Lesson 4 -Part A Forecasting Quantitative Approaches to Forecasting Components of a Time Series...
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Lesson 4 -Part AForecastingQuantitative Approaches to ForecastingComponents of a Time SeriesMeasures of Forecast AccuracySmoothing MethodsTrend Projection Trend and Seasonal ComponentsRegression Analysis
Forecasting MethodsForecastingMethodsQuantitativeQualitativeCausalTime SeriesSmoothingTrendProjectionTrend ProjectionAdjusted forSeasonal Influence
Quantitative Approaches to ForecastingQuantitative methods are based on an analysis of historical data concerning one or more time series.A time series is a set of observations measured at successive points in time or over successive periods of time.If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method.If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method.
Time Series MethodsThree time series methods are:smoothingtrend projectiontrend projection adjusted for seasonal influence
Components of a Time SeriesThe pattern or behavior of the data in a time series has several components.The four components we will study are:
Components of a Time SeriesTrend ComponentThe 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.
Components of a Time SeriesAny regular pattern of sequences of values above and below the trend line lasting more than one year can be attributed to the cyclical component.Cyclical ComponentUsually, this component is due to multiyear cyclical movements in the economy.
Components of a Time SeriesThe seasonal component accounts for regular patterns of variability within certain time periods, such as a year.Seasonal ComponentThe 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.
Components of a Time SeriesThe irregular component is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series.Irregular ComponentIt is unpredictable.This component is the residual, or catch-all, factor that accounts for unexpected data values.
Measures of Forecast AccuracyThe average of the squared forecast errors for the historical data is calculated. The forecasting method or parameter(s) that minimize this mean squared error is then selected.The mean of the absolute values of all forecast errors is calculated, and the forecasting method or parameter(s) that minimize this measure is selected. (The mean absolute deviation measure is less sensitive to large forecast errors than the mean squared error measure.)Mean Squared ErrorMean Absolute Deviation
Smoothing MethodsThree common smoothing methods are:In cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular component of the time series.
Smoothing MethodsThe moving averages 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
Sales of Comfort brand headache medicine forthe past ten weeks at Rosco Drugsare shown on the next slide. If Rosco Drugs uses a 3-periodmoving average to forecast sales,what is the forecast for Week 11?Example: Rosco DrugsSmoothing Methods: Moving Averages
Example: Rosco DrugsSmoothing Methods: Moving Averages12345678910110115125120125120130115110130WeekWeekSalesSales
Time Series Plot using MINITAB
Time series appears to be stable over time, though random variability is present.Thus, smoothing methods are applicable.
Smoothing Methods: Moving Averages1234567891011110115125120125120130115110130116.7120.0123.3121.7125.0121.7118.3118.3123.3121.7125.0121.7118.3118.3116.7120.0WeekSales3MAForecast(110 + 115 + 125)/3
Moving Averages using MINITAB
You can specify this i.e. 4, 6, 12, & its depend on the problem# of future forecasts needed# of data points
Moving Average for Sales MovingAverage: Length 3Time Sales MA Predict Error1 110 * * *2 115 * * *3 125 116.667 * *4 120 120.000 116.667 3.33335 125 123.333 120.000 5.00006 120 121.667 123.333 -3.33337 130 125.000 121.667 8.33338 115 121.667 125.000 -10.00009 110 118.333 121.667 -11.666710 130 118.333 118.333 11.6667ForecastsPeriod Forecast 95% CI11 118.333 101.954 134.713
MSE# of values to be included in the moving average is chosen such that itminimizes the MSE
Smoothing MethodsWeighted Moving AveragesThe more recent observations are typically given more weight than older observations.For convenience, the weights usually sum to 1.To use this method we must first select the number of data values to be included in the average.Next, we must choose the weight for each of the data values.
Smoothing MethodsWeighted Moving AveragesAn example of a 3-period weighted moving average (3WMA) is:3WMA = .2(110) + .3(115) + .5(125) = 119Most recent of thethree observationsWeights (.2, .3,and .5) sum to 1
Smoothing MethodsExponential SmoothingThis method is a special case of a weighted moving averages method; we select only the weight for the most recent observation.The weights for the other data values are computed automatically and become smaller as the observations grow older.The exponential smoothing forecast is a weighted average of all the observations in the time series.
