Extrapolation in Space (EVA Theory)

The extrapolation methods aim at statistical inference about extreme wall loss/pit depth on the whole HX (or stratum), or on the not inspected part. For this purpose, two methods are used, the first is the return-level method, the other the extrapolated distribution method.

Return Level Method

This plot estimates the most likely value of the max wall loss in the whole HX (or stratum) together with the confidence bounds. In essence, this plot is the linear version of the Variate Plot.

Return Level Method.For a given value of p, where 0<p<1, we determine such a value of Z_p that:

Where F is the fitted CDF (Cumulative Distribution Function).

For the GEV model:

For the Gumbel mode:

Where

This implies that for any tube the max wall loss exceed Z_p with probability p. Z_p is called the return level associated with the return period N = 1/p. The level is expected to be exceeded on average once every 1/p blocks. If N = total number of tubes, then Z_p is the wall loss expected to be exceeded once.

Confidence bounds can be calculated by the profile likelihood method or the Delta method. IMS uses the Delta method.

Extrapolated Distribution Method

The Probability Density Function (PDF) of the Gumbel can also be used to extrapolate in space.

Gumble PDF.

 

Given the estimated distribution model for the max wall loss of one (single) tube, F, the cumulative distribution function of the maximum wall loss for N HX tubes can be derived according to the following transformation:

For the GEV model, the extrapolated CDF reads:

Where

This is the link between the original parameters and the parameters of the extrapolated distribution. The extrapolated PDF function can be easily derived from the provided extrapolated CDF. When assuming the Gumbel model, the extrapolation of the CDF & PDF is done in a similar way.

The peak of the plot is the Most Likely max Wall Loss.