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3.3 Model validation

A model validation was performed on the Steady State model to evaluate the

significance of the values that were predicted. The model used a Runge-

Kutte method with a time step of 2 months. The techniques applied to

validate the model are considered a “standard” for evaluating whether

modeled values correlate with measured values. There must be three criteria

satisfied for validation of the model. The three criteria include; Residuals of 

the predicted concentrations must be randomly distributed, the slope of the

regression line must be significantly close to 1, and the correlationcoefficient must be statistically significant at the 95%confidence level. The

first statistical technique applied was a linear regression analysis. Figure __ 

below shows a linear regression analysis. For the model to be validated the

slope of the line must be significantly close to 1.

 YearMeasured

Calculated

ChiSquare

  (mg/m3) (mg/m3) χ^29/15/19

70 50 44.70.62168

1359/15/19

72 58.9 38.910.2730

3639/15/19

73 43 37.30.86983

1939/15/19

74 55 36.19.87419

2059/15/19

85 19 31.4

4.90862

5929/15/19

86 36.1 31.30.75237

2469/15/19

95 30 30.30.00235

5919/15/19

96 3 30.224.4924

9059/15/19 25 30.0 0.83430

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99 3739/15/20

00 24.8 29.90.88460

8769/15/20

01 35.5 29.91.05128

586

Totalχ^2

54.5647847

Figure __: Comparison of Predicted Values to Measured Values

Using a linear least squares regression to fit a trendline to the measured

data, the data is shown as non-linear due to a low r2 value of -1.644. Since

the slope of the line is 0.702 (below 1) the model fails one step in the

validation. A slope less than 1 tells us that the model is under predicting

concentrations.

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Figure ___: Comparison of Predicted Values to Measured Values’

Chi Square analysis of the measured results versus the modeled results

shows that the actual results differ too greatly from the calculated results for

the two to be proven correlated. A sum of the chi squares results in a value

of 54.56. This value isn’t even less than the Chi Square value for a .1%

confidence level (29.59 Chi Square value). The biggest factor in this large

value is an outlier concentration of 3 mg/m3 in the measured data. Without

this outlier, the Chi Square sum drops down to ~30 which is a much more

reasonable number, but still larger than any reasonable confidence level.

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  These results indicate that the model does not do a very good job of 

approximating the actual concentrations in the records.

A plot of the residuals was the second criteria performed for validation. For

the criteria to be satisfied the residuals most be randomly distributed. As

seen in Figure __ above, the residuals are not randomly distributed. Since the

values are not randomly distributed the model fails the second criteria of 

validation.

 The third criteria were not performed because the model is assumed to be

invalidated. The Runge-Kutte model was not used to predict concentrations

when selecting different remedial options.