河海学习预测与决策part fundamentals of quantitative forecasting

1、 Forecasting and Decision Making刘 钢 2/1 Explanatory versus Time-Serise Forecasting2/2 Least Squares Estimates（最小二乘法）2/3 Discovering and Describing Existing Relationships of Patterns2/3/1 Time-Series Pattern2/3/2 Explanatory or Causal Patterns2/4 Useful Descriptive Statistics2/4/1 Univariate Data（单变量

2、数据）2/4/2 Bivariate Date（双变量数据）2/4/3 A Single Time Series Lagged on itself2/5 The Accuracy of Forecasting Methods 2/5/1 Standard Statistical Measures 2/5/2 Relative Measures 2/5/3 Theils U-Statistic 2/5/4 Other Measures2 FUNDAMENTALS OF QUANTITATIVE FORECASTINGExplanatory Forecasting Explanatory fore

3、casting assumes a cause and effect relationship between the inputs to the system and its output,as shown in Figure 2-1. FIGURE 2-1 explanatory or causal relationship.2/1 Explanatory versus Time-Series ForecastingExample: Boyles Law （波义耳定律）Boyles law:Where:P is pressureN is the number of moleculesV i

4、s the volume is a proportionality factor2/1 Explanatory versus Time-Series ForecastingTime-Series ForecastingUnlike explanatory forecasting ,time-series forecasting treats the system as a black box and makes no attempt to discover the factors affecting its behavior. FIGURE 2-2 time-series relationsh

5、ip 2/1 Explanatory versus Time-Series ForecastingExample: GNP (gross national product)Explanatory forecasting: GNP=f( monetary and fiscal policies,inflation,capital spending,imports,exports). (2-2)If the only purpose is to forecast future values of GNP without concern as to why a certain level of GN

6、P will be realized,a time-series approach would be appropriate: GNP t+1=f(GNP t,GNPt-1,GNPt-2,GNPt-3 ) (2-3) where t is the present month, t+1 is the next month, t -1 is the last month, t-2 is two months ago, and so on.2/1 Explanatory versus Time-Series Forecasting 2/2 Least Squares EstimatesGNP=f(m

7、onetary and fiscal policies, inflation, capital spending, imports,exports,u). (2-4)GNP t+1=f(GNP t,GNPt-1,GNPt-2,GNPt-3 ,ut) (2-5)2/2 Least Squares EstimatesRandom effectsFIGURE 2-3 explanatory or causal relationship with random noiseDefinitionData = pattern + error ( 2-6 ) The critical task in fore

8、casting is to separate the pattern from the error component so that the former can be used for forecasting.Least squares is based on the fact that this estimation procedure seeks to minimize the sum of the squared errors in equation( 2-6 ).2/2 Least Squares EstimatesExample: Sample expenditures for

9、supermarket clients2/2 Least Squares EstimatesFigure 2-5 shows the resulting MSEs for all estimates from 0 through 20,and it can be seen that the MSEs form a parabola.The minimum MSE will be achieved when the value of the estimate is 10,and we say that 10 is the least squared estimate of customer sp

10、ending.2/2 Least Squares Estimates In fact ,the value can be found mathematically with the help of differentiation. error = data pattern (2-7) (2-8)(2-9)For convenience, the error will be denoted by e ,the data by X and the pattern by .In addition,the subscript i(i=1,2,3,12) will be added to denoted

11、 the ith customer:2/2 Least Squares EstimatesSumming these values(squared errors) for all 12 customers yields:(2-10)So thator(2-11)Which impliesApplying (2-11) to the store managers data of Table 2-1 gives:2/2 Least Squares EstimatesNoticeMinimize Is not popular！See p242/2 Least Squares Estimates2/3

12、 Discovering and Describing Existing Relationship or Patterns2/3/1 Time-Series PatternTable 2-3 gives the population of France (in millions) for the years 1961 to 1970 and the resulting errors when the mean is used as the estimate for the pattern.2/3 Discovering and Describing Existing Relationship

13、or PatternsThe result is shown in Figure 2-6:Error 2/3 Discovering and Describing Existing Relationship or PatternsThe results shows the error is not random, but exhibits some systematic variation .This variation could perhaps be included as part of the pattern.Since the errors go from negative to p

