Geoscience Reference
In-Depth Information
within 12 classes, each with a width of 30 degrees.
h = histogram(data_degrees_1,0:30:360);
counts = h.Values;
h e expected number of observations is 40/12, where 40 is the total number
of observations and 12 is the number of classes.
expect = 40/12 * ones(1,12);
h e ˇ
2
-test explores the squared dif erences between the observed and
expected frequencies. h e quantity ˇ
2
is dei ned as the sum of these squared
dif erences divided by the expected frequencies.
chi2 = sum((counts - expect).^2 ./expect)
chi2 =
102.8000
h e critical ˇ
2
value can be calculated using
chi2inv
. h e ˇ
2
-test requires
the degrees of freedom ʦ. In our example we test the hypothesis that the
data are uniformly distributed, i.e., we estimate one parameter, which
is the number of possible values
N
. Since the number of classes is 12, the
number of degrees of freedom is ʦ=12-1-1=10. We test our hypothesis
on a
p
=95% signii cance level. h e function
chi2inv
computes the inverse
of the cumulative distribution function (CDF) of the ˇ
2
distribution with
parameters specii ed by ʦ for the corresponding probabilities in
p
.
chi2inv(0.95,12-1-1)
ans =
18.3070
Since the critical ˇ
2
of 18.3070 is well below the measured ˇ
2
of 102.8000 we
reject the null hypothesis and conclude that our data do not follow a uniform
distribution, i.e., they are not randomly distributed.
10.6 Test for the Signii cance of a Mean Direction
Having measured a set of directional data in the i eld, we may wish to know
whether there is a prevailing direction documented in the data. We use
the Rayleigh's test for the signii cance of a mean direction (Mardia 1972,
Mardia and Jupp 2000). h is test uses the mean resultant length introduced
in Section 10.3, which increases as the preferred direction becomes more
signii cant.
