The ATAforecasting’s seasonality diagnostics described before in this paper are implemented into a function named ATA.Seasonality that can calculate all of them respectively. The function syntax is

    ATA.Seasonality(input = train_data, ppy = frequency(train_data)
     , attr_set = seas_attr_set)

The ATA.Seasonality function works with many different types of inputs. The inputs are below.

Here is an another simple example, applying ATA.SeasAttr and ATA.Seasonality to the M3 data:

    library(ATAforecasting)
    library(Mcomp)
    seas_attr_set <- ATA.SeasAttr(suroot.test = "correlogram"
     , corrgram.tcrit = 1.28, uroot.pkg="tseries"
     , uroot.test="adf", uroot.type = "trend"
     , uroot.alpha = 0.05, uroot.maxd = 1)
    
    is.season <- ATA.Seasonality(M3[[1899]]$x
     , frequency(M3[[1899]]$x)
     , seas_attr_set)

Seasonal decomposition

A substantial aim in time series analysis is the decomposition of a time series into latent parts that can be incorporated with dissimilar versions of temporal variations. Persons (1919) was the first to state the assumptions of latent parts particularly. Persons indicated that time series was constituted of four types of fluctuations (Dagum and Bianconcini 2016): residual variations, seasonal movement, secular trend, and cyclical movements. Further, research in that direction included Copeland (1915) and Persons (1919), who introduced for extracting the seasonal component until Macaulay (1931) suggested a technique which turned into "classical" in the log run.

Macaulay (1931) developed a computer program which is significantly simplified the calculations (Shiskin 1957). Extensively used various techniques and features which have such as ARIMA extensions, regressors, calendar effects, robustness, and extensive diagnostics in literature are X-11 (Shiskin et al. 1967), X-11-ARIMA (Dagum 1988), X-12-ARIMA (Findley et al. 1998) and X-13ARIMA-SEATS (Monsell et al. 2003; Findley 2005; Monsell 2007). X-13ARIMA-SEATS contains a type of the TRAMO/SEATS procedure which was improved by the Bank of Spain for seasonal adjustment.

Cleveland et al. (1990) recommended a different approach and developed STL (Seasonal Trend decomposition using LOESS) based on local regression familiar as moving regression which is a generalization of moving average and polynomial regression. LOESS is a connected nonparametric method that assembles multiple regression models in a metamodel based on the k-nearest neighbor. Burman (1980) discussed plenty of seasonal adjustment techniques and remarked that all but one were ad hoc techniques. Since this study, several model-based methods for seasonal decomposition have been evolved, including the TRAMO/SEATS procedure, assorted structural time series models (Harvey 1990; Commandeur et al. 2011) and the BATS and TBATS models of (Livera et al. 2011).

Conventionally, the four variations suppose to be mutually independent of one another and signify by means of an additive decomposition model. If there is dependency among the hidden parts, this relation is signified via a multiplicative decomposition model. In some cases, combined additive and multiplicative models can be employed. See (Dagum and Bianconcini 2016) for further details.

The ATA.Decomposition function for seasonality

Automatic seasonal decomposition for the ATA method is called ATA.Decomposition function in the ATAforecasting package. The function returns seasonally adjusted data constructed by removing the seasonal component. The methodology is fully automatic. The ATA.Decomposition function works with many different types of inputs. The inputs are below.

ATA.Decomposition function returns four outputs. The outputs are below.

As an example, let us compute seasonal decomposition on the real life tsibbledata dataset shown in the following seven figures (Figures 2, 3, 4, 5, 6, and 7).

    best_fit_seas <- ATA(train_data, start.phi = 0.80, end.phi = 0.99
    , size.phi = 0.01, train_test_split = 18, seasonal.test = TRUE
    , seasonal.model = c("decomp","stl", "stlplus","tbats", "stR")
    , negative.forecast = FALSE, plot.out = TRUE)
    best_fit_seas$is.season
graphic without alt text
Figure 2: Best forecasts for the Australian Retail Turnover using ATA seasonal trended methods.
    library(ggplot2)
    autoplot(train_data) +
    autolayer(fit_decomp$forecast, series="ATA-decomp") +
    autolayer(fit_stl$forecast, series="ATA-stl") +
    autolayer(fit_stlplus$forecast, series="ATA-stlplus") +
    autolayer(fit_stR$forecast, series="ATA-stR") +
    autolayer(fit_tbats$forecast, series="ATA-tbats") +
    ggtitle("Forecasts from ATA seasonal trended methods") + xlab("Year") +
    ylab("Monthly Retail Trade Turnover of Australian States") +
    guides(colour=guide_legend(title="Forecast"))
graphic without alt text
Figure 3: Forecasts from ATA seasonal trended methods.

There are five different techniques for seasonal decomposition in the package. We use the following techniques

    fit_decomp <- ATA(train_data, seasonal.test = TRUE
      , seasonal.model = "decomp" , negative.forecast = FALSE)
graphic without alt text
Figure 4: The Ata method with classical decomposition.
    fit_stl <- ATA(train_data,  model.type = "A", seasonal.type = "M"
    , seasonal.test = TRUE, seasonal.model = "stl", negative.forecast = FALSE)
graphic without alt text
Figure 5: The Ata method with STL decomposition.
    fit_stlplus <- ATA(train_data, model.type = "A", seasonal.type = "M"
    , seasonal.test = TRUE, seasonal.model = "stlplus", negative.forecast = FALSE)
graphic without alt text
Figure 6: The Ata method with STLplus decomposition.
    fit_tbats <- ATA(train_data, seasonal.test = TRUE, seasonal.model = "tbats"
    , level.fixed = TRUE, negative.forecast = FALSE, plot.out = TRUE)
graphic without alt text
Figure 7: The Ata method with TBATS decomposition.
    fit_stR <- ATA(train_data, seasonal.test = TRUE, seasonal.model = "stR"
    , negative.forecast = FALSE, plot.out = TRUE)
graphic without alt text
Figure 8: The Ata method with stR decomposition.

Univariate time series forecasting with the Ata method

Ata method is an innovative new forecasting technique where the forms of the models are similar to ES models. Still, the smoothing parameters depend on the sample size, are optimized on a discrete space. Initialization is both easier as it is done simultaneously when the parameters are optimized and is less influential since the weights assigned to initial values approach zero quickly. The Ata method can easily be applied to all time series settings and provides better forecasting performance due to its flexibility. ATA-damped is a version of the Ata method that mainly focuses on the trend component, allowing it to range both in magnitude and form.

