# THE ANALYSIS OF TIME SERIES AN INTRODUCTION PDF

time series/The Analysis of Time Series - An Introduction 6ed_Chris sppn.info Loading latest commit This file is too big to show. Sorry! Desktop version. ANALYSIS. OF. TIME SERIES. AN INTRODUCTION. Fifth edition. Chris Chatfield. Reader in Statistics. The University of Bath. United Kingdom. CHAPMAN &. The Analysis of Time Series - An Introduction 6ed_Chris sppn.info - Ebook download as PDF File .pdf), Text File .txt) or read book online.

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An Introduction to State Space Time Series Analysis. Read more New Introduction to Multiple Time Series Analysis. Read more. Introduction. A time series is a set of observations xt, each one being recorded at a specific time t. Definition A time series model for the observed data {xt}. Time-series analysis is an area of statistics which is of particular interest at the Search within book. Front Matter. Pages i-xiv. PDF · Introduction. C. Chatfield.

Understanding a dataset, called time series analysis, can help to make better predictions, but is not required and can result in a large technical investment in time and expertise not directly aligned with the desired outcome, which is forecasting the future.

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In descriptive modeling, or time series analysis, a time series is modeled to determine its components in terms of seasonal patterns, trends, relation to external factors, and the like. Time series analysis involves developing models that best capture or describe an observed time series in order to understand the underlying causes. This often involves making assumptions about the form of the data and decomposing the time series into constitution components.

The quality of a descriptive model is determined by how well it describes all available data and the interpretation it provides to better inform the problem domain. The primary objective of time series analysis is to develop mathematical models that provide plausible descriptions from sample data Time Series Forecasting Making predictions about the future is called extrapolation in the classical statistical handling of time series data.

More modern fields focus on the topic and refer to it as time series forecasting. Forecasting involves taking models fit on historical data and using them to predict future observations. Descriptive models can borrow for the future i. An important distinction in forecasting is that the future is completely unavailable and must only be estimated from what has already happened.

The purpose of time series analysis is generally twofold: to understand or model the stochastic mechanisms that gives rise to an observed series and to predict or forecast the future values of a series based on the history of that series — Page 1, Time Series Analysis: With Applications in R.

The skill of a time series forecasting model is determined by its performance at predicting the future. This is often at the expense of being able to explain why a specific prediction was made, confidence intervals and even better understanding the underlying causes behind the problem. Stop learning Time Series Forecasting the slow way! Take my free 7-day email course and discover how to get started with sample code. Click to sign-up and also get a free PDF Ebook version of the course.

Components of Time Series Time series analysis provides a body of techniques to better understand a dataset. Perhaps the most useful of these is the decomposition of a time series into 4 constituent parts: Level. The baseline value for the series if it were a straight line. The optional and often linear increasing or decreasing behavior of the series over time. The optional repeating patterns or cycles of behavior over time.

The optional variability in the observations that cannot be explained by the model. Estimation in the Time Domain.

Stationary Processes in the Frequency Domain. Spectral Analysis. Bivariate Processes. Linear Systems. Some other Topics. Back Matter Pages About this book Introduction Time-series analysis is an area of statistics which is of particular interest at the present time. Time series arise in many different areas, ranging from marketing to oceanography, and the analysis of such series raises many problems of both a theoretical and practical nature.

I first became interested in the subject as a postgraduate student at Imperial College, when I attended a stimulating course of lectures on time-series given by Dr.

The subject has fascinated me ever since. Several books have been written on theoretical aspects of time-series analysis. However, it may be helpful to introduce here the idea of stationarity from an intuitive point of view.

Broadly speaking a time series is said to be stationary if there is no systematic change in mean no trend , if there is no systematic change in variance and if strictly periodic variations have been removed.

In other words, the properties of one section of the data are much like those of any other section. However, the phrase is often used for time-series data meaning that they exhibit characteristics that suggest a stationary model can sensibly be fitted. Much of the probability theory of time series is concerned with stationary time series, and for this reason time-series analysis often requires one to transform a non-stationary series into a stationary one so as to use this theory.

For example, it may be of interest to remove the trend and seasonal variation from a set of data and then try to model the variation in the residuals by means of a stationary stochastic process. However, it is also worth stressing that the non-stationary components, such as the trend, may be of more interest than the stationary residuals. This graph, called a time plot, will show up important features of the series such as trend, seasonality, outliers and discontinuities.