Smoothing MethodsExponential Smoothing ModelFt+1 = aYt + (1 a)FtwhereFt+1 = forecast of the time series for period t + 1Yt = actual value of the time series in period tFt = forecast of the time series for period ta = smoothing constant (0 < a < 1)To start thecalculations,we let F1 = Y1
With some algebraic manipulation, we can rewrite Ft+1 = aYt + (1 a)Ft as:Smoothing MethodsExponential Smoothing ModelFt+1 = Ft + a(Yt Ft)We see that the new forecast Ft+1 is equal to the previous forecast Ft plus an adjustment, which is a times the most recent forecast error, Yt Ft.
Sales of Comfort brand headache medicine forthe past ten weeks at Rosco Drugsare shown on the next slide. IfRosco Drugs uses exponentialsmoothing to forecast sales, whichvalue for the smoothing constant ,.1 or .8, gives better forecasts?Example: Rosco DrugsSmoothing Methods: Exponential Smoothing
Example: Rosco Drugs12345678910110115125120125120130115110130WeekWeekSalesSalesSmoothing Methods: Exponential Smoothing
Smoothing Methods: Exponential SmoothingExponential Smoothing ( = .1, 1 - = .9)F1 = 110F2 = .1Y1 + .9F1 = .1(110) + .9(110) = 110 F3 = .1Y2 + .9F2 = .1(115) + .9(110) = 110.5F4 = .1Y3 + .9F3 = .1(125) + .9(110.5) = 111.95F5 = .1Y4 + .9F4 = .1(120) + .9(111.95) = 112.76F6 = .1Y5 + .9F5 = .1(125) + .9(112.76) = 113.98F7 = .1Y6 + .9F6 = .1(120) + .9(113.98) = 114.58F8 = .1Y7 + .9F7 = .1(130) + .9(114.58) = 116.12F9 = .1Y8 + .9F8 = .1(115) + .9(116.12) = 116.01F10= .1Y9 + .9F9 = .1(110) + .9(116.01) = 115.41
Smoothing Methods: Exponential SmoothingExponential Smoothing ( = .8, 1 - = .2)F1 = 110F2 = .8(110) + .2(110) = 110F3 = .8(115) + .2(110) = 114F4 = .8(125) + .2(114) = 122.80F5 = .8(120) + .2(122.80) = 120.56F6 = .8(125) + .2(120.56) = 124.11F7 = .8(120) + .2(124.11) = 120.82F8 = .8(130) + .2(120.82) = 128.16F9 = .8(115) + .2(128.16) = 117.63F10= .8(110) + .2(117.63) = 111.53
Smoothing Methods: Exponential Smoothing[(Y2-F2)2 + (Y3-F3)2 + (Y4-F4)2 + . . . + (Y10-F10)2]/9Mean Squared ErrorIn order to determine which smoothing constant gives the better performance, we calculate, for each, the mean squared error for the nine weeks of forecasts, weeks 2 through 10.
Smoothing Methods: Exponential Smoothing2345678910115125120125120130115110130Week = .1 = .8FtFt(Yt - Ft)2(Yt - Ft)2110.00110.50111.95112.76113.98114.58116.12116.01115.41110.00114.00122.80120.56124.11120.82128.16117.63111.5325.00210.2564.80149.9436.25237.731.2636.12212.8725.00121.007.8419.7116.9184.23173.3058.26341.27Sum 974.22Sum 847.52Sum/9 108.25Sum/9 94.17MSEYt
Exponential Smoothing using MINITAB
# of future forecasts needed# of data pointsvalue for the smoothing constant
Single Exponential Smoothing for Sales SmoothingConstant Alpha 0.1Time Sales Smooth Predict Error1 110 110.000 110.000 0.00002 115 110.500 110.000 5.00003 125 111.950 110.500 14.50004 120 112.755 111.950 8.05005 125 113.980 112.755 12.24506 120 114.582 113.980 6.02057 130 116.123 114.582 15.41848 115 116.011 116.123 -1.12349 110 115.410 116.011 -6.011110 130 116.869 115.410 14.5901ForecastsPeriod Forecast Lower Upper11 116.869 96.5445 137.193
Single Exponential Smoothing for Sales SmoothingConstant Alpha 0.8Time Sales Smooth Predict Error1 110 110.000 110.000 0.00002 115 114.000 110.000 5.00003 125 122.800 114.000 11.00004 120 120.560 122.800 -2.80005