For a time series \(\{y_{1},\ldots,y_{n}\}\), the Ata method can be given in additive form as below:

\[\label{ata1} l_t=\left(\frac{p}{t}\right) y_t+\left(\frac{t-p}{t}\right)(l_{t-1}+\phi b_{t-1}), \tag{1}\]

\[\label{ata2} b_t=\left(\frac{q}{t}\right)(l_{t}-l_{t-1})+\left(\frac{t-q}{t}\right) \left(\phi b_{t-1}\right), \tag{2}\]

where \(p\) is the smoothing parameter for level, \(q\) is the smoothing parameter for trend, \(\phi\) is the dampening parameter and \(l_t=y_t\) for \(t \le p\), \(b_{t}=y_{t}-y_{t-1}\) for \(t \le q\), \(b_1=0\), \(p\in\{1,2,\ldots,n\},\) \(q\in \{0,1,2,\ldots,p\}\), \(\phi \in (0,1]\), and \(p \ge q\). Then, the \(h\) step ahead forecasts can be obtained by:

\[\label{ata3} \hat{y}_{t+h|t}=l_t+\left( \phi+\phi^2+ \ldots +\phi^h \right) b_t. \tag{3}\]

Similarly for a time series \(\{y_{1},\ldots,y_{n}\}\), the Ata method can be given in multiplicative form as below:

\[\label{ata4} l_t=\left(\frac{p}{t}\right) y_t+\left(\frac{t-p}{t}\right) \left(l_{t-1} b^{\phi}_{t-1}\right), \tag{4}\]

\[\label{ata5} b_t=\left(\frac{q}{t}\right) \left( \frac{l_{t}}{l_{t-1}} \right)+\left(\frac{t-q}{t}\right) \left( b^{\phi}_{t-1}\right), \tag{5}\]

where, again, \(p\) is the smoothing parameter for level, \(q\) is the smoothing parameter for trend, \(\phi\) is the dampening parameter and \(l_t=y_t\) for \(t \le p\), \(b_t=\frac {y_t}{y_{t-1}}\) for \(t \le q\), \(b_1=1\), \(p\in\{1,2,\ldots,n\},\) \(q\in \{0,1,2,\ldots,p\}\), \(\phi \in (0,1]\), and \(p \ge q\). Then, the \(h\) step ahead forecasts can be obtained by:

\[\label{ata6} \hat{y}_{t+h|t}=l_t+b^{\left( \phi+\phi^2+ \ldots +\phi^h \right)}_t. \tag{6}\]

Since both versions of the method require three parameters, we will distinguish between them by using the notation \(ATA_{add} (p,q, \phi)\) for the additive form and \(ATA_{mult} (p,q, \phi)\) for the multiplicative form.

Notice that when \(q=0\), both forms of ATA are reduced to the simple form \(ATA(p,0, \phi)\) which can be written as:

\[\label{ata7} l_t=\left(\frac{p}{t}\right) y_t+\left(\frac{t-p}{t}\right) l_{t-1}, \tag{7}\]

where \(p\in \{1,2,\ldots,n\}\) and \(l_t=y_t\) for \(t \le p\). Forecasts then can be obtained by \(\hat{y}_{t+h|t}=l_t\).

When \(q\neq 0\) and \(\phi=1\), the additive and multiplicative forms of ATA are reduced to the trended versions \(ATA_{add} (p,q,1)\) and \(ATA_{mult} (p,q,1)\), which are given below, respectively:

\[\label{ata8} l_t=\left(\frac{p}{t}\right) y_t+\left(\frac{t-p}{t}\right)(l_{t-1}+ b_{t-1}), \tag{8}\]

\[\label{ata9} b_t=\left(\frac{q}{t}\right)(l_{t}-l_{t-1})+\left(\frac{t-q}{t}\right) \left( b_{t-1}\right), \tag{9}\]

\[\label{ata10} \hat{y}_{t+h|t}=l_t+h b_t, \tag{10}\]

and

\[\label{ata11} l_t=\left(\frac{p}{t}\right) y_t+\left(\frac{t-p}{t}\right) \left(l_{t-1} b_{t-1}\right), \tag{11}\]

\[\label{ata12} b_t=\left(\frac{q}{t}\right) \left( \frac{l_{t}}{l_{t-1}} \right)+\left(\frac{t-q}{t}\right) \left( b_{t-1}\right), \tag{12}\]

\[\label{ata13} \hat{y}_{t+h|t}=l_t+b^{h}_t. \tag{13}\]

To sum up, ATA can be given in 7 forms, namely the additive damped form \(ATA_{add} (p,q,\phi)\) (equations (1-3)), multiplicative damped form \(ATA_{mult} (p,q,\phi)\) (equations (4-6)), simple form
\(ATA(p,0,\phi)\) (equation (7)), additive trend form \(ATA_{add} (p,q,1)\) (equations (8-10)), and finally multiplicative trend form \(ATA_{mult} (p,q,1)\) (equations (11-13)).

Another distinction can be made based on the parameter optimization process used for these forms. Unless otherwise stated, the parameter values that minimized the in-sample one step ahead using selected accuracy measures such as sMAPE, MASE, or OWA are used as optimum values, and optimization is carried out for all the parameters simultaneously. However, in some cases, we realized that fixing the smoothing parameter for the level and then optimizing the trend parameter can be beneficial. We call these the “level-fixed” versions of ATA. The optimization is carried out for these models as follows:

  1. Find the value of \(p\) that minimized the in-sample one step ahead \(sMAPE\) for \(q = 0\) and \(\phi = 1\). Call this value \(p^*\).

  2. Holding \(p = p^*\) fixed optimize \(q\) (and \(\phi\) if needed) ,again, by minimizing the in-sample one step ahead \(sMAPE\).

Models where the parameter optimization is carried out using the algorithm in 1. and 2. will receive the superscript (lf) an abbreviation for “level-fixed” such as \(ATA_{add}^{lf} (p,q,\phi)\) or \(ATA_{mult}^{lf} (p,q,\phi)\).

Obtaining prediction interval

For forecasting horizon \(h\), the prediction interval is obtained by:

\[y_{n+h|n} \pm C_h,\]

where \(C_h=\sqrt{h} Z_{\alpha/2} S_e\), \(Z_{\alpha/2}\) is the Normal deviate corresponding to \((1-\alpha)\%\) confidence interval, and \(S_e\) is the standard deviation of the one step ahead errors of model fitting. If any lower bounds are found to be negative, they are set equal to zero.

