Time series In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the Nile River at Aswan. Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values.

Time series are very frequently plotted via line charts. Time series data have a natural temporal ordering. This makes time series analysis distinct from other common data analysis problems, in which there is no natural ordering of the observations (e. g. explaining people’s wages by reference to their education level, where the individuals’ data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e. . accounting for house prices by the location as well as the intrinsic characteristics of the houses). A time series model will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility. Methods for time series analyses may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and recently wavelet analysis; the latter include auto-correlation and cross-correlation analysis. ————————————————- Models Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are theautoregressive (AR) models, the integrated (I) models, and the moving average (MA) models.

These three classes depend linearly[3] on previous data points. Combinations of these ideas produceautoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial “V” for “vector”.

An additional set of extensions of these models is available for use where the observed time-series is driven by some “forcing” time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter’s control. For these models, the acronyms are extended with a final “X” for “exogenous”. Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a chaotic time series.

However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models, over those from linear models, as for example in nonlinear autoregressive exogenous models. Among other types of non-linear time series models, there are models to represent the changes of variance along time (heteroskedasticity). These models represent autoregressive conditional heteroskedasticity (ARCH) and the collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc. ).

Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a doubly stochastic model. In recent work on model-free analyses, wavelet transform based methods (for example locally stationary wavelets and wavelet decomposed neural networks) have gained favor. Multiscale (often referred to as multiresolution) techniques decompose a given time series, attempting to illustrate time dependence at multiple scales.

Notation A number of different notations are in use for time-series analysis. A common notation specifying a time series X that is indexed by the natural numbers is written X = {X1, X2, … }. Another common notation is Y = {Yt: t ? T}, where T is the index set. Conditions There are two sets of conditions under which much of the theory is built: * Stationary process * Ergodicity However, ideas of stationarity must be expanded to consider two important ideas: strict stationarity and second-order stationarity.

Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified. In addition, time-series analysis can be applied where the series are seasonally stationary or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in time-frequency analysis which makes use of a time–frequency representation of a time-series or signal. [4] Models The general representation of an autoregressive model, well-known as AR(p), is here the term ? t is the source of randomness and is called white noise. It is assumed to have the following characteristics: * * * With these assumptions, the process is specified up to second-order moments and, subject to conditions on the coefficients, may be second-order stationary. If the noise also has a normal distribution, it is called normal or Gaussian white noise. In this case, the AR process may be strictly stationary, again subject to conditions on the coefficients. A Time Series (TS) is a sequence of observations ordered in time.

Mostly these observations are collected at equally spaced, discrete time intervals. When there is only one variable upon which observations are made then we call them a single time series or more specifically a univariate time series. A basic assumption in any time series analysis/modeling is that some aspects of the past pattern will continue to remain in the future. Also under this set up, often the time series process is assumed to be based on past values of the main variable but not on explanatory variables which may affect the variable/ system.

So the system acts as a black box and we may only be able to know about ‘what’ will happen rather than ‘why’ it happens. So if time series models are put to use, say, for instance, for forecasting purposes, then they are especially applicable in the ‘short term’. Here it is tacitly assumed that information about the past is available in the form of numerical data. Ideally, at least 50 observations are necessary for performing TS analysis/ modeling, as propounded by Box and Jenkins who were pioneers in TS modeling.

As far as utility of time series modeling in agriculture is concerned, its application in the area of statistical forecast modeling needs hardly any emphasis. Lack of timely forecasts of, say, agricultural production, especially short? term forecasts has often proved to be a major handicap to planners. Various statistical approaches viz. regression, time series and stochastic approaches are in vogue for arriving at crop forecasts. Every approach has its own advantages and limitations. Time series models have advantages in certain situations.

They can be used more easily for forecasting purposes because historical sequences of observations upon study variables are readily available from published secondary sources. These successive observations are statistically dependent and time series modeling is concerned with techniques for the analysis of such dependencies. Thus in time series modeling, the prediction of values for the future periods is based on the pattern of past values of the variable under study, but not generally on explanatory variables which may affect the system. There are two main reasons for resorting to such time series models.

First, the system may not be understood, and even if it is understood it may be extremely difficult to measure the cause and effect relationship, second, the main concern may be only to predict what will happen and not to know why it happens. Many a time, collection of information on causal factors (explanatory variables) Decomposition models are among the oldest approaches to time series analysis albeit a number of theoretical weaknesses from a statistical point of view. These were followed by the crudest form of forecasting methods called the moving averages method.

As an improvement over this method which had equal weights, exponential smoothing methods came into being which gave more weights to recent data. Exponential smoothing methods have been proposed initially as just recursive methods without any distributional assumptions about the error structure in them, and later, they were found to be particular cases of the statistically sound AutoRegressive Integrated Moving Average (ARIMA) models. In early 1970’s, Box and Jenkins pioneered in evolving methodologies for time series modeling in the univariate case often referred to as Univariate Box-Jenkins (UBJ) ARIMA modeling.

