Density probability plot for von Mises distribution fitted to circular data
circdpvm varname [if exp] [in range] [ , a(#) param(numlist) generate(newvar1 newvar2) line(line_options) graph_options ]
circdpvm gives a density probability plot for the fit of a von Mises (a.k.a. circular normal) distribution to a circular variable on a scale between 0 and 360 degrees. By default circvm is used to fit the distribution, estimating the two parameters vector mean mu and concentration parameter kappa. Both observed and expected densities are rotated so that each set is centred on the vector mean. Each is nevertheless labelled in terms of varname.
To establish notation, and to fix ideas with a concrete example: consider an observed variable theta, whose distribution we wish to compare with a von Mises distributed variable phi. That variable has density function f(phi), distribution function P = F(phi) and quantile function Q(P). (The distribution function and the quantile function are inverses of each other.) Clearly, this notation is fairly general and also covers other distributions, at least for continuous variables.
The particular density function f(theta | parameters) most pertinent to comparison with data for phi can be computed given values for its parameters, either estimates from data on theta, or parameter values chosen for some other good reason. In the case of a von Mises distribution, these parameters would usually be the vector mean and the concentration parameter kappa.
The density function can also be computed indirectly via the quantile function as f(Q(P)). In the case of von Mises distributions, it is convenient to centre variables at the vector mean, so that P = 0.5 corresponds to the vector mean. In practice P is calculated as so-called plotting positions p_i attached to values of a sample of size n which have rank i. One simple rule uses p_i = (i - 0.5) / n. Most other rules follow one of a family (i - a) / (n - 2a + 1) indexed by a.
Plotting both f(theta | parameters) and f(Q(P = p_i)), calculated using plotting positions, versus observed theta gives two curves. In our example, the first is von Mises by construction and the second would be a good estimate of a von Mises density if theta were truly von Mises with the same parameters. The match or mismatch between the curves allows graphical assessment of goodness or badness of fit. What is more, we can use experience from comparing frequency distributions, as shown on histograms, dot plots or other similar displays, in comparing or identifying location and scale differences, skewness, tail weight, tied values, gaps, outliers and so forth.
Such density probability plots were suggested by Jones and Daly (1995). See also Jones (2004). For further discussion see dpplot.
a() specifies a family of plotting positions, as explained above. The default is 0.5. Choice of a is rarely material unless the sample size is very small, and then the exercise is moot whatever is done.
param() specifies two parameter values which give an alternative reference distribution. The first is the vector mean and the second is the concentration parameter kappa. (By default these parameters are estimated from the data.)
generate() specifies two new variable names to hold the results of densities estimated from the data directly (as f() given parameters) and indirectly (as f(Q(P)) given parameters).
line(line_options) are options of twoway mspline and twoway line, which may be used to control the rendition of the density function curve.
show() specifies a numlist of axis labels to be shown, overriding the default.
graph_options are options of twoway.
. circdpvm wallasp . circdpvm wallasp, show(0(45)315)
Jones, M.C. 2004. Hazelton, M.L. (2003), "A graphical tool for assessing normality," The American Statistician 57: 285-288: Comment. The American Statistician 58: 176-177.
Jones, M.C. and F. Daly. 1995. Density probability plots. Communications in Statistics, Simulation and Computation 24: 911-927.
Nicholas J. Cox, University of Durham, U.K. email@example.com
On-line: help for circvm, circpvm, dpplot (if installed)