1. Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies
1.4.2.3. Random Walk

1.4.2.3.3.

Develop A Better Model

Lag Plot Suggests Better Model

Since the underlying assumptions did not hold, we need to develop a better model.

The lag plot showed a distinct linear pattern. Given the definition of the lag plot, Y_i versus Y_i-1, a good candidate model is a model of the form

\( Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} \)

Fit Output

The results of a linear fit of this model generated the following results.

      Coefficient    Estimate     Stan. Error   t-Value
          A0         0.050165      0.024171       2.075
          A1         0.987087      0.006313     156.350 
 
      Residual Standard Deviation = 0.2931
      Residual Degrees of Freedom = 497 

The slope parameter, A₁, has a t value of 156.350 which is statistically significant. Also, the residual standard deviation is 0.2931. This can be compared to the standard deviation shown in the summary table, which is 2.078675. That is, the fit to the autoregressive model has reduced the variability by a factor of 7.

Time Series Model

This model is an example of a time series model. More extensive discussion of time series is given in the Process Monitoring chapter.