Statistical Analysis of Geographical Information

1
Statistical Analysis of Geographical Information

• Dr. Marina Gavrilova

2
Topics

• Introduction
• Distribution Descriptors One Variable
• Relationship Descriptors Two Variables
• Point Pattern Descriptors
• Point Pattern Analyzers
• Autocorrelation

3
Introdution quantitative measures to describe
data

• Statistics classification
• Classified by function
• Description statistics
• Inferential statistics
• Classified by areas of application
• Classical statistics sociology, political
  science, medicine and engineering.
• Spatial statistics based on classical and
  extended to the spatially referenced data.
• Geostatistics one kind of Spatial statistics and
  originated in geo-science.

4
Random and Systematic process

• A certain phenomenon occurs Random process or
  Systematic Process?
• Soil Example
• Hypothesis soil fertility of a farm is low
• To test the hypothesis, gather more data about
  the soil.
• Collect a sample of soil for further examination
  instead of the entire population.
• Observation each examined location Sample size
  number of observations selected.

5
Features about spatial data(1)

• A region can be partitioned in many ways based
  on the given criteria. USA States boundaries,
  census geography. Modifiable Area Unit Problem
  (MAUP) include
• Scale effect Analyze data at multiple levels of
  spatial resolution results in inconsistency.
• Zoning effect Analyze data derived from
  different zonal systems with similar number of
  areal units results in inconsistency.

6
Features about spatial data(2)

• Spatial autocorrelation represents the nature of
  geography and, consequently, will almost always
  be present in spatial data.
• Tober First Law of Geography
• All things related to each other, but closer
  things are related more.
• Butterfly Effect Butterfly flapping in China
  may cause a hurricane landfall in the US due to
  spatial propagation of air disturbances.

7
Distribution descriptorsone variable
8
Measure of central tendency

• Mode The value that occurs most frequently in a
  set of data or called the modal value. If two or
  more categories have the highest frequency, then
  data is bimodal or multimodal.
• Median The middle value after all values are
  sorted in ascending or descending order.
• Mean or Average n observation, each with an
  observed value xi then the simple arithmetic
  mean is defined as

9
Measure of central tendency

• Grouped or weighted mean if data values are
  grouped into classes, then all data within each
  group are represented by on value as the overall
  value in that class. A mean derived from the
  grouped data is called a grouped mean or a
  weighted mean.
• If xi is the midpoint of the i th class (k
  classes together) with fi as the number of data
  values in that class (frequency), the weighted
  mean

10
Measures of dispersion (1)

• While mean is a good measure of the central
  tendency of a set of data, it captures no
  information about how the values are concentrated
  or scattered around the mean.
• Range, Minimum, Maximum, and Percentiles
• Range Maximum-Minumum
• Percentiles are the corresponding data values
  that have certain percentages of the data smaller
  than these values. Data Xa and Xb have the same
  median 7, different 25th (3 for Xa and -5 for Xb
  ) Xa 1 3 5 7 9 11 13
• Xb -11
  -5 1 7 13 19 25

11
Measures of dispersion (2)

• Mean Deviation unlike the dispersion measures
  discussed so far using one or a few data values
  in the series, the mean deviation takes into
  account all data values. It is calculated by
  summing all the differences that individual data
  values have from the mean and then dividing this
  sum by the number of observation.

12
Measures of dispersion (3)

• Variance and Standard Deviation Another way to
  avoid the offsets caused by adding positive and
  negative deviations from the mean together is to
  square all deviations from the mean before
  summing them.

13
Measures of dispersion (4)

• Weighted Variance and Weighted Standard
  Deviation.
• fi is the frequency for the i th group or class,
  
• xi is the midpoint value in the i th group,
• is the weighted mean, and
• k is the number of groups.

14

• Relationship Descriptors
• Two Variables

15
One Variables

• The mean and its variations address the issue of
  location, where the observations distribute along
  the continuous value line. Median and mode
  consider this central tendency issue. Variance,
  standard deviation, and percentiles address the
  issue of dispersion. Skewness deals with
  direction clustering. Kurtosis addresses the
  issue of concentration. All these measures focus
  on the distribution of the values using one
  variable at a time.

