TIMES SERIES INTRODUCTION INTRODUCTION. Page 1. A time series is a set of observations made sequentially through time

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1 TIMES SERIES INTRODUCTION A time series is a set of observations made sequentially through time A time series is said to be continuous when observations are taken continuously through time, or discrete when observations are taken at discrete time, usually equally spaced, even when the measured variable is continuous. A continuous series is always sampled at equal intervals to give a discrete series INTRODUCTION If future values can be predicted exactly from past values then the series is said to be deterministic. However most series are stochastic in that the future is only partly determined by past values. An observed time series may be regarded as a realisation from a stochastic process, which is a family of random variables indexed over time. A stationary process has constant mean and variance and its other properties also do not change with time. Page 1 1

2 INTRODUCTION The special feature of time series analysis is that successive observations are usually not independent and so the analysis must take into account the order of the observations. The main (possible) objectives are: 1.to describe the data 2. to find a suitable model 3. to forecast values 4. to control the future behavior of the series TIME SERIES ANALYSIS Description -First Steps The first step in the analysis is to construct a time plot of each series. Features such as trend (long term changes in the mean), seasonal variation, outliers, smooth changes in structures and sudden discontinuity will usually be evident. Simple descriptive statistics may also be calculated to help in summarising the data and model formulation. EXPLANATION When observations are taken on two or more variables, it may be possible to use variation in one time series to explain the variation in another. This may lead to a greater understanding of the mechanism which generated a given time series. The analysis of linear systems entails examining the properties of the input and output of a system to assess the properties of the linear system for example it is of interest to see how sales are affected by price and economic conditions. Input Linear System Output Page 2 2

3 PREDICTION Given an observed time series it is often required to predict future values of the series. This is important in sales forecasting inventory forecasting, and in the analysis of economic and industrial time series. Prediction is closely related to control problems in many situations. For example if it is possible to predict that a manufacturing process is going to move off target, then appropriate action can be taken. A TEST FOR RANDOMNESS Before discussing techniques for dealing with time series data exhibiting non-randomness. First consider a non-parametric test for testing for randomness called the runs test. The runs test is particularly easy to perform: Data is separated into two equally likely types (odd, even, empty, full etc) or for numerical data above and below the median. A run is a sequence of like elements that are preceded and followed by different elements or no elements at all. A TEST FOR RANDOMNESS For Example a sequence of 16 flips of a coin are unlikely to occur as follows: H T H T H T H T H T H T H T H T There are 16 runs in a sequence of 16 possibilities - this is the maximum possible! Another unlikely example may contain all heads: H H H H H H H H H H H H H H H H Page 3 3

4 A TEST FOR RANDOMNESS It is obvious that a truly random sequence will lie somewhere between these two extreme cases for example: H T H H T H H T T T H H T T H H By referring to a cumulative distribution function for a series of 16 observations with 9 runs the probability under the null hypothesis of finding 9 or fewer runs is Therefore the null hypothesis of randomness can only be rejected against the alternative of positive association between adjacent observations at the 59.9% level of significance A TEST FOR RANDOMNESS (formal definition) Suppose that we have a time series of n observations. (assume n is even). A sequence of signs, with + denoting a value above the median and - a value below, is formed from these data. Let R denote the number of runs in the sequence. The null hypothesis to be tested is of randomness of the time series. The table gives the smallest significance level against which this null hypothesis can be rejected against a positive association between observations that are adjacent in time. A TEST FOR RANDOMNESS (formal definition) If the alternative is the two-sided hypothesis of randomness, the significance level must be doubled if less than 0.5. Alternatively, if the significance level a read from the table is bigger than 0.5 the appropriate significance test against a two sided alternative is 2 (1 - a ). For a large number of observations (> 20) a normal approximation can be used. The test is quite weak for small sample sizes and only checks one aspect of nonrandomness Page 4 4

