LECTURE 16
GEOS 430
1
Joint probability distributions and covariance
Up until now we have been dealing with univariate statistics (1 variable).
After the break we will be doing multivariate statistics. This will require us to begin
using matrices (lecture

LECTURE 2
GEOS 430
1
Types of data
1. Nominal: Classification Scheme, may be numeric but could as easily be A,B,C
1 - limestone
2 - sandstone in a geol. column
no relationship between 1 and 2 i.e. sandstone is not twice as (whatever) as limestone
2. Or

LECTURE 38
Examples of PCA
GEOS 430
1
LECTURE 38
GEOS 430
2
LECTURE 38
GEOS 430
3
LECTURE 38
GEOS 430
4
LECTURE 38
5
GEOS 430
Examples of factor analysis:
An Example from Davis using Boxes (BOXES.TXT) See figure 6-13 for pictures of the boxes
Label
Long a

LECTURE 26
GEOS 430
Spectral, Harmonic or Fourier Analysis
Fourier analysis attempts to decompose a time series
into a suite of wave forms. These can be displayed as
harmonics or as some graph of periods or
frequencies.
Basically the idea is that any time

LECTURE 5
1
GEOS 430
Univariate statistics Continued
Measures of dispersion
OK, so we have defined central tendency, what about dispersion about that value (the spread)?
n
X
Absolute deviation of the mean:
i =1
i
X
n
Can be used with the median or the mea

LECTURE 23
1
GEOS 430
Forced regressions through 0
Sometimes a line passes through the origin, and we know that it passes through the origin.
In fact sometimes that is the best point that we know.
This line through the origin has the form:
Y = 1X
What is

LECTURE 20
GEOS 430
1
Bivariate Regression:
2 variables: separate but related to correlation
Regression attempts to fit a line (linear) or a curve (non-linear) to data in an attempt to explain or
model the variance seen. Is there some underlying control a

LECTURE 17
1
GEOS 430
Matrix Algebra
My analysis is very much applied and we will not get into linear algebra theory
Remember that a matrix is a table (like the data table from the Steese district).
Definitions:
A matrix has data in rows and columns
The n

LECTURE 32
GEOS 430
1
Multivariate regression analysis
Trend surface analysis is, in fact the easiest type of multivariate regression analysis. We were
explaining the variance on one dependent variable (Y) in terms of two independent variables (X1
and X2)

LECTURE 15
1
GEOS 430
NONPARAMETRIC ANOVA AND POST HOC TESTS
Nonparametric ANOVA
Kruskal-Wallis test: many group extension of Mann-Whitney uses chi-square tables for most
samples. As with the Mann-Whitney, samples for each group are ranked:
Lets assume th

LECTURE 11
1
GEOS 430
Error estimation and propagation of
errors:
f+ f
f(x)
The Theory:
Consider a function
of a variable: f(x)
x
If there is a small change in the value of x:
x + x this will result in a new value of the
function: f + f.
The value f can b

LECTURE 7
GEOS 430
1
Hypothesis testing:
We have addressed the question:
What is the probability of finding a shell (ore sample, etc.) smaller than a certain size
from a population with known parameters?
However what we really want to ask is:
Is my shell

LECTURE 4
GEOS 430
Distributions
For almost all geologic data, the data are distributed among a range of values. Again, it is this
variability that we want to investigate. We will use various statistics to describe this variability (the
location, spread,

LECTURE 39
1
GEOS 430
Q-mode factor analysis and Canonical Correlation
Q-mode factor analysis
R-mode factor analysis uses a set of specimens to examine the relationship between variables,
while Q-mode factor analysis uses a set of variables to examine the

LECTURE 14
1
GEOS 430
Analysis of variance
So far we have looked at data and noticed that there is variability. We have quantified this
variance and have done some tests to compare two samples that have some variance.
Now we want to do a more careful exam

LECTURE 30
1
GEOS 430
8
Kriging: The example from Swan and
Sandilands (Introduction to Geological Data
Analysis, 1995).
0
14.00
12.00
Here are the basic data again (swan1.txt on the
CD.
0
13
2
2
0
11
13
8
gamma
8
3
7
2.00
4.00
8
3
2.00
6.00
0
8.00
10.00
1