Model selection

There are plenty of measures and criteria available in the forecasting literature for interpreting the achievements and accuracy of forecasting methods. In the M-Competitions, some of these measures were employed without any obvious consensus as to the pros and cons of each.

A forecast error is a difference between an observed value and its forecast. Forecast errors are different from residuals in two aspects. Firstly, residuals are computed on the training set, while forecast errors are computed on the test set. Secondly, residuals are based on one-step-ahead forecasts, while forecast errors can contain multi-step forecasts (Hyndman and Athanasopoulos 2019).

Let \(Y_{t}\) indicates the observation at time \(t\), and \(F_{t}\) indicates the forecast of \(Y_{t}\). The forecast error \(e_{t} = Y_{t} - F_{t}\) is calculated. The forecasts are calculated from a common base time and are of varying forecast horizons. Hence, we calculate out of sample forecasts \(F_{n + 1} , \dots , \quad F_{n + m}\) based on data from times \(t = 1 , \ldots , n\). Optionally, the forecasts can be from varying base times and be of a coherent forecast horizon. Namely, we can calculate forecasts \(F_{1 + h} , \dots , F_{m + h}\) where each \(F_{j + h}\) is based on data from times \(t = 1 , \ldots , n\). The in-sample forecasts in the examples above were based on the second scenario with \(h = 1\). A third scenario shows up when we request to compare the accuracy of methods across many series at a forecast horizon. Then we calculate a single \(F_{n + h}\) based on data from times \(t = 1 , \ldots , n\) for each of m different series (Hyndman and Koehler 2006). In this study, we adapt M4 and prior M-Competitions’ accuracy measures pool.

Automatic forecasting

We unite the prior concepts to obtain a robust and widely appropriate automatic forecasting algorithm. The concept is summarized below.

  1. Identify and correct for seasonality in time series, respectively.

    • Detect stationarity and seasonality in time series.
    • Decompose time series.

  2. For the selected time series data, apply all models that are applicable, optimizing the parameters of the ATA model in each case.

  3. Select the best of the ATA models according to the selected accuracy measure (SMAPE is default for the ATAforecasting package).

  4. Generate point forecasts using the best model (with optimized parameters).

  5. Obtain prediction intervals for the best model.

ATAforecasting in practice

This section introduces an overview of how the package is structured.

This software enables both numerical and graphical outputs to be displayed for all methods described in the previous section. This software is intended to be used with the R statistical program (R Core Team 2016). Our package is composed of 13 functions that allow users to obtain estimates for all proposed methods. Details on the usage of the functions (described in Table 1) can be obtained with the corresponding help pages.

Returns ATA(p,q,\(\phi\)) applied to X, based on the modified simple ES as described in (Yapar 2018). The Ata method is a new univariate time series forecasting method that provides innovative solutions to issues faced during the initialization and optimization stages of existing methods. The ATA’s forecasting performance is superior to existing methods both in terms of easy implementation and accurate forecasting. It can be applied to non-seasonal or deseasonalized time series, where the deseasonalization can be performed via any preferred decomposition method. This methodology performed extremely well on the M3 and M4-Competition data.

Functions of ATAforecasting package

Many functions, including ATA, ATA.Forecast, ATA.Plot, ATA.Print, ATA.Accuracy, ATA.Seasonality, ATA.Transform, ATA.BackTransform produce output in the form of a ATAforecasting object (i.e., an object of class "ata"). This package needs some R packages for unit root tests, seasonal unit root tests, seasonal decompositions, M3 dataset, M4 dataset, and benchmark forecast models to work consistently across a range of forecasting models. These R package are Rcpp (Eddelbuettel et al. 2020a), RcppArmadillo (Eddelbuettel et al. 2020b), tseries (Trapletti and Hornik 2020), forecast (Hyndman et al. 2020), urca (Pfaff et al. 2016), uroot (López-de-Lacalle 2020), seasonal (Sax and Eddelbuettel 2020), stR (Dokumentov and Hyndman 2018), stlplus (Hafen 2016), xts (Ryan et al. 2020), timeSeries (Wuertz et al. 2020), TSA (Chan and Ripley 2020), Mcomp (Hyndman et al. 2018), M4Comp2018 (BenTaieb 2018).

Objects of class "ata" contain information about the forecasting method, the data used, the point forecasts obtained, prediction intervals, residuals, and fitted values. There are several functions designed to work with these objects, including ATA.Forecast, ATA.Accuracy, ATA.Plot andATA.Print.

Description of the ATA function

ATA function produces a ata object directly. If the first argument is of class ts (time series object) or msts (multi seasonal time series objects), it returns forecasts from the automatic ATA algorithm discussed in this chapter. The definition of ATA function is below.

ATA(X, Y = NULL, parP = NULL, parQ = NULL, parPHI = NULL
, start.phi = NULL, end.phi = NULL, size.phi = NULL
, model.type = NULL, seasonal.test = NULL, seasonal.model = NULL
, seasonal.period = NULL, seasonal.type = NULL, find.period = NULL
, seasonal.test.attr = NULL, accuracy.type = NULL
, level.fixed = FALSE, trend.fixed = FALSE, trend.search = FALSE
, initial.level = NULL, initial.trend = NULL, h = NULL
, train_test_split = NULL, holdout = FALSE
, holdout.adjustedP = TRUE, holdout.set_size = NULL
, transform.order = "before", transform.method = NULL
, transform.attr = NULL, lambda = NULL, shift = NULL
, ci.level = 95, negative.forecast = TRUE
, plot.out = TRUE, print.out = TRUE)

Inputs of ATA function

The ATA function works with many different types of inputs. It generally takes a time series data or time series model as its main argument, and produces forecasts appropriately. It always returns objects of class "ata".

If the first argument is of class ts or msts, it returns forecasts from the automatic ATA algorithm discussed in this chapter before.

Level Parameters :

Trend Parameters :

Seasonal Parameters :

Accuracy Parameters :

Transform Parameters :

Holdout Parameters :

Output of ATA function

Returns an object of class "ata", containing the generic access or functions ATA.Forecast, and ATA.Accuracy extracts the useful features of the value returned by "ata" and associated functions.

