The Effect of Data Transformation on Singular Spectrum Analysis for Forecasting

Abstract

Data transformations are an important tool for improving the accuracy of forecasts from time series models. Historically, the impact of transformations have been evaluated on the forecasting performance of different parametric and nonparametric forecasting models. However, researchers have overlooked the evaluation of this factor in relation to the nonparametric forecasting model of Singular Spectrum Analysis (SSA). In this paper, we focus entirely on the impact of data transformations in the form of standardisation and logarithmic transformations on the forecasting performance of SSA when applied to 100 different datasets with different characteristics. Our findings indicate that data transformations have a significant impact on SSA forecasts at particular sampling frequencies.

1. Introduction

Amidst the emergence of Big Data and Data Mining techniques, forecasting continues to remain an important tool for planning and resource allocation in all industries. Accordingly, researchers, academics, and forecasters alike invest time and resources into methods for improving the accuracy of forecasts from both parametric and nonparametric forecasting models. One approach to improving the accuracy of forecasts is via data transformations prior to fitting time series models. For example, it is noted in [1] that data transformations can simplify the forecasting task, whilst evidence from other research indicates that, in economic analysis, taking logarithms can provide forecast improvements if it results in stabilising the variance of a series [2]. However, studies also indicate that data transformations will not always improve forecasts [3] and that they could complicate time series analysis models [4,5].

In fact, a key challenge for forecasting under data transformation is to transform the data back to its original scale, a process which could result in a forecasting bias [6,7]. Historically, most studies have focused on the impact of data transformations on parametric models such as Regression and Autoregressive Integrated Moving Average (ARIMA) models [8,9]. More recently, authors have resorted to evaluating the impact of data transformations on several other forecasting models [10,11], further highlighting the relevance and importance of the topic. Our interest is focused on the evaluation of the impact of data transformations on a time series analysis and forecasting technique called Singular Spectrum Analysis (SSA).

In brief, the SSA technique is a popular denoising, forecasting, and missing value prediction technique with both univariate and multivariate capabilities [12,13]. Recently, its diverse applications have focused on forecasting solutions for varied industries and fields, from tourism [14,15] and economics [16,17,18] to fashion [19], climate [20,21], and several other sectors [22,23,24]. Regardless of its wide and varied applications, researchers have yet to explore the effect of data transformations on the forecasting performance of this nonparametric forecasting technique. Previously, in [25], the authors evaluated the forecasting performance of the two basic SSA algorithms under different data structures. However, their work did not extend to evaluating the impact of data transformations to provide empirical evidence for future research. Accordingly, through this paper, we aim to contribute to the existing research gap by studying the effects of different data transformation options on the forecasting behaviour of SSA.

Logarithmic transformation is the most commonly used transformation in time series analysis. It has been used to convert multiplicative time series structures to additive structures or to reduce the time series skewness volatility and increase stability [2,26]. The autocorrelation structure in the time series may change under different transformations that may affect the model, and different transformations may result in different specifications for ARIMA models [6,27]. Like ARIMA models, SSA too can be greatly influenced by transformations. For instance, if data transformation makes noise uncorrelated or reduces the complexity of the time series, it can improve SSA performance [21,26]. As data standardisation and logarithmic transformations are the easiest in terms of interpretability and back-transformation to the original scale, we explore the effect of these data transformations on the forecasting performance of SSA.

The remainder of this paper is organised as follows. In Section 2, we provide a detailed exposition of SSA and its recurrent and vector forecasting algorithms. In Section 3, we present data transformation techniques and their effect on forecasting accuracy. Procedures for examining the effect of transformation based on different characteristics of time series are presented in Section 4. In Section 5, we analyse different datasets of varied characteristics and present our results for an evidence-based exploration of the effect of data transformations on SSA forecasts. Finally, we present our concluding remarks in Section 6.

2. SSA Forecasting

There are two different algorithms for forecasting with SSA, namely recurrent forecasting and vector forecasting [12,28]. Those interested in a comparison of the performance of both algorithms are referred to [25]. Both of these forecasting algorithms require that one follows two common steps of SSA, the decomposition and reconstruction of a time series [12,28]. In what follows, we provide a brief description of forecasting processes in SSA.

2.1. Decomposition and Reconstruction of Time Series

In SSA, we embed the time series $\{x_1,x_2,\dots ,x_N\}$ into a high-dimensional space by constructing a Hankel structured trajectory matrix of the form:

$$\mathit{X}=\left(\begin{array}{ccccc}{x}_1& x_2& x_3& \dots & x_n\\ x_2& x_3& x_4& \dots & x_{n+1}\\ ⋮& ⋮& ⋮& \dots & ⋮\\ x_m& x_{m+1}& x_{m+2}& \dots & x_N\end{array}\right)=[{\mathit{x}}_1\dots {\mathit{x}}_i\dots {\mathit{x}}_n],$$

(1)

where m is the window length, the $m-$ lagged vector ${\mathit{x}}_i={(x_i,x_{i+1},\dots ,x_{i+m-1})}^{\prime }$ is the ith column of the trajectory matrix $\mathit{X}$, $n=N-m+1$, and $m\le n$.

The singular value decomposition (SVD) of the trajectory matrix $\mathit{X}$ can be expressed as

$$\mathit{X}={\mathit{S}}_k+{\mathit{E}}_k=\sum _{j=1}^k\sqrt{{\lambda }_j}{\mathit{u}}_j{\mathit{v}}_j^{\prime }+\sum _{j=k+1}^m\sqrt{{\lambda }_j}{\mathit{u}}_j{\mathit{v}}_j^{\prime }$$

(2)

where ${\mathit{u}}_j$ is the jth eigenvector of $\mathit{X}{\mathit{X}}^{\prime }$ corresponding to the eigenvalue ${\lambda }_j$ and ${\mathit{v}}_j={\mathit{X}}^{\prime }{\mathit{u}}_j/\sqrt{{\lambda }_j}$.

If k is the number of signal components, ${\mathit{S}}_k={\sum }_{j=1}^k\sqrt{{\lambda }_j}{\mathit{u}}_j{\mathit{v}}_j^{\prime }$ represents a matrix of signal, and ${\mathit{E}}_k={\sum }_{j=k+1}^m\sqrt{{\lambda }_j}{\mathit{u}}_j{\mathit{v}}_j^{\prime }$ is the matrix of noise. We apply the diagonal averaging procedure to ${\mathit{S}}_k$ to reconstruct the signal series ${\tilde{x}}_t$ such that the observed series can be expressed as

$$x_t={\tilde{x}}_t+{\tilde{e}}_t,$$

(3)

where ${\tilde{x}}_t$ is the less noisy, filtered series. A detailed explanation of decomposition in Equation (3) can be found in [28,29].

To construct the trajectory matrix $\mathit{X}$ in Equation (1) and to conduct the SVD in Equation (2), we have to select the Window Length m and the number of signal components k. Since our aim is not to demonstrate the selection of SSA choices (m and k), we opt not to reproduce the selection procedures for SSA choices, as these are already covered in depth in [12,28]. As our interest is in examining the effect of transformation on the forecasting performance of SSA, we select m and k such that the Root Mean Squared Error (RMSE) in forecasting is minimised.

2.2. Recurrent Forecasting

Recurrent forecasting in SSA is also known as R-forecasting, and the findings in [25] indicate that R-forecasting is best when dealing with large samples. If ${\mathit{u}}_j^{\nabla }={(u_{1j},\dots ,u_{(m-1)j})}^{\prime }$ is the vector of the first $m-1$ elements of the jth eigenvector ${\mathit{u}}_j$, and $u_{mj}$ is the last element of ${\mathit{u}}_j$. The coefficients of linear recurrent equation can be estimated as

$$\mathit{a}={(a_{(m-1)},\dots ,a_1)}^{\prime }=\frac{1}{1-{\sum }_{j=1}^k{u}_{mj}^2}\sum _{j=1}^k{u}_{mj}{\mathit{u}}_j^{\nabla }.$$

(4)

With the parameters in Equation (4), a linear recurrent equation of the form

$${\tilde{x}}_t=\sum _{i=1}^{m-1}{a}_{(m-i)}{\tilde{x}}_{t-m+i}$$

(5)

is used to obtain a one-step-ahead recursive forecast [29]. This linear recurrent formula in Equation (5) is used to forecast the signal at time $t+1$ given the signal at time $t,t-1,\dots ,t-m+2$ [28] (Section 2.1, Equations (1)–(6)), and the one-step-ahead recursive forecast of $x_{N+j}$ is

$${\widehat{x}}_{N+j}=\{\begin{array}{cc}\sum _{i=1}^{j-1}{a}_i{\widehat{x}}_{N+j-i}+\sum _{i=1}^{m-j}{a}_{m-i}{\tilde{x}}_{N+j-m+i}\hfill & \mathrm{for}j\le m-1;\hfill \\ \sum _{i=1}^{m-1}{a}_i{\widehat{x}}_{N+j-i}\hfill & \mathrm{for}j>m-1.\hfill \end{array}$$

(6)

We apply the recursive forecasting method in Equation (6) to obtain a one-step-ahead forecast.

