CFD Shape and

Criteria AssessmentTMAW – 6/15/04

Elgin Perry

A simulation of the CFD shows that the level of sampling affects the shape of the curve. Sampling error tends to make the observed curves (green) less steep that the true curve (red).

Simulation

• Assume segment has 1000 cells.

• Simulate values for the 1000 cells each day for 3 years.

• Sample 10 cells once a month.

• Interpolate from the 10 to the 1000.

• Compare the CFD from sampling to the true CFD.

Solutions

• Follow precedent and ignore problem.

• Require equal sampling for Reference and Assessment CFD.

• Explore analytical tools to remove bias.

Proposed Model

Xij = U + ai + bij

i = 1, 2, . . . M and j = 1, 2, . . . N.

a is temporal variance termb is spatial variance term

U a b c

5 1 1 5 Red

4 1 1 5 Orange

3 1 1 5 Yellow

2 1 1 5 Green

1 1 1 5 Blue

Plot Parameters

Decreasing Mean

As the mean decreases, percent of noncompliance decreases in both the spatial and temporal dimension.

Increasing Temporal Variance

U a b c

3 1 1 5 Red

3 2 1 5 Orange

3 3 1 5 Yellow

3 4 1 5 Green

3 5 1 5 Blue

Increasing Temporal Variance

As temporal variance increases, there is a higher frequency of events where a large portion of the segment is out of compliance.

Increasing Spatial Variance

U a b c

3 1 1 5 Red

3 1 2 5 Orange

3 1 3 5 Yellow

3 1 4 5 Green

3 1 5 5 Blue

Increasing Spatial Variance

As spatial variance increases, there is a higher frequency of a small portion of the segment being out of compliance.

Increasing Both

U a b c

3 1 1 5 Red

3 2 2 5 Orange

3 3 3 5 Yellow

3 4 4 5 Green

3 5 5 5 Blue

Increasing Both

If both spatial and temporal variance increase, the effect is about the same as increasing the mean toward the criterion.

Sampling Distribution

U a b c stations dates

4 1 1 5 N M Red

4 4 4 5 10 M Green

Sampling Distribution

Under this nested model, the sampling distribution (green step function) is flatter than the true CFD (red curve). This analytical finding seems to confirm the findings of the simulation (compare to slide 3).

Summary

What we have:• Tool that links CFD to

mean, temporal variance and spatial variance

• Tool to adjust CFD for sampling.

What we need:• Review of tools by a

Math Stat type• Model for effect of

interpolation

Addendum

Oct. 7, 2004

Elgin Perry

New Developents

• Narcheel Nagaraj and Bimal Sinha of UMBC are looking at problem.

• They are not impressed with my nested model.

• The feeling is mutual.

• Nagaraj suggests a post Kriging bootstrap approach to quantifying the sampling distribution of the CFD.

Post Kriging Bootstrap

• For each sampling date, estimate a Krig surface.

• In the cells where there are observed data, compute the residual.

• Pool these residuals over dates, to form a large population of residuals.

Post Kriging Bootstrap

• For each date, resample from the pooled residuals once for each cell. Compute a bootstrap estimate for each cell as the sum of the original estimate and a residual.

• Repeat the bootstrap procedure for each sample date. The compute a bootstrap CFD.

• Repeat this 1000 times to form a population of CFD’s.

• From this population, obtain a confidence envelope for the CFD.

Implementation Problems

• Most implementations of Kriging “honor the data”. More like interpolation.

• There are Kriging equations that do not require this. Would require special code.

Conclusion(s)

• Needs more work.