Here are quick start examples using "aus_retail" dataset monthly retail turnover (in million AUD) in Australian states from April 1982 to December 2018 in the tsibbledata package.

    library(tsibble)
    library(tsibbledata)
    library(lubridate)
    library(dplyr)
    library(tsbox)
    library(ATAforecasting)
    
    main_data <- aus_retail %>% 
        filter(State == "New South Wales", 
        Industry == "Department stores", 
        `Series ID`== "A3349790V") 
    train_data <- tsbox::ts_ts(train_data)
    test_data <- tail(train_data, 24)
    train_data <- window(train_data, start = 1983, end = 2016.917)
    ata_fit <- ATA(train_data, test_data, h=24)
    ata_fit$is.season
    unlist(ata_fit$accuracy$sMAPE)
    unlist(ata_fit$accuracy$sMAPE$inSample)
    unlist(ata_fit$accuracy$sMAPE$outSample)
    unlist(ata_fit$accuracy$fits)
    ata_fit$fitted
    ata_fit$forecast
    ata_fit$residuals

Here are some outputs for the above example from the ATAforecasting Package whose results are shown in Figure 9 and 10. 40 properties of the ATA module, including all results of the automatic forecasting using the Ata method are able to be obtained by using the "\(\$\)" command as shown in the above example.

graphic without alt text
Figure 9: Forecasts from automatic ATA seasonal damped trended methods.
graphic without alt text
Figure 10: The default model output from the automatic ATA seasonal damped trended methods.

Another sample data is Makridakis Competitions 2000 (also known as the M-Competitions) monthly data in the Mcomp package (Hyndman et al. 2018).

    atafit <- ATA(M3[[1899]]$x, M3[[1899]]$xx, parQ = 1, parPHI = 1
     , model.type = "A", seasonal.type = "M", seasonal.test = TRUE
     , seasonal.model = "decomp", level.fixed = FALSE, transform.method = "Box_Cox"
     , negative.forecast = FALSE)

Here are some outputs for the above example from the ATAforecasting Package. The results are shown in Figure 11.

graphic without alt text
Figure 11: Forecasts from the automatic ATA seasonal trended methods for M3 sample.

The object atafit is of class "ata" and contains all of the necessary information about the fitted model including model parameters, residuals, and so on. Printing the atafit object presents the main items of interest.

    ATA.Forecast(atafit, h = 18, ci.level = 99
    , negative.forecast = TRUE)

Some goodness-of-fit measures of forecast accuracy are obtained based on only the fitting data using ATA.Accuracy, we use the following commands.

    ATA.Accuracy(atafit)

Fable modeling wrappers for ATAforecasting

We also developed a wrapper software (called fable.ata (Taylan et al. 2021b) to add the Ata method into the fable ecosystems using the fabletools (O’Hara-Wild et al. 2021b) package, which provides tools, helpers, and data structures for developing algorithms for the fable ecosystems (O’Hara-Wild et al. 2021a). Here are the quick start examples using the "aus_retail" dataset.

    library(fable)
    library(fable.ata)
    
    fit <- aus_retail %>%
    filter(State %in% c("New South Wales", "Victoria"),
    Industry == "Department stores") %>%
    model(
      ets = ETS(Turnover),
      arima = ARIMA(Turnover),
      snaive = SNAIVE(Turnover),
      ata = AutoATA(Turnover~trend("M") + season(type="M",method="stR"))
     ) %>% 
    mutate(mixed = (ets + arima + snaive + ata) / 4)
    fc <- fit %>% forecast(h = 12)
    fc %>% autoplot(filter(aus_retail, year(Month) > 2010), level = NULL)

Here are some outputs for the above example from the fable ecosystem functions (fable and fable.ata packages). The results are shown in Figure 12 and Figure 13.

graphic without alt text
Figure 12: Forecasts from fable models for aus_retail dataset.
    
    fit %>%
    accuracy() %>%
    group_by(.model) %>%
    summarise(
        RMSE = mean(RMSE),
        MAE = mean(MAE),
        MASE = mean(MASE)
        ) %>%
    arrange(MASE)
graphic without alt text
Figure 13: Comparison of fable models accuracy measures.

Holdout forecasting

Using holdout samples is substantial implementation to fit a model where the epoch of fit is dissimilar to the epoch of assessment. According to this model evaluation procedure, the epoch of fit completes at any moment before the last observation, and the rest of the data are held out as a non-overlapping epoch of assessment. In concern with the epoch of fit, the holdout sample is an epoch in the future, used to compare the forecasting accuracy of model fits to past data.

The concept of a holdout sample is to split the in-sample data into two parts. The last few data points are taken out from the in-sample data. The leftover data is called the training set, and the removed data is called the validation set or holdout set. Assume k periods have been taken out as holdout samples from a total of T periods. The parameters are optimized by minimizing the fit accuracy measure for the first part of the data. After the parameters are optimized, for each model, computed multi-step forecasts over the period covered by the second part, or holdout sample. The models are then evaluated, comparing accuracy measures for these out-of-sample multi-step predictions of the holdout sample. The model whose out-of-sample predictions best fit the holdout sample is chosen. The selected model is refitted using all the data to get the final forecasting model.

The ATAforecasting package makes it easy to use the holdout sample method of model selection. The time range used to fit models and the time range used for model evaluation are able to independently controlled. To use holdout samples, the period of evaluation range to that last part of the data, and the period of fit range to the remainder of the data are set. The automatic model selection feature is able to be used to select the model whose multiperiod out-of-sample predictions best fit the holdout sample.

Now, a quick start example of how to call the holdout method in the package.

    ata_holdout <- ATA(train_data, test_data, h=24, holdout = TRUE
     , holdout.set_size = 24, holdout.adjustedP = TRUE  
     , seasonal.test = TRUE, seasonal.model = "decomp")

3 Applications

Ata method was proposed as an alternative to ES, and it is not a special case of it. The details on the method and how it helps solve some issues that ES suffers from can be found in Yapar et al. (2019), Yapar (2018), Yapar et al. (2018), Yapar et al. (2017). ATA can be adapted to all types of time series data and will always outperform its counter ES models.

There are many studies on the numerical and theoretical comparison of Box-Jenkins and ES methods. Several empirical studies have been published in turn by Reid (1969), Newbold and Granger (1974), Makridaki and Hibon (1979), Makridakis et al. (1982), Makridakis et al. (1993), Makridakis and Hibon (2000), Makridakis et al. (2018).

