# Time Series Data Analysis And Theory Pdf

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*Psychological research has increasingly recognized the importance of integrating temporal dynamics into its theories, and innovations in longitudinal designs and analyses have allowed such theories to be formalized and tested. However, psychological researchers may be relatively unequipped to analyze such data, given its many characteristics and the general complexities involved in longitudinal modeling.*

- Time Series Analysis and Forecasting
- David R. Brillinger Time Series Data Analysis and Theory 2001
- Time series analysis for psychological research: examining and forecasting change

## Time Series Analysis and Forecasting

Psychological research has increasingly recognized the importance of integrating temporal dynamics into its theories, and innovations in longitudinal designs and analyses have allowed such theories to be formalized and tested.

However, psychological researchers may be relatively unequipped to analyze such data, given its many characteristics and the general complexities involved in longitudinal modeling. The current paper introduces time series analysis to psychological research, an analytic domain that has been essential for understanding and predicting the behavior of variables across many diverse fields.

First, the characteristics of time series data are discussed. Second, different time series modeling techniques are surveyed that can address various topics of interest to psychological researchers, including describing the pattern of change in a variable, modeling seasonal effects, assessing the immediate and long-term impact of a salient event, and forecasting future values.

To illustrate these methods, an illustrative example based on online job search behavior is used throughout the paper, and a software tutorial in R for these analyses is provided in the Supplementary Materials.

Although time series analysis has been frequently used many disciplines, it has not been well-integrated within psychological research. In part, constraints in data collection have often limited longitudinal research to only a few time points. However, these practical limitations do not eliminate the theoretical need for understanding patterns of change over long periods of time or over many occasions.

Psychological processes are inherently time-bound, and it can be argued that no theory is truly time-independent Zaheer et al. Further, its prolific use in economics, engineering, and the natural sciences may perhaps be an indicator of its potential in our field, and recent technological growth has already initiated shifts in data collection that proliferate time series designs.

For instance, online behaviors can now be quantified and tracked in real-time, leading to an accessible and rich source of time series data see Stanton and Rogelberg, As a leading example, Ginsberg et al. Importantly, this work was based in prior research showing how search engine queries correlated with virological and mortality data over multiple years Polgreen et al. Furthermore, although experience sampling methods have been used for decades Larson and Csikszentmihalyi, , nascent technologies such as smartphones allow this technique to be increasingly feasible and less intrusive to respondents, resulting in a proliferation of time series data.

As an example, Killingsworth and Gibert presented an iPhone Apple Incorporated, Cupertino, California application which tracks various behaviors, cognitions, and affect over time. At the time their study was published, their database contained almost a quarter of a million psychological measurements from individuals in 83 countries.

Finally, due to the growing synthesis between psychology and neuroscience e. Due to these overarching trends, we expect that time series data will become increasingly prevalent and spur the development of more time-sensitive psychological theory. Mindful of the growing need to contribute to the methodological toolkit of psychological researchers, the present article introduces the use of time series analysis in order to describe and understand the dynamics of psychological change over time.

In contrast to these current trends, we conducted a survey of the existing psychological literature in order to quantify the extent to which time series methods have already been used in psychological science. This search yielded a small sample of 36 empirical papers that utilized time series modeling. Further investigation revealed the presence of two general analytic goals: relating a time series to other substantive variables 17 papers and examining the effects of a critical event or intervention 9 papers; the remaining papers consisted of other goals.

Thus, this review not only demonstrates the relative scarcity of time series methods in psychological research, but also that scholars have primarily used descriptive or causal explanatory models for time series data analysis Shmueli, The prevalence of these types of models is typical of social science, but in fields where time series analysis is most commonly found e.

As a result, the statistical time series literature is dominated by models that are aimed toward prediction , not explanation Shmueli, , and almost every book on applied time series analysis is exclusively devoted to forecasting methods McCleary et al.

Although there are many well-written texts on time series modeling for economic and financial applications e. Thus, a psychologist looking to use these methodologies may find themselves with resources that focus on entirely different goals. The current paper attempts to amend this by providing an introduction to time series methodologies that is oriented toward issues within psychological research.

