Probability Model Fitting Steps For a given data x1, x2, …, xN Plot the histogram with the default bin width >hist(x, probability=T, …, …) Also plot the boxplot Select candidate PDFs based on the histogram and boxplot Fit the PDFs – i.e., compute the parameters of the PDFs Evaluate the PDF at the observational points or on a finer grid (for better plotting) E.g., to fit a Normal PDF to the data >theta1 = mean(x) >theta2 = sd(x) >fitnormpdf = dnorm(sort(x), theta1, theta2) Similarly fit all the candidate PDFs to the data – using the appropriate commands

Goodness of Fit The goodness of the fitted probability model can be evaluated in two ways – visual and Quantitative Visual Overlay the fitted PDF on the histogram >lines(sort(x), fitnormpdf, col=“red”) Quantile plots Compute the quantiles from the fitted PDFs and plot them agains the empirical quantiles. If they fall on a straight line then the model is a good fit. Empirical quantile P_i = (i- a)/(N + 1 – 2a) i is the ‘rank’ of the observation x_i and a = 0 If the observations are sorted then their ranks are simply the sequential order >N = length(x) >empquant = (1:N)/(N+1) >fitnormquant = qnorm(empquant, theta1, theta2) >plot(fitnormquant, sort(x), xlab=“Model Quantiles”, ylab= “Empirical Quantiles”)

Goodness of Fit Quantitative Perform a Kolmogrov-Smirnov (K-S) test on the empirical CDF and the fitted model CDF >ksnorm = ks.test(x, “pnorm”, mean=theta1, sd=theta2) This compares the empirical CDF with the fitted Normal PDF If ksnorm$p.value is ‘greater than or equal’ to 0.05 it implies that At 95% confidence level the empirical and the fitted distribution are not different Use the quantitative and visual metrics to decide on the best model

Boxplot One step = 1.5*IQR IQR = 33 –26 = 7 °F One step = 1.5*7 = 10.5 °F Lower inner fence = 26 – 10.5 = 15.5 °F Upper inner fence = 33 + 10.5 = 43.5 °F The whiskers are drawn to the most extreme temperatures inside the inner fences, 37 and 17 °F. The whiskers are therefore shortened to extend only to the last observation within one step beyond either end of the box (“adjacent values”). One step = 1.5*IQR

Histogram Histogram of Ithaca temperature, January 1987.

Histogram and Probability Density Function