The plot is vital, both to describe the data and to help in formulating a sensible model, and several examples have already been given in Chapter 1. Plotting a time series is not as easy as it sounds. The choice of scales, the size of the intercept and the way that the points are plotted e. Nowadays, graphs are usually produced by computers. Some are well drawn but packages sometimes produce rather poor graphs and the reader must be prepared to modify them if necessary or, better, give the computer appropriate.

Page 14 instructions to produce a clear graph in the first place. For example, the software will usually print out the title you provide, and so it is your job to provide a clear title. It cannot be left to the computer. Further advice and examples are given in Chapter The three main reasons for making a transformation are as follows. In particular, if the standard deviation is directly proportional to the mean, a logarithmic transformation is indicated.

On the other hand, if the variance changes through time without a trend being present, then a transformation will not help. Instead, a model that allows for changing variance should be considered. The seasonal effect is then said to be additive.

In particular, if the size of the seasonal effect is directly proportional to the mean, then the seasonal effect is said to be multiplicative and a logarithmic transformation is appropriate to make the effect additive. However, this transformation will only stabilize the variance if the error term is also thought to be multiplicative see Section 2.

The logarithmic and square-root transformations, mentioned above, are special cases of a general class of transformations called the Box-Cox transformation. Page 15 This is effectively just a power transformation when , as the constants are introduced to make yt a continuous function of at the value.

It is instructive to note that Nelson and Granger found little improvement in forecast performance when a general Box-Cox transformation was tried on a number of series.

There are problems in practice with transformations in that a transformation, which makes the seasonal effect additive, for example, may fail to stabilize the variance. Thus it may be impossible to achieve all the above requirements at the same time. In any case a model constructed for the transformed data may be less than helpful.

This can introduce biasing effects. My personal preference is to avoid transformations wherever possible except where the transformed variable has a direct physical interpretation. For example, when percentage increases are of interest, then taking logarithms makes sense see Example Further general remarks on transformations are given by Granger and Newbold , Section It is much more difficult to give a precise definition of trend and different authors may use the term in different ways.

The trend in Equation 2. In practice, this generally provides an unrealistic model, and nowadays there is more emphasis on models that allow for local linear trends. One possibility is to fit a piecewise linear model where the trend line is locally linear but with change points where the slope and intercept change abruptly. It is usually arranged that the lines join up at the change points, but, even so, the sudden changes in slope often seem unnatural.

Thus, it often seems more sensible to look at models that allow a smooth transition between the different submodels. Some examples of suitable models, under. Page 16 the general heading of state-space models, are given in Chapter Another possibility, depending on how the data look, is that the trend has a non-linear form, such as quadratic growth. Exponential growth can be particularly difficult to handle, even if logarithms are taken to transform the trend to a linear form.

Even with present-day computing aids, it can still be difficult to decide what form of trend is appropriate in a given context see Ball and Wood and the discussion that followed.

It also depends on whether the data exhibit seasonality see Section 2. With seasonal data, it is a good idea to start by calculating successive yearly averages, as these will provide a simple description of the underlying trend.

An approach of this type is sometimes perfectly adequate, particularly if the trend is fairly small, but sometimes a more sophisticated approach is desired. We now describe some different general approaches to describing trend. The global linear trend in Equation 2. Fitting the curves to data may lead to non-linear simultaneous equations. For all curves of this type, the fitted function provides a measure of the trend, and the residuals provide an estimate of local fluctuations, where the residuals are the differences between the observations and the corresponding values of the fitted curve.

Page 17 2. Moving averages are discussed in detail by Kendall et al. The simple moving average is not generally recommended by itself for measuring trend, although it can be useful for removing seasonal variation.

As q gets large, the weights approximate to a normal curve. A fourth example, called the Henderson moving average, is described by Kenny and Durbin and is widely used, for example, in the X and X seasonal packages see Section 2.

This moving average aims to follow a cubic polynomial trend without distortion, and the choice of q depends on the degree of irregularity. The symmetric nine-term moving average, for example, is given by The general idea is to fit a polynomial curve, not to the whole series, but to a local set of points. Whenever a symmetric filter is chosen, there is likely to be an end-effects problem e.