2.3. Vector Forecasting

In contrast, the SSA Vector forecasting algorithm has proven to be more robust than the R-forecasting algorithm in most cases [25]. Let us define ${\mathit{U}}_k^{\nabla }=[{\mathit{u}}_1^{\nabla }\dots {\mathit{u}}_k^{\nabla }]$ as the $(m-1)\times k$ matrix consisting of the first $m-1$ elements of k eigenvectors. The vector forecasting algorithm computes $m-$ lagged vectors $\widehat{{\mathit{z}}_i}$ and constructs a trajectory matrix $\mathit{Z}=[{\widehat{\mathit{z}}}_1\dots {\widehat{\mathit{z}}}_n{\widehat{\mathit{z}}}_{n+1}\dots {\widehat{\mathit{z}}}_{n+h}]$ such that

$${\widehat{\mathit{z}}}_i=\{\begin{array}{ll}{\mathit{s}}_i\hfill & \mathrm{for}i=1,2,\dots ,n;\hfill \\ \left(\begin{array}{c}({\mathit{U}}_k^{\nabla }{\mathit{U}}_k^{{\nabla }^{\prime }}+(1-\sum _{j=1}^k{u}_{mj}^2)\mathit{a}{\mathit{a}}^{\prime }){\widehat{\mathit{z}}}_{i-1}^{▵}\\ {\mathit{a}}^{\prime }{\widehat{\mathit{z}}}_{i-1}^{▵}\end{array}\right)& \mathrm{for}i=n+1,\dots ,n+h.\hfill \end{array}$$

(7)

where ${\mathit{s}}_i$ is the ith column of the reconstructed signal matrix ${\mathit{S}}_k={\sum }_{j=1}^k\sqrt{{\lambda }_j}{\mathit{u}}_j{\mathit{v}}_j^{\prime }$, and ${\widehat{\mathit{z}}}_i^{▵}$ is the last $m-1$ elements of the vector ${\widehat{\mathit{z}}}_i$.

After a diagonal averaging of the matrix $\mathit{Z}=[{\widehat{\mathit{z}}}_1\dots {\widehat{\mathit{z}}}_n{\widehat{\mathit{z}}}_{n+1}\dots {\widehat{\mathit{z}}}_{n+h}]$ constructed by employing Equation (7), we obtain a time series $\{{\widehat{z}}_1,\dots ,{\widehat{z}}_N,{\widehat{z}}_{N+1},\dots ,{\widehat{z}}_{N+h}\}$, as has also been explained in [28] (Section 2.3). Thus, ${\widehat{x}}_{N+j}={\widehat{z}}_{N+j}$ produces a forecast corresponding to $x_{N+j}$ for $j=1,\dots ,h$.

3. Transformation of Time Series

Data transformation is useful when the variation increases or decreases with the level of the series [1]. Whilst logarithmic transformation and standardisation are the most commonly used data transformation techniques in time series analysis, it is noteworthy that there are other transformations from the family of power transformation such as square root and cube root transformations. However, the interpretability is not as simple and common as that for standardisation and logarithmic transformation.

3.1. Standardisation

Standardisation of time series $\left\{x_t\right\}$ is formulated as

$$y_t=\frac{x_t-\overline{x}}{{\sigma }_x},$$

(8)

where $\overline{x}$ and ${\sigma }_x$ are the mean and standard deviation of the series $\left\{x_t\right\}$, respectively. Data standardisation is another common data transformation in preprocessing. Standardisation is mostly common in machine learning techniques to reduce training time and error. In time series forecasting, standardisation has proven advantages when we are using machine learning algorithms (e.g., neural networks and deep neural networks) [30]. In terms of SSA, the theoretical literature does not investigate the effect of standardisation on SSA forecasts in detail. However, in Golyandina and Zhigljavski [26], the authors addressed the effect of centering the time series as preprocessing. In theory, if the time series can be expressed as an oscillation around a linear trend, centering will increase the SSA’s accuracy [26].

3.2. Logarithmic Transformation

In this paper, the following logarithmic transformation is applied on time series $\left\{x_t\right\}$:

$$\begin{array}{c}\hfill y_t=log(C+x_t),\end{array}$$

(9)

where C is a constant value, large enough to guarantee that the term inside the logarithm is positive. As mentioned before, log-transform is a common preprocessing to handle variance instability or right skewness. Furthermore, one may use log-transform as a form of preprocessing to convert a time series with a multiplicative structure to an additive one. Given that SSA can be applied to time series with both additive and multiplicative structures, it does not necessarily need log-transform pre-processing [26]. However, the authors in Golyandina and Zhigljavski [26] showed that using log-transform could affect SSA’s forecasting accuracy. In fact, SSA’s forecasting accuracy will increase if the rank of a transformed series is smaller than the original one.

4. Comparison between Transformations

Time series with different characteristics will behave differently after transformation. For instance, forecasting accuracy in time series, with positive skewness, non-stationarity, and non-normality, may improve with logarithmic transformation. Furthermore, in time series with large observations or large variance, standardisation can improve the forecasting accuracy. Sampling frequency is another potential factor affecting forecasting accuracy. Time series with high sampling frequency (e.g., hourly or daily) usually have an oscillation frequency close to its noise frequency and consequently show instable and noisy behaviour. On the other hand, time series with larger sampling frequency are smoother. These characteristics of time series may affect forecasting and accuracy as well. As such, to investigate the practical effect of data transformation in SSA forecasting, we should consider “Sampling Frequency,” “Skewness,” “Normality,” and “Stationarity” as control factors.

To observe the effectiveness of data transformation prior to the application of SSA, we may compare the forecasting performance of SSA under different transformations and control factors: firstly, by comparing the Root Mean Squared Forecast Error (RMSFE), and secondly, by employing a nonparametric test to examine the treatment effect (data transformation).

4.1. Root Mean Squared Forecast Error (RMSFE)

The most commonly adopted approach for comparing the predictive accuracy between forecasts is to compute and compare the RMSFE from out of sample forecasts. The RMSFE can be defined as

$$RMS{E}_h=\sqrt{\frac{1}{h}\sum _{t=N+1}^{N+h}{(x_t-{\widehat{x}}_t)}^2},$$

(10)

where h is the forecast horizon, N is the number of observations, $x_t$ is observed value of time series, and ${\widehat{x}}_t$ is the forecasted value.

The application of data transformation prior to forecasting with SSA may significantly affect the forecasting outcome and the affect may vary based on the properties of a time series. Thus, we need to examine the effect of data transformation on RMSFE along with the differing properties of time series. Comparisons between the RMSFE of the original and transformed time series can be used to learn about the forecasting performance of a model for a given time series. However, comparison of RMSFE for a pool of time series with different characteristics is not straightforward. We compute $RMS{E}_h$ for $h=1,3,6,12$ ($h=1$ for a short-term forecast, $h=12$ for a long-term forecast, and $h=3,6$ as a medium-term forecasting horizon) for each of the time series in the pool and examine the effect of transformation by using statistical tests.

4.2. Nonparametric Repeated Measure Factorial Test

Treatment effects in the presence of factors can be examined by employing the nonparametric repeated measure factorial test [31,32] for a pool of time series of different characteristics. Thus, the effect of data transformation (treatment) can be examined by using this test under different characteristics of a time series.

