c.For each data point i=1,2,..,n, find ziusing Equation (10.3), then find the average of z. d.Use the average z in ŷ = LV + z * DVL=2. This is the Nonlinear Regression function.Example 10.1: Nonlinear Multiple Variable Regression Using Z-Theory Consider Example 9.2 with two independent variables, x1and x2, and one dependent variable, y. Information for the past seven periods is also given in Table 10.1. Compare the quality of the solution obtained by Z-theory versus the solution obtained by the conventional Quadratic Regression method. The quadratic solution obtained in Example 9.2 was, b0= 33.003, b1,1= -0.554, b1,2= 0.082, b2,1= -1.467, and b2,2= 0.028. Therefore,LV= 33.003 + (– 0.554 x1+ 0.082 x12) + (– 1.467 x2+ 0.028x22) where v1(x1) = (– 0.554 x1+ 0.082 x12) , and v2(x2) = (– 1.467 x2+ 0.028x22) DVL=2= The following table shows LV and DV for each data point. For example, for point i=1,LV1=33.003 + (– 0.554*9 + 0.082*92) + (– 1.467*31 + 0.028*312) = 16.09DV1== 37.54.z1= (16.03-15.72)/37.54= -0.008. The average z= 0.004. Therefore, the regression function is:N.B. The information contained within is the sole intellectual property of Professor Behnam Malakooti of Case Western Reserve University, Cleveland, Ohio. All rights reserved. No part may be reproduced without prior written permission. Page 51of 66
Professor Behnam MalakootiForecastingRevision 49-5June 10, 2012ŷ = LV + 0.004* DVL=2ix1,ix2,iyiQuadratic Regression- (Prob.9.2), ŷi= LVv1(x1)v2(x2)DVzŷError193115.7216.031.66-18.5737.47-0.00816.190.47254220.1620.16-0.72-12.2238.530.00020.320.16313515.315.3-0.47-17.0535.990.00015.450.15463114.3414.01-0.37-18.5735.610.00914.160.18544824.8424.84-0.90-5.9040.100.00025.010.17675131.5731.030.14-1.9945.220.01231.220.35724523.4722.82-0.78-9.3239.870.01622.990.48-Avg z=0.004MAD=0.28Table 10.1: Example 10.1 data and Z-nonlinear regression solutionNow, we can compare Nonlinear Z-utility function to Quadratic function solution by comparing their MAD. To compare the two methods, find the MAD. The following results show that Z-utility function provided a better solution. FunctionMAD52
Professor Behnam MalakootiForecastingRevision 49-5June 10, 2012Quadratic RegressionLV=33.003 – 0.554 x1+ 0.082 x12– 1.467 x2+ 0.028x220.48Z-utility Nonlinear Regression ŷ =LV +0.004DV0.2810.2 Nonlinear Time Series by Z-Theory* †The approach for solving time series forecasting is based on Section 10.1 Nonlinear Regression. For this problem, only two variables are considered. The first independent variable is, t, time, where xt= t for t=1, 2,…, n. The second independent variable is predicted Ftfor t = 1, 2,…, n, by using any of given Time-Series methods (e.g. using Exponential Smoothing (Section 4.1) or Linear Regression (Section 7.2) or Quadratic Regression Section 9.1). These two independent variables are evaluated by the actual output, yt, e.g. the actual demand yt=Dt. First, we find Ftfor t = 1, 2,…, n using a forecasting method. Then, we find the Quadratic Regression (10.8) for the given two independent variables, xtand Ft, i.e. LV = b0+( b11*t + b12*t2)+( b21*Ft+ b22*Ft2) = b0+v1(t) +v2(Ft)(10.8)Example 10.2: Nonlinear Regression Time Series Using Linear Regression Consider Example 7.2 as shown in Table 7.2. Solving this problem by Linear Regression method of Section 7.2 (alternatively, one can use Quadratic Regression method of Section 9.1), the solution is: Ft= 1320 + 510*t. Table 10.2 shows the details.