16
Relationship Descriptors

• Mean, standard variable cannot measure the
  relationships between different distributions
  quantitatively.
• One of statistics is based on the concept
  correlation measures statistically the direction
  and strength of the relationship between two sets
  of data or two variables for a number of
  observation. Regression measures the dependence
  of one variable on another.

17
Correlation Analysis (1)

• Education is traditionally regarded as an asset.
  It enriches a persons life in many ways. We
  usually believe that education and income are
  somewhat related and change in the same
  direction. If we recognize the value of education
  in eventually achieving a higher income, it would
  be nice to know how strong this relationship is,
  that is, how these aspects of life are related or
  correlated.

18
Correlation Analysis (2)

• Each relationship has two important aspects the
  direction and strength of the relationship.
  Between two related variable, the relationship is
  typically measured as correlation a statistical
  measure indicating how values in one variable are
  related to values in the other variable.
• Positive or direct correlation
• Negative or inverse correlation

19
Trend Analysis

• Trend analysis is a technique measuring the
  trend, while correlation is a statistical measure
  of two variables.
• Trend analysis addresses the dependence of one
  variable on another.
• Going beyond the strength and direction of the
  relationship, trend analysis allow us to model
  the relationship and to estimate likely value of
  one variable based on the value of another
  variable.
• Models that are constructed with this technique
  are known as regression models.

20
Simple Linear Regression Model

• Simple linear regression model or bivariate
  regression model Using a straight line to model
  the relationship between tow variables. Here are
  an example. A regression between median household
  income and median house value for 51 states.

21
Regression model

• Some phenomena may be modeled by the regression
  reasonable well, and others may not.
• Regression model assumes a linear relationship
  between the variable. If the relationship is not
  linear or if the two variables have weak or no
  relationship, then the model will perform poorly.
  
• A multivariate regression model, which can
  accommodate multiple independent variables. Under
  either circumstance, we may have committed a
  model specification error.

22
Point Pattern descriptors and analyzers
23
Point Pattern

• Point Pattern Descriptors
• Central Tendency
• Dispersion and Orientation
• Point Pattern Analyzers
• Quadrant Analysis
• Nearest-Neighbor Analysis
• Spatial Autocorrelation of Points
• K-Function

24
The Nature of Point Features

• Point pattern descriptors cover
• The methods for determining the overall patterns
  of a given set of points.
• Measures used to describe the magnitude of
  spatial dispersion of a given set of points.
• How the direction bias of a set of points can be
  extracted statistically.

25
Central Tendency of Point Distributions

• A set of point descriptors provide certain
  descriptive information on the distribution of a
  set of points.
• Central tendency information, mean centers,
  weighted mean centers, and median centers provide
  a good summary of how a set of points distributes
  in the geographic space.
• To describe the spatial dispersion
  characteristics of a set of points, the measures
  of standard distance and standard ellipse will be
  discussed. These measures indicate the spatial
  variation and orientation of a point distribution.

26
Mean Center

• The mean center, or spatial mean, is a central
  or average location of a set of points. For n
  points xmc and ymc are the coordinates of the
  mean center, xi and yi are the coordinates of
  point i, and n is the number of points.

27
Weighted Mean Center

• The weighted mean center of a distribution of
  points can be found by multiplying the x- and y-
  coordinates of each point by the weight assigned
  to each observation or location.
• wi is the weight at point i

28
Dispersion and Orientation of Point Distributions

• Two sets of points may occupy the same geographic
  space and may be interrelated.
• For example, one set of points represents the
  location of forest fires and the other the
  locations of camping cabins in a wildlife region.
  They may have the same overall locations, but
  forest fire have a more dispersed spatial pattern
  than cabins.
• In additional to spatial central tendency, it may
  be interesting to evaluate the magnitude of
  dispersion of locations and the orientation of
  the spatial distribution.