Efforts for better forecasting and the competitions in which the outcomes of these efforts are tested and measured will never cease. Better forecasting is crucial to every science and business field. The most important platforms in which the performance of the studies for accurate forecasting is measured are the M-Competitions (Hyndman 2020). The most recent of these competitions, M4, has ended (Makridakis et al. 2018). The aim of the M4-Competition, like the competitions held before it, was to learn how to improve the forecasting accuracy, and how such learning can be applied to advance the theory and practice of forecasting, and are there any new methods that could really make a difference.

M-Competitions are very important and prestigious platforms for forecasting researchers since they provide researchers and developers of new forecasting methods opportunities to test and prove themselves. Another benefit of these competitions is that they usually lead to both the destruction of many taboos known in the forecasting literature and the discovery of new methods that help increase forecasting accuracy. The M-Competition was established by Spyros G. Makridakis in 1982 in a paper that studied the post-sample accuracy of several time series forecasting methods (Makridakis et al. 1982). The number of series was increased to 1001, and the data were subdivided into various categories (micro, macro, industry, demography, finance, other). The participants tested the accuracy of 24 methods on 1001 series with various horizons which were six for yearly data, eight for quarterly data, and eighteen for monthly data. The competition’s goal was to explore how different procedures differ from each other and how information can be ensured that forecasters can make convenient choices under various conditions (Makridakis et al. 1984).

In Makridakis and Hibon (2000), the M3-Competition reports the reasons for conducting the competition and summarizes its outcomes. In the M3-Competition, 3003 series, composed of 6 different types of series and 4 different time intervals between successive observations. The three prior competitions have played a very major role in the forecasting literature. Their results ensured a basis for future forecasting research. Consequently, Makridakis et al. initiated the fourth competition. As per the Makridakis’ team, the goal of the M4-Competition is to further study the utility and accuracy of various forecasting methods. Thus, the categories and number of the series and the forecasting methods are increased.

The M4-Competition is the progression of three previous competitions that began more than 45 years ago, whose objective was to learn how to evolve forecasting accuracy and how such learning can be implemented to proceed with the theory and performance of forecasting.

The purpose of M4 was to replicate the consequences of the prior ones and expand them into three aspects:

  1. Substantially enhanced the number of series,

  2. Contained machine learning forecasting methods,

  3. Interpret both point forecasts and prediction intervals.

The some substantial outcomes of the M4-Competition are:

  1. 12 of the 17 most accurate methods were "combinations" of mostly statistical approaches.

  2. "hybrid" approach was a significant finding that use both statistical and machine learning features.

In the M4-Competition, the number of data from the previous M3-Competition (Makridakis and Hibon 2000) was increased from 3,000 to 100,000. There were numerous applications (248), but only 49 of the applicants were able to provide forecasts for the entire 100,000 series. With the addition of 10 benchmarks and 2 standard methods, 61 methods were considered (Makridakis et al. 2020). Only 17 out of 49 valid applications outperformed the benchmark set by the competition committee. Of these 17 successful methods, 12 are combinations of known statistical methods obtained by using different weighting techniques.

The M3-Competition data set consists of 645 yearly, 756 quarterly, 1428 monthly, and 174 other series. The M4-Competition data set consists of 23000 yearly, 24000 quarterly, 48000 monthly, 359 weekly, 4227 daily, and 414 hourly series. The original data sets, as well as the forecasts of the methods that participated in the competitions, are available in the R packages Mcomp (Hyndman et al. 2018) and M4comp2018 (BenTaieb 2018).

In order to test and to apply this approach’s forecasting performance on real data and compare it to the benchmarks and especially counter ES models, forecasts obtained from five versions of it given the shortcode numbers Model-M3 and Model-M4 are fitted to the M3 and M4 competitions.

Therefore, in this implementation, predetermined model parameters as defined in the following list items are used to obtain accurate forecasts. Results from seven different applications of the Ata method will be considered here.

  1. \(ATA(p,0,1)\) is an alternative to SES method where \(p\) is the optimum value for \(q=0\) with fixed damped trend (\(\phi=1\)), and is where a simple model selection of the two models in \(ATA^{add}(p,0,1)\) and \(ATA^{mult}(p,0,1)\) is carried out based on selected in-sample accuracy measure.

        # --- Code for Makridakis Competition 2018 
    # --- (M4 Forecasting Competition) Dataset.

    # Load packages for creating plots
    library(ATAforecasting)
    library(M4comp2018)

    fit1 <- ATA(M4[[1]]$x, M4[[1]]$xx, h = M4[[1]]$h, parQ = 0
    , parPHI = 1 , seasonal.test = TRUE
    , seasonal.model = "decomp", accuracy.type = "sMAPE"
    , negative.forecast = FALSE)

  2. \(ATA^{add}(p,1,1)\) where p is optimized for \(q=1\) with fixed damped trend (\(\phi=1\))

        fit2 <- ATA(M4[[1]]$x, M4[[1]]$xx, h = M4[[1]]$h, parQ = 1
    , parPHI = 1, seasonal.test = TRUE, seasonal.model = "decomp"
    , model.type = "A", accuracy.type = "sMAPE"
    , negative.forecast = FALSE)

  3. \(ATA-comb\) where a simple average of the forecasts from the two models in (1) and (2) is used as a forecast.

        fit3 <- (fit1 + fit2) / 2

  4. \(ATA^{add}(p,1,\phi)\) is an alternative to damped trend method where \(q\) is optimized for \(p=p^*\) with damped trend.

        fit4 <- ATA(M4[[1]]$x, M4[[1]]$xx, h = M4[[1]]$h, parQ = 1
    , start.phi = 0.80, end.phi = 1, size.phi = 0.01
    , seasonal.test = TRUE, seasonal.model = "decomp"
    , model.type = "A", accuracy.type = "sMAPE"
    , negative.forecast = FALSE)

Model encoded by Model-M4 fits the \(ATA^{add}(p,1,\phi)\) to the yearly data sets and uses the \(ATA-comb\), a simple average of the forecasts obtained from the models \(ATA^{add}(p,1,1)\) and \(ATA(p,0,1)\), for the other data sets in M4-Competitions. Model-M3 fits \(ATA^{add}(p,0,1)\), \(ATA(p,0,1)\), and uses the \(ATA-comb\) for the data sets in M3-Competitions.

