Page 106 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 4

Note that the matrix is symmetrical and the diagonal elements remain unchanged,
within the limits of rounding error. The off-diagonal elements are the expected
frequencies of transitions within the embedded sequence, assuming independence
between successive states. If the diagonal elements are stripped from the matrix,
it may be compared directly to the observed transition frequency matrix because
the row and column totals of the two are the same, again within rounding limits.
The comparison by x2 methods yields a test statistic of x2 = 172. The test has
v = (m - 1)2 - m degrees of freedom, where m is the number of states, or in this
example, v = 11. The critical value of x2 for 11 degrees of freedom and an o( = 0.05
level of significance is 19.68, which is far exceeded by the test statistic. Therefore,
we must conclude that successive lithologies encountered in the Scottish well are
not independent, but rather exhibit a strong first-order Markovian property.
If tests determine that a sequence exhibits partial dependence between succes-
sive states, the structure of this dependence may be investigated further. Simple
graphs of the most significant transitions may reveal repetitive patterns in the suc-
cession. Modified x2 procedures are available to test the significance of individual
transition pairs. Some authors have found that the eigenvalues extracted from the
transition probability matrix are useful indicators of cyclicity. (It should be noted,
however, that extracting the eigenvectors from an asymmetric matrix such as the
transition probability matrix may not be an easy task!) These topics will not be
pursued further in this book; the interested reader should refer to the texts by Ke-
meny (1983) and Norris (1997), as well as the book on quantitative sedimentology
by Schwarzacher (1975). Chi-square tests appropriate for embedded sequences
are discussed by Goodman (1968). In a geological context, the articles by Dove-
ton (1971) and Doveton and Skipper (1974), plus the comment by Tiirk (1979), are
recommended.

Series of Events
An interesting type of time series we will now consider is called a series of events.
Geological examples of this type of data sequence include the historical record
of earthquake occurrences in California, the record of volcanic eruptions in the
Mediterranean area, and the incidence of landslides in the Tetons. The character-
istics of these series are (a) the events are distinguishable by when they occur in
time; (b) the events are essentially instantaneous; and (c) the events are so infre-
quent that no two occur in the same time interval. A series of events is therefore
nothing more than a sequence of the intervals between occurrences. Our data may
consist of the duration between successive events, or the cumulative length of time
over which the events occur. One form may be directly transformed into the other.
Series-of-events models may be appropriate for certain types of spatially dis-
tributed data. We might, for example, be interested in the occurrence of a rare
mineral encountered sporadically on a traverse across a thin section or in the ap-
pearance of bentonite beds in a vertical succession of sedimentary rocks. Justifica-
tion for applying series-of-events models to spatial data may be tenuous, however,
and depends on the assumption that the spatial sequence has been created at a
constant rate. This assumption probably is reasonable in the first example, but
the second requires that we assume that the sedimentation rate remained constant
through the series.
The historic record of eruptions of the volcano Aso in Kyushu, Japan, has been
kept since 1229 (Kuno, 1962), and is given in Table 4-5 and file ASO.TXT. Aso is

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