This is accomplished by first introducing the basic characteristics of time series data: the four components of variation trend, seasonality, cycles, and irregular variation , autocorrelation, and stationarity.

Then, various time series regression models are explicated that can be used to achieve a wide range of goals, such as describing the process of change through time, estimating seasonal effects, and examining the effect of an intervention or critical event. Not to overlook the potential importance of forecasting for psychological research, the second half of the paper discusses methods for modeling autocorrelation and generating accurate predictions—viz.

The final section briefly describes how regression techniques and ARIMA models can be combined in a dynamic regression model that can simultaneously explain and forecast a time series variable.

Thus, the current paper seeks to provide an integrative resource for psychological researchers interested in analyzing time series data which, given the trends described above, are poised to become increasingly prevalent. In order to better demonstrate how time series analysis can accomplish the goals of psychological research, a running practical example is presented throughout the current paper. For this particular illustration, we focused on online job search behaviors using data from Google Trends, which compiles the frequency of online searches on Google over time.

We were particularly interested in the frequency of online job searches in the United States 2 and the impact of the economic crisis on these rates. Our primary research hypothesis was that this critical event resulted in a sharp increase in the series that persisted over time.

The monthly frequencies of these searches from January to June were recorded, constituting a data set of 90 total observations. Importantly, the values of the series do not represent the raw number of Google searches, but have been normalized 0— in order to yield a more tractable data set; each monthly value represents its percentage relative to the maximum observed value 3.

A plot of the original Google job search time series and the series after seasonal adjustment. Conceptual expositions of new analytical methods can often be undermined by the practical issue of software implementation Sharpe, To preempt this obstacle, for each analysis we provide accompanying R code in the Supplementary Material, along with an intuitive explanation of the meanings and rationale behind the various commands and arguments.

On account of its versatility, the open-source statistical package R R Development Core Team, remains the software platform of choice for performing time series analyses, and a number of introductory texts are oriented solely toward this program, such as Introductory Time Series with R Cowpertwait and Metcalfe, , Time Series Analysis with Applications in R Cryer and Chan, , and Time Series Analysis and Its Applications with R Examples Shumway and Stoffer, In recent years, R has become increasingly recognized within the psychological sciences as well Muenchen, We believe that psychological researchers with even a minimal amount of experience with R will find this tutorial both informative and accessible.

Before introducing how time series analyses can be used in psychological research, it is necessary to first explicate the features that characterize time series data.

At its simplest, a time series is a set of time-ordered observations of a process where the intervals between observations remain constant e. Time series data is often distinguished from other types of longitudinal data by the number and source of the observations; a univariate time series contains many observations originating from a single source e.

The length of time series can vary, but are generally at least 20 observations long, and many models require at least 50 observations for accurate estimation McCleary et al.

More data is always preferable, but at the very least, a time series should be long enough to capture the phenomena of interest.

Due to its unique structure, a time series exhibits characteristics that are either absent or less prominent in the kinds of cross-sectional and longitudinal data typically collected in psychological research. In the next sections, we review these features that include autocorrelation and stationarity. However, we begin by delineating the types of patterns that may be present within a time series. That is, the variation or movement in a series can be partitioned into four parts: the trend, seasonal, cyclical , and irregular components Persons, Trend refers to any systematic change in the level of a series—i.

Both the direction and slope rate of change of a trend may remain constant or change throughout the course of the series. However, there are sections in this particular series that do not exhibit the same rate of increase. The beginning of the series displays a slight negative trend, and starting approximately at , the series significantly rises until , after which a small downward trend may even be present.

Because a trend in the data represents a significant source of variability, it must be accounted for when performing any time series analysis. That is, it must be either a modeled explicitly or b removed through mathematical transformations i.

The former approach is taken when the trend is theoretically interesting—either on its own or in relation to other variables. Conversely, removing the trend through methods discussed later is performed when this component is not pertinent to the goals of the analysis e.

The decision of whether to model or remove systematic components like a trend represents an important aspect of time series analysis. The various characteristics of time series data are either of theoretical interest—in which case they should be modeled—or not, in which case they should be removed so that the aspects that are of interest can be more easily analyzed.