LECTURE 18
1
GEOS 430
Determinants:
The determinant is another property of a square matrix. It is a number calculated for a matrix
and is unique for that matrix.
a
For a 2 by 2 matrix: [ A] = 11
a 21
|A| = a11a22 a12a21
The determinant of [A]
1
det
3
a12

LECTURE 24
GEOS 430
Circular and Spherical Statistics continued
Distributions on a circle.
There are several different models for describing a
univariate distribution on a circle. The most
common is called the von Mises distribution. It
contains two param

LECTURE 37
GEOS 430
1
Principal component analysis (PCA) and Factor Analysis
R-Mode Factor Analysis
The standard way of doing factor analysis is to look for relationships between variables (Rmode). We can then identify these relationships and see how indi

LECTURE 28
GEOS 430
1
SEMIVARIANCE
If we have a more regular pattern of data we could use more points
to estimate our value. In a random sequence, there is no
relationship between the points so interpolation doesnt work.
When we do an interpolation, we sa

LECTURE 9
GEOS 430
1
Comparing one sample to another: the unpaired t-test
When we compare samples we are really looking at both location (are the mean values the same)
and scale (are the variances or spreads the same).
We need to consider both scale and l

LECTURE 3
1
GEOS 430
Box plots (schematic plot)
from letter value table can construct a box
a graphical representation of data
Skeletal box (and whisker)
5 number summary, uses median, extremes and hinges
pH
Upper extreme = 5.78
5.0
Upper hinge = 4.82
Med

CP3311 Practical Week 6
Using the Unity NavMesh system
Working with static obstacles
o Briefly reiterate the basic purpose of a NavMesh, and how it saves
the need to manually implement pathfinding algorithms (see the
lecture slides for details.)
o Create

LECTURE 34
GEOS 430
1
Cluster Analysis
Often times we do not have any preconceived information about the grouping of our data, instead
we want the data to define its own groups. To do this, cluster analysis has been developed to
group data. Cluster analys

LECTURE 27
1
GEOS 430
Sequences of Data and Time Series
We will now look at data in sequence in which there is a spatial or temporal relationship in
how the data are collected or analyzed
We will work with bivariate data with one independent variable (X),

LECTURE 35
GEOS 430
1
Hierarchical cluster analysis by Pearsons correlation coefficient using average linkage of the Lime
Peak Data set.
Most of these clusterings can be explained by a quick glance at a periodic table.
Using a phenon line at 15 gives us 4

LECTURE 27
GEOS 430
Estimating the value at a point: a prelude to semivariance and geostatistics
The next topic that we are going to want to address is estimating the value at a point This is
useful in one dimension (transects or time series) or 2 dimensi

LECTURE 10
GEOS 430
1
Chi-Square Test 2:
There are several applications of the Chi-Square statistic. The most common is the goodness of
fit test where observations are compared to a distribution. This test is applied to discrete data
and is a non-parametr

LECTURE 6
GEOS 430
NORMAL DISTRIBUTION
If we consider random variations about some
mean value, this is the same as the binomial
equation with p = 0.5, but with a different mean
and a measure of the amount of variation
As N becomes large, the function beco

LECTURE 10
GEOS 430
1
Chi-Square Test 2:
There are several applications of the Chi-Square statistic. The most common is the goodness of
fit test where observations are compared to a distribution. This test is applied to discrete data
and is a non-parametr

1
LECTURE 22
GEOS 430
Residuals
The difference between the observed and expected values of Y in least squares regression
is called the residual. We can plot the residuals versus the X variable to look for trends:
If we have a good regression, we would exp

LECTURE 13
GEOS 430
1
TESTING DATA DISTRIBUTION ON MAPS
Geology often deals with maps and sample locations. The underlying geology may control the
distribution of sample locations (such as oil wells). We will use the ideas of map distribution
throughout t

LECTURE 19
GEOS 430
1
Eigenvectors and Eigenvalues
We can talk about this bivariate distribution
as having variance in the two variables and
a covariance. The variables are
independent, or orthogonal.
The covariance tells us about the
relationship of the