Let us assume that we have K time series in the pool with series code $A_k,k=1,\dots ,K$ and for each of the series $RMS{E}_h$ for $h=1,3,6,12$ are computed. If the interest lies on exploring the effect of transformation for the skewness property of time series, we essentially perform the test for treatment effect (transformation) for categories of skewness properties of these time series. There are three factor levels of the factor Skewness, namely Skew Negative, Skew Positive, and Skew Symmetric. Similarly, we will have two levels for the factor Normality (Yes = normal; No = not normal) and two levels for the factor Stationarity (Yes = stationary; No = nonstationary). To test the effect of transformation (No transformation, Standardisation, and Logarithmic transformation), we follow the procedures described below.

First, we learn some basic characteristics of a time series such as normality, stationarity, skewness, and frequency. For example, the frequency of a time series can be learnt by examining the time of measurement: hourly, daily, weekly, monthly, or annually. We also classify time series into different categories via a series of statistical tests such as the Jarque-Bera test for normality [33], the KPSS test for stationarity [34], and the D’Agostino test for skewness [35].

Secondly, the nonparametric repeated measure factorial test [31,32] is used to test the effect of the transformation on RMSFE, across different categories where categories are defined based on Frequency, Normality, Skewness, and Stationarity.

5. Data Analysis

We used the same set of time series employed by Ghodsi et al. [25] to test the effect of data transformation on SSA forecasting accuracy, with different characteristics. The dataset contains 100 real time series with different sampling frequencies and stationarity, normality, and skewness characteristics, representing various fields and categories, obtained via the Data Market (http://datamarket.com).

Table 1. Number of time series with each feature.

Factor	Levels
Sampling Frequency	Annual	Monthly	Quarterly	Weekly	Daily	Hourly
	5	83	4	4	2	2
Skewness	Positive Skew		Negative Skew		Symmetric
	61		21		18
Normality	Normal			Non-normal
	18			82
Stationarity	Stationary			Non-Stationary
	14			86

presents the number of time series with each feature. It is evident that the real data includes data recorded at varying frequencies (annual, monthly, weekly, daily, and hourly) alongside varying distributions (normal distribution, skewed, stationary, and non-stationary). Interestingly, the majority of the data are non-stationary overtime, which resonates with expectations within real-life scenarios.

The name and description of each time series and their codes assigned to improve presentation are presented in

Table A1. List of 100 real time series.

Code	Name of Time Series
A001	US Economic Statistics: Capacity Utilization.
A002	Births by months 1853–2012.
A003	Electricity: electricity net generation: total (all sectors).
A004	Energy prices: average retail prices of electricity.
A005	Coloured fox fur returns, Hopedale, Labrador, 1834–1925.
A006	Alcohol demand (log spirits consumption per head), UK, 1870–1938.
A007	Monthly Sutter county workforce, Jan. 1946–Dec. 1966 priesema (1979).
A008	Exchange rates—monthly data: Japanese yen.
A009	Exchange rates—monthly data: Pound sterling.
A010	Exchange rates—monthly data: Romanian leu.
A011	HICP (2005 = 100)—monthly data (annual rate of change): European Union (27 countries).
A012	HICP (2005 = 100)—monthly data (annual rate of change): UK.
A013	HICP (2005 = 100)—monthly data (annual rate of change): US.
A014	New Homes Sold in the United States.
A015	Goods, Value of Exports for United States.
A016	Goods, Value of Imports for United States.
A017	Market capitalisation—monthly data: UK.
A018	Market capitalisation—monthly data: US.
A019	Average monthly temperatures across the world (1701–2011): Bournemouth.
A020	Average monthly temperatures across the world (1701–2011): Eskdalemuir.
A021	Average monthly temperatures across the world (1701–2011): Lerwick.
A022	Average monthly temperatures across the world (1701–2011): Valley.
A023	Average monthly temperatures across the world (1701–2011): Death Valley.
A024	US Economic Statistics: Personal Savings Rate.
A025	Economic Policy Uncertainty Index for United States (Monthly Data).
A026	Coal Production, Total for Germany.
A027	Coke, Beehive Production (by Statistical Area).
A028	Monthly champagne sales (in 1000’s) (p. 273: Montgomery: Fore. and T.S.).
A029	Domestic Auto Production.
A030	Index of Cotton Textile Production for France.
A031	Index of Production of Chemical Products (by Statistical Area).
A032	Index of Production of Leather Products (by Statistical Area).
A033	Index of Production of Metal Products (by Statistical Area).
A034	Index of Production of Mineral Fuels (by Statistical Area).
A035	Industrial Production Index.
A036	Knit Underwear Production (by Statistical Area).
A037	Lubricants Production for United States.
A038	Silver Production for United States.
A039	Slab Zinc Production (by Statistical Area).
A040	Annual domestic sales and advertising of Lydia E, Pinkham Medicine, 1907 to 1960.
A041	Chemical concentration readings.
A042	Monthly Boston armed robberies January 1966-October 1975 Deutsch and Alt (1977).
A043	Monthly Minneapolis public drunkenness intakes Jan.’66–Jul’78.
A044	Motor vehicles engines and parts/CPI, Canada, 1976–1991.
A045	Methane input into gas furnace: cu. ft/min. Sampling interval 9 s.
A046	Monthly civilian population of Australia: thousand persons. February 1978–April 1991.
A047	Daily total female births in California, 1959.
A048	Annual immigration into the United States: thousands. 1820–1962.
A049	Monthly New York City births: unknown scale. January 1946–December 1959.
A050	Estimated quarterly resident population of Australia: thousand persons.
A051	Annual Swedish population rates (1000’s) 1750–1849 Thomas (1940).
A052	Industry sales for printing and writing paper (in Thousands of French francs).
A053	Coloured fox fur production, Hebron, Labrador, 1834–1925.
A054	Coloured fox fur production, Nain, Labrador, 1834–1925.
A055	Coloured fox fur production, oak, Labrador, 1834–1925.
A056	Monthly average daily calls to directory assistance Jan.’62–Dec’76.
A057	Monthly Av. residential electricity usage Iowa city 1971–1979.
A058	Montly av. residential gas usage Iowa (cubic feet) * 100 ’71–’79.
A059	Monthly precipitation (in mm), January 1983–April 1994. London, United Kingdom.
A060	Monthly water usage (ml/day), London Ontario, 1966–1988.
A061	Quarterly production of Gas in Australia: million megajoules. Includes natural gas from July 1989. March 1956–September 1994.
A062	Residential water consumption, Jan 1983–April 1994. London, United Kingdom.
A063	The total generation of electricity by the U.S. electric industry (monthly data for the period Jan. 1985–Oct. 1996).
A064	Total number of water consumers, January 1983–April 1994. London, United Kingdom.
A065	Monthly milk production: pounds per cow. January 62–December 75.
A066	Monthly milk production: pounds per cow. January 62–December 75, adjusted for month length.
A067	Monthly total number of pigs slaughtered in Victoria. January 1980–August 1995.
A068	Monthly demand repair parts large/heavy equip. Iowa 1972–1979.
A069	Number of deaths and serious injuries in UK road accidents each month. January 1969–December 1984.
A070	Passenger miles (Mil) flown domestic U.K. Jul. ’62–May ’72.
A071	Monthly hotel occupied room av. ’63–’76 B.L.Bowerman et al.
A072	Weekday bus ridership, Iowa city, Iowa (monthly averages).
A073	Portland Oregon average monthly bus ridership (/100).
A074	U.S. airlines: monthly aircraft miles flown (Millions) 1963–1970.
A075	International airline passengers: monthly totals in thousands. January 49–December 60.
A076	Sales: souvenir shop at a beach resort town in Queensland, Australia. January 1987–December 1993.
A077	Der Stern: Weekly sales of wholesalers A, ’71–’72.
A078	Der Stern: Weekly sales of wholesalers B, ’71–’72’
A079	Der Stern: Weekly sales of wholesalers ’71–’72.
A080	Monthly sales of U.S. houses (thousands) 1965–1975.
A081	CFE specialty writing papers monthly sales.
A082	Monthly sales of new one-family houses sold in USA since 1973.
A083	Wisconsin employment time series, food and kindred products, January 1961–October 1975.
A084	Monthly gasoline demand Ontario gallon millions 1960–1975.
A085	Wisconsin employment time series, fabricated metals, January 1961–October 1975.
A086	Monthly empolyees wholes./retail Wisconsin ’61–’75 R.B.Miller.
A087	US monthly sales of chemical related products. January 1971–December 1991.
A088	US monthly sales of coal related products. January 1971–December 1991.
A089	US monthly sales of petrol related products. January 1971–December 1991.
A090	US monthly sales of vehicle related products. January 1971–December 1991.
A091	Civilian labour force in Australia each month: thousands of persons. February 1978–August 1995.
A092	Numbers on Unemployment Benefits in Australia: monthly January 1956–July 1992.
A093	Monthly Canadian total unemployment figures (thousands) 1956–1975.
A094	Monthly number of unemployed persons in Australia: thousands. February 1978–April 1991.
A095	Monthly U.S. female (20 years and over) unemployment figures 1948–1981.
A096	Monthly U.S. female (16–19 years) unemployment figures (thousands) 1948–1981.
A097	Monthly unemployment figures in West Germany 1948–1980.
A098	Monthly U.S. male (20 years and over) unemployment figures 1948–1981.
A099	Wisconsin employment time series, transportation equipment, January 1961–October 1975.
A100	Monthly U.S. male (16–19 years) unemployment figures (thousands) 1948–1981.