In some situations this may not. Page 18 be important, as, for example, in carrying out some retrospective analyses. The analyst can project the smoothed values by eye or, alternatively, can use an asymmetric filter that only involves present and past values of xt.

For example, the popular technique known as exponential smoothing see Section 5. Having estimated the trend, we can look at the local fluctuations by examining. How do we choose the appropriate filter? The answer to this question really requires considerable experience plus a knowledge of the frequency aspects of time-series analysis, which will be discussed in later chapters.

As the name implies, filters are usually designed to produce an output with emphasis on variation at particular frequencies. For example, to get smoothed values we want to remove the local fluctuations that constitute what is called the high-frequency variation.

In other words we want what is called a low-pass filter. To get Res xt , we want to remove the long-term fluctuations or the low-frequency variation. In other words we want what is called a high-pass filter. The Slutsky or Slutsky-Yule effect is related to this problem. Slutsky showed that by operating on a completely random series with both averaging and differencing procedures one could induce sinusoidal variation in the data.

Slutsky went on to suggest that apparently periodic behaviour in some economic time series might be accounted for by the smoothing procedures used to form the data. We will return to this question later.

Filters in series A smoothing procedure may be carried out in two or more stages. Page 19 As an example, two filters in series may be represented as on the opposite page. It is easy to show that a series of linear operations is still a linear filter overall. We will see that this method is an integral part of the so-called BoxJenkins procedure. For non-seasonal data, first-order differencing is usually sufficient to attain apparent stationarity.

Occasionally second-order differencing is required using the. First differencing is widely used and often works well. For example, Franses and Kleibergen show that better out-of-sample forecasts are usually obtained with economic data by using first differences rather than fitting.

For series showing little trend.

The analysis of time series. Seasonal differencing. With monthly data. Model A describes the additive case. For series that do contain a substantial trend. January by finding the average of each January observation minus the corresponding yearly average in the additive case.

The time plot should be examined to see which model is likely to give the better description. In model C the error term is also multiplicative. The indices are usually normalized so that they sum to zero in the additive case. A mixed additive-multiplicative seasonal model is described by Durbin and Murphy A simple moving average cannot be used. For quarterly data. It is a fairly complicated procedure that employs a series of linear filters and adopts a recursive approach.

On mainland Europe. These smoothing procedures all effectively estimate the local deseasonalized level of the series. Preliminary estimates of trend are used to get preliminary estimates of seasonal variation.

Two general reviews of methods for seasonal adjustment are Butter and Fase and Hylleberg Without going into great detail.

A simple moving average over 13 months cannot be used. Bell and Hillmer.

A check should be made that the seasonals are reasonably stable. The new software for X gives the user more flexibility in handling outliers. A seasonal effect can also be eliminated by a simple linear filter called seasonal differencing.

X also allows the user to deal with the possible presence of calendar effects. Given N observations x1. We assume that the reader is familiar with the ordinary correlation coefficient.

If the two variables are independent. Regarding the first observation in each pair as one variable. It can easily be shown that the value does not depend on the units in which the two variables are measured. In Chapter 4. They measure the correlation. We then compute 2. In practice the autocorrelation coefficients are usually calculated by computing the series of autocovariance coefficients.

As the coefficient given by Equation 2. In a similar way. This gives the even simpler formula 2. Note that some authors prefer to use rather than Equation 2.

The correlogram may alternatively be called the sample autocorrelation function ac. Of course. See also Section 4. A visual inspection of the correlogram is often very helpful.

Examples are given in Figures 2. Values of rk for longer lags tend to be approximately zero. Random series A time series is said to be completely random if it consists of a series of independent observations having the same distribution.

An example of such a correlogram is shown in Figure 2. Here we offer some general advice. As a result. In fact we will see later that. We note. This spotlights one of the difficulties in interpreting the correlogram.

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Short-term correlation Stationary series often exhibit short-term correlation characterized by a fairly large value of r1 followed by one or two further coefficients.

A time series that gives rise to such a correlogram is one for which an observation above the mean tends to be followed by one or more further observations above the mean.

Figure 2. Alternating series If a time series has a tendency to alternate. With successive values on opposite sides of the mean. An alternating time series together with its correlogram is shown in Figure 2. The sinusoidal pattern of the correlogram is clearly evident.