14、ositive,an estimate of the pattern in the form of a trend line as shown in Figure 2-7 will give much smaller errors than using the mean as an estimate.One possible trend line is of the form Population = 46.133 + 0.48X2/3 Discovering and Describing Existing Relationship or Patterns2/3/1 Time-Series P

15、atternTrend line2/3 Discovering and Describing Existing Relationship or Patterns2/3 Discovering and Describing Existing Relationship or PatternsFor forecasting purposes,either Figure 2-6 or 2-7 can be used to predict the population for the year 1971.In terms of Figure 2-6,the forecast is 48.776 mill

16、ion.From Figure 2-7,the forecast is 51.4 million.From historical data,one would expect the actual population value to be closer to 51.4 million than to 48.776.(The actual 1971 population was 51.25 million.)Thus in practice it is clear that Figure 2-7 will be chosen,since it fits the pattern of the d

17、ata much better than Figure 2-6.2/3 Discovering and Describing Existing Relationship or Patterns2/3/1 Time-Series Pattern2/3/2 Explanatory or Causal PatternsIn this section we will attempt to determine the pattern of an output variable using a causal relationship involving at least one other variabl

18、e.Table 2-5 and Figure 2-8 shows the GNP of France (in billions of francs) during the year 1961 through 1970.2/3 Discovering and Describing Existing Relationship or PatternsLike population,GNP is not accurately estimated using the mean value,as illustrated in Table 2-5.2/3 Discovering and Describing

19、 Existing Relationship or PatternsAs shown in Figure 2-8,a trend line gives a time-series forecast for the year 1971 of 817.2 billion francs.2/3 Discovering and Describing Existing Relationship or PatternsTwo different patterns have been fitted to the data in Figure 2-9.One is linear (a straight lin

20、e) of the form GNP = -4478.92 + 102.79P (2-15) where P is population.The second is a nonlinear one called an exponential pattern (see dotted line on Figure 2-9),and is of the form GNP = e-3.376 + 0.1972P (2-16) 2/3 Discovering and Describing Existing Relationship or Patterns2/3/2 Explanatory or Caus

21、al Patterns2/3 Discovering and Describing Existing Relationship or PatternsForecasting using two modelGNP = -4478.92 + 102.79P (2-15)GNP1971= -4478.92 + 102.79P1971 =804.486The percentage error of this forecast was 11%.GNP = e-3.376 + 0.1972P (2-16)GNP1971 =862.7The percentage error of this forecast

22、 was 4.8%.The forecasting value of 1971:First:Second:2/3 Discovering and Describing Existing Relationship or Patterns2/4 Useful Descriptive Statistics2/4 Useful Descriptive StatisticsIn economic and behavioral systems there is always uncertainty (randomness), and we must turn to the field of statist

23、ics for help in describing such random variables.For a single date set (univariate date) or a single time series the most common descriptive statistics are the mean, the standard deviation, and the variance. We would also like to mention a measure of skewness. For a pair of random variables (bivaria

24、te date) or for paired time series date it is of interest to describe how the two series relate to each other. The most widely used summary numbers (statistics) for this purpose are the covariance and the correlation.Example: The age of 10 employees in a firm2/4 Useful Descriptive Statistics2/4/1 Un

25、ivariate Data2/4/1 Univariate DataConsider the data set in Table 2-6,representing the age of 10 employees in the firm.Using the letter A to denote age and a subscript i (i=1,2,3,10) to denote the ith employee,the mean age can be written Next,for each employee it is possible to find out how far their

26、 age is from the mean age .Deviation:Mean:2/4 Useful Descriptive Statistics The mean of the absolute deviations is denoted MAD,and for the age data If the squared deviations are summed,we get what is often designated SS (or SSD,for sum of squared deviations): If the mean of these squared deviations

27、is designated MS (or MSD,for mean squared deviations):2/4 Useful Descriptive Statistics2/4/1 Univariate Data平均绝对偏差偏差平方和均方差 Closely related to MSD is the variance, which is defined as the sum of squared deviations divided by the degrees of freedom. For the age data the variance of age isBy taking the

28、 squared root of these two summary numbers, we get two additional summary statistics,as follows: Where RMS = root mean squared and S = standard deviation2/4 Useful Descriptive Statistics2/4/1 Univariate Data均方根标准差方差2/4 Useful Descriptive Statistics To summarize,the univariate statistics (summary num