.

Table A2. Descriptives for the 100 time series.

Code	F	N	Mean	Med.	SD	CV	Skew.	SW(p)	ADF	Code	F	N	Mean	Med.	SD	CV	Skew.	SW(p)	ADF
A001	M	539	80	80	5	6	−0.55	<0.01	−0.60 ${}^{†}$	A002	M	1920	271	249	88	33	0.16	<0.01	−1.82 ${}^{†}$
A003	M	484	2.59 × 10${}^5$	2.61 × 10${}^5$	6.88 × 10${}^5$	27	0.15	<0.01	−0.90 ${}^{†}$	A004	M	310	7	7	2	28	−0.24	<0.01	0.56 ${}^{†}$
A005	D	92	47.63	31.00	47.33	99.36	2.27	<0.01	−3.16	A006	Q	207	1.95	1.98	0.25	12.78	−0.58	<0.01	0.46 ${}^{†}$
A007	M	252	2978	2741	1111	37.32	0.79	<0.01	−0.80 ${}^{†}$	A008	M	160	128	128	19	15	0.34	<0.01	−0.59 ${}^{†}$
A009	M	160	0.72	0.69	0.10	13	0.66	<0.01	0.53 ${}^{†}$	A010	M	160	3.41	3.61	0.83	24	−0.92	<0.01	1.58 ${}^{†}$
A011	M	201	4.7	2.6	5.0	106	2.24	<0.01	−2.66	A012	M	199	2.1	1.9	1.0	49	0.92	<0.01	−0.79 ${}^{†}$
A013	M	176	2.5	2.4	1.6	66	−0.52	<0.01	−2.27 ${}^{†}$	A014	M	606	55	53	20	35	0.79	<0.01	−1.41 ${}^{†}$
A015	M	672	3.39	1.89	3.48	103	1.09	<0.01	2.46 ${}^{†}$	A016	M	672	5.18	2.89	5.78	111	1.13	<0.01	1.91 ${}^{†}$
A017	M	249	130	130	24	19	0.35	<0.01	0.24${}^{†}$	A018	M	249	112	114	25	22	−0.01	0.01*	0.06 ${}^{†}$
A019	M	605	10.1	9.6	4.5	44	0.05	<0.01	−4.77	A020	M	605	7.3	6.9	4.3	59	0.04	<0.01	−6.07
A021	M	605	7.2	6.8	3.3	46	0.13	<0.01	−4.93	A022	M	605	10.3	9.9	3.8	37	0.04	<0.01	−4.19
A023	M	605	24	24	10	40	−0.02	<0.01	−7.15	A024	M	636	6.9	7.4	2.6	38	−0.29	<0.01	−1.18 ${}^{†}$
A025	M	343	108	100	33	30	0.99	<0.01	−1.23 ${}^{†}$	A026	M	277	11.7	11.9	2.3	20	−0.16	0.06 *	−0.40 ${}^{†}$
A027	M	171	0.21	0.13	0.19	88	1.26	<0.01	−1.81 ${}^{†}$	A028	M	96	4801	4084	2640	54.99	1.55	<0.01	−1.66 ${}^{†}$
A029	M	248	391	385	116	30	−0.03	0.08 *	−1.22 ${}^{†}$	A030	M	139	89	92	12	13	−0.82	<0.01	−0.28 ${}^{†}$
A031	M	121	134	138	27	20	0.05	<0.01	1.51 ${}^{†}$	A032	M	153	113	114	10	9	−0.29	0.45 *	−0.52 ${}^{†}$
A033	M	115	117	118	17	15	−0.29	0.03 *	−0.46 ${}^{†}$	A034	M	115	110	111	11	10	−0.53	0.02 *	0.30 ${}^{†}$
A035	M	1137	40	34	31	78	0.56	<0.01	5.14${}^{†}$	A036	M	165	1.08	1.10	0.20	18.37	−1.15	<0.01	−0.59 ${}^{†}$
A037	M	479	3.04	2.83	1.02	33.60	0.46	<0.01	0.61 ${}^{†}$	A038	M	283	9.39	10.02	2.27	24.15	−0.80	<0.01	−1.01 ${}^{†}$
A039	M	452	54	52	19	36	−0.15	<0.01	0.08 ${}^{†}$	A040	Q	108	1382	1206	684	49.55	0.83	<0.01	−0.80 ${}^{†}$
A041	H	197	17.06	17.00	0.39	2.34	0.15	0.21 *	0.09 ${}^{†}$	A042	M	118	196.3	166.0	128.0	65.2	0.45	<0.01	0.41 ${}^{†}$
A043	M	151	391.1	267.0	237.49	60.72	0.43	<0.01	−1.17 ${}^{†}$	A044	M	188	1344	1425	479.1	35.6	−0.41	<0.01	−1.28 ${}^{†}$
A045	H	296	−0.05	0.00	1.07	−1887	−0.05	0.55 *	−7.66	A046	M	159	11890	11830	882.93	7.42	0.12	<0.01	5.71
A047	D	365	41.98	42.00	7.34	17.50	0.44	<0.01	−1.07 ${}^{†}$	A048	A	143	2.5 × 10${}^5$	2.2 × 10${}^5$	2.1 × 10${}^5$	83.19	1.06	<0.01	−2.63
A049	M	168	25.05	24.95	2.31	9.25	−0.02	0.02 *	0.07 ${}^{†}$	A050	Q	89	15274	15184	1358	8.89	0.19	<0.01	9.72 ${}^{†}$
A051	A	100	6.69	7.50	5.88	87.87	−2.45	<0.01	−3.06	A052	M	120	713	733	174	24.39	−1.09	<0.01	−0.78 ${}^{†}$
A053	A	91	81.58	46.00	102.07	125.11	2.80	<0.01	−3.44	A054	A	91	101.80	77.00	92.14	90.51	1.43	<0.01	−3.38
A055	A	91	59.45	39.00	60.42	101.63	1.56	<0.01	−3.99	A056	M	180	492.50	521.50	189.54	38.48	−0.17	<0.01	−0.65 ${}^{†}$
A057	M	106	489.73	465.00	93.34	19.06	0.92	<0.01	−1.21 ${}^{†}$	A058	M	106	124.71	94.50	84.15	67.48	0.52	<0.01	−3.88
A059	M	136	85.66	80.25	37.54	43.83	0.91	<0.01	−1.88 ${}^{†}$	A060	M	276	118.61	115.63	26.39	22.24	0.86	<0.01	−0.47 ${}^{†}$
A061	Q	155	61728	47976	53907	87.33	0.44	<0.01	0.06 ${}^{†}$	A062	M	136	5.72 × 10${}^7$	5.53 × 10${}^7$	1.2 × 10${}^7$	21.51	1.13	<0.01	−0.84${}^{†}$
A063	M	142	231.09	226.73	24.37	10.55	0.52	0.01	−0.39 ${}^{†}$	A064	M	136	31388	31251	3232	10.30	0.25	0.22 *	−0.16 ${}^{†}$
A065	M	156	754.71	761.00	102.20	13.54	0.01	0.04 *	0.04 ${}^{†}$	A066	M	156	746.49	749.15	98.59	13.21	0.08	0.04 *	−0.38 ${}^{†}$
A067	M	188	90640	91661	13926	15.36	−0.38	0.01 *	−0.38 ${}^{†}$	A068	M	94	1540	1532	474.35	30.79	0.38	0.05 *	0.54 ${}^{†}$
A069	M	192	1670	1631	289.61	17.34	0.53	<0.01	−0.74 ${}^{†}$	A070	M	119	91.09	86.20	32.80	36.01	0.34	<0.01	−1.93 ${}^{†}$
A071	M	168	722.30	709.50	142.66	19.75	0.72	<0.01	−0.52 ${}^{†}$	A072	W	136	5913	5500	1784	30.17	0.67	<0.01	−0.68 ${}^{†}$
A073	M	114	1120	1158	270.89	24.17	−0.37	<0.01	0.76 ${}^{†}$	A074	M	96	10385	10401	2202	21.21	0.33	0.18 *	−0.13 ${}^{†}$
A075	M	144	280.30	265.50	119.97	42.80	0.57	<0.01	−0.35 ${}^{†}$	A076	M	84	14315	8771	15748	110	3.37	<0.01	−0.29 ${}^{†}$
A077	W	104	11909	11640	1231	10.34	0.60	<0.01	−0.16 ${}^{†}$	A078	W	104	74636	73600	4737	6.35	0.64	<0.01	−0.59 ${}^{†}$
A079	W	104	1020	1012	71.78	7.03	0.60	0.01 *	−0.41 ${}^{†}$	A080	M	132	45.36	44.00	10.38	22.88	0.17	0.15 *	−0.81 ${}^{†}$
A081	M	147	1745	1730	479.52	27.47	−0.39	<0.01	−1.15 ${}^{†}$	A082	M	275	52.29	53.00	11.94	22.83	0.18	0.13 *	−1.30 ${}^{†}$
A083	M	178	58.79	55.80	6.68	11.36	0.93	<0.01	−0.92 ${}^{†}$	A084	M	192	1.62 × 10${}^5$	1.57 × 10${}^5$	41661	25.71	0.32	<0.01	0.25 ${}^{†}$
A085	M	178	40.97	41.50	5.11	12.47	−0.07	<0.01	1.45 ${}^{†}$	A086	M	178	307.56	308.35	46.76	15.20	0.17	<0.01	1.51 ${}^{†}$
A087	M	252	13.70	14.08	6.13	44.73	0.16	<0.01	1.13 ${}^{†}$	A088	M	252	65.67	68.20	14.25	21.70	−0.53	<0.01	−0.53 ${}^{†}$
A089	M	252	10.76	10.92	5.11	47.50	−0.19	<0.01	−0.05 ${}^{†}$	A090	M	252	11.74	11.05	5.11	43.54	0.38	<0.01	−0.88 ${}^{†}$
A091	M	211	7661	7621	819	10.70	0.03	<0.01	3.27 ${}^{†}$	A092	M	439	2.21 × 10${}^5$	5.67 × 10${}^4$	2.35 × 10${}^5$	106.32	0.77	<0.01	1.61 ${}^{†}$
A093	M	240	413.28	396.50	152.84	36.98	0.36	<0.01	−1.60 ${}^{†}$	A094	M	211	6787	6528	604.62	8.91	0.56	<0.01	2.69 ${}^{†}$
A095	M	408	1373	1132	686.05	49.96	0.91	<0.01	0.60 ${}^{†}$	A096	M	408	422.38	342.00	252.86	59.87	0.65	<0.01	−1.95 ${}^{†}$
A097	M	396	7.14 × 10${}^5$	5.57 × 10${}^5$	5.64 × 10${}^5$	78.97	0.79	<0.01	−2.51 ${}^{†}$	A098	M	408	1937	1825	794	41.04	0.64	<0.01	−1.15 ${}^{†}$
A099	M	178	40.60	40.50	4.95	12.19	−0.65	<0.01	−0.10 ${}^{†}$	A100	M	408	520.28	425.50	261.22	50.21	0.64	<0.01	−1.65 ${}^{†}$

