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help for circdpvm
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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 ]

Description

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.

Remarks

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).

Options

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.

Examples

. circdpvm wallasp

. circdpvm wallasp, show(0(45)315)

References

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.

Author

Nicholas J. Cox, University of Durham, U.K.
n.j.cox@durham.ac.uk

Also see

On-line: help for circvm, circpvm, dpplot (if installed)

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