This is because an observation on one side of the overall mean tends to be followed by a large number of further observations on the same side of the mean because of the trend. A typical nonstationary time series together with its correlogram is shown in Figure 2. Seasonal series If a time series contains seasonal variation.

Little can be inferred from a correlogram of this type as the trend dominates all other features. In particular if xt follows a sinusoidal pattern. In fact the sample ac. This indicates short-term correlation in that a month.

The dotted lines in b are at. Values outside these lines are said to be significantly different from zero. Note that it is generally wise to look at coefficients covering at least three seasons.

The seasonal variation was removed from the air temperature data by the simple. The correlogram of the resulting series Figure 2.

Outliers If a time series contains one or more outliers. If the seasonal variation is removed from seasonal data. Chapter 2. It is also necessary to investigate the sampling properties of rk. If the series really is random. Tests of the above types will not be described here. This may show up trend. Various tests exist for this purpose as described. Section One type of approach is to carry out what is called a test of randomness in which one tests whether the observations x1.

It is convenient to briefly mention such tests here. An alternative type of test is based on runs of observations. A converse definition applies to local minima. In addition it is necessary to study the probability theory of stationary series and learn about the classes of models that may be appropriate.

One type of test is based on counting the number of turning points. General remarks Considerable experience is required to interpret sample autocorrelation coefficients.

This may show up shortterm correlation. These topics will be covered in the next two chapters and we will then be in a better position to interpret the correlogram of a given time series.

This can often be done visually. The latter test can also be used when assessing models by means of a residual analysis. Under the null hypothesis of randomness. After cleaning the data. If so. It really is important to avoid being driven to bad conclusions by bad data. The context of the problem is crucial in deciding how to modify data. The analyst should also deal with any other known peculiarities.

Data cleaning often arises naturally during a simple preliminary descriptive analysis. The process of checking through data is often called cleaning the data.

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Assessing the structure and format of the data is a key step. This can sometimes be done using fairly crude devices.

In reality. Practitioners will tell you that these types of questions often take longer to sort out than might be expected. In my experience. It is an essential precursor to attempts at modelling data. This explains why it is essential to get background knowledge about the problem. An even more basic question is whether the most appropriate variables have been measured in the first place. A corollary is that it is difficult to make any general remarks or give general recommendations on data cleaning.

Testing residuals for randomness will be discussed in Section 4. Data cleaning could include modifying outliers. We close by giving the following checklist of possible actions. You will need to use the trigonometrical results listed in Section 7. Exercises 2. Using Equation 7. Does the operator transform Xt to stationarity? If not. Show that the seasonal difference operator acts on Xt to produce a stationary series.

Is there any evidence of non-randomness? As stationarity is not formally defined until Chapter 3. Examples include the length of a queue. Grimmett and Stirzaker Some stochastic processes. The properties of the latter are typically determined by the investigator. Papoulis In this chapter we concentrate on those aspects particularly relevant to timeseries analysis. The theory of stochastic processes has been extensively developed and is discussed in many books including Cox and Miller Most statistical problems are concerned with estimating the properties of a population from a sample.

Some tools for describing the properties of such models are specified and the important notion of stationarity is formally defined.

In time-series analysis. A time series is said to be strictly stationary if the joint distribution of X t1. These functions will now be defined for continuous time. Time-series analysis is essentially concerned with evaluating the properties of the underlying probability model from this observed time series. Every member of the ensemble is a possible realization of the stochastic process.

More generally. The variance function is a special case of the acv. A heuristic idea of stationarity was introduced in Section 2. From this model.

When applied to a sequence of random variables. N if time is discrete. X tk for any set of times t1. Many models for stochastic processes are expressed by means of an algebraic formula relating the random variable at time t to past values of the process. Higher moments of a stochastic process may be defined in an obvious way.

The observed time series can be thought of as one particular realization. This infinite set of time series is sometimes called the ensemble. The variance function alone is not enough to specify the second moments of a sequence of random variables. Nevertheless we may regard the observed time series as just one example of the infinite set of time series that might have been observed. A simpler. Of course the conditional distribution of X t2 given that X t1 has taken a particular value.

The size of an autocovariance coefficient depends on the units in which X t is measured. Thus once such a process has been running for some time. Thus the acv.

