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## Time Series Forecasting– Part I

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**Time Series Forecasting– Part I**• What is a Time Series ? • Components of Time Series • Evaluation Methods of Forecast • Smoothing Methods of Time Series • Time Series Decomposition by Duong Tuan Anh Faculty of Computer Science and Engineering September 2011 1**29**28 27 26 25 24 23 0 50 100 150 200 250 300 350 400 450 500 What is a Time series ? A time series is a collection of observations made sequentially in time. A study on random sample of 4000 graphics from 15 of the the world’s news papers published between 1974 and 1989 found that more than 75% of all graphics were time series. Examples: Financial time series, scientific time series 2**Time series models**• Regression models • Predict the response over time of the variable under study to changes in one or more of the explanatory variables. • Deterministic models of time series • Stochastic models of time series All the three kinds of models can be used for forecasting. 3**Components of a time series**• The pattern or behavior of the data in a time series has several components. • Theoretically, any time series can be decomposed into: • Trend • Cyclical • Seasonal • Irregular • However, this decomposition is often not straight-forward because these factors interact. 4**Trend component**• The trend component accounts for the gradual shifting of the time series to relatively higher or lower values over a long period of time. • Trend is usually the result of long-term factors such as changes in the population, demographics, technology, or consumer preferences. 5**Seasonal component**• The seasonal component accounts for regular patterns of variability within certain time periods, such as a year. • The variability does not always correspond with the seasons of the year (i.e. winter, spring, summer, fall). • There can be, for example, within-week or within-day “seasonal” behavior. 6**Cyclical component**• Any regular pattern of sequences of values above and below the trend line lasting more than one year can be attributed to the cyclical component. • Usually, this component is due to multiyear cyclical movements in the economy. 7**Evaluating Methods of forecasts**• Forecasting method is selected - many times by intuition, previous experience, or computer resource availability • Divide the data into two sections - an initialization part and a test part • Use the forecast technique to determine the fitted values for the initialization data set • Use the forecast technique to forecast the test data set and determine the forecast errors • Evaluate errors (MAD, MPE, MSD, MAPE) • Use the technique, modify, or develop new model 8**Evaluation Methods of Forecasts**• There are three measures of accuracy of the fitted models: MAPE, MAD and MSD for each of the sample forecasting and smoothing methods. • For all three measures, the smaller the value, the better the fit of the model. • Use these statistics to compare the fit of the different methods. • MAPE (Mean Absolute Percentage Error) measure the accuracy of fitted time series values. It expresses accuracy as a percentage. |(yt-yt’)/yt| MAPE = -------------- 100 (yt 0) n 9**MAPE, MAD, and MSD**where yt is the actual value, yt’ is the fitted value and n is the number of observations. • MAD (Mean Absolute Deviation) expresses accuracy in the same units as the data, which help conceptualize the amount of error. |yt-yt’| MAD = ---------- n where yt is the actual value, yt’ is the fitted value and n is the number of observations. 10**MAPE, MAD, and MSD**• MSD(Mean Squared Deviation) is a more sensitive measure of an unusually large forecast error than MAD. (yt-yt’)2 MSD = ---------- n where ytis the actual value, yt’ is the fitted value and n is the number of observations. 11**Methods of smoothing time series**• Arithmetic Moving Average • Exponential Smoothing Methods • Holt-Winters method for Exponential Smoothing Smoothing a time series: to eliminate some of short-term fluctuations. Smoothing also can be done to remove seasonal fluctuations, i.e., to deseasonalize a time series. These models are deterministic in that no reference is made to the sources or nature of the underlying randomness in the series. The models involves extrapolation techniques. 