29
Standard Distance

• Similar to those in classical statistics, the
  population standard deviation, ,or the
  sample standard deviation, S, can be computed as

30
Weighted Standard Distance

• Points in a distribution may have different
  attribute values that reflect the relative
  importance of different point observation.
• Wi is the weight for point i, and
• (xwmc, ywmc) is the weighted spatial mean.

31
Standard Deviational Ellipses

• The standard distance circle is a very effective
  visualization tool to show the spatial spread of
  a set of point location.
• A logical extension of the standard distance
  circle is the standard deviational ellipse. It
  can capture the directional bias in a point
  distribution. Three components are needed to
  describe it
• An angle of rotation
• Deviation along the major axis
• Deviation along the minor axis

32
Elements defining a standard deviational ellipse
33
Standard deviational ellipses for men-only and
women-only shelters
34
Point Pattern Analyzers

• To fully understand the various states and
  dynamics of a particular geographic phenomenon,
  an analyst must be able to detect spatial
  patterns from the point distributions and to
  track the changes in point patterns at different
  time.

35
Point Pattern Analyzers

• Quadrant Analysis allows analysts to determine if
  a point distribution is similar to a random
  pattern using a spatial sampling framework.
• Nearest Neighbor Analysis compares the average
  distance between nearest neighbors in a set of
  points to that of a theoretical pattern.
• Spatial autocorrelation coefficients measure how
  similar neighboring points are.
• K-function analysis can identify and evaluate the
  clustering of points at different spatial scales,
  or extents.

36
Quadrant Analysis

• Quadrant Analysis evaluates a point distribution
  by examining how its density changes over space.
• The density measured by Quadrant Analysis is then
  compared with the density of a theoretically
  constructed random pattern to see if the point
  distribution in question is more clustered or
  more dispersed than the random pattern.

37
General Concept in Quadrant Analysis (1)

• A regular square grid and a number of points
  falling in some squares.
• The square are referred to as quadrants, which
  are essentially sampling units in spatial
  statistical jargon.
• Circle is the most geometrically compact shape,
  however circles cannot cover the entire
  geographic space unless they overlap.
• In an extremely clustered point pattern, all or
  most of the points fall inside one or a few
  squares only. In an extremely dispersed pattern
  referred to as a uniform pattern or a triangular
  lattice, all squares contain similar number of
  points.

38
Observed pattern of Ohio cities and hypothetical
clustering and dispersed pattern
39
General Concept in Quadrant Analysis (2)

• Statistically, Quadrant Analysis will achieve a
  fair evaluation of the density across the study
  area if it applies a large enough number of
  randomly generated quadrants.
• An optimal size of quadrant can be calculated by
  2A/r . A is the area of study area, and r is the
  number of points in the distribution.
• Once the quadrant size for a point distribution
  is determined, Quadrant Analysis can proceed to
  establish the frequency distribution of the
  number of points for all quadrant.

40
Examples of systematic and random quadrants
41
Comparing Observed and Expected Patterns

• Besides using K-S statistics to test if the
  observed pattern is different from a random
  pattern, one may perform the Variance-Mean Ratio
  Test by taking advantage of a specific
  statistical property of the Position
  distribution.

42
Ordered Neighbor Analysis

• Quadrant Analysis is useful in comparing an
  observed point pattern to a random or
  theoretically known distribution. However, it has
  certain limitations.
• The analysis captures information on the points
  within each quadrant, but no information on
  points between quadrants is used in the analysis.
  As a result, Quadrant Analysis may be
  insufficient to distinguish between certain point
  pattern in the following figures.

43
Spatial Configurations

• Visually, the two patterns are different. Using
  Quadrat Analysis, however, the two patterns yield
  the same result.

44
Nearest Neighbor Statistic

• Nearest Neighbor Statistic is derived from the
  average distance between points and each of their
  nearest neighbors.
• The second-ordered neighbor statistic uses the
  distance of the second nearest neighbors.
  Higher-ordered neighbors can be defined in
  similar ways.
• Ordered Statistics can evaluate the pattern at
  different spatial scales.