Note: * indicates data is normally distributed based on a Shapiro-Wilk test at p = 0.01. † indicates a nonstationary time series based on the augmented Dickey-Fuller test at p = 0.01. A indicates annual, M indicates monthly, Q indicates quarterly, W indicates weekly, D indicates daily and H indicates hourly. N indicates series length.

presents descriptive statistics for all time series to enable the reader to obtain a rich understanding of the nature of the real data. This also includes skewness statistics, and results from the normality (Shapiro-Wilk) and stationarity (Augmented Dickey-Fuller) tests. As visible in Table A1, the data comes from different fields such as energy, finance, health, tourism, housing market, crime, agriculture, economics, chemistry, ecology, and production.

Figure 1 shows the time series for a selection of 9/100 series used in this study. This enables the reader to obtain a further understanding of the different structures underlying the data considered in the analysis. For example, A007 is representative of an asymmetric non-stationary time series for the labour market in a U.S. county. This monthly series shows seasonality with an increasing non-linear trend. In contrast, A022 is related to a meteorological variable that is asymmetric, yet stationary and highly seasonal in nature. An example of a time series that is both asymmetric and non-stationary is A038, which represents the production of silver. Here, structural breaks are visible throughout. A055 is an annual time series, which is stationary and asymmetric, and relates to the production of coloured fox fur. An example of a quarterly time series representing the energy sector is shown via A061. This time series is non-stationary and asymmetric with a non-linear trend and an increasing seasonality over time. Another example focuses on the airline industry (A075) and is also asymmetric and non-stationary in nature. It appears to showcase a linear and increasing trend along with seasonality. A skewed and non-stationary sales series is shown via A081, with the trend indicating increasing seasonality with major drops in the time series between each season. A time series for house sales (A082) can be found to be normally distributed and non-stationary over time. It also shows a slightly curved non-linear trend and a sine wave that is disrupted by noise. Finally, the labour market is drawn on again via A094, but this is an example of a time series affected by several structural breaks leading to a non-stationary, asymmetric series, which also has seasonal periods and a clear non-linear trend.

R packages “Rssa” [36,37,38] and “nparLD” [39] are employed to implement SSA forecasting and the nonparametric repeated measure factorial test, respectively. We apply SSA to three versions of a dataset: a dataset without any transformation, a standardised dataset, and a log-transformed dataset. For each of the three datasets, we obtain RMSFE from out-of-sample forecasting at forecast horizons $h=1,3,6,12$. It is noteworthy that our aim in this paper is to examine the effect of transformation in SSA forecasting. Thus, we consider the best forecast based on the RMSFE of the last 12 data points regardless of whether the forecast is from the recurrent or vector-based approach.

We also know that the window length m, the number of components k, and the forecasting methods (recurrent and vector) affect the forecasting outcome. Thus, we adopt a computationally intensive approach by considering combinations of m and k, and methods that provide the minimum RMSFE for the out-of-sample forecast for the last 12 data points. The RMSFEs obtained from the computationally intensive approach are given in

Table A3. Out-of-sample forecasting RMSFE.

