##### Document Text Contents

Page 2

Statistics and Data Analysis

Third Edition

John C. Davis

Kansas Geological Survey

The University of Kansas

John Wiley & Sons

New York Clxchester Brisbane Toronto Singapore

Page 128

Spatial Analysis

0

0

0 0

0

0 0

0

5 4 O O

0 0

0

0 (1

0

U

0

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0

C

% 1 5 -

L -

0 z

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0 000 0

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00

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Easting

Figure 5-5. Locations of 123 exploratory holes drilled to top of Ordovician rocks (Arbuckle

Group) in central Kansas. Map has been divided into 12 cells of equal size.

orientation of subareas. However, the test is most efficient if the number of subar-

eas is a maximum (this increases the degrees of freedom), subject to the restriction

that no subarea contain fewer than five points. The expected number of points in

each subarea is

N

k (5 .8)

E = -

where N is the total number of data points and k is the number of subareas. A

x 2 test of goodness of fit of the observed distribution to the expected (uniform)

distribution is

where Oi is the observed number of data points in subarea i and E is the expected

number. The test has v = k - 2 degrees of freedom, where k is the number of

subareas.

As an example of the application of this test, consider the data-point distribu-

tion shown in Figure 5-5. These are the locations of 123 holes drilled in the search

for oil in the Ordovician Arbuckle stratigraphic succession in central Kansas. These

data are listed in file ARBUCKLE.TXT. In Figure 5-5, the map area has been divided

into 12 equal subareas, each of which we expect to contain about ten points, if the

points are uniformly distributed. The observed number of points in each subarea

and the computations necessary to find the test value are given in Table 5-1. This

test has v = 10 degrees of freedom, so the critical value of x 2 at the 5% (a = 0.05)

significance level is 18.3. The computed test value of x 2 = 17.0 does not exceed

301

Page 129

Statistics and Data Analysis in Geology - Chapter 5

this, so we conclude that there is no evidence suggesting that the quadrats are

unevenly populated. Note that the test applies only to the uniformity of point den-

sities between areas of a specified size and shape. It is possible that we could

select quadrats of different sizes or shapes that might not be uniformly populated,

especially if they were smaller than those used in this test.

Table 5-1. Number of wells in 12 subareas of central Kansas.

Observed Number (0 - E ) *

of Points E

10 0.006

5 2.689

5 2.689

11 0.055

13 0.738

5 2.689

12 0.299

16 3.226

16 3.226

9 0.152

13 0.738

8 0.494

TOTAL = 123

aTest value is not significant at the a = 0.05 level.

x 2 = 16.995"

Random patterns

Establishing that a pattern is uniform does not specify the nature of the unifor-

mity, for both regular and random patterns are expected to be homogeneous. For

many purposes, verifying uniformity is sufficient; but, if we desire more informa-

tion about the pattern, we must turn to other tests. If points are distributed at

random across a map area, even though the coverage is uniform, we do not expect

exactly the same number of points to lie within each subarea. Rather, there will

be some preferred number of points that occur in most subareas and there will

be progressively fewer subareas that contain either more points or fewer. This is

apparent in the example we just worked: although our hypothesis of uniformity

specified that we expect about ten observations in each subarea, we actually found

some areas that contained more than ten and some that contained fewer.

You will recall that the Poisson probability distribution is the limiting case

of the binomial distribution when p , the probability of a success, is very small

and (1 - p ) approaches 1.0. The Poisson distribution can be used to model the

occurrence of rare, random occurrences in time, as it was used in Chapter 4, or

it can be used to model the random placement of points in space. Although the

Poisson distribution, like the binomial, uses the numbers of successes, failures,

and trials in the calculation of probabilities, it can be rewritten so that neither the

number of failures nor the total number of trials is required. Rather, it uses the

number of points per quadrat and the density of points in the entire area to predict

how many quadrats should contain specified numbers of points. These predicted

302

Page 256

Index Terms Links

This page has been reformatted by Knovel to provide easier navigation.

West Texas (USA) 36

WHITE.TXT 93

Whitewater Bay 93 96 111

whorl 502

Wilburton gas field (Oklahoma) 391

Wilcoxon test 105

Williston Basin (North Dakota) 239

Windfall Reef (Devonian) 406

Wind River Basin 73

Wisconsinan 274

Wishart’s modification 498

witherite 494

within-cluster similarity 498

within-groups covariance matrix 573 576

WLYONS.TXT 113

Wolf River (Kansas) 350

Woodford shale (Devonian) 282

Wulff net 338 446

Wyoming (USA) 70 72 125 153 279 286 397

406 446

X

xenocryst 286

Xian province (China) 113

X-ray fluorescence 206

Y

Yellowcraigs (Scotland) 108

Yuma (Arizona) 446

YUMA.PIC 447

YUMA.TIF 447

Z

Page 257

Index Terms Links

This page has been reformatted by Knovel to provide easier navigation.

zeolites, sedimentary 114 592

ZEOLITES.TXT 592

zero isopach problem 391 449

zircon 102

zonation 234

z-score 57 95 110 476

z-statistic 57 61 63 66 310

Statistics and Data Analysis

Third Edition

John C. Davis

Kansas Geological Survey

The University of Kansas

John Wiley & Sons

New York Clxchester Brisbane Toronto Singapore

Page 128

Spatial Analysis

0

0

0 0

0

0 0

0

5 4 O O

0 0

0

0 (1

0

U

0

10

0

C

% 1 5 -

L -

0 z

I

o c

I

" 0 %

m o

0 0

0 0 00

0

0 "

