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Spatial Data Analysis: Understanding Autocorrelation, Quadrat Counts, and Geostatistics, Exams of Environmental Science

University of Idaho (U of I)Environmental Science

An introduction to spatial data analysis, focusing on spatial autocorrelation, quadrat counts, and geostatistics. Spatial autocorrelation refers to the correlation of a data point with itself in space. Quadrat counts involve gridding a study area and counting the occurrences of variables in each cell. Geostatistics is a commonly used method for spatial data analysis, which includes determining a variogram and using it to carry out kriging for predicting values of variables in between data points.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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Module 9: Spatial Statistics

9.1 Spatial Data Analysis

8/15/2003 Module 9.1 2

Spatial Autocorrelation



Spatial autocorrelation is similar to temporal

autocorrelation



Data that are positively autocorrelated in

space are more similar the closer together

they are



As data points become further and further

apart, they are less alike



Data that are negatively autocorrelated in

space are more similar the more distant they

are. That is much less common.

Discover Exams of Environmental Science University of Idaho (U of I)

Partial preview of the text

Download Spatial Data Analysis: Understanding Autocorrelation, Quadrat Counts, and Geostatistics and more Exams Environmental Science in PDF only on Docsity!

Module 9: Spatial Statistics

9.1 Spatial Data Analysis

8/15/2003 Module 9.1 2

Spatial Autocorrelation

Spatial autocorrelation is similar to temporal

autocorrelation

Data that are positively autocorrelated in

space are more similar the closer together

they are

As data points become further and further

apart, they are less alike

Data that are negatively autocorrelated in

space are more similar the more distant they

are. That is much less common.

8/15/2003 Module 9.1 3

Spatial Autocorrelation

Examples of positively autocorrelated

data:

soil characteristics
makeup of plant communities
contamination that has spread from a point

source

geology
etc

8/15/2003 Module 9.1 4

Spatial Autocorrelation

Like time series analysis, the field of

spatial data analysis is large and

complex

This module will give an overview of

some of the concepts and techniques

as an introduction to the topic

8/15/2003 Module 9.1 7

Quadrat Counts

Then

should equal

1 if the counts are randomly distributed
less than 1 if they are uniformly distributed
greater than 1 if they are clustering

R X

s

8/15/2003 Module 9.1 8

Quadrat Counts

The null hypothesis is that the data are

randomly distributed over the area of study

The test statistic is

which has a t distribution with n-1 degrees of

freedom

−

8/15/2003 Module 9.1 9

Quadrat Counts - Example

Distance East/West 30 0 0 8 4 1 0 0 Distance 20 1 2 13 7 3 1 0 North/South 10 2 6 20 14 6 1 0 0 0 4 12 18 8 2 1 -10 1 3 9 10 15 2 0 -20 0 1 3 6 7 9 3 -30 0 2 1 2 4 3 6 -30 -20 -10 0 10 20 30

Mean 4. Variance 25. R 5. T (0.05, 48) 22. Reject null hypothesis. Counts are not randomly distributed - clusters exist.

8/15/2003 Module 9.1 10

Spatial Data Analysis

Calculating the correlation between two

spatial data variables is difficult and requires

specialized software

There are several ways to do spatial data

analysis that involve computer intensive

methods (similar to bootstrapping).

These are outlined in Manly and won’t be

covered here.

8/15/2003 Module 9.1 13

Variograms

There are several types of variograms

The variogram cloud simply plots the points.

An empirical variogram creates a curve

through the points using smoothing or

averaging techniques

A model variogram fits a mathematical

function to the data and estimates its

parameters

At each step, the variogram becomes

smoother and smoother

8/15/2003 Module 9.1 14

Variograms - Example

Our example data set:

Distance East/West 30 0 0 8 4 1 0 0 Distance 20 1 2 13 7 3 1 0 North/South 10 2 6 20 14 6 1 0 0 0 4 12 18 8 2 1 -10 1 3 9 10 15 2 0 -20 0 1 3 6 7 9 3 -30 0 2 1 2 4 3 6 -30 -20 -10 0 10 20 30

8/15/2003 Module 9.1 15

Variogram Clouds - Example

See Manly Figures 9.4 and 9.5 on

pages 242 and 243.

It is often difficult to see the underlying

relationship from a variogram cloud.

Some sort of smoothing is needed.

8/15/2003 Module 9.1 16

Empirical Variograms - Example

The empirical variogram for our

example data set is:

Semivariogram

0 10 20 30 40 50 60 70 Distance

30 Co effi cie nt

8/15/2003 Module 9.1 19

Model Variograms

Gaussian Function

γ(h) = c + (S - c)(1 - exp(-3h 2 /a^2 ))

where h is the distance, c is the nugget,

S is the sill, and a is the range of

influence.