12**Averaging Methods**• Simple Averages - quick, inexpensive (should only be used on stationary data) • Moving Average method consists of computing an average of the most recent n data values for the series and using this average for forecasting the value of the time series for the next period. • Moving averages are useful if one can assume item to be forecast will stay steady over time. • Series of arithmetic means – used only for smoothing, provides overall impression of data over time (most recent n data items) Moving Average = ------------------------------------------ • n 13**Moving average methods**• Works best with stationary data. • The smaller the number, the more weight given to recent periods. • A smaller number is desirable when there are sudden shifts in the level of the series. • The greater the number, less weight is given to more recent periods. • The larger the order of the moving average, the greater the smoothing effect. Larger n when there are wide, infrequent fluctuations in the data. • By smoothing recent actual values, removes randomness. 14**Weighted Moving Averages**• Weighted Moving Average - place more weight on recent observations. Sum of the weights needs to equal 1. • Used when trend is present • Older data usually less important (weight for period n)(Value in period n) WMA = -------------------------------------------------------- weights 15**Notes on Moving Averages**• MA models do not provide information about forecast confidence. • We can not calculate standard errors. • We can not explain the stochastic component of the time series. This stochastic component creates the error in our forecast. 16**Exponential Smoothing Methods**• Single Exponential Smoothing (Averaging) • Double Exponential Smoothing & Holt’s Method • Winter’s Model. Note: - Single Exponential Smoothing is for series without trend and without seasonal component. - Double Exponential Smoothing is for series with trend and without seasonal component. - Winter’s model is for for series with trend and seasonal component. 17**Single Exponential Smoothing**• Continually revising a forecast in light of more recent experiences. Averaging (smoothing) past values of a series in a decreasing (exponential) manner. The observations are weighted with more weight being given to the more recent observations At = αYt-1 + (1 – α) At-1 (S1) New forecast = α (old observation) + (1- α) old forecast Here we denote the original series by yt and the smoothed series by At. The equation can be rewritten as: At = At-1 + α(Yt –At-1) 18**Single Exponential Smoothing**• When looking at the formula – new forecast is really the old forecast plus a times the error in the old forecast • To get started, we need a smoothing constant a, an initial forecast, and an actual value. We can use the first actual as the forecast value or we can average the first n observations. • The smoothing constant serves as the weighting factor. When a is close to 1, the new forecast will include a substantial adjustment for any error that occurred in the preceding forecast. When a is close to 0, the new forecast is very similar to the old forecast. 19**Single Exponential Smoothing (cont.)**• The smoothing constant α is not an arbitrary choice - but generally falls between 0.1 and 0.5. If we want predictions to be stable and random variation smoothed, use a small a. If we want a rapid response, a larger a value is required. 20**Why Exponential?**At = Yt-1 + (1- )At-1 At-1 = Yt-2 + (1- )At-2 At-2 = Yt-3 + (1- )At-3 … At = Yt-1 + (1- ) Yt-2 + (1- ) 2Yt-3 + …. + (1 - ) kYt-k+1 k decreases exponentially. 21**The large a in this example responds quickly to the data.**23**Tracking**• Use a tracking signal (measure of errors over time) and setting limits. For example, if we forecast n periods, count the number of negative and positive errors. If the number of positive errors is substantially less or greater than n/2, then the process is out of control. • Can also use 95% prediction interval (1.96 * sqrt (MSE)). If the forecast error is outside of the interval, use a new optimal a. • Looking back at the .1 single exponential smoothing: 1.96*sqrt(24261) = +-305 Observation #21 is out-of-control. We need to re-evaluate alpha level because this technique is biased. 24**Exponential Smoothing Adjusted for Trend: Holt’s method**• In some situations, the observed data are trending and contain information that allows the anticipation of future upward movement. • In that case, a linear trend forecast function is needed. • Holt’s smoothing method allows for evolving local linear trend in a time series and can be used to forecast. • When there is a trend, an estimate of the current slope and the current level is required. 25**Holt’s Method**• Holt’s method uses two coefficients. • a is the smoothing constant for the level • b is the trend smoothing constant - used to remove random error. • Advantage of Holt’s method: it provides flexibility in selecting the rates at which the level and trend are tracked. 