At first sight it may seem surprising to suggest that there are processes for which the distribution of X t should be the same for all t. The above definition holds for any value of k. Indeed if the initial conditions are specified to be identical to the equilibrium distribution. Its empirical counterpart was introduced in Section 2.

The definition also implies that both the variance and the mean must be finite. This weaker definition of stationarity will generally be used from now on. A process is called second-order stationary or weakly stationary if its mean is constant and its acv. This section investigates the general properties of the ac. Property 1: The ac. X tk is multivariate normal for all t1. The multivariate normal distribution is completely characterized by its first and second moments.

One important class of processes where this is particularly true is the class of normal processes where the joint distribution of X t1. Similarly the theoretical ac. It is usually possible to find many normal and non-normal processes with the same ac. It is proved by noting that for any constants 1.

Although a given stochastic process has a unique covariance structure. We normally further assume that the random variables are normally distributed with mean zero and variance. Even for stationary normal processes. Property 3: Jenkins and Watts From the definition it follows that the process has constant mean and variance. This is adequate for linear. The possibility of defining a continuous-time purely random process is discussed in Section 3.

In fact the independence assumption implies that the process is also strictly stationary. The best-known examples of time series. A model. Note that a purely random process is sometimes called white noise. Purely random processes are useful in many situations. As the mean and variance change with t. We find immediately that since the Zs are independent.

Thus we cannot identify an MA process uniquely from a given ac. The imposition of the invertibility condition ensures that there is a unique MA process for a given ac. Are you surprised? It can be shown that an MA q process is invertible if the roots of the equation all lie outside the unit circle. Autoregressive processes are introduced below in Section 3.

The invertibility condition for an MA process of any order is best expressed by using the backward shift operator. This effectively means that the process can be rewritten in the form of an autoregressive process. Consider the following first-order MA processes: It can easily be shown that these two different processes have exactly the same ac.

It turns out that an estimation procedure for an MA process. Note that an arbitrary constant. Thus Equation 3. Such events will not only have an immediate effect but may also affect economic indicators to a lesser extent in several subsequent periods.

Rather than use successive substitution to explore this duality. First-order process For simplicity. This means that the roots. MA processes have been used in many areas. By successive substitution into Equation 3. The possibility that AR processes may be written in MA form. Then 3. This does not affect the ac. If we then multiply through Equation 3.

The above method of obtaining the ac. Then E Xt must be a constant. This may be done by successive substitution. Note how quickly the ac.

General-order process As in the first-order case. Figure 3. Having expressed Xt as an MA process. The variance is finite provided that for stationarity. The acv.

## Time Series

The alternative simpler way is to assume the process is stationary. Yule and Sir Gilbert Walker. It can be shown that an equivalent way of expressing the stationarity condition is to say that the roots of the equation 3. Of particular interest is the AR 2 process.

We can in principle find the ac. AR processes have been applied to many situations in which it is reasonable to assume that the present value of a time series depends linearly on the immediate past values together with a random error.

Example 3. For simplicity we have only considered processes with mean zero. A2 are also real and When the roots are real. For complex roots. See Example 3. In order to answer the first question we find the roots of Equation 3.

In order to find the ac. It is given by 3. Since is less than zero.

Equation 3. Note the values of the ac. As the rnodulus of both roots exceeds one. See Exercise 3. By convention we write AR model is in the form. This is an early example of what is often called the Principle of Parsimony. It is straightforward in principle. This says that we want to find a model with as few parameters as possible.

By comparison with Equation 3. The weights. Section 3. If the observed time series is non-stationary in the mean. It follows that the process is stationary and invertible.

The model for Xt is clearly non-stationary.

## Time Series and Forecasting Methods in NCSS

If Xt is replaced by a model capable of describing certain types of non-stationary series. In order to fit a stationary model. By analogy with a discrete-time purely random process. ARIMA models can be generalized to include seasonal terms. Here we are mainly concerned with indicating some of the mathematical problems that arise when time is continuous.

The latter have been used in some applications. This subsection2 gives a brief introduction to processes in continuous time. In practice.Unfortunately this cannot be done and so explicit least squares estimates cannot be found. Another variation of ES that deserves mention here is the use of a damped trend Gardner and McKenzie. This book devotes a greater amount of space to the more advanced techniques. Hand and C. There are analogous formulae for damped Holt-Winters. By convention we write AR model is in the form.