Table 2: Average forecasting errors for various data types and overall ranks with respect to sMAPE.
Team Method Type Yearly Quarterly Monthly Weekly Daily Hourly Total Rank
Smyl Hybrid 13.176 9.679 12.126 7.817 3.170 9.328 11.374 1
Montero-Manso, et al. Combination (S & ML) 13.528 9.733 12.639 7.625 3.097 11.506 11.720 3
Pawlikowski, et al. Combination (S) 13.943 9.796 12.747 6.919 2.452 9.611 11.845 5
Jaganathan. & Prakash Combination (S & ML) 13.712 9.809 12.487 6.814 3.037 9.934 11.695 2
Fiorucci & Louzada Combination (S) 13.673 9.816 12.737 8.627 2.985 15.563 11.836 4
Petropoulos & Svetunkov Combination (S) 13.669 9.800 12.888 6.726 2.995 13.167 11.887 6
Shaub Combination (S) 13.679 10.378 12.839 7.818 3.222 13.466 12.020 9
Legaki & Koutsouri Statistical 13.366 10.155 13.002 9.148 3.041 17.567 11.986 8
Doornik, et al. Combination (S) 13.910 10.000 12.780 6.728 3.053 8.913 11.924 7
Pedregal, et al. Combination (S) 13.821 10.093 13.151 8.989 3.026 9.765 12.114 13
Model-M4 Statistical 13.930 10.292 12.936 8.540 3.095 12.851 12.098 11
Spiliotis & Assimakopoulos Statistical 13.804 10.128 13.142 8.990 3.027 17.756 12.148 15
Roubinchtein Combination (S) 14.445 10.172 12.911 8.435 3.270 12.871 12.183 17
Ibrahim Statistical 13.677 10.089 13.321 9.089 3.071 18.093 12.198 18
Tartu M4 seminar Combination (S & ML) 14.096 11.109 13.290 8.513 2.852 13.851 12.496 23
Waheeb Combination (S) 14.783 10.059 12.770 7.076 2.997 12.047 12.146 14
Darin & Stellwagen Statistical 14.663 10.155 13.058 6.582 3.077 11.683 12.279 19
Dantas & Cyrino Oliveira Combination (S) 14.746 10.254 13.462 8.873 3.245 16.941 12.553 25
The M4 Team (Theta) Statistical 14.593 10.311 13.002 9.093 3.053 18.138 12.309 20
The M4 Team (Com) Statistical 14.848 10.175 13.434 8.944 2.980 22.053 12.555 27
The M4 Team (Arima) Statistical 15.168 10.431 13.443 8.653 3.193 12.045 12.661 29
The M4 Team (Damped) Statistical 15.198 10.237 13.473 8.866 3.064 19.265 12.661 30
The M4 Team (ETS) Statistical 15.356 10.291 13.525 8.727 3.046 17.307 12.725 31
The M4 Team (Holt) Statistical 16.354 10.907 14.812 9.708 3.066 29.249 13.775 43
The M4 Team (SES) Statistical 16.396 10.600 13.618 9.012 3.045 18.094 13.087 37

The forecasting performance of the Model-M4 that competed in the M4-Competition are given in the following three tables (Tables 2, 3, and 4) with respect to the error criteria sMAPE, MASE, and OWA, respectively.

According to sMAPE (Table 2), the Model-M4 of the ATA models is ranked in the first 20. The Model-M4 performs much better than ETS despite the fact that only sMAPE was used for optimizing the ATA approaches for the in-sample data, and these approaches only considered limited numbers of candidate models to choose from, unlike ETS.