26**Equations in Holt’s method**• The exponentially smoothed series, or the current level estimate: At = Yt + (1- )(At-1 + Tt-1) (S2) • The trend estimate: Tt = (At – At-1)+(1- )Tt-1 (S3) • Forecast p periods into the future: Y’t+p = At + pTt where At = new smoothed value (estimate of current level) Yt = new actual value at time t. Tt = trend estimate Y’t+p = forecast for p periods into the future. = smoothing constant for the level = smoothing constant for trend estimate 27**How to initiate Holt’s method**• To get started, initial values for A and T in equation (S2) and (S3) must be determined. • One approach is to set A1 to Y1 and T1 to zero. • The second approach is to use the average of the first five or six observations as A1. T1 is then estimated by the slope of a line that is fit to these five or six observations.**Holt’s method**Holt exponential smoothing with parameters = 1.0 and = 0.099 for time series of electricity consumption.**Winter’s Method**• Winters’ method is an easy way to account for seasonality when data have a seasonal pattern. • It extends Holt’s Method to include an estimate for seasonality. • a is the smoothing constant for the level • b is the trend smoothing constant - used to remove random error. • g smoothing constant for seasonality • This formula removes seasonal effects. The forecast is modified by multiplying by a seasonal index. 30**Winter’s Method**The four equations used in Winters’ (multiplication) smoothing are: • The smoothed series or level estimate: At = Yt /St-s+ (1- )(At-1 + Tt-1) • The trend estimate: Tt = (At – At-1)+(1- ) Tt-1 • The seasonality estimate: St = Yt/At + (1- )St-s • Forecast p periods into the future: Y’t+p = (At + pTt)St-s+p where At = new smoothed value (estimate of current level) Yt = new actual value at time t. Tt = trend estimate Y’t+p = forecast for p periods into the future. Tt = trend estimate = smoothing constant for the level = smoothing constant for trend estimate = smoothing constant for seasonality estimate p = periods to be forecast into the future s = length of seasonality WINTERS’ METHOD Is also called TRIPLE EXPONENTIAL SMOOTHING ) 31**How to initiate Winter’s method**• To begin the Winter’s method, the initial values for the smoothed series At, the trend Tt and the seasonal indices St must be set. • One approach is to set the first estimate of At to Y1. The trend is estimated to 0 and the seasonal indices are each set to 1.0.**Decomposition**• Decomposition is a procedure to identify the component factors of a time series. • How the components relate to the original series: a model that expresses the time series variable Y in terms of the components T (trend), C (cycle), S (seasonal) and I (iregular). • Additive components model & multiplicative components model. • It is difficult to deal with cyclical component of a time series. To keep things simple we assume that any cycle in the data is part of the trend. • Additive model: Yt = Tt + St + It • Multiplicative model: Yt = Tt St It**Additive and multiplicative models**• The additive model works best when the time series has roughly the same variability through the length of the series. • That is, all the values of the series fall within a band with constant width centered on the trend. • The multiplicative model works best when the variability of the time series increased with the level. • That is the values of the series become larger as the trend increases. • See the figure in the next slide. • Most economic time series have seasonal variation that increases with the level of the series. So multiplicative model is suitable to them.**(a) A time series with constant variability**(b) A time series with variability increasing with level**Trend equations**• Trend can be described by a straight line or a smooth line. • Linear trend: T’t = a + bt • Here T’t is the predicted value for the trend at time t. The symbol t used for the variable represents time and takes integer values 1,2,3,… The slope b is the average increase or decrease in T for each one-period increase in time. • Time trend equations can be fit to the data using the method of least squares. • Recall that this method selects the values of coefficients in the trend equation (e.g. a and b) so that the estimated trend values T’t are close to the actual value Yt as measured by the sum of squared errors criterion SSE = (Yt – T’t)2 (See Appendix of this chapter for how to find a and b)**Additional trend curves**• The life cycle of a new product has 3 stages: introduction, growth, and maturity and saturation. • A curve is needed to model the trend over a new product. • A simple function that allows for curvature is the quadratic trend • T’t = b0 + b1t + b2t2 • When a time series starts slowly and then appears to be increasing at an increasing rate Exponential trend: • T’t = b0b1t • The coefficient b1 is related to the growth rate.**The increase in the number of salespeople is not constant.