Ex8 and

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Statistics!
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Statistical Functions

• Correlation
• Validate
• Crosstab
• Trend
• Regression
• Linear
• Multiple Linear
• Logistic

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Cross Correlation

• Answers the question How similar are two vectors
  of numbers
• Or to the degree of linear association
  prevailing between a pair of variables.
• Range 1.0 to 1.0

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The calculation

• Sum of the cross products divided by the RMS of
  the cross products
• Example from the book

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CC -1.0
CC 0.1082
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Autocorrelation

• How well is something correlated with its self?
• What?
• Consider
• 1 2 3 4 5 6 1 2 3 4 5 6

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Autocorrelation
Usually the autocorrelation for things like the
stock market decrease with time things near one
another in time are more likely to be similar
than things distant in time
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Autocorrelation

• Sometimes the same is true for stuff in space
  like pollution levels, elevation, population
  density (to a point), etc
• Things close in space tend to be correlated
• But that depends of scale Albany vs. Syracuse

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Spatial Autocorrelation

• The basic idea is the same as for time
  autocorrelation ( or any series of numbers)
  except that it is complicated by the fact that
  it has to take place in two directions, X and Y
• This gets very messy if you have to go beyond 1
  space lag
• Thus spatial autocorrelation is based on our old
  friend the 3x3 kernel that is passed over the
  image
• IDRISI will do two kinds of autocorrelation

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AUTOCORR
Kernels

• These are the two types of kernels used to
  compute the autocorrelation
• Kernels are passed over the map as in filtering

Rooks
Kings.70.707
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Complications

• By comparing the central cell of a 3x3 kernel
  with its eight surrounding cells you run into a
  problem
• the deviation of the central cell from its mean
  is zero
• The linear autocorrelation equation wont work.

Dividing by zero
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Complications

• So equations like this were invented by folks
  like Moran

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Complications

• Weights dij

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Morans I for elev

• Rooks case 0.9782
• Kings case 0.9708
• Not the same
• And the standard equations dont have the weights
  that IDRISI uses for Kings case

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That was Morans I

• If each cell in the kernel is very much like its
  neighbors then I will be nearly one (1.0000)
• If there is no relation the I will be 0.
• If the I is negative there is a clustering of
  dissimilar values.

I 0.5
I -0.875
I -0.250
Similar
Random
Dissimilar
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Cramers V and Kappa k

• Correlation between two images on a cell by cell
  basis
• Cramer's V ranges from 0.0, indicating no
  correlation, to 1.0 indicating perfect
  correlation.
• Kappa is similar except that the two images must
  have the same classes

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Cross tabulation

• Cramers V and Kappa are both produced by Crosstab

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Cross tabulation
Cross tabulation the categories of one image are
compared with those of a second image and a
tabulation is kept of the number of cells in
each combination.
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11 Arnot on Slope class 1 0 to 5
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Trend

• Statistical process (least squares) of fitting a
  mathematical surface to an image.
• So Trend applied to a distance surface gives a
  result like this

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Distance from Roads
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Ortho dist roads
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Trend surfaces
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Cubic Trend Surface
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Cubic Surface
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Trend

• I have never used Trend for anything
• But you never know
• It does really smooth things

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Other stuff on Stats Menu

• Other regression functions
• Multiple linear regression
• Logistic regression
• CENTER
• CRATIO
• VALIDATE
• SAMPLE
• RANDOM
• STANDARD
• PATTERN

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CENTER

• CENTER calculates the weighted or unweighted mean
  center and standard radius of a point data set
  expressed as cell frequencies.
• The mean center is the average position of the
  points, while the weighted mean center is closer
  in concept to a "center of gravity."
• The standard radius is the direct spatial
  analogue of the standard deviation, and thus
  expresses the dispersion of points about a mean
  center.
• The Coefficient of Relative Dispersion is also
  calculated.

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CRATIO

• CRATIO computes the compactness ratio of polygons
  
• compares the area of a polygon to that of a
  circle having the same perimeter as the polygon,
• For all cells identified (using GROUP) as part of
  the polygon (same integer value).

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CRATIO
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CRATIO
0.625788
0.194337
0.700624
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VALIDATE

• VALIDATE provides a method to measure agreement
  between two categorical (integer or byte) images,
  a "comparison" map and a "reference" map. Each
  image can contain up to 32,000 categories. The
  comparison map will usually be the result of a
  simulation or classification model whose validity
  is being assessed against a reference map that
  depicts reality. map.

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VALIDATE

• Actually calculates various kappas
• These statistics indicate how well the two images
  agree

• Agree
• Due to chance
• Due to quantity
• Due to location
• Cell level
• Stratified level

• Disagree
• Due to quantity Due to location
• Cell level
• Stratified level

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VALIDATE

• Actually calculates various kappas
• These statistics indicate how well the two images
  agree

COMPLEX! AND IT DISAPPEARS INTO A CLOUD OF
MATHEMATICS!

• Agree
• Due to chance
• Due to quantity
• Due to location
• Cell level
• Stratified level

• Disagree
• Due to quantity Due to location
• Cell level
• Stratified level

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SAMPLE

• SAMPLE produces a vector file of point locations
  for use in sampling problems.
• The points may be selected according to a random,
  systematic or stratified random sampling scheme.

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SAMPLE
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RANDOM
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RANDOM
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Standard (Normalization)

• STANDARD converts a quantitative image to a new
  image expressed as standard scores.
• It calculates the mean and standard deviation of
  the cells in the image.
• It then subtracts the mean from each cell and
  divides each of these results by the standard
  deviation.
• The usual application of STANDARD is to
  standardize images to a consistent numeric scale
  before comparison in some way

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PATTERN

• PATTERN uses variability in a 3 x 3 pixel window
  or a 5 x 5 or 7 x 7 octagonal pixel window to
  assess several different measures used in
  Landscape Ecology.
• It has an additional option for measuring the
  frequency with which an input value occurs within
  a 3 x 3, 5 x 5 or 7 x 7 pixel window.

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PATTERN
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PATTERN
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SO

• There are quite a few statistical analysis
  modules in IDRISI
• And they will let you do
• Sampling
• Regressions
• Image comparisons
• Correlations
• Ecological pattern analysis