Series’	h = 1			h = 3
Code	NT	Std	Log	NT	Std	Log
A001	1.283	0.542	1.144	1.884	1.157	1.715
A002	36.275	35.019	28.844	36.991	35.900	30.741
A003	12,521.688	13,643.067	13,616.737	16,041.250	16,584.228	17,449.138
A004	0.250	0.150	0.139	0.792	0.354	0.333
A005	61.625	61.548	60.476	53.906	53.268	58.074
A006	0.068	0.063	0.067	0.100	0.107	0.099
A007	338.358	511.055	288.753	511.033	560.970	331.925
A008	7.129	5.667	7.505	19.200	16.096	17.845
A009	0.042	0.040	0.042	0.051	0.051	0.051
A010	0.122	0.107	0.155	0.268	0.306	0.417
A011	0.338	0.229	0.286	0.831	0.407	0.560
A012	0.984	0.963	1.049	1.374	1.410	1.386
A013	1.345	1.101	1.395	3.141	2.971	7.484
A014	8.096	6.829	6.410	9.515	9.810	9.638
A015	7.24 × 10${}^9$	6.45 × 10${}^9$	6.31 × 10${}^9$	1.1 × 10${}^{10}$	8.45 × 10${}^9$	7.08 × 10${}^9$
A016	1.28 × 10${}^{10}$	1.46 × 10${}^{10}$	1.56 × 10${}^{10}$	1.76 × 10${}^{10}$	1.74 × 10${}^{10}$	1.81 × 10${}^{10}$
A017	12.423	9.066	Inf	19.782	15.435	Inf
A018	7.950	8.093	10.205	15.132	12.983	16.137
A019	1.429	1.425	1.375	1.531	1.510	1.469
A020	1.319	1.389	1.669	1.363	1.482	1.429
A021	1.070	1.076	1.051	1.129	1.147	1.122
A022	1.133	1.209	1.152	1.280	1.270	1.275
A023	6.097	5.936	5.309	6.551	6.674	5.980
A024	0.959	0.771	0.954	1.067	0.971	1.096
A025	22.689	26.924	56.529	26.056	43.196	49.542
A026	1.174	1.212	2.490	1.686	1.787	3.475
A027	0.050	0.100	0.064	0.114	0.509	0.226
A028	4137.576	4218.129	4038.143	4474.756	4199.967	4183.622
A029	59.124	44.474	52.390	62.490	69.349	78.321
A030	15.207	31.175	16.755	24.388	51.218	32.464
A031	8.783	5.662	8.633	80.118	8.464	18.103
A032	9.779	10.315	9.972	12.431	13.093	12.748
A033	5.820	5.432	5.791	9.729	8.527	10.148
A034	3.061	2.785	3.320	5.796	5.286	6.157
A035	0.965	1.455	5.973	1.536	2.155	6.234
A036	0.151	0.175	0.186	0.169	0.279	0.249
A037	0.293	0.310	0.308	0.417	0.395	0.368
A038	1.923	1.243	3.462	2.427	1.370	2.474
A039	4.853	3.508	5.107	7.494	6.099	9.125
A040	489.909	614.577	717.710	815.463	785.927	929.787
A041	0.329	0.322	0.328	0.390	0.408	0.389
A042	68.459	82.182	67.108	132.417	212.367	118.468
A043	33.081	33.066	33.750	41.996	40.189	43.350
A044	420.634	389.750	545.116	538.590	552.070	726.264
A045	0.522	0.522	0.886	0.999	0.998	1.297
A046	15.552	1.906	1.169	18.721	5.275	3.773
A047	8.206	8.222	11.116	8.679	8.640	10.166
A048	3.15 × 10${}^5$	1.66 × 10${}^5$	1.79 × 10${}^7$	3.82 × 10${}^5$	1.95 × 10${}^5$	595,729.790
A049	1.189	1.248	1.199	1.277	1.377	1.285
A050	18.038	128.254	17.562	37.219	295.980	35.731

NT = No Transformation, Std = Standardisation, and Log = Logarithmic.

,

Table A4. Out-of-sample forecasting RMSFE (Continuation).

Series’	h = 1			h = 3
Code	NT	Std	Log	NT	Std	Log
A051	3.983	3.976	4.003	5.694	5.612	5.605
A052	272.279	276.113	574.713	268.784	271.246	445.832
A053	35.559	39.680	36.963	26.795	32.500	31.927
A054	124.519	89.800	125.412	110.606	88.796	107.684
A055	43.121	37.090	44.808	34.715	37.302	40.039
A056	266.333	99.502	1.43E+12	287.931	214.556	9.42 × 10${}^{88}$
A057	125.600	84.462	126.023	131.253	92.122	129.780
A058	38.474	35.384	71.104	119.964	99.107	139.656
A059	44.950	41.240	45.696	45.079	40.224	45.094
A060	7.598	8.085	7.845	8.248	9.090	8.709
A061	6819.116	7597.052	23,730.348	10,097.877	11,645.535	16,058.889
A062	8.44 × 10${}^6$	7.04 × 10${}^6$	1.37 × 10${}^7$	1.42 × 10${}^7$	8.94 × 10${}^6$	1.76 × 10${}^7$
A063	21.829	21.831	13.583	26.600	26.655	10.258
A064	4393.038	3077.310	4376.077	5016.437	2925.211	4980.827
A065	28.982	11.405	27.430	30.717	16.662	30.903
A066	12.033	10.131	15.854	19.196	16.703	28.192
A067	11,923.554	11,039.522	10,617.132	17,077.208	13,448.762	13,328.422
A068	362.752	357.340	369.231	462.893	433.739	473.690
A069	160.579	203.037	208.287	203.002	208.562	230.166
A070	14.483	13.741	14.152	29.635	26.206	29.278
A071	26.793	27.217	23.647	27.381	33.930	25.245
A072	1379.200	1382.348	1472.325	1565.464	1624.687	1401.969
A073	69.327	69.141	68.699	122.183	114.652	115.324
A074	3294.883	2015.225	3445.829	3741.524	2288.009	3749.168
A075	48.901	59.574	58.507	41.848	117.860	64.366
A076	25,153.667	29,044.831	19,684.339	35,607.579	58,525.282	21,322.355
A077	394.752	387.456	395.114	873.390	813.589	836.260
A078	701.741	1275.259	790.650	1805.609	4921.674	1802.354
A079	35.709	34.064	35.661	45.108	43.559	45.010
A080	8.947	7.183	9.725	13.505	11.505	19.930
A081	498.376	530.862	473.551	380.003	447.889	438.681
A082	9.233	7.292	5.204	11.262	9.342	6.710
A083	1.291	1.137	1.225	1.621	1.477	1.518
A084	21,495.185	9111.162	11,832.143	32,355.027	9641.016	11,414.744
A085	0.883	0.862	0.641	2.054	1.640	1.273
A086	3.725	2.874	3.613	5.016	4.500	4.665
A087	1.035	1.273	0.768	1.408	1.958	1.148
A088	7.109	7.672	6.258	5.385	7.010	5.581
A089	0.862	1.170	1.025	2.248	2.282	2.331
A090	2.164	2.428	2.081	2.755	2.609	2.373
A091	240.568	124.286	129.086	1376.708	148.271	160.964
A092	3.35 × 10${}^{31}$	31,233.891	16,483.627	2.38× 10${}^{32}$	71,880.798	40,209.373
A093	63.119	5.79× 10${}^{25}$	54.632	300.893	1.35× 10${}^{26}$	76.301
A094	44,254.670	66,245.621	66,414.588	76,182.034	86,009.422	91,714.035
A095	136.663	139.571	144.039	287.480	311.372	265.696
A096	58.558	80.578	67.889	65.715	79.429	70.496
A097	1.42× 10${}^5$	144,364.409	143,654.990	192,501.733	182,442.168	192,581.617
A098	441.676	476.749	173.231	691.051	595.127	372.177
A099	3.199	3.168	4.478	3.236	3.075	5.052
A100	79.931	90.467	79.684	132.074	118.099	109.238

NT = No Transformation, Std = Standardisation, and Log = Logarithmic.

,

Table A5. Out-of-sample forecasting RMSFE (Continuation).