**It appears as if increasingly larger numbers of people are being added in the later years. An exponential trend curve fit to the salepeople data has the equation: T’t = 10.016(1.313)t**Seasonality**• Several methods for measuring seasonal variation. • The basic idea: • first estimate and remove the trend from the original series and then smooth out the irregular component. This leaves data containing only seasonal variation. • The seasonal values are collected and summarized to produce a number for each observed interval of the year (week, month, quarter, and so on)**Identification of seasonal component**• The identification of seasonal component in a time series differs from trend analysis in two ways: • The trend is determined directly from the original data, but the seasonal component is determined indirectly after eliminating the other components from the data. • The trend is represented by one best-fitting curve, but a separate seasonal value has to be computed for each observed interval. • If an additive decomposition is employed, estimates of the trend, seasonal components are added together to produce the original series. • If an multiplicative decomposition is employed, estimates of individual components must be multiplied together to produce the original series**Seasonal indices**• The seasonal indices measure the seasonal variation in the series. • Seasonal indices are percentages that show changes over time. • Ex: • With monthly data, a seasonal index of 1.0 for a particular month means the expected value for that month is 1/12 the total for the year. • An index of 1.25 for a different month implies the observation for that month is expected to be 25% more than 1/12 of the annual total. • A monthly index of 0.80 indicates that the expected level of that month is 20% less than 1/12 the total for the year.**Seasonal adjustment**• After the seasonal component has been isolated, it can be used to calculate seasonally adjusted data. • Seasonal adjustment techniques are ad hoc methods of computing seasonal indices and use those indices to deseasonalize the series by removing those seasonal variation. • For an multiplicative decomposition, the seasonally adjusted data are computed by dividing the original data by the seasonal component (i.e. seasonal index) deseasonalized data = raw data/seasonal index**Seasonal adjustment technique**• Seasonal adjustment techniques are based on the idea that a time series yt can be represented as the product of 4 components: yt = T S C I • The objective is to eliminate the seasonal component S. • First, we try to isolate the combined trend and cyclical components T C. This cannot be done exactly; instead an ad-hoc smoothing procedure is used to remove T C from the original time series. • For example, supposed that ytconsists of monthly data. Then a 12-month average ymt is computed: ymt = (yt+6+… + yt + yt-1 + … + yt-5)/12 • Presumably ymt is relatively free of seasonal and irregular fluctuations and is thus as estimate of T C. • Now, we divide the original data by this estimate of T C to obtain an estimate of the combined seasonal and irregular components S I.**Seasonal adjustment technique (cont.)**S I = yt/ ymt = zt • The next step is to eliminate the irregular component I in order to obtain the seasonal index. To do this, we average the values of S I corresponding to the same month. • In other words, suppose that y1 (and hence z1) corresponds to January, y2 to February, etc., and there are 48 months of data. We thus compute zm1 = (z1 + z13 + z25 + z37) zm2 = (z2 + z14 + z26 + z38) …………………………… zm12 = (z12 + z24 + z36 + z48)**Seasonal adjustment technique (cont.)**• The rationale here is that when the seasonal-irregular percentages zt are averaged for each month (each quarter if the data are quarterly), the irregular fluctuations will be largely smoothed out. • The 12 averages zm1,…, zm12 will then be estimates of the seasonal indices. They should sum close to 12. • The deseasonalization of the original series yt is now straightforward; just divide each value in the series by its corresponding seasonal index. • Thus, the seasonally adjusted yat is obtained from ya1=y1/ zm1, ya2 =y2/ zm2 …, ya12 =y12/ zm12, etc.**Appendix: Least-square parameter estimates**• Our goal is to minimize (Yt – Y’t)2 where Y’t = a + bXi is the fitted value of Y corresponding to a particular observation Xi. • We minimize the expression by taking the partial derivatives with respect to a and to b, setting each equal to 0, and solving the resulting pair of simultaneous equations: =-2 (A.1) (A.2) =-2**Least-square parameter estimates**• Equating these derivatives to zero and dividing by -2, we get (Yi – a – bXi) = 0 (A.3) Xi(Yi – a – bXi) = 0 (A.4) • Finally by rewriting Eqs. (A.3) and (A.4), we obtain the pair of simultaneous equations: Yi = aN + bXi (A.5) XiYi = aXi +bXi2 (A.6) • Now we can solve for a and b simultaneously by multiplying (A.5) by Xi and Eq. (A.6) by N: XiYi = aNXi + b(Xi)2 (A.7) NXiYi = aNXi +bN(Xi)2 (A.8)