45
Quadrant Analysis and Nearest Neighbor Analysis

• While both Quadrant Analysis and Nearest Neighbor
  Analysis test point distribution, they utilize
  different spatial concepts.
• Quadrant Analysis tests a point distribution with
  the points per area concept using quadrants as
  sampling units.
• Nearest Neighbor Analysis uses the concept of
  area per point.
• Both methods are similar in sense that the
  observed pattern is compared with some know
  distribution (random pattern).
  

46
Nearest Neighbor statistics

• How Nearest Neighbor Analysis works.
• In a homogeneous region, the most uniform pattern
  formed by a set of points occurs when this region
  is partitioned into a set of identical hexagons
  with a point at its center. The distance between
  points will be
• , where A is the area of the region and n is the
  number of points.

47
R statistic or R scale

• R statistic is the ratio of the observed average
  distance between nearest neighbors of a point
  distribution and the expected average nearest
  neighbor distance. It is also the nearest
  neighbor statistic.
• robs is the observed average distance between
  nearest neighbors and rexp is the expected
  average distance between nearest neighbors as
  determined by the theoretical pattern.

48
Calculation of the observed nearest neighbor
distance

• d1d13 d2d23 d3d32 d4d43
• (For point 1, the nearest neighbor is 3)

49
Cities in Ohio

• By selecting the seven largest cities in Ohio,
  we can compute their nearest neighbor distance
  and the observed average nearest neighbor
  distance robs 51.82miles.

50
Higher-order neighbor statistics

• Nearest Neighbor Analysis has been extended to
  accommodate the second, third, and other
  higher-order neighbor definitions. When two
  points are not immediate nearest neighbors but
  rather the second nearest neighbors, the way
  distances are computed between them will need to
  be adjusted accordingly.

51
Second-order nearest neighbor distance

• The second-order nearest neighbor statistic R2 is
  robs/rexp .
• di is the distance between i and its second
  nearest neighbor.
• The expected nearest neighbor distance in the
  denominator of the R2 statistic is similar to the
  first-order expected distance, the constant
  change from 0.5 to 0.75.

52
Observed and expected high-order nearest neighbor
distance

• Standard error estimate for second-order nearest
  neighbor distance
• Generally, for k-order neighbor statistic,
• are the constants for
  expected distance and standard error,
  respectively.
  

53
K-Function Analysis Steps (1)

• Another statistic that can offer some insights
  and is more parsimonious to evaluate if the
  magnitude of clustering is uniform over different
  spatial scales is K-function analysis. It is an
  extension of the ordered neighbor statistics. For
  a set of point in a region, the K-function
  analysis involves following steps
• Select a distance increment or spatial lab, d,
  that is analogous to the unit reflecting the
  change in the spatial scale.
• Set the iteration number g1 to begin the
  process.

54
K-Function Analysis Steps (2)

• Around each point i in a region, create a
  circular buffer with a radius of h, where hdg.
  Therefore, the buffer will have a size d in the
  first iteration and 2d in the second and so on.
• For each point, count the number of points
  falling within its buffer of size h and denote
  that count as n(h).
• Increase the radius of the buffer by d.
• Repeat steps 3, 4, and 5 by increasing h until
  gr or gD/d.

55
Estimation of the K-function

• Figure in next slide uses only four points to
  illustrate the procedure.
• Only three rings or buffers were created instead
  of the full range up to D. For a give h, we count
  the number of points within the buffers centered
  at all points. Point A is rather dispersed from
  other points, and therefore the counts are
  relatively low for buffers with small h. For
  point B, the point is in the middle of the
  cluster, and therefore the point count are
  relatively high with the small buffers, but the
  increases in point counts are substantial with
  large hs. For Point C and D, the points
  themselves are apart from the cluster.

56
Estimation of the K-function
57
Relationship between point counts and the spatial
lag h

• The relationship between point counts and the
  spatial lag from empirical observation can be
  compared with a known patter, most likely a
  random pattern.
• In a random pattern, point counts increase with
  increasing h but in no particular pattern.
• K-function detect clustering at different scales
  by comparing the relationship between point
  counts and the size of h to that in a random
  distribution.