Series’	h = 6			h = 12
Code	NT	Std	Log	NT	Std	Log
A001	3.083	2.326	2.919	5.593	4.355	5.503
A002	37.593	36.769	33.318	39.847	38.221	37.346
A003	16,770.672	17,357.863	16,657.420	15,925.414	18,493.303	16,868.789
A004	0.709	0.455	0.446	0.715	0.639	0.585
A005	63.208	61.157	63.065	61.792	60.274	61.740
A006	0.140	0.144	0.138	0.209	0.194	0.204
A007	642.282	522.970	388.967	613.790	550.802	482.934
A008	36.757	22.657	32.054	31.678	25.325	31.028
A009	0.063	0.064	0.063	0.091	0.092	0.091
A010	0.381	0.489	0.515	0.492	0.908	2.268
A011	0.964	0.817	0.689	0.929	1.592	0.977
A012	1.856	1.994	1.782	2.536	2.947	2.197
A013	4.561	3.983	142.109	3.901	3.624	2.37 × 10${}^7$
A014	10.397	9.917	10.106	13.580	12.915	13.602
A015	1.92 × 10${}^{10}$	1.12 × 10${}^{10}$	8.94 × 10${}^9$	2.86 × 10${}^{10}$	1.65 × 10${}^{10}$	1.14 × 10${}^{10}$
A016	2.44 × 10${}^{10}$	2.09 × 10${}^{10}$	2.15 × 10${}^{10}$	4.10 × 10${}^{10}$	2.70 × 10${}^{10}$	2.80 × 10${}^{10}$
A017	30.286	23.902	Inf	46.368	28.383	Inf
A018	21.450	19.146	20.342	34.721	21.988	28.244
A019	1.555	1.436	1.447	1.517	1.476	1.511
A020	1.330	1.391	1.435	1.387	1.440	1.557
A021	1.138	1.134	1.092	1.126	1.134	1.166
A022	1.265	1.239	1.287	1.321	1.265	1.273
A023	6.861	6.813	6.278	7.870	7.750	7.283
A024	1.198	1.293	1.283	1.396	1.943	1.555
A025	29.947	44.077	78.266	33.726	57.839	467.347
A026	2.515	3.076	4.651	2.847	4.475	5.937
A027	0.152	13.916	0.486	0.180	12187.788	0.889
A028	4436.727	4208.136	3995.665	2687.645	3283.876	2860.657
A029	70.063	104.764	108.981	80.046	153.812	222.842
A030	40.923	103.102	82.010	50.163	1302.044	200.370
A031	1557.631	12.751	9.338	2.16E+25	16.890	348,877.932
A032	15.136	13.364	14.781	20.471	11.586	19.519
A033	16.619	11.811	14.619	338.296	212.221	31,730.543
A034	10.100	9.151	11.136	27.066	16.203	24.326
A035	2.554	3.283	6.623	4.415	5.513	7.378
A036	0.190	0.179	0.199	0.259	0.241	0.237
A037	0.542	0.494	0.467	0.706	0.795	0.771
A038	2.077	1.588	2.504	4.153	2.112	3.248
A039	9.958	7.750	15.538	12.330	9.615	27.556
A040	1185.420	967.918	1187.496	1781.242	1087.955	1476.007
A041	0.437	0.491	0.437	0.537	0.630	0.536
A042	282.364	652.016	211.125	1844.972	4.31 × 10${}^6$	488.603
A043	68.250	65.163	82.580	114.347	100.176	263.026
A044	467.834	637.165	587.869	511.228	585.946	626.670
A045	1.422	1.419	1.661	1.334	1.329	1.570
A046	23.722	11.922	9.536	35.328	28.669	19.088
A047	8.883	8.557	10.191	9.115	8.849	9.983
A048	6.35 × 10${}^5$	2.25 × 10${}^5$	Inf	2.73 × 10${}^6$	2.71 × 10${}^5$	Inf
A049	1.353	1.320	1.355	1.326	1.424	1.338
A050	59.765	528.428	56.831	103.999	935.576	99.881

NT = No Transformation, Std = Standardisation, and Log = Logarithmic.

and

Table A6. Out-of-sample forecasting RMSFE (Continuation).

Series’	h = 6			h = 12
Code	NT	Std	Log	NT	Std	Log
A051	6.645	6.646	6.689	8.259	39.667	8.384
A052	327.886	333.472	349.271	519.432	455.743	539,109.574
A053	55.656	71.070	56.373	77.760	88.068	80.306
A054	135.441	107.388	114.467	121.277	114.368	111.057
A055	47.052	47.075	49.962	44.947	49.411	44.323
A056	318.035	442.579	Inf	369.935	1397.180	Inf
A057	111.700	97.869	110.430	76.521	123.163	78.379
A058	93.679	73.906	161.798	76.617	72.198	33.833
A059	47.999	43.077	49.706	47.200	38.382	50.650
A060	9.065	9.929	8.915	9.775	11.311	9.225
A061	21,401.308	21,029.664	34,763.978	47,497.769	42,578.592	43,718.789
A062	1.53 × 10${}^7$	9.22 × 10${}^6$	1.43 × 10${}^7$	1.04 × 10${}^7$	9.77 × 10${}^6$	1.33 × 10${}^8$
A063	28.561	28.558	10.218	25.355	25.598	9.908
A064	4121.217	2945.853	3866.332	21881.052	3065.284	6688.179
A065	31.507	24.464	27.822	31.161	39.859	30.524
A066	33.907	24.501	26.523	85.870	39.738	25.640
A067	24,790.111	14,696.437	16,013.912	40,325.312	12,620.240	18,179.972
A068	490.894	450.505	499.947	327.430	426.795	335.514
A069	233.149	233.000	217.660	261.487	235.576	212.738
A070	21.055	17.474	36.299	18.092	15.445	16.239
A071	30.033	30.922	28.063	23.335	37.390	28.972
A072	991.918	1186.083	1013.985	1022.795	1148.504	1004.258
A073	191.317	173.546	170.028	371.023	236.600	288.816
A074	4012.015	2191.142	3290.446	8470.587	2279.084	3402.155
A075	40.891	115.551	33.467	44.112	228.585	43.708
A076	69,298.230	2.26 × 10${}^5$	24,927.158	2.63 × 10${}^5$	3.64 × 10${}^6$	7571.182
A077	1714.226	1532.812	1561.112	3608.945	2515.070	3097.836
A078	4173.555	654,416.059	3581.874	1.25 × 10${}^4$	1.01 × 10${}^9$	7095.955
A079	58.260	52.793	58.023	97.730	97.056	96.553
A080	16.268	13.189	14.747	12.158	13.096	15.346
A081	450.450	494.004	450.436	523.863	609.279	614.195
A082	10.665	10.620	7.961	10.362	7.757	10.242
A083	1.871	1.698	1.958	7.386	1.967	3.098
A084	74,374.861	11,949.864	15,030.189	4.54 × 10${}^5$	15,064.148	35,040.170
A085	2.972	2.375	2.443	5.394	3.867	4.246
A086	6.089	5.903	5.641	7.324	9.107	7.144
A087	1.517	2.552	1.521	2.522	3.060	2.358
A088	5.616	6.772	5.806	4.916	7.063	5.706
A089	3.882	2.942	3.045	5.597	3.709	4.223
A090	2.866	3.398	2.659	2.830	3.763	2.913
A091	28,312.543	254.885	326.947	1.39 × 10${}^7$	369.785	724.020
A092	7.46 × 10${}^{32}$	1.41 × 10${}^5$	73816.394	9.10 × 10${}^{32}$	3.81 × 10${}^5$	1.36 × 10${}^5$
A093	7814.412	1.30 × 10${}^{26}$	95.842	7.95 × 10${}^6$	2.84 × 10${}^{25}$	128.056
A094	1.02 × 10${}^5$	9.46 × 10${}^4$	1.10 × 10${}^5$	1.41 × 10${}^5$	1.30 × 10${}^5$	1.75 × 10${}^5$
A095	406.105	441.419	404.087	503.204	588.870	604.329
A096	78.448	90.126	75.130	100.969	104.199	83.458
A097	2.06 × 10${}^5$	1.92 × 10${}^5$	2.06 × 10${}^5$	2.44 × 10${}^5$	2.43 × 10${}^5$	2.42 × 10${}^5$
A098	858.043	625.258	751.271	1077.612	849.184	1205.582
A099	3.761	3.337	4.914	4.370	3.253	5.528
A100	140.073	132.262	141.576	188.195	173.609	194.600

NT = No Transformation, Std = Standardisation, and Log = Logarithmic.

.

Given that the best forecasting results are achieved by util ising a computationally intensive approach, we seek to identify the factors that can affect the RMSFE. In order to address this, we employ statistical tests described in Section 4.2. For each of the series with RMSFE reported in Table A3, Table A4, Table A5 and Table A6, we examine the characteristics of the time series by employing a statistical test, as described in Section 4.2. At this stage, we are ready with the inputs for nonparametric repeated measure factorial test to conduct testing on the treatment effect (data transformation) under different characteristics of these time series. Results obtained from the Wald type tests are provided in

Table 2. Wald-type test results.