Table 3: Average forecasting errors for various data types and overall ranks with respect to MASE.
Team Method Type Yearly Quarterly Monthly Weekly Daily Hourly Total Rank
Smyl Hybrid 2.980 1.118 0.884 2.356 3.446 0.893 1.536 1
Montero-Manso, et al. Combination (S & ML) 3.060 1.111 0.893 2.108 3.344 0.819 1.551 3
Pawlikowski, et al. Combination (S) 3.130 1.125 0.905 2.158 2.642 0.873 1.547 2
Jaganathan. & Prakash Combination (S & ML) 3.126 1.135 0.895 2.350 3.258 0.976 1.571 6
Fiorucci & Louzada Combination (S) 3.046 1.122 0.907 2.368 3.194 1.203 1.554 4
Petropoulos & Svetunkov Combination (S) 3.082 1.118 0.913 2.133 3.229 1.458 1.565 5
Shaub Combination (S) 3.038 1.198 0.929 2.947 3.479 1.372 1.595 7
Legaki & Koutsouri Statistical 3.009 1.198 0.966 2.601 3.254 2.557 1.601 8
Doornik, et al. Combination (S) 3.262 1.163 0.931 2.302 3.284 0.801 1.627 11
Pedregal, et al. Combination (S) 3.185 1.164 0.943 2.488 3.232 1.049 1.614 10
Model-M4 Statistical 3.117 1.231 0.962 2.578 3.277 2.238 1.631 13
Spiliotis & Assimakopoulos Statistical 3.184 1.178 0.959 2.488 3.232 1.808 1.628 12
Roubinchtein Combination (S) 3.244 1.159 0.921 2.290 3.632 1.129 1.633 15
Ibrahim Statistical 3.075 1.185 0.977 2.583 3.894 2.388 1.644 16
Tartu M4 seminar Combination (S & ML) 3.091 1.250 1.002 2.375 3.025 1.058 1.633 14
Waheeb Combination (S) 3.400 1.160 1.029 2.180 3.321 0.861 1.706 27
Darin & Stellwagen Statistical 3.406 1.168 0.924 2.107 4.128 0.856 1.693 25
Dantas & Cyrino Oliveira Combination (S) 3.294 1.170 0.952 2.534 3.436 1.598 1.657 17
The M4 Team (Theta) Statistical 3.382 1.232 0.970 2.637 3.262 2.455 1.696 26
The M4 Team (Com) Statistical 3.280 1.173 0.966 2.432 3.203 4.582 1.663 18
The M4 Team (Arima) Statistical 3.402 1.165 0.930 2.556 3.410 0.943 1.666 19
The M4 Team (Damped) Statistical 3.379 1.173 0.972 2.404 3.236 2.956 1.683 23
The M4 Team (ETS) Statistical 3.444 1.161 0.948 2.527 3.253 1.824 1.680 21
The M4 Team (Holt) Statistical 3.550 1.198 1.009 2.420 3.223 9.356 1.772 34
The M4 Team (SES) Statistical 3.981 1.340 1.019 2.685 3.281 2.385 1.885 39
Table 4: Average forecasting errors for various data types and overall ranks with respect to OWA.
Team Method Type Yearly Quarterly Monthly Weekly Daily Hourly Total Rank
Smyl Hybrid 0.778 0.847 0.836 0.851 1.046 0.440 0.821 1
Montero-Manso, et al. Combination (S & ML) 0.799 0.847 0.858 0.796 1.019 0.484 0.838 2
Pawlikowski, et al. Combination (S) 0.820 0.855 0.867 0.766 0.806 0.444 0.841 3
Jaganathan & Prakash Combination (S & ML) 0.813 0.859 0.854 0.795 0.996 0.474 0.842 4
Fiorucci & Louzada Combination (S) 0.802 0.855 0.868 0.897 0.977 0.674 0.843 5
Petropoulos & Svetunkov Combination (S) 0.806 0.853 0.876 0.751 0.984 0.663 0.848 6
Shaub Combination (S) 0.801 0.908 0.882 0.957 1.060 0.653 0.860 7
Legaki & Koutsouri Statistical 0.788 0.898 0.905 0.968 0.996 1.012 0.861 8
Doornik, et al. Combination (S) 0.836 0.878 0.881 0.782 1.002 0.410 0.865 9
Pedregal, et al. Combination (S) 0.824 0.883 0.899 0.939 0.990 0.485 0.869 11
Model-M4 Statistical 0.818 0.916 0.901 0.930 1.008 0.817 0.872 12
Spiliotis & Assimakopoulos Statistical 0.823 0.889 0.907 0.939 0.990 0.860 0.874 13
Roubinchtein Combination (S) 0.850 0.885 0.881 0.873 1.091 0.586 0.876 14
Ibrahim Statistical 0.805 0.890 0.921 0.961 1.098 0.991 0.880 15
Tartu M4 seminar Combination (S & ML) 0.820 0.960 0.932 0.892 0.930 0.598 0.888 17
Waheeb Combination (S) 0.880 0.880 0.927 0.779 0.999 0.507 0.894 18
Darin & Stellwagen Statistical 0.877 0.887 0.887 0.739 1.135 0.496 0.895 19
Dantas & Cyrino Oliveira Combination (S) 0.866 0.892 0.914 0.941 1.057 0.794 0.896 20
The M4 Team (Theta) Statistical 0.872 0.917 0.907 0.971 0.999 1.006 0.897 21
The M4 Team (Com) Statistical 0.867 0.890 0.920 0.926 0.978 1.556 0.898 22
The M4 Team (Arima) Statistical 0.892 0.898 0.903 0.932 1.044 0.524 0.902 23
The M4 Team (Damped) Statistical 0.890 0.893 0.924 0.917 0.997 1.141 0.907 25
The M4 Team (ETS) Statistical 0.903 0.891 0.915 0.931 0.996 0.852 0.908 26
The M4 Team (Holt) Statistical 0.947 0.932 0.988 0.966 0.995 2.749 0.971 37
The M4 Team (SES) Statistical 1.003 0.970 0.951 0.975 1.000 0.990 0.975 39

The forecasting performance of the Model-M3 that competed in the M4-Competition are given in the Table 5 with the error criteria sMAPE.

Table 5: Average sMAPE across different forecast horizons: all 3003 series.
Forecasting horizons Averages
Method 1 2 3 4 5 6 8 12 15 18 1-4 1-6 1-8 1-12 1-15 1-18
Naive2 10.5 11.3 13.6 15.1 15.1 15.9 14.5 16.0 19.3 20.7 12.62 13.57 13.76 14.24 14.81 15.47
Single 9.5 10.6 12.7 14.1 14.3 15.0 13.3 14.5 18.3 19.4 11.73 12.71 12.84 13.13 13.67 14.32
Holt 9.0 10.4 12.8 14.5 15.1 15.8 13.9 14.8 18.8 20.2 11.67 12.93 13.11 13.42 13.95 14.60
Winter 9.1 10.5 12.9 14.6 15.1 15.9 14.0 14.6 18.9 20.2 11.77 13.01 13.19 13.48 14.01 14.65
Dampen 8.8 10.0 12.0 13.5 13.8 14.3 12.5 13.9 17.5 18.9 11.07 12.05 12.17 12.45 12.98 13.64
Comb (S-H-D) 8.9 10.0 12.0 13.5 13.7 14.2 12.4 13.6 17.3 18.3 11.10 12.04 12.13 12.4 12.91 13.52
ETS 8.8 9.8 12.0 13.5 13.9 14.7 13.0 14.1 17.6 18.9 11.04 12.13 12.32 12.66 13.14 13.77
\(ATA(p,0,1)\) 8.9 10.0 12.1 13.7 13.9 14.7 12.8 13.9 17.3 18.9 11.16 12.21 12.34 12.64 13.13 13.77
\(ATA(p,1,1)\) 8.4 9.7 11.5 12.9 13.6 14.2 12.9 15.4 18.9 20.9 10.64 11.72 11.94 12.66 13.32 14.09
\(ATA(p,q,\phi=0.5)\) 8.6 9.6 11.6 13.2 13.5 14.2 12.4 13.7 17.0 18.6 10.76 11.77 11.92 12.24 12.75 13.39
Model-M3 8.5 9.6 11.4 12.8 13.0 13.6 12.0 13.1 16.3 17.4 10.56 11.47 11.58 11.94 12.40 12.94

These results should motivate users to consider ATA instead of ES-based forecasting. An important result from the M4-Competition was that combining forecasts improved accuracy. This improvement will become even stronger if the set of initial candidate models are chosen wisely and more meaningful if the combination can be obtained faster. Speed is an undeniable factor when choosing a forecasting method due to the need to obtain forecasts for the streaming and big data sets. The results obtained by using a simple combination of ARIMA and ATA for the M4-Competition data set are given in Table 6. For all error metrics considered, ATA approaches provide much better forecasts, and since the optimization is much faster than ETS, these more satisfying forecasts are obtained much faster.