58
Computation of K-Function

• The number of points within the buffer with a lag
  h, as follows
• i and j are the indices of points.
• dij is the distance between the two points i, j.
• Ih is an indicator function such that Ih1 if
  dijlth and Ih0 otherwise

59
Boundary Problems in K-Function

• Sharing similar problems with other spatial
  statistical and analytical techniques, the
  K-function is also subject to the boundary
  problems.
• Image that a point is located rather close to the
  edge of the study region. When buffers are formed
  around the point, a significant proportion of
  buffers will be outside of the study area and
  thus will distort the probability of finding a
  point within the vicinity of h.

60
Spatial Autocorrelation of Points

• Spatial autocorrelation coefficients measure and
  test how clustered/dispersed the point locations
  are with respect to their attribute values.
• Spatial autocorrelation of a set of points refers
  to the degree of similarity between points or
  events occurring at these points and points or
  evens in nearby locations.
• With the spatial autocorrelation coefficient, we
  can measure
• The proximity of location
• The similarity of the characteristics of these
  locations.

61
Measures for Spatial Autocorrelation

• Two popular indices for measuring spatial
  autocorrelation applicable to a point
  distribution Gearys Ratio and Morans I Index.
• sij representing the similarity of point i s and
  point j s attributes.
• wij representing the proximity of point i s and
  point j s locations, wii0 for all points.
• xi representing the value of the attribute of
  interest for point i .
• n representing the total number of points.

62
SAC (1)

• The spatial autocorrelation coefficient (SAC) is
  proportional to the weighted similarity of the
  point attribute values.

63
SAC (2)

• The spatial weights in the computations of the
  spatial autocorrelation coefficient may take on a
  form other than a distance-based format. For
  example
• wij can take a binary form of 1 or 0, depending
  on whether point i and point j are spatially
  adjacent.
• If tow regions share a common boundary, the two
  centroids of these regions can be defined as
  spatially adjacent wij 1 otherwise wij 0.

64
Gearys Ratio

• In Gearys Ratio, the similarity attribute
  values between two points is defined
• The computation of Gearys Ratio

65
Morans I Index

• In Morans I Index, the similarity attribute
  values between two points is defined
• The computation of Morans I Index

66
Gearys Ratio vs. Morans I Index
Numerical scales of Gearys Ratio and Morans I Numerical scales of Gearys Ratio and Morans I Numerical scales of Gearys Ratio and Morans I
Spatial Patterns Gearys C Morans I
Clustered pattern in which adjacent or nearby points show similar characteristics 0ltClt1 I gt E(I)
Random pattern in which points do not show particular patterns of similarity C 1 I E(I)
Dispersed pattern in which adjacent or nearby points show different characteristics 1ltClt2 I lt E(I)
E(I) (-1)/(n-1), which n denoting the number of points in distribution E(I) (-1)/(n-1), which n denoting the number of points in distribution E(I) (-1)/(n-1), which n denoting the number of points in distribution
67
Scales of Gearys Ratio and Morans I Index

• The indexs scale for Gearys Ratio does not
  correspond to our conventional impression of the
  correlation coefficient of the (-1, 1) scale,
  while the scale of Morans I resembles more
  closely the scale conventional correlation
  measure
• The value for no spatial autocorrelation is not
  zero but -1/n-1
• The values of Morans I Index in some empirical
  studies are not bounded by (-1,1), especially the
  upper bound of 1.

68
Conclusions

• Distribution Descriptors using single variable
  and Relationship Descriptors using two (or more)
  variables are typical statistical tools.
• Point Pattern Descriptors and Point Pattern
  Analyzers can be used to study more deep patterns
  of the data, in combination with various
  representations (spatial, grid, k-mean, ellipse
  etc)
• Autocorrelation analysis is sued to understand
  further data relationship in respect to distance
  between spatial locations