Model	Factor	P-Value
		h = 1	h = 3	h = 6	h = 12
RMSFE ∼ Tr + Skew	Skew	0.0037	0.0043	0.0056	0.0131
+ Tr × Skew	Tr	0.0718	0.1447	0.4186	0.2098
	Tr × Skew	0.4177	0.5106	0.2120	0.1482
RMSFE ∼ Tr + Stationarity	Stationarity	0.0997	0.053	0.0501	0.0248
+ Tr × Stationarity	Tr	0.2351	0.3754	0.7607	0.5276
	Tr × Stationarity	0.5160	0.6808	0.7678	0.3792
RMSFE ∼ Tr + Normality	Normality	0.5052	0.5320	0.4954	0.5820
+ Tr × Normality	Tr	0.0747	0.1152	0.5849	0.4892
	Tr × Normality	0.2492	0.3576	0.4042	0.4549
RMSFE ∼ Tr + Freq	Freq	0.0000	0.0000	0.0000	0.0000
+ Tr × Freq	Tr	0.0841	0.1194	0.1355	0.1143
	Tr × Freq	0.0000	0.0000	0.0000	0.0000
RMSFE ∼ Tr	Tr	0.4271	0.6740	0.9535	0.4860

Here, Freq, Skew, and Tr represent frequency, skewness, and transformation, respectively. Bold values show the significant effects at the $\alpha =0.05$ significance level.

.

Based on the Wald-type test results in Table 2, we may conclude that, at the $\alpha =0.05$ significance level,

• normality does not affect SSA forecasting performance;
• stationarity affects SSA forecasting performance in long-term forecasting (h = 12) but not at shorter horizons;
• skewness and sampling (observation) frequency affect SSA forecasting performance;
• transformation does not affect SSA forecasting performance, but the interaction between sampling frequencies and transformation is significant, which means the SSA performance is affected by transformation at some sampling frequencies.

The above findings are important in the practice for several reasons. First, in the real world, it is well known that most time series do not meet the assumption of normality. However, as the effect of normality and its interactions with transformations are not significant, when faced with normally distributed data, our findings indicate that there is no impact on the forecasting accuracy of SSA with or without data transformations. Furthermore, these findings also indicate that data transformations do not improve the forecast accuracy in non-normal data either. Secondly, we find that, when series are stationary, it affects the long-term forecasting accuracy of SSA. However, when generating short-term forecasts, the forecasting accuracy of SSA is not affected by stationarity. Thirdly, in reality, as most time series are skewed and increasingly found at varying frequencies (especially following the emergence of Big Data), these findings show that forecasters should remember that varying skewness and frequency of data are features indicative of the need for careful exploration of the use of SSA as the forecasts are sensitive to these features. In general, transformations are not required when forecasting with SSA, as there is no evidence of transformations impacting the SSA forecasting performance; however, there could be a significant impact at certain sampling frequencies. This indicates that, when modelling data with different frequencies, the sensitivity of SSA forecasts to such frequencies could potentially be controlled by transforming the input data.

Since the interaction between sampling frequency and transformation is significant, we explore the relative effect of frequencies on RMSFE. Figure 2 shows the effect plot of treatment (transformation) for different forecast horizons $h=1,3,6,12$.

To explore the relative effects of sampling frequency for different forecast horizons, we plot the relative effect of frequencies in Figure 3 and Figure 4.

Sampling frequencies under investigation are hourly (F H), daily (F D), weekly (F W), monthly (F M), and annual (F A). When the relative effect plots in Figure 3 and Figure 4 are compared with the effect plots in Figure 2, we can evaluate how the hourly (F H), weekly (F W), quarterly (F Q), and annual (F A) sampling frequencies are affecting the forecasting performance of SSA. Moreover, the change in shape of the transform’s relative effects (e.g., see the difference between the shapes of “F Q” and “F H” lines in Figure 3 and Figure 4) suggests an interaction between transformation and sampling frequency.

We analyse the results by forecasting horizon. It can be seen in Figure 3 that, in very short-term forecasting ($h=1$), the standardisation produces a comparatively large RMSFE in quarterly frequencies, while the log transformation reports a slightly larger RMSFE at daily, quarterly, hourly, and annual frequencies. This indicates that users should certainly avoid transforming data with quarterly frequencies when forecasting at $h=1$ step ahead with SSA. In the short-term forecasting horizon ($h=3$) (see Figure 3), the smallest RMSFE belongs to standardisation for monthly frequencies, while standardisation has the largest RMSFE at quarterly frequencies. In mid- and long-term forecasting horizons ($h=6$ and 12), which are visible in Figure 4, the following can be seen. At $h=6$ steps ahead, standardisation produces the lowest RMSFE at monthly sampling frequencies, whilst it has the largest RMSFE in quarterly and weekly time series data. The log transformation produces higher RMSFEs at daily, hourly, and annual frequencies. Accordingly, the only instance when standardisation could produce better forecasts with SSA at this horizon is when faced with monthly data. At $h=12$ steps ahead, standardisation leads to better forecasts at daily frequencies, whilst log transformations can provide better forecasts with SSA at weekly frequencies.

Finally, these findings indicate that standardisation should only be used to transform data when forecasting with SSA at $h=12$ steps ahead at the daily frequency, at $h=3$ or $h=6$ steps ahead when dealing with a monthly frequency, and at $h=1$ step ahead when forecasting data with monthly or weekly frequencies. At the same time, standardisation should not be employed when forecasting quarterly data at any frequency, as it worsens the forecasting accuracy by comparatively larger margins. Interestingly, log transformations are only suggested when dealing with forecasting weekly data at $h=6$ or $h=12$ steps ahead. In the majority of the instances, SSA is able to provide superior forecasts without the need for data transformations when compared with time series following varied frequencies.

6. Concluding Remarks

This paper focused on evaluating the impact of data transformations on the forecasting performance of SSA, a nonparametric filtering and forecasting technique. Following a concise introduction, the paper introduces the SSA forecasting approaches followed by the transformation techniques considered here. Regardless of its popularity (and in contrast to other methods such as ARIMA and neural networks), there has been no empirical attempt to quantify the impact of data transformations on the forecasting capabilities of SSA. Accordingly, we consider the impact of standardisation and logarithmic transformations on the forecasting performance of both vector and recurrent forecasting in SSA. In order to ensure robustness within the analysis, we not only compare the forecasts using the RMSFE but also rely on a nonparametric repeated measure factorial test.

The forecast evaluation is based on 100 time series with varying characteristics in terms of frequencies, skewness, normality, and stationarity. Following the application of SSA to three versions of the same dataset, i.e. the original data, standardised data, and log transformed data, we generate out-of-sample forecasts at horizons of 1, 3, 6, and 12 steps ahead. Our findings indicate that, in general, data transformations do not affect SSA forecasts. However, the interaction between sampling frequency and transformations are found to be significant, indicating that data transformations are significant at certain sampling frequencies.

According to the results of this study, in time series with a higher sampling frequency (i.e. daily or hourly data), standardisation can improve SSA forecasting accuracy in the very long term at daily frequencies only. On the other hand, in time series with low sampling frequencies (i.e. quarterly and annual), neither logarithmic transformation nor standardisation is suitable across all horizons. In other time series’ sampling frequencies (weekly and monthly), data transformation with standardisation can affect all forecasting horizons (except $h=12$) when faced with monthly data and at $h=1$ step ahead when faced with weekly data. The results also show improvement in forecasting accuracy in weekly data with logarithmic transformations at $h=6$ and $h=12$ steps ahead. These findings provide additional guidance to forecasters, researchers, and practitioners alike in terms of improving the accuracy of forecasts when modelling data with SSA.

Future research should consider the relative gains of suggested data transformations at different sampling frequencies in relation to other benchmark forecasting models as well as theories explaining the mechanism of these effects in detail. Moreover, the development of automated SSA forecasting algorithms could be informed by the findings of this paper to ensure that data transformations are conducted prior to forecasting at selected sample frequencies.