Geoscience Reference
In-Depth Information
probability density function
y
, and display the result.
x = 2 : 1/20 : 10;
y = normpdf(x,phat(:,1),phat(:,2));
y = trapz(v,n) * y/trapz(x,y);
bar(v,n), hold on, plot(x,y,'r'), hold off
In earth sciences we ot en encounter mixed distributions. Examples are
multimodal grain size distributions (Section 8.8), multiple preferred
paleocurrent directions (Section 10.6), and multimodal chemical ages in
monazite rel ecting multiple episodes of deformation and metamorphism
in a mountain belt. Fitting Gaussian mixture distributions to the data aims
to determine the means and variances of the individual distributions that
combine to produce the mixed distribution. h e methods described in this
section help to determine the episodes of deformation in a mountain range,
or to separate the dif erent paleocurrent directions caused by tidal l ow in an
ocean basin.
As a synthetic example of Gaussian mixture distributions we generate
two sets of 100 random numbers
ya
and
yb
with means of 6.4 and 13.3,
respectively, and standard deviations of 1.4 and 1.8, respectively. We then
vertically concatenate the series using
vertcat
and store the 200 data values
in the variable
data
.
clear
rng(0)
ya = 6.4 + 1.4*randn(100,1);
yb = 13.3 + 1.8*randn(100,1);
data = vertcat(ya,yb);
Plotting the histogram reveals a bimodal distribution. We can also determine
the frequency distribution
n
using
histogram
.
e = -0.5 : 30.5;
h = histogram(data,e);
v = h.BinWidth * 0.5 + h.BinEdges(1:end-1);
n = h.Values;
We use the function
mgdistribution.fit(data,k)
to i t a Gaussian mixture
distribution with
k
components to the data. h e function i ts the model by
maximum likelihood, using the
Expectation-Maximization (EM) algorithm
.
h e EM algorithm introduced by Arthur Dempster, Nan Laird and Donald
Rubin (1977) is an iterative method alternating between performing an
expectation step and a maximization step. h e expectation step computes an
expectation of the logarithmic likelihood with respect to the current estimate
