Spatial statistics

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Spatial statistics

• Lecture 3

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What are spatial statistics

• Not like traditional, a-spatial or non-spatial
  statistics
• But specific methods that use distance, space,
  and spatial relationships as part of the math for
  their computations
• It is a spatial distribution and pattern analysis
  tool
• Identifying characteristics of a distribution
  tools used to answer questions like where is the
  center, or how are feature distributed around the
  center? (Measuring Geographic Distributions)
• Quantifying or describing spatial pattern are
  our features random, clustered, or evenly
  dispersed across our study area? (Analyzing
  Patterns and mapping clusters)
• Mainly deal with point, line, polygon (vector)
• Why use spatial statistics?
• To help assess patterns, trends, and
  relationships
• Better understanding of geographic phenomena
• Pinpoint causes of specific geographic patterns
• Make decision with high level of confidence
• Summarize the distribution in a single number

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1. Measuring geographic (spatial) distribution

• Not only crime analysts but also GIS
  practitioners in many research areas, such as
  epidemiology, archaeology, wildlife biology, and
  retail analysis, will benefit from the spatial
  statistics tools in ArcGIS 9. These tools can be
  easily modified or extended because most were
  written using the Python scripting language. The
  source code for the statistical tools can be
  accessed from ArcToolbox and serve as samples and
  templates for further customization

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1.1
Mean center of population distribution and
pattern, Track changes in the distribution
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Average of x, y coordinates
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Median center

• Identifies the location that minimizes overall
  Euclidean distance to the features in a dataset
• While the Mean_Center tool returns a point at the
  average X and average Y coordinate for all
  feature centroids, the median center uses an
  iterative algorithm to find the point that
  minimizes Euclidean distance to all features in
  the dataset.
• Both the Mean_Center and Median Center are
  measures of central tendency. The algorithm for
  the Median Center tool is less influenced by data
  outliers.

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1.2
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Distances from each feature centroid to every
other feature centroid in the dataset are
calculated and summed. Then the feature
associated with the shortest accumulative
distance to all other features (weighted if a
weight is specified) is selected and copied to a
newly created output feature class
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Central feature
Mean center
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How feature disperse around center
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Mean center and central feature tools tell about
the center of a distribution But do not tell the
overall distribution. Following tools tell how
dispersed our features are around that center

• Standard distance
• Directional distribution (standard deviational
  ellipse)
• Linear directional mean

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Showing those locations are within one standard
deviation of the central feature
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Showing those locations are within one standard
deviational ellipse of the central feature, in a
north-west to south-east direction
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• The trend of a set of line features is measured
  by calculating the average angle of the lines.
  The statistic used to calculate the trend is
  known as the directional mean. While the
  statistic itself is termed the "directional
  mean", it is used to measure either direction
  (such as hurricanes) or orientation (faults).

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Python Script
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2. Analyzing spatial patterns

• Give us ways to measure the degree to which our
  features are clustered, dispersed, or randomly
  distributed across the study area
• 2.1 Analyzing Patterns
• Global calculations
• Identifies the patterns/overall trends of data
• Are features clustered and what is the overall
  pattern?
• Spatial Autocorrelation tool
• 2.2 Mapping Cluster
• Local calculations
• Identifies the extent and location of clustering
  or dispersion
• Where are the clusters (or where are the hot
  spots)?
• Hot Spot Analysis tool

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2.1 Analyzing patterns

• Average nearest neighbor
• High/low clustering
• Multi-distance spatial cluster analysis
• Spatial autocorrelation

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• The Average Nearest Neighbor tool returns five
  values Observed Mean Distance, Expected Mean
  Distance, Nearest Neighbor Index, z-score, and
  p-value

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Nearest neighbor index, gt1 (dispersion)
lt1 (clustering)
Very sensitive to the area
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• How Spatial Autocorrelation Moran's I (Spatial
  Statistics) works
• This tool measures spatial autocorrelation
  (feature similarity) based on both feature
  locations and feature values simultaneously.
  Given a set of features and an associated
  attribute, it evaluates whether the pattern
  expressed is clustered, dispersed, or random. The
  tool calculates the Moran's I Index value and
  both a Z score and p-value evaluating the
  significance of that index. In general, a Moran's
  Index value near 1.0 indicates clustering while
  an index value near -1.0 indicates dispersion.
  However, without looking at statistical
  significance you have no basis for knowing if the
  observed pattern is just one of many, many
  possible versions of random.
• In the case of the Spatial Autocorrelation tool,
  the null hypothesis states that "there is no
  spatial clustering of the values associated with
  the geographic features in the study area". When
  the p-value is small and the absolute value of
  the Z score is large enough that it falls outside
  of the desired confidence level, the null
  hypothsis can be rejected. If the index value is
  greater than 0, the set of features exhibits a
  clustered pattern. If the value is less than 0,
  the set of features exhibits a dispersed pattern.

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• Z score is a measure of standard deviation. If
  you have s is (-1.96, 1.96), z score is falling
  between them, you are seeing a pattern of random
  pattern. If z score falls outside, like -2.5 or
  5.4, then you have a pattern thats too unusual
  to be a pattern of random chance

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2.2 Mapping cluster

• Cluster and outlier analysis
• Hot spot analysis

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Example for park-served population (congestion)
Red Gi_Z-score (gt1.96), with plt0.05
Red Morans I_Z-score (gt1.96), with plt0.05 Blue
Morans I_Z-score (lt-1.96), with plt0.05
Source Yunbo Bis Masters thesis, 2012
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references

• understanding Spatial Statistics in ArcGIS 9 by
  Sandi Schaefer and Lauren Scott.
• ArcGIS desktop help