8/15/2003 Module 9.1 20

Model Variograms

Spherical Function

γ(h) = c + (S - c)(1.5(h/a) - 0.5(h/a)3)^ h<=a

c otherwise

where h is the distance, c is the nugget,

S is the sill, and a is the range of

influence.

8/15/2003 Module 9.1 21

Model Variograms - Example

Semivariogram

0 10 20 30 40 50 60 70 Distance

Coefficient

The spherical model variogram for

our example data set is:

8/15/2003 Module 9.1 22

Kriging

A common problem in spatial data analysis is to

determine the value of a variable at a point in space

where it wasn’t measured

Kriging is a technique to do this

It relies, in part, on the fitted model variogram

The kriging estimate is a weighted linear combination

of the known values where the weights are chosen so

that the prediction errors have the minimum variance

possible

Ordinary kriging allows the true mean to vary over the

study area

8/15/2003 Module 9.1 25

Spatial Data Analysis: Understanding Autocorrelation, Quadrat Counts, and Geostatistics, Exams of Environmental Science

Related documents

Partial preview of the text

Download Spatial Data Analysis: Understanding Autocorrelation, Quadrat Counts, and Geostatistics and more Exams Environmental Science in PDF only on Docsity!

Module 9: Spatial Statistics

9.1 Spatial Data Analysis

Spatial Autocorrelation

 Spatial autocorrelation is similar to temporal

autocorrelation

 Data that are positively autocorrelated in

space are more similar the closer together

they are

 As data points become further and further

apart, they are less alike

 Data that are negatively autocorrelated in

space are more similar the more distant they

are. That is much less common.

Spatial Autocorrelation

 Examples of positively autocorrelated

data:

source

Spatial Autocorrelation

 Like time series analysis, the field of

spatial data analysis is large and

complex

 This module will give an overview of

some of the concepts and techniques

as an introduction to the topic

Quadrat Counts

 Then

 should equal

s

Quadrat Counts

 The null hypothesis is that the data are

randomly distributed over the area of study

 The test statistic is

 which has a t distribution with n-1 degrees of

freedom

Quadrat Counts - Example

 Calculating the correlation between two

spatial data variables is difficult and requires

specialized software

 There are several ways to do spatial data

analysis that involve computer intensive

methods (similar to bootstrapping).

 These are outlined in Manly and won’t be

covered here.

Variograms

 There are several types of variograms

 The variogram cloud simply plots the points.

 An empirical variogram creates a curve

through the points using smoothing or

averaging techniques

 A model variogram fits a mathematical

function to the data and estimates its

parameters

 At each step, the variogram becomes

smoother and smoother

Variograms - Example

 Our example data set:

Variogram Clouds - Example

 See Manly Figures 9.4 and 9.5 on

pages 242 and 243.

 It is often difficult to see the underlying

relationship from a variogram cloud.

 Some sort of smoothing is needed.

 The empirical variogram for our

example data set is:

Model Variograms

 Gaussian Function

γ(h) = c + (S - c)(1 - exp(-3h 2 /a^2 ))

 where h is the distance, c is the nugget,

S is the sill, and a is the range of

influence.

Model Variograms

 Spherical Function

γ(h) = c + (S - c)(1.5(h/a) - 0.5(h/a)3)^ h<=a

c otherwise

 where h is the distance, c is the nugget,

S is the sill, and a is the range of

Spatial autocorrelation is similar to temporal

Data that are positively autocorrelated in

As data points become further and further

Data that are negatively autocorrelated in

Examples of positively autocorrelated

Like time series analysis, the field of

This module will give an overview of

Then

should equal

The null hypothesis is that the data are

The test statistic is

which has a t distribution with n-1 degrees of

Calculating the correlation between two

There are several ways to do spatial data

These are outlined in Manly and won’t be

There are several types of variograms

The variogram cloud simply plots the points.

An empirical variogram creates a curve

A model variogram fits a mathematical

At each step, the variogram becomes

Our example data set:

See Manly Figures 9.4 and 9.5 on

It is often difficult to see the underlying

Some sort of smoothing is needed.

The empirical variogram for our

Gaussian Function

where h is the distance, c is the nugget,

Spherical Function

where h is the distance, c is the nugget,

A common problem in spatial data analysis is to

Kriging is a technique to do this

It relies, in part, on the fitted model variogram

The kriging estimate is a weighted linear combination

Ordinary kriging allows the true mean to vary over the