In course of time, various organisations/ workers in India and abroad have done modeling/ forecasting (of course, not necessarily for agricultural systems) based on time series data using the different methodologies viz. time series decomposition models, exponential smoothing models, ARIMA models and their variations such as seasonal ARIMA models, vector ARIMA models using multivariate time series, ARMAX models i. e. ARIMA with explanatory variables etc affecting the study variable(s) may be cumbersome /impossible and hence availability of long series data on explanatory variables is a problem.

In such situations, the time series models are a boon for forecasters. A good account on exponential smoothing methods is given in Makridakis. A practical treatment on ARIMA modeling along with several case studies can be found in Pankratz (1983). A reference book on ARIMA and related topics with a more rigorous theoretical flavour is by Box et al. 2. Time Series Components and Decomposition An important step in analysing TS data is to consider the types of data patterns, so that the models most appropriate to those patterns can be utilized.

Four types of time series components can be distinguished. They are Horizontal ? when data values fluctuate around a constant value Trend ? when there is long term increase or decrease in the data Seasonal ? when a series is influenced by seasonal factor and recurs on a regular periodic basis Cyclical ? when the data exhibit rises and falls that are not of a fixed period Note that many data series include combinations of the preceding patterns. After separating out the existing patterns in any time series data, the pattern that remains unidentifiable, form the ‘random’ or ‘error’ component.

Time plot (data plotted over time) and seasonal plot (data plotted against individual seasons in which the data were observed) help in visualizing these patterns while exploring the data. A crude yet practical way of decomposing the original data (ignoring cyclical pattern) is to go for a seasonal decomposition either by assuming an additive or multiplicative model viz. Yt = Tt + St + Et or Yt = Tt . St . Et A good account on exponential smoothing methods is given in Makridakis et al. (1998).

A practical treatment on ARIMA modeling along with several case studies can be found in Pankratz (1983). A reference book on ARIMA and related topics with a more rigorous theoretical flavour is by Box et al. 2. Time Series Components and Decomposition An important step in analysing TS data is to consider the types of data patterns, so that the models most appropriate to those patterns can be utilized. Four types of time series components can be distinguished. They are Horizontal ? when data values fluctuate around a constant value Trend ? when there is long term increase or decrease in the data

Seasonal ? when a series is influenced by seasonal factor and recurs on a regular periodic basis Cyclical ? when the data exhibit rises and falls that are not of a fixed period Note that many data series include combinations of the preceding patterns. After separating out the existing patterns in any time series data, the pattern that remains unidentifiable, form the ‘random’ or ‘error’ component. Time plot (data plotted over time) and seasonal plot (data plotted against individual seasons in which the data were observed) help in visualizing these patterns while exploring the data.

A crude yet practical way of decomposing the original data (ignoring cyclical pattern) is to go for a seasonal decomposition either by assuming an additive or multiplicative model viz. Yt = Tt + St + Et or Yt = Tt . St . Et Double moving averages The simple moving average is intended for data of constant and no trend nature. If the data have a linear or quadratic trend, the simple moving average will be misleading. In order to correct for the bias and develop an improved forecasting equation, the double moving average can be calculated.

To calculate this, simply treat the moving averages Mt[1] over time as individual data points and obtain a moving average of these averages. Simple Exponential smoothing(SES) Let the time series data be denoted by Y1, Y2,…,Yt. Suppose we wish to forecast the next value of our time series Yt+1 that is yet to be observed with forecast for Yt denoted by Ft. Then the forecast Ft+1 is based on weighting the most recent observation Yt with a weight value ? and weighting the most recent forecast Ft with a weight of (1-? ) where ? s a smoothing constant/ weight between 0 and 1 Thus the forecast for the period t+1 is given by Ft +1 = Ft +? (Yt ? Ft ) Note that the choice of ? has considerable impact on the forecast. A large value of ? (say 0. 9) gives very little smoothing in the forecast, whereas a small value of ? (say 0. 1) gives considerable smoothing. Alternatively, one can choose ? from a grid of values (say ? =0. 1,0. 2,…,0. 9) and choose the value that yields the smallest MSE value. If you expand the above model recursively then Ft+1 will come out to be a function of ? past yt values and F1. So, having known values of ? and past values of yt our point of concern relates to initializing the value of F1. One method of initialization is to use the first observed value Y1 as the first forecast (F1=Y1) and then proceed. Another possibility would be to average the first four or five values in the data set and use this as the initial forecast. However, because the weight attached to this user-defined F1 is minimal, its effect on Ft+1 is negligible. Double exponential smoothing (Holt) This is to allow forecasting data with trends.

The forecast for Holt’s linear exponential smoothing is found by having two more equations to SES model to deal with – one for level and one for trend. The smoothing parameters (weights) ? and ? can be chosen from a grid of values (say, each combination of ? =0. 1,0. 2,…,0. 9 and ? =0. 1,0. 2,…,0. 9) and then select the combination of ? and ? which correspond to the lowest MSE. In SPSS, the weights ? and ? can be chosen from a grid of values (say, each combination of ? =0. 1,0. 2,…,0. 9 and ? =0. 1,0. 2,…,0. 9) and then select the combination of ? and ? which correspond to the lowest MSE.