Laserfiche WebLink
<br />tv <br />~ <br />~ <br />~ <br /> <br />z(l)i - alz(2)i + a2z(3)i + a38(4)i + a4Z(5)i~ l-R2. e(l)i (9) <br /> <br />Here, z(2) through z(5) are the residual values at neighboring sta- <br />tions wHich influence the residual, z(l) at time i. The a's are <br />regression coefficients determined from the multiple regression <br />analysi~. R2 is the decimal percent explained variance of z(l) by <br />this eq~ation (not to be confused with the R2 of equation (8). <br /> <br />To pres~rve the total variance of z(l), the unexplained vari&nce, <br />l-R2, mqst be included. This is done by adding the quantity <br /> <br />~ ., e to the z(l) value from the regression equation, Here e is <br />~ ;~~~O$ variable with mean of 0.0 and variance of 1.0. When generating, <br />e is obuained by transforming a random number from a uniform distribu- <br />, <br />tion fu~ction into a normal distribution and then through a polynomial <br />equatiorl to the distribution Which the data analysis exposed at the <br />particu~ar station, This procedure generates values Which are related <br />to adjacent stations yet also maintain their own unique properties. <br /> <br />In actu41 operation of the synthetic data generation model, residual <br />values 4t flow stations are developed one at a time. The first sta- <br />tion is iin4ependent of the others and is generated only through its <br />unique dis~ribution function. Once the residual is generated at the <br />first s~ation, it can be used as a single specially independent vari- <br />able fot g~nerating a value at a second station to maintain their <br />correlaUion. The third station's residual is based on the first two; <br />and in . similar manner, the fourth uses the previous three. Beyond <br />this, oUher residuals may be generated based upon four specially <br />independent values, as shown in equation (9). <br /> <br />2.1.5 <br /> <br />Distribution of Random Component <br />r <br /> <br />The final task is to discover the characteristics of the distribution <br />of the ~esidual series. The distribution's characteristics can <br />often be described by a small number of statistics. such as the mean, <br />standar~ deviation, skew, and kurtosis. The mean or average indi- <br />cates t~e central tendency of the group. The standard deviation <br />measure. the spread about the mean. Another quantity, the skew, <br />indicat~s the symmetry of the distribution. The sharpness or flatness <br />is meas~red by kurtosis. The entire distribution itself can be <br />describ~d by the equation of a line, or a mathematical function, <br />relatin~ the value of a quantity to the probability it will be <br />exceeded. Of the many such functions which have been developed, <br />probably the normal distribution is best known. <br /> <br />The dis~ribution function of the residuals can be adequately defined <br />if a function is found which closely represents the sample data, In <br /> <br />11 <br /> <br />.-,"- <br /> <br />