Table 6: Average forecasting errors for various data types and error metrics using simple combinations of forecasts.
Yearly Quarterly Monthly Weekly Daily Hourly Total
sMAPE
ETS & ARIMA 14.691 10.027 12.917 8.439 3.076 14.377 12.205
Model-M3 & ARIMA 13.847 9.987 12.653 7.607 2.998 11.942 11.859
MASE
ETS & ARIMA 3.334 1.132 0.909 2.476 3.259 1.249 1.627
Model-M3 & ARIMA 3.093 1.148 0.908 2.345 3.255 1.436 1.575
OWA
ETS & ARIMA 0.869 0.868 0.875 0.906 1.002 0.652 0.875
Model-M3 & ARIMA 0.813 0.872 0.866 0.837 0.989 0.625 0.849

Just by using the simple combination of ATA and ARIMA, forecasts that are more accurate than most of the methods that competed in the M4-Competition and that can compete with the more accurate methods considering the computation complexity and time as important factors can be obtained. The results are given along with the ranks when all the methods are ranked according to OWA in Table 7. The three simple combinations of ATA and ARIMA are ranked in the top 10 when all other methods are considered.

Table 7: Average forecasting errors (OWA) for various data types along with the ranks.
Team Yearly Quarterly Monthly Weekly Daily Hourly Total Rank
Smyl 0.778 0.847 0.836 0.851 1.046 0.440 0.821 1
Montero-Manso, et al. 0.799 0.847 0.858 0.796 1.019 0.484 0.838 2
Pawlikowski, et al. 0.820 0.855 0.867 0.766 0.806 0.444 0.841 3
Jaganathan. & Prakash 0.813 0.859 0.854 0.795 0.996 0.474 0.842 4
Fiorucci & Louzada 0.802 0.855 0.868 0.897 0.977 0.674 0.843 5
Petropoulos & Svetunkov 0.806 0.853 0.876 0.751 0.984 0.663 0.848 6
Model-M4 & ARIMA 0.813 0.872 0.866 0.837 0.989 0.625 0.849 8
Shaub 0.801 0.908 0.882 0.957 1.060 0.653 0.860 10
Legaki & Koutsouri 0.788 0.898 0.905 0.968 0.996 1.012 0.861 11
Doornik, et al. 0.836 0.878 0.881 0.782 1.002 0.410 0.865 12

4 Conclusion

In this study, we have introduced a novel method of bagging for the Ata method using power family transformations and various seasonal decomposition techniques. Ata method is a new and simple forecasting method that is an alternative to exponential smoothing. Although the Ata method’s form is analogous to exponential smoothing, its weighting and parameterization schemes are utterly particular. Therefore, it is not a specific case of ES. It can be adapted to all types of time series data, much like ES and ARIMA, in addition to providing more accurate forecasts. Also, ATA can be optimized faster than exponential smoothing since its parameters can take on a limited number of discrete values only.

The goal of this manuscript is to introduce a new package for a new univariate time series forecasting method that provides innovative solutions to issues faced during the initialization and optimization stages of existing methods. The ATAforecasting package implements several different routines, most of which are related to the Ata method. Nevertheless, its modular structure enables the user to customize and complement the included functionality by means of custom algorithms or even other R packages. The ATAforecasting package provides a more general-purpose development as a comprehensive toolkit for automatic time series forecasting without any expertise on the R program. It focuses on modeling all types of time series components with any preferred Ata method and handling seasonality patterns by utilizing some popular decomposition techniques. Also, it combines several stationarity and seasonality tests, Box–Cox transformations, seasonal decomposition techniques with the Ata method. ATAforecasting performance is superior to existing methods both in terms of easy implementation, accurate, and flexible forecasting framework.

The ATAforecasting package categorizes some of the best-known techniques into three groups: (a) power transformation-based methods, (b) decomposition-based methods, and (c) time series forecasting-based methods. The package is also designed to assist research along with the whole modeling process: data preparation, model selection, prediction and forecasting, and interpretation of outcomes handling summaries and demonstrating functionalities. Providing these combinations of methods to the users is considered to introduce a new decomposition-based approach to time series forecasting with the Ata method, to provide automation, optimization, and bagging of the Ata method, which is an innovative and accurate univariate time series analysis method without any expertise by R program. Specifically, a proposed analytical methodology of the time series method with theATAforecasting R package combines several stationarity and seasonality tests, power family transformations, and various seasonal decomposition techniques with the Ata method. In addition to this theoretical model, we focus on the computational implementation of all considered Ata methods in the ATAforecasting package. In particular, simulation and estimation have been demonstrated. Besides, the ATAforecasting package is aligned to many worthy R packages, such as forecast, urca, uroot, seasonal, stR, stlplus, xts, timeSeries, TSA, tseries.

In the future, the package should be extended to provide a comprehensive set of tools for three common issues in forecast combination prior to estimation, fast optimization of model parameters, missing values, and modeling with regressor variables. Users would have the option to automate the selection algorithm so that a good combination method is found based on the training set fit. Finally, the package offers specialized functions for summarizing and visualizing the combination results. Along this vein, a class for model specifications should be added alongside the actual implementations via arguments for the fitting functions. In that way, the package can be aligned to M-Competition benchmark time series models and useful R package. Furthermore, the package could benefit from robust estimation methods, another focus for future research.

5 Acknowledgement

This article was produced from Ali Sabri Taylan’s PhD dissertation.

CRAN packages used

ATAforecasting, forecast, stats, stlplus, stR, tsibbledata, ucra, tseries, seasonal, Rcpp, RcppArmadillo, urca, uroot, xts, timeSeries, TSA, Mcomp, fable.ata, fabletools, fable

CRAN Task Views implied by cited packages

Econometrics, Environmetrics, Finance, HighPerformanceComputing, MissingData, NumericalMathematics, SpatioTemporal, TimeSeries

Note

This article is converted from a Legacy LaTeX article using the texor package. The pdf version is the official version. To report a problem with the html, refer to CONTRIBUTE on the R Journal homepage.

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Citation

For attribution, please cite this work as

Taylan, et al., "Automatic Time Series Forecasting with Ata Method in R: ATAforecasting Package", The R Journal, 2021

BibTeX citation

@article{RJ-2021-101,
  author = {Taylan, Ali Sabri and Yapar, Guckan and Selamlar, Hanife Taylan},
  title = {Automatic Time Series Forecasting with Ata Method in R: ATAforecasting Package},
  journal = {The R Journal},
  year = {2021},
  note = {https://rjournal.github.io/},
  volume = {13},
  issue = {2},
  issn = {2073-4859},
  pages = {507-541}
}

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