29、bers) that will be used in this text are defined (generally) as follows:MEAN(2-17)MEAN ABSOLUTE DEVIATION (2-18)SUM OF SQUARED DEVIATIONS(2-19)MEAN SQUARED DEVIATION(2-20)2/4 Useful Descriptive StatisticsVARIANCE(2-21)ROOT MEAN SQUARE(2-22)STANDARD DEVIATION(2-23)SKEWNESS(2-24)2/4 Useful Descriptive

30、 Statistics2/4/2 Bivariate DataTable 2-8 show the heights (in inches) and the weights (in pounds) for 10 employees of a firm.2/4 Useful Descriptive StatisticsA Positive Relationship A statistic which indicates how two variable “co-vary”is called the covariance and is defined as follow:Where X and Y

31、are the two variables and are the means of X and Y,respectively. xi and yi are the respective deviations for X and Y.and n is the number of paired observation.(2-25) The correlation coefficient, designated r,is a special covariance measure that takes care of the scale problem . (2-26)2/4 Useful Desc

32、riptive Statistics协方差相关系数For the data in Table 2-8 the computations involved in getting to the correlation coefficient are included in Table 2-9:2/4 Useful Descriptive Statistics2/4/3 A Single Time Series Lagged on itselfIn Table 2-10 there is a single time series over 20 time periods.2/4 Useful Des

33、criptive Statistics1.Autocovariance:(Lag k)(2-29)2.Autocorrelation:(Lag k)(2-30)2/4 Useful Descriptive Statistics自协方差自相关系数 By way of illustration,consider the data in Table 2-10 and the calculations of autocovariance and autocorrelation in Table 2-11 .2/4 Useful Descriptive Statistics(Lag 1) The cal

34、culation of autocovariance,with lag one period,using equation (2-29) is as follow And the autocorrelation,for lag one,is computed as follows:(Lag 1) Using exactly similar procedures,the autocorrelations for lags two,three,and four could be obtained,and the results for the data in Table 2-10 are as f

35、ollows:auto-r (lag 2) = -0.07,auto-r (lag 3) = -0.11,auto-r (lag 4) = -0.13.2/4 Useful Descriptive Statistics2/5 The Accuracy of Forecasting MethodsHow to measure the suitability of a particular forecasting method for a given data set?In explanatory modeling, goodness-of-fit measures predominate.In

36、time-series modeling it is possible to use a subset of the known date to forecast the rest of the known data.2/5 The Accuracy of Forecasting Methods2/5/1 Standard Statistical MeasuresTable 2-12 contains a set of observed sales values,Xi,for each of ten time periods (i =1,2,10),and the forecasted (or

37、 fitted) values,Fi,for the same periods.2/5 The Accuracy of Forecasting MethodsMEAN ERRORMEAN ABSOLUTE ERRORSUM OF SQUARED ERRORMEAN SQUARED ERRORSTANDARD DEVIATION OF ERROR(2-31)(2-32)(2-33)(2-34)(2-35)2/5 The Accuracy of Forecasting Methods Table 2-13 illustrates the computation of these standard

38、statistical measures.2/5 The Accuracy of Forecasting Methods the objective of statistical optimization is very often to choose a model so as to minimize MSE( or SSE), but this measure has two drawbacks.First, overfitting a model to a data series, which is equivalent to including randomness as part o

39、f the generating process, is as bad as failing to identify the nonrandom pattern in the data.Second, different methods use different procedures in the fitting phase. Thus parison of such methods on a single criterion is of limited value.2/5 The Accuracy of Forecasting Methods2/5/2 Relative MeasuresP

40、ERCENTAGE ERRORMEAN PERCENTAGE ERRORMEAN ABSOLUTE PERCENTAGE ERROR(2-36)(2-37)(2-38)2/5 The Accuracy of Forecasting Methods误差百分率 Table 2-14 shows how how to compute the PE,MPE,and MAPE measures.2/5 The Accuracy of Forecasting MethodsTwo different naive methods of forecasting for use as a basis in evaluating other methods in a given situation:(2-39)(2-40)2/5 The Accuracy of Forecasting Methods Table 2-15 shows how how to c