Evaluating wind speed probability distribution models with a novel goodness of fit metric: a Trinidad and Tobago case study

Abstract

Wind energy has been explored as a viable alternative to fossil fuels in many small island developing states such as those in the Caribbean for a long time. Central to evaluating the feasibility of any wind energy project is choosing the most appropriate wind speed model. This is a function of the metric used to assess the goodness of fit of the statistical models tested. This paper compares a number of common metrics then proposes an alternative to the application-blind statistical tools commonly used. Wind speeds at two locations are considered: Crown Point, Tobago; and Piarco, Trinidad. Hourly wind speeds over a 15-year period have been analyzed for both sites. The available data is modelled using the Birnbaum-Saunders, Exponential, Gamma, Generalized Extreme Value, Generalized Pareto, Nakagami, Normal, Rayleigh and Weibull probability distributions. The distributions were compared graphically and their parameters were estimated using maximum likelihood estimation. Goodness of fit was assessed using the normalised mean square error testing, Chi-squared testing, Kolmogorov-Smirnov, R-squared, Akaike information criteria and Bayesian information criteria tests and the distributions ranked. The distribution ranking varied widely depending on the test used highlighting the need for a more contextualized goodness of fit metric. With this in mind, the concept of application-specific information criteria (ASIC) for testing goodness of fit is introduced. This allows distributions to be ranked by secondary features which are a function of both the primary data and the application space.

Pedagogy of chi‐square goodness of fit test for continuous distributions

“…These densities are important in statistical modeling in engineering and medical signal analysis involving wireless systems, communications, lifetimes, images etc. [11][12][13]22 Appendix I contains the densities and appropriate moments. The hypothesis testing could be done in Microsoft ® Excel.…”

“…The procedure described so far is simple and straightforward to see whether a particular density is a reasonable fit or not. However, often any one of several densities may be reasonable fits and our interest is to determine the best fit among these densities . With each density, we now have a null hypothesis and we may choose the closest fit by simply picking the density with the lowest value of the test statistic in Equation .…”

“…However, often any one of several densities may be reasonable fits and our interest is to determine the best fit among these densities. [11][12][13] With each density, we now have a null hypothesis and we may choose the closest fit by simply picking the density with the lowest value of the test statistic in Equation (3). This might not suffice when the number of degrees of freedom may be different as it is the case when we have two densities, Rayleigh and gamma.…”

“…The availability of a density to model of the data makes it possible to develop signal processing algorithms to extract additional information and suppression of variability in data mitigated through diversity techniques. Therefore, interest exists in checking whether a sample of data came from a population having a specific density or multiple densities and chi‐square testing provides a means to accomplish this . Another reason for introducing chi‐square tests is that it offers a number of didactic aspects of probability and random variables because it requires students to acquire familiarity with binning, estimation of parameters of the densities, degrees of freedom, P values, and so on, spanning a broad spectrum of topics.…”

“…Therefore, interest exists in checking whether a sample of data came from a population having a specific density or multiple densities and chi-square testing provides a means to accomplish this. [9][10][11][12][13] Another reason for introducing chi-square tests is that it offers a number of didactic aspects of probability and random variables because it requires students to acquire familiarity with binning, estimation of parameters of the densities, degrees of freedom, P values, and so on, spanning a broad spectrum of topics. One of the problems with teaching goodness of fit testing is the fact that textbooks offer only a limited discussion, often covering normal or uniform densities, and very few books explore ways of obtaining the best fit among a number of densities.…”