5 little does he know -I m using time series forecasting to predict his future COMPONENTS OF A TIME SERIES One way of thinking about the behavior of an actual observed series is to regard it as being made up of various components. Traditionally, four possible components have been considered, with the notion that any or all might be present in any particular series. These components are as follows: 1. Trend component 2. Seasonality Component. 3. Cyclical Component 4. Irregular Component COMPONENTS OF A TIME SERIES Trends are characterized in a time series by the tendency to grow or decrease steadily over quite long periods. Seasonal time series consist of quarterly or monthly patterns which repeat from year to year. Cyclical patterns are oscillations in the time series which are not connected with seasonal behaviour. The irregular element of a time series is induced by a multitude of factors influencing the behavior of any practical time series and whose pattern looks unpredictable. Page 5 5

6 THE ADDITIVE MODEL The conceptual breakdown into trend, seasonal, cyclical and irregular components provides a very useful vocabulary for describing a time series. It is often convenient to think of a time series as the sum of its components: X t = T t + S t + C t + I t where T t = Trend Component S t = Seasonal Component C t = Cyclical Component I t = Irregular Component THE MULTIPLICITIVE MODEL Alternatively in some circumstances, it might be appropriate to view a time series as the product of its components: X t = T t S t C t I t It is not necessary to restrict attention to just these two models. In some circumstances, it may be convenient to treat some factors as additive and others has multiplicitive TIME PLOTS Simple graphical displays are extremely useful in revealing the major characteristics of a time series. Although more sophisticated techniques are necessary for a fuller analysis, a time plot is invariably a sensible first step in any analysis of data. Page 6 6

7 IRREGULAR COMPONENT stochastic CYCLICAL COMPONENT deterministic TREND COMPONENT (deterministic) Page 7 7

8 IRREGULAR + TREND IRREGULAR + CYCLICAL IRREGULAR + CYCLICAL + TREND Page 8 8

9 MOVING AVERAGES X X MOVING AVERAGES The irregular component in some time series may be so large that it obscures any underlying regularities, thus making it difficult for any visual interpretation of the time plot. In these circumstances, the actual plot will appear rather jagged, and it may be necessary to smooth it to achieve a clearer picture. This smoothing can be achieved through the method of moving averages which is based on the ideas that any large component at any point in time will exert a smaller effect if the observation at that point is averaged with its immediate neighbours SIMPLE CENTRED (2m+1) POINT MOVING AVERAGE Let X 1, X 2,..X n be n observations on a time series. A smoothed series can be obtained through the use of a simple centred (2m+1) - point moving average yielding: x 1 2m + 1 m * t = x t+ j j= m = X t-m + X t-m X t +..+ X t+m-1 + X t+m 2m + 1 The moving average X * is centred on X t Page 9 9

10 A 5 POINT MOVING AVERAGE Setting m = 2 yields: X t * = X t-2 + X t-1 + X t + X t+1 + X t+2 5 MOVING AVERAGE 5 point moving average MOVING AVERAGE 10 point moving average Page 10 10

11 MOVING AVERAGE 20 point moving average FORECASTING FORECASTING There are many techniques available to forecast time series; the choice of technique depends on a variety of practical considerations including objectives, prior information, the properties of the data etc. Univariate forecasting procedures are based only on the present and past values of the time series to be forecast. In the absence of trend and Seasonality the objective is to estimate the current level of the time series. This estimate is then used to forecast all future values Page 11 11

12 EXPONENTIAL SMOOTHING Simple exponential smoothing provides a forecast based on a weighted average of current and past values. In forming this average, most weight is given to the most recent observation, rather less to the immediately preceding value, less to the one before and so on. An extension is the Holt-Winters Model in which, the local level, local trend and local seasonal factor are all updated by exponential smoothing. Moving Average and Exponential Weights weight Five Period Moving Average Exponential Weighted Average (λ = 0.2) time EXPONENTIAL SMOOTHING Exponential smoothing is one of the more popular and frequently used forecasting techniques for a variety of reasons. Exponential smoothing requires minimal data. Only the forecast for the current period, the actual demand for the current period and a weighting factor called a smoothing constant are necessary. Exponential smoothing has a good track record of success. It has been employed over the years by many companies that have found it an accurate method of forecasting demand. Page 12 12