Thus, it is incumbent upon the analyst to establish the goals of the analysis and determine which components of a time series are of interest and treat them accordingly. This topic will be revisited throughout the forthcoming sections. Unlike the trend component, the seasonal component of a series is a repeating pattern of increase and decrease in the series that occurs consistently throughout its duration. For instance, restaurant attendance may exhibit a weekly seasonal pattern such that the weekends routinely display the highest levels within the series across weeks i.

Retail sales often display a monthly seasonal pattern, where each month across yearly periods consistently exhibits the same relative position to the others: viz. Importantly, the pattern represented by a seasonal effect remains constant and occurs over the same duration on each occasion Hyndman and Athanasopoulos, Although its underlying pattern remains fixed, the magnitude of a seasonal effect may vary across periods.

Seasonal effects can also be embedded within overarching trends. After February, they continue to rise until about July or August, after which the series significantly drops for the remainder of the year, representing the effects of seasonal employment.

Notice the consistency of both the form i. The fact that online job search behavior exhibits seasonal patterns supports the idea that this behavior and this example in particular is representative of job search behavior in general.

In the United States, thousands of individuals engage in seasonal work which results in higher unemployment rates in the beginning of each year and in the later summer months e.

One may be interested in the presence of seasonal effects, but once identified, this source of variation is often removed from the time series through a procedure known as seasonal adjustment Cowpertwait and Metcalfe, , p.

This is in keeping with the aforementioned theme: Once a systematic component has been identified, it must either be modeled or removed.

The popularity of seasonal adjustment is due to the characteristics of seasonal effects delineated above: Unlike other more dynamic components of a time series, seasonal patterns remain consistent across periods and are generally similar in magnitude Hyndman and Athanasopoulos, Their effects may also obscure other important features of time series—e.

Unemployment rates are often seasonally adjusted to remove the fluctuations due to the effects of weather, harvests, and school schedules that remain more or less constant across years. In our data, the seasonal effects of job search behavior are not of direct theoretical interest relative to other features of the data, such as the underlying trend and the impact of the economic crisis.

Thus, we may prefer to work with the simpler seasonally adjusted series. It can be seen that the trend is made notably clearer after removing the seasonal effects. Despite the spike at the very end, the suspected downward trend in the later part of the series is much more evident. This insight will prove to be important when selecting an appropriate time series model in the upcoming sections.

A cyclical component in a time series is conceptually similar to a seasonal component: It is a pattern of fluctuation i. However, unlike seasonal effects whose duration is fixed across occurrences and are associated with some aspect of the calendar e. Put simply, cycles are any non-seasonal component that varies in a recognizable pattern e.

In contrast to seasonal effects, cycles generally occur over a period lasting longer than 2 years although they may be shorter , and the magnitude of cyclical effects is generally more variable than that of seasonal effects Hyndman and Athanasopoulos, Furthermore, just as the previous two components—trend and seasonality—can be present with or without the other, a cyclical component may be present with any combination of the other two.

For instance, a trend with an intrinsic seasonal effect can be embedded within a greater cyclical pattern that occurs over a period of several years. Alternatively, a cyclical effect may be present without either of these two systematic components. This is expected, as there are no strong theoretical reasons to believe that online or job search behavior is significantly influenced by factors that consistently manifest across a period of over one year.

We have significant a priori reasons to believe that causal factors related to seasonality exist e. While the previous three components represented three systematic types of time series variability i.

It constitutes any remaining variation in a time series after these three systematic components have been partitioned out. In time series parlance, when this component is completely random i.

## David R. Brillinger Time Series Data Analysis and Theory 2001

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## Time series analysis for psychological research: examining and forecasting change

Many types of data are collected over time. Stock prices, sales volumes, interest rates, and quality measurements are typical examples. Because of the sequential nature of the data, special statistical techniques that account for the dynamic nature of the data are required.

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*Sign in. Whether we wish to predict the trend in financial markets or electricity consumption, time is an important factor that must now be considered in our models.*

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