0 0

0 0 0 0

0 0

0 0

0

0

0

0 0 3 0

0 0

0 254

0

0 o o O

U

1 5 20

0 000 0

m

U

0

0

0

O;I 0 0

00

b

0

b

2 5 3 0 3 5 40

Easting

Figure 5-5. Locations of 123 exploratory holes drilled to top of Ordovician rocks (Arbuckle

Group) in central Kansas. Map has been divided into 12 cells of equal size.

orientation of subareas. However, the test is most efficient if the number of subar-

eas is a maximum (this increases the degrees of freedom), subject to the restriction

that no subarea contain fewer than five points. The expected number of points in

each subarea is

N

k (5 .8)

E = -

where N is the total number of data points and k is the number of subareas. A

x 2 test of goodness of fit of the observed distribution to the expected (uniform)

distribution is

where Oi is the observed number of data points in subarea i and E is the expected

number. The test has v = k - 2 degrees of freedom, where k is the number of

subareas.

As an example of the application of this test, consider the data-point distribu-

tion shown in Figure 5-5. These are the locations of 123 holes drilled in the search

for oil in the Ordovician Arbuckle stratigraphic succession in central Kansas. These

data are listed in file ARBUCKLE.TXT. In Figure 5-5, the map area has been divided

into 12 equal subareas, each of which we expect to contain about ten points, if the

points are uniformly distributed. The observed number of points in each subarea

and the computations necessary to find the test value are given in Table 5-1. This

test has v = 10 degrees of freedom, so the critical value of x 2 at the 5% (a = 0.05)

significance level is 18.3. The computed test value of x 2 = 17.0 does not exceed

301

Page 129

Statistics and Data Analysis in Geology - Chapter 5

this, so we conclude that there is no evidence suggesting that the quadrats are

unevenly populated. Note that the test applies only to the uniformity of point den-

sities between areas of a specified size and shape. It is possible that we could

select quadrats of different sizes or shapes that might not be uniformly populated,

especially if they were smaller than those used in this test.

Table 5-1. Number of wells in 12 subareas of central Kansas.

Observed Number (0 - E ) *

of Points E

10 0.006

5 2.689

5 2.689

11 0.055

13 0.738

5 2.689

12 0.299

16 3.226

16 3.226

9 0.152

13 0.738

8 0.494

TOTAL = 123

aTest value is not significant at the a = 0.05 level.

x 2 = 16.995"

Random patterns

Establishing that a pattern is uniform does not specify the nature of the unifor-

mity, for both regular and random patterns are expected to be homogeneous. For

many purposes, verifying uniformity is sufficient; but, if we desire more informa-

tion about the pattern, we must turn to other tests. If points are distributed at

random across a map area, even though the coverage is uniform, we do not expect

exactly the same number of points to lie within each subarea. Rather, there will

be some preferred number of points that occur in most subareas and there will

be progressively fewer subareas that contain either more points or fewer. This is

apparent in the example we just worked: although our hypothesis of uniformity

specified that we expect about ten observations in each subarea, we actually found

some areas that contained more than ten and some that contained fewer.

You will recall that the Poisson probability distribution is the limiting case

of the binomial distribution when p , the probability of a success, is very small

and (1 - p ) approaches 1.0. The Poisson distribution can be used to model the

occurrence of rare, random occurrences in time, as it was used in Chapter 4, or

it can be used to model the random placement of points in space. Although the

Poisson distribution, like the binomial, uses the numbers of successes, failures,

and trials in the calculation of probabilities, it can be rewritten so that neither the

number of failures nor the total number of trials is required. Rather, it uses the

number of points per quadrat and the density of points in the entire area to predict

how many quadrats should contain specified numbers of points. These predicted

302

Page 256

Index Terms Links

This page has been reformatted by Knovel to provide easier navigation.

West Texas (USA) 36

WHITE.TXT 93

Whitewater Bay 93 96 111

whorl 502

Wilburton gas field (Oklahoma) 391

Wilcoxon test 105

Williston Basin (North Dakota) 239

Windfall Reef (Devonian) 406

Wind River Basin 73

Wisconsinan 274

Wishart’s modification 498

witherite 494

within-cluster similarity 498

within-groups covariance matrix 573 576

WLYONS.TXT 113

Wolf River (Kansas) 350

Woodford shale (Devonian) 282

Wulff net 338 446

Wyoming (USA) 70 72 125 153 279 286 397

406 446

X

xenocryst 286

Xian province (China) 113

X-ray fluorescence 206

Y

Yellowcraigs (Scotland) 108

Yuma (Arizona) 446

YUMA.PIC 447

YUMA.TIF 447

Z

Page 257

Index Terms Links

This page has been reformatted by Knovel to provide easier navigation.

zeolites, sedimentary 114 592

ZEOLITES.TXT 592

zero isopach problem 391 449

zircon 102

zonation 234

z-score 57 95 110 476

z-statistic 57 61 63 66 310