distribution analysis - finding the best distribution that explains … group... · 2016. 3....

Distribution AnalysisReal dataSummary

References

Distribution AnalysisFinding the best distribution that explains your data

Sébastien Casault

ENMAX Energy Corporation

8 October, 2015

Sébastien Casault Distribution Analysis


References

IntroductionStatistical testsGoodness of fit

Introduction

We often fit observations to a model (e.g., lognormal distribution).How can we ensure that the model is appropriate? Is there a modelthat would provide more accurate predictions?

Goodness of fitMeasures of goodness of fit typically summarize the discrepancybetween observed values and the values expected under the modelin question. – Wikipedia



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Kolmogorov-Smirnov

The K-S statistic, D, is defined as:

Dn = supx|Fn(x)− F (x)|

for the hypothesized distribution is F , and empirical (sample)cumulative distribution function is Fn.



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Anderson-Darling

There are many fit tests - they are mostly variations of the KS test.

For example, the AD statistic, A, is defined as:

A = n∫ ∞−∞

(Fn(x)− F (x))2

F (x) (1− F (x)) dF (x)

and is a weighted sum of the quadratic difference between thehypothesized distribution and the sample one, placing more weighton observations in the tails.



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P value statistics

The P value is the answer to this question:

If the two samples were randomly sampled from identicalpopulations, what is the probability that the two cumulativefrequency distributions would be as far apart as observed?More precisely, what is the chance that the value of the teststatistic would be as large or larger than observed?

If the P value is small, conclude that the two groups were sampledfrom populations with different distributions. The populations maydiffer in median, variability or the shape of the distribution.



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ContextCoal plant outagesDistribution fitting

Electricity prices in Alberta

One of the most volatile commodities traded in wholesale markets.



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Sources of volatility

An increasing portion of the supply portfolio is stochastic:

Alberta has an installed wind capacity of 8.3%Coal-fired power plants undergo forced outages



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Case study - Sundance 2

TransCanada’s Sundance A and B Power Purchase Arrangements

entitle TransCanada to more than 900 megawatts (MW) of capacity

from the Sundance Power Plant. TransCanada sells this electricity

under long-term contracts and into the spot market. The Sundance

Power Plant has a total of six generating units and is owned and

operated by TransAlta.

Sundance A & B Power Purchase Agreement

Power Purchase Arrangement Highlights

Sundance A PPA:

100 per cent of the output from units 1 & 2 = 560 MW. Term expires in 2017.

Sundance B PPA:

50 per cent of the output from Units 3 & 4 = 353 MW. Term expires in 2020.

Location:

The plant is located 70 kilometres (about 45 miles) west of Edmonton, Alberta on the south shore of Lake Wabamun.

In-Service Date:

Unit 1 - 1970; Unit 2 - 1973; Unit 3 - 1976; Unit 4 - 1977.

Capacity:

2,029 MW.

Fuel:

Coal from TransAlta’s Highvale mine.

Environmental Features:

Meets ISO 14001 standards; Regulated by Alberta Environment and the Alberta Electric Utilities Board.

Owner:

TransAlta Utilities Corporation.

Operator:

TransAlta Utilities Corporation.



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Outage statistics

The Sundance 2 unit has undergone several forced outages in 2015 -often coinciding with wholesale market price spikes.



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Distribution fitting

PROC UNIVARIATE DATA = WORK.ON ;VAR ON ;HISTOGRAM ON / NORMAL LOGNORMAL EXP WEIBULL ;CDFPLOT ON / WEIBULL ;

RUN ;



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Distribution statistics and goodness of fit

Both fits provide an accurate description of the observed data.There may be a more theoretical reason to choose the Weibulldistribution.



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Summary

Finding the right predictive model is importantThere are several tests that can quantify how well a certainmodel fits empirical dataUsing these tests, we can obtain GOF statisticsBuild more reliable models using the right fit



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Bibliography

1 Base SAS(R) 9.2 Procedures Guide: Statistical Procedures. UNIVARIATE Procedure, Goodness-of-FitTests.

2 Clauset, A., Shalizi, C. R., and Newman, M. E. J. (2007). Power-Law Distributions in Empirical Data.SIAM Review, 51, 661-703.

3 Hagiwara, Y. (1974). Probability of earthquake occurrence as obtained from a Weibull distribution analysisof crustal strain. Tectonophysics, 23, 313-318.


Distribution AnalysisIntroductionStatistical testsGoodness of fit

Real dataContextCoal plant outagesDistribution fitting

SummaryReferences

distribution analysis - finding the best distribution that explains … group... · 2016. 3....

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