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Fig. 7. Steps in the data analysis for testing chaotic properties of a dynamic process.

Figure 7 Steps in the data analysis for testing chaotic properties of a dynamic process.

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Fig. 1. Illustration of the dynamics of balancing between exploitation and exploration using a logistic difference equation x,,, = kx, (1—x,), x, =0.45, t= 1,2,3,...,123. (A) Stable time path, k = 1.80. (B) Phase plot for the stable time path. (C) Periodic time path, k= 3.20. (D) Phase plot for the periodic time path. (E) Chaotic time path, k = 3.65. (F) Phase plot for the chaotic time path (strange attractor). (G) Random time path (data in E is randomly shuffled). (H) Phase for the random time path.
Fig. 1. Illustration of the dynamics of balancing between exploitation and exploration using a logistic difference equation x,,, = kx, (1—x,), x, =0.45, t= 1,2,3,...,123. (A) Stable time path, k = 1.80. (B) Phase plot for the stable time path. (C) Periodic time path, k= 3.20. (D) Phase plot for the periodic time path. (E) Chaotic time path, k = 3.65. (F) Phase plot for the chaotic time path (strange attractor). (G) Random time path (data in E is randomly shuffled). (H) Phase for the random time path.
Fig. 2. Illustration of the chaotic property of sensitive dependence to initial conditions. (A) Time paths for x,,, = kx, —x,), k = 3.65, t= 1,2,3,...,123. (B) Time path for x} — x,,x, = 0.45000, x} = 0.45001.
Fig. 2. Illustration of the chaotic property of sensitive dependence to initial conditions. (A) Time paths for x,,, = kx, —x,), k = 3.65, t= 1,2,3,...,123. (B) Time path for x} — x,,x, = 0.45000, x} = 0.45001.
Fig. 3. Transition from Negative Metal Oxide Semiconductor (NMOS) technology to Complementary Metal Oxide Semiconductor (CMOS) technology in the wafer fabrication plant during the study period. (% Technology refers to the percentage of wafers released in a given week to be processed by NMOS and CMOS technologies, respectively).
Fig. 3. Transition from Negative Metal Oxide Semiconductor (NMOS) technology to Complementary Metal Oxide Semiconductor (CMOS) technology in the wafer fabrication plant during the study period. (% Technology refers to the percentage of wafers released in a given week to be processed by NMOS and CMOS technologies, respectively).
Fig. 4. 123-Week yield data.
Fig. 4. 123-Week yield data.
Fig. 6. 123-Week activities data. can be easily confused with chaos. The ‘residual test’ documented in Brock (1986) and Brock and Sayers (1988) is a means to address this problem. This involves fitting a plausible model to the time series data, and analyzing the residuals using the diagnostics for detecting chaotic behavior. According to Brock and Sayers (1988) (p. 78), ‘‘if a time series {a,}, t=1,...,T, has a deterministic explanation and you fit a smooth time series model, then the residuals {e,}, t=1,...,T, of that model must have the same largest Lyapunov exponent and the same correlation dimension as {a,}’’. The application of residual test has now become a routine procedure in studies aimed at detecting chaos in time series data (see Sayers, 1990; Cheng and Van de Ven, 1996). Accordingly, we initiate the data analysis by fitting time series models to the data sets on yield, cycle time, and activities, respectively. For each one of the best-fit- ting time series models, we examined the residuals (which have been declared to be white noise, i.e.,
Fig. 6. 123-Week activities data. can be easily confused with chaos. The ‘residual test’ documented in Brock (1986) and Brock and Sayers (1988) is a means to address this problem. This involves fitting a plausible model to the time series data, and analyzing the residuals using the diagnostics for detecting chaotic behavior. According to Brock and Sayers (1988) (p. 78), ‘‘if a time series {a,}, t=1,...,T, has a deterministic explanation and you fit a smooth time series model, then the residuals {e,}, t=1,...,T, of that model must have the same largest Lyapunov exponent and the same correlation dimension as {a,}’’. The application of residual test has now become a routine procedure in studies aimed at detecting chaos in time series data (see Sayers, 1990; Cheng and Van de Ven, 1996). Accordingly, we initiate the data analysis by fitting time series models to the data sets on yield, cycle time, and activities, respectively. For each one of the best-fit- ting time series models, we examined the residuals (which have been declared to be white noise, i.e.,
Fig. 5. 123-Week cycle time data.
Fig. 5. 123-Week cycle time data.
Estimation of the best-fitting ARIMA models (t-statistics in parentheses) Table 1
Estimation of the best-fitting ARIMA models (t-statistics in parentheses) Table 1
Estimates of correlation exponents for embedding dimensions, m= 1 to 5, for the original residuals from the best-fitting ARIMA models, and for the randomly shuffled residuals
Estimates of correlation exponents for embedding dimensions, m= 1 to 5, for the original residuals from the best-fitting ARIMA models, and for the randomly shuffled residuals
Estimates of dominant (largest) Lyapunov exponents for the origi- nal residuals from the best-fitting ARIMA models Table 3 outlined in Fig. 7. Following step 2, all three original time series data sets were made stationary by taking first logarithmic difference, Alnx, = Inx,—Inx,_,, where x, is the original observation at time t, and In is the natural logarithm. We then used the ARIMA procedure of SAS Institute (1988) to obtain the following best-fitting models: ARIMA (0,1,1) for cycle time; ARIMA (2,1,3) for yield; ARIMA (2,1,0) for activities. The residuals of each of the fitted models exhibited properties of white noise, and Ljung-Box Q-statistics at various lags implied an adequate model fit for all three cases. Table 1 con- tains the equations, parameter estimates, and the relevant statistics for the three best-fitting ARIMA models.
Estimates of dominant (largest) Lyapunov exponents for the origi- nal residuals from the best-fitting ARIMA models Table 3 outlined in Fig. 7. Following step 2, all three original time series data sets were made stationary by taking first logarithmic difference, Alnx, = Inx,—Inx,_,, where x, is the original observation at time t, and In is the natural logarithm. We then used the ARIMA procedure of SAS Institute (1988) to obtain the following best-fitting models: ARIMA (0,1,1) for cycle time; ARIMA (2,1,3) for yield; ARIMA (2,1,0) for activities. The residuals of each of the fitted models exhibited properties of white noise, and Ljung-Box Q-statistics at various lags implied an adequate model fit for all three cases. Table 1 con- tains the equations, parameter estimates, and the relevant statistics for the three best-fitting ARIMA models.
Related topics:
EngineeringChaos TheoryDifference equation
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