The analysis of experimental data that have been observed at different points in time leads to new and unique problems in statistical modeling and inference. The obvious correlation introduced by the sampling of adjacent points in time can severely restrict the applicability of the many conventional statistical methods traditionally dependent on the assumption that these adjacent observations are independent and identically distributed. The systematic approach by which one goes about answering the mathematical and statistical questions posed by these time correlations is commonly referred to as time series analysis.

Necessity for Time Series Analysis

Time series analysis is required in many situations: deciding whether to build another power generation plant in the next five years requires forecasts of future demand; scheduling staff in a call centre next week requires forecasts of call volumes; stocking an inventory requires forecasts of stock requirements. Forecasts can be required several years in advance (for the case of capital investments), or only a few minutes beforehand (for telecommunication routing). Whatever the circumstances or time horizons involved, forecasting is an important aid to effective and efficient planning.The impact of time series analysis on scientific applications can be partially documented by producing an abbreviated listing of the diverse fields in which important time series problems may arise. For example, many familiar time series occur in the field of economics, where we are continually exposed to daily stock market quotations or monthly unemployment figures. Social scientists follow population series, such as birthrates or school enrollments. An epidemiologist might be interested in the number of influenza cases observed over some time period. In medicine, blood pressure measurements traced over time could be useful for evaluating drugs used in treating hypertension. Functional magnetic resonance imaging of brain-wave time series patterns might be used to study how the brain reacts to certain stimuli under various experimental  conditions.

Many of the most intensive and sophisticated applications of time series methods have been to problems in the physical and environmental sciences. This fact accounts for the basic engineering flavor permeating the language of time series analysis. One of the earliest recorded series is the monthly sunspot numbers studied by Schuster (1906). More modern investigations may center on whether a warming is present in global temperature measurements or whether levels of pollution may influence daily mortality in Los Angeles. The modeling of speech series is an important problem related to the efficient transmission of voice recordings. Common features in a time series characteristic known as the power spectrum are used to help computers recognize and translate speech. Geophysical time series such as those produced by yearly depositions of various kinds can provide long-range proxies for temperature and rainfall. Seismic recordings can aid in mapping fault lines or in distinguishing between earthquakes and nuclear explosions.

The above series are only examples of experimental databases that can be used to illustrate the process by which classical statistical methodology can be applied in the correlated time series framework. In our view, the first step in any time series investigation always involves careful scrutiny of the recorded data plotted over time.

There are two main goals of time series analysis: (a) identifying the nature of the phenomenon represented by the sequence of observations, and (b) forecasting (predicting future values of the time series variable). Both of these goals require that the pattern of observed time series data is identified and more or less formally described. Once the pattern is established, we can interpret and integrate it with other data (i.e., use it in our theory of the investigated phenomenon, e.g., seasonal commodity prices). Regardless of the depth of our understanding and the validity of our interpretation (theory) of the phenomenon, we can extrapolate the identified pattern to predict future events. 

What can be Forecasted?

Some things are easier to forecast than others. The time of the sunrise tomorrow morning can be forecast very precisely. On the other hand, tomorrow's lotto numbers cannot be forecast with any accuracy. The predictability of an event or a quantity depends on several factors including:

For example, forecasts of electricity demand can be highly accurate because all three conditions are usually satisfied. We have a good idea on the contributing factors: electricity demand is driven largely by temperatures, with smaller effects for calendar variation such as holidays, and economic conditions. Provided there is a sufficient history of data on electricity demand and weather conditions, and we have the skills to develop a good model linking electricity demand and the key driver variables, the forecasts can be remarkably accurate.

On the other hand, when forecasting currency exchange rates, only one of the conditions is satisfied: there is plenty of available data. However, we have a very limited understanding of the factors that affect exchange rates, and forecasts of the exchange rate have a direct effect on the rates themselves. If there are well-publicized forecasts that the exchange rate will increase, then people will immediately adjust the price they are willing to pay and so the forecasts are self-fulfilling. In a sense the exchange rates become their own forecasts. This is an example of the "efficient market hypothesis". Consequently, forecasting whether the exchange rate will rise or fall tomorrow is about as predictable as forecasting whether a tossed coin will come down as a head or a tail. In both situations, you will be correct about 50% of the time, whatever you forecast. In situations like this, forecasters need to be aware of their own limitations, and not claim more than is possible.