13 FORECASTING THROUGH EXPONENTIAL SMOOTHING The exponential smoothing forecast is computed using the recursive formula: F 1 = D 1 F t+1 = αd t + (1 - α)f t where F t+1 = the forecast for the next perod D t = the actual demand in the present period F t = the previously determined foreacst α = a weighting factor referred to as the smoothing constant The smoothing constant α is between 0 and 1.0. It reflects the weight given to the most recent demand data. For example if α = 0.2 F t+1 = 0.2D t + 0.8F t which means that the forecast for the next period is based on 20% of recent demand and 80 % of past demand (in the form of forecasts since these are derived from pervious demand). The higher α is, the more sensitive the forecasts will be to changes in recent demand, and the smoothing will be less. As α approaches zero, the forecast will react and adjust more slowly to differences between actual demand and forecast demand. FORECASTING USING SIMPLE EXPONENTIAL SMOOTHING observations forecasts time Page 13 13

14 CHOICE OF SMOOTHING CONSTANT In practice the choice of the smoothing constant may be based on subjective or objective grounds. Visual inspection of a graph of the available data can provide useful information. Alternatively a more objective approach is to try several different values and see which would be the most successful at predicting historical movements in the time series. Whatever the value of the smoothing constant, simple exponential smoothing can be regarded as an updating mechanism where the most recent estimate is a weighted average of the previous estimate and the new observation EXPONENTIAL SMOOTHING a = 0.7 EXPONENTIAL SMOOTHING a = 0.5 Page 14 14

15 EXPONENTIAL SMOOTHING a = 0.2 MEAN ABSOLUTE DEVIATION One measure of the overall error for a model is the mean absolute deviation (MAD). This value is computed by taking the absolute values of the individual forecast errors and dividing them by the number of forecast data periods: MAD = Σ forecast errors n forecast error = observed value - forecast MEAN SQUARED ERROR The Mean Squared Error (MSE) is another way of measuring overall forecast error. MSE is the average of the squared differences between the forecasted and observed values: MSE = Σ ( forecast errors ) 2 n forecast error = observed value - forecast Page 15 15

16 Adjusted Exponential Smoothing The adjusted exponential smooting forecast consists of the exponential smoothing forecast with a trend adjustment factor added to it: AF t+1 = F t+1 + T t+1 where T = an exponentially smoothed trend factor The trend factor is computed much the same as the exponentially smoothed forecast. It is, in effect a forecast model for trend: T t+1 = β(f t+1 - F t ) + (1- β)t t Adjusted Exponential Smoothing Where T t = the last period s trend factor β = a smoothing constant. β is a value between 0 and 1. It reflects the weight given to the most recent trend data. A high β reflects trend changes more than a low β. It is not uncommon for β to equal α in this method. Notice that this formula for the trend factor reflects a weighted measure of the increase (or decrease) between the next period F t+1 and the current forecast F t TIME PLOTS Simple graphical displays are extremely useful in revealing the major characteristics of a time series. Although more sophisticated techniques are necessary for a fuller analysis, a time plot is invariably a sensible first step in any analysis of data. Page 16 16

17 Interpreting Time Series Irregular (stochastic) Stationary Irregular + Trend Non - Stationary (Increasing Mean) Page 17 17

18 Non-Stationary (Increasing Variance) Two Series With Different Means Two Series With Different Variances Page 18 18

19 Series With Outliers Errors? Use Intervention Analysis Series with Large Effect & Shock Use Intervention Analysis Random (First Order Autoregressive f = 0) Page 19 19

20 First Order Autoregressive f = 0.5 First Order Autoregressive f = 0.9 First Order Autoregressive f = 0.99 Page 20 20