Often in forecasting, a key step is knowing when something can be forecast accurately, and when forecasts will be no better than tossing a coin. Good forecasts capture the genuine patterns and relationships which exist in the historical data, but do not replicate past events that will not occur again. In this course, we will learn how to tell the difference between a random fluctuation in the past data that should be ignored, and a genuine pattern that should be modelled and extrapolated.

Many people wrongly assume that forecasts are not possible in a changing environment. Every environment is changing, and a good forecasting model captures the way in which things are changing. Forecasts rarely assume that the environment is unchanging. What is normally assumed is that the way in which the environment is changing will continue into the future. That is, a highly volatile environment will continue to be highly volatile; a business with fluctuating sales will continue to have fluctuating sales; and an economy that has gone through booms and busts will continue to go through booms and busts. A forecasting model is intended to capture the way things move, not just where things are.

Forecasting situations vary widely in their time horizons, factors determining actual outcomes, types of data patterns, and many other aspects. Forecasting methods can be very simple such as using the most recent observation as a forecast (which is called the "naïve method''), or highly complex such as neural nets and econometric systems of simultaneous equations. Sometimes, there will be no data available at all. For example, we may wish to forecast the sales of a new product in its first year, but there are obviously no data to work with. In situations like this, we use judgemental forecasting. The choice of method depends on what data are available and the predictability of the quantity to be forecast.

Approaches to Time Series Analysis

This scrutiny often suggests the method of analysis as well as statistics that will be of use in summarizing the information in the data. Before looking more closely at the particular statistical methods, it is appropriate to mention that two separate, but not necessarily mutually exclusive, approaches to time series analysis exist, commonly identified as the time domain approach and the frequency domain approach.

The time domain approach is generally motivated by the presumption that correlation between adjacent points in time is best explained in terms of a dependence of the current value on past values. The time domain approach focuses on modeling some future value of a time series as a parametric function of the current and past values. In this scenario, we begin with linear regressions of the present value of a time series on its own past values and on the past values of other series. This modeling leads one to use the results of the time domain approach as a forecasting tool and is particularly popular with economists for this reason.

One approach, develops a systematic class of models called autoregressive integrated moving average (ARIMA) models to handle time-correlated modeling and forecasting. The approach includes a provision for treating more than one input series through multivariate ARIMA or through transfer function modeling. The defining feature of these models is that they are multiplicative models, meaning that the observed data are assumed to result from products of factors involving differential or difference equation operators responding to a white noise input.

A more recent approach to the same problem uses additive models more familiar to statisticians. In this approach, the observed data are assumed to result from sums of series, each with a specified time series structure; for example, in economics, assume a series is generated as the sum of trend, a seasonal effect, and error. The statespace model that results is then treated by making judicious use of the celebrated Kalman filters and smoothers, developed originally for estimation and control in space applications. 

Conversely, the frequency domain approach assumes the primary characteristics of interest in time series analyses relate to periodic or systematic sinusoidal variations found naturally in most data. These periodic variations are often caused by biological, physical, or environmental phenomena of interest. A series of periodic shocks may influence certain areas of the brain; wind may affect vibrations on an airplane wing; sea surface temperatures caused by El Niño oscillations may affect the number of fish in the ocean. The study of periodicity extends to economics and social sciences, where one may be interested in yearly periodicities in such series as monthly unemployment or monthly birth rates.

In spectral analysis, the partition of the various kinds of periodic variation in a time series is accomplished by evaluating separately the variance associated with each periodicity of interest. This variance profile over frequency is called the power spectrum. In our view, no schism divides time domain and frequency domain methodology, although cliques are often formed that advocate primarily one or the other of the approaches to analyzing data. In many cases, the two approaches may produce similar answers for long series, but the comparative performance over short samples is better done in the time domain. In some cases, the frequency domain formulation simply provides a convenient means for carrying out what is conceptually a time domain calculation.