Geostatistic Analysis

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Geostatistic Analysis

• Lecture 6-8
• Feb25-Mar10, 2008

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Purposes

• Estimating or interpolating values at unsampled
  sites within the area covered by existing
  observations
• Visiting every location is usually difficult or
  expensive.
• Assumption spatially distributed objects are
  spatially correlated. in other words, things are
  close together tend to have similar
  characteristics (first law of geography, also
  called spatial autocorrelation).

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example

• Few sample points to fill all cells

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Example Point elevation to surface
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Global interpolation

• global interpolators determine a single function
  which is mapped across the whole region
• a change in one input value affects the entire
  map
• global algorithms tend to produce smoother
  surfaces with less abrupt changes are used when
  there is an hypothesis about the form of the
  surface, e.g. a trend

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Local interpolation

• Local interpolators apply an algorithm repeatedly
  to a small portion of the total set of points. On
  average, values at points closer in space are
  more likely to be similar than point further
  apart (spatial autocorrelation)
• A change in an input value only affects the
  result within the window
• Two important steps for local interpolation
• Define sampling neighborhood
• Find points (samples) in the neighborhood

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If no directional influence

• If there are no directional influences in the
  data, you want to give equal weight to sample
  points regardless of their direction from the
  prediction location. This means that you probably
  want your neighborhood to be a circle

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If directional influence

• if there is directional influence in your data
  (such as might be caused by water draining
  downhill), then you may want to make an ellipse
  with the major axis running uphill/downhill

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Sample points

• Once a shape is specified, you can restrict which
  sample points within the neighborhood are used.
  You do this by specifying the maximum and minimum
  numbers of points to use and by dividing the
  neighborhood into sectors. If the neighborhood is
  sectored, then the maximum and minimum
  constraints are applied to each part

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1. Explore data before interpolation

• Before creating a surface, explore data (ED) tool
  enables you to gain a deeper understanding of the
  phenomena you are investigating so that you can
  make better decisions on issues relating to your
  data.
• Exploring the distribution of the data, looking
  for global and local outliers, looking for global
  trends, examining spatial autocorrelation, and
  understanding the covariation among multiple
  datasets.
• Tools includes Histogram, Voronoi Map, Normal
  QQPlot, Trend Analysis, Semivariogram/Covariance
  Cloud, General QQPlot, and Crosscovariance Cloud.
• Only works on point and polygon layers

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1.1 Histogram

• provides a univariate (one-variable) description
  of your data, displays the frequency distribution
  for the dataset of interest and calculates
  summary statistics.
• measures of location
• - mean, median, and quartiles
• measures of spread
• - standard deviation, variance
• measures of shape
• - skewness, kurtosis

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1.2 QQPlot

• The quantile-quantile (q-q) plot is a graphical
  technique for determining if two data sets come
  from populations with a common distribution.
• A q-q plot is a plot of the quantiles of the
  first data set against the quantiles of the
  second data set. By a quantile, we mean the
  fraction (or percent) of points below the given
  value. That is, the 0.3 quantile (or 30) is the
  point at which 30 of the data fall below and 70
  fall above that value
• 0.3 quantile is also called 30 percentile, 0.4
  quantitle is 40 percentile
• 25 percentile is called first quantitle
• 50 percentile is called second quantitle (equal
  to median value)
• 75 percentile is called third quantitle
• 100 percentile is called fourth quantile
• The q-q plot is used to answer the following
  questions
• Do two data sets come from populations with a
  common distribution?
• Do two data sets have common location and scale?
• Do two data sets have similar distributional
  shapes?
• Do two data sets have similar tail behavior?

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standard

• QQPlots are graphs on which quantiles from two
  distributions are plotted relative to each other.
  
• a cumulative distribution is produced by ordering
  the data and producing a graph of the ordered
  values versus cumulative distribution values

Normal QQPlot
General QQPlot
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1.3 Voronoi map

• Voronoi maps are constructed from a series of
  polygons (thiessen polygon) formed around the
  location of a sample point.
• The value for each polygon can be calculated
  using any of methods
• - simple, mean, mode, cluster, entropy, median
  standard deviation, IQR (interquartile range)

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1.4 Trend analysis

• You may be interested in mapping a trend, or you
  may wish to remove a trend from the dataset
  before using kriging. The Trend Analysis tool can
  help identify global trends in the input dataset.

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globe trend and anisotropy

• globe trend is an overriding process that affects
  all measurements, can be represented by a
  mathematical formula (polynomial)
• anisotropy is a random process that shows higher
  autocorrelation in one direction than in another
  (directional autocorrelation or directional
  influence). the reason for directional influence
  may not be known, but they can be statistically
  quantified.

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1.5 Semivariogram/covariance cloud

• The semivariogram/covariance cloud shows the
  empirical semivariogram (half of the difference
  squared) and covariance for all pairs of
  locations within a dataset and plots them as a
  function of the distance between the two
  locations.
• the empirical semivariogram for the (i,j)th pair
  is simply
• 0.5(z(si)-z(sj))2, and the empirical
  covariance is the cross-product
• where is the
  average of the data. The semivariogram/covariance
  cloud can be used to examine the local
  characteristics of spatial autocorrelation within
  a dataset and look for outliers.

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Creating Variography
Semivariogram depicts the spatial autocorrelation
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Understanding a semivariogram -range, sill, and
nugget
The distance where the model first flattens out
is known as the range The value at which the
model attains the range is called the sill The
value at which the model intercepts the y-axis is
called the nugget
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Fitting the semivariogram

• Circular, spherical, exponential, Gaussian, and
  linear.

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Making a prediction

• prediction based on the semivarioogram model and
  the measured values that are nearby (using search
  radius)
• fixed search radius requires a distance and
  minimum number of points
• variable search radius number of points needs to
  be specified. You can also specify a maximum
  distance (radius), that the search radius cannot
  exceed.

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To explore a directional influence in the
semivariogram cloud, we use the Search Direction
tools

• the direction the pointer is facing determiners
  which pairs of data locations are plotted on the
  semivariogram.
• lag size is the size of a lag distance, used to
  reduce the larger number of possible
  combinations. it is the size of the cells in the
  semivariogram surface.
• number of cells is called number of lags, counted
  as the number of adjacent cells in a straight
  horizontal or vertical line from the center to
  the edge of the figure.

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1.6 Crosscovariance cloud

• The crosscovariance cloud shows the empirical
  crosscovariance for all pairs of locations
  between two datasets and plots them as a function
  of the distance between the two locations.
• Let z(si) denote the value at the i th location
  in dataset 1, and let y(tj) denote the value at
  the j th location in dataset 2.

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Relation between variogram and covariance
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2. Inverse Distance Weighted (IDW)

• Each sample point has a local influence that
  diminishes with distance.
• Weights the points closer to the processing cell
  more heavily than those farther away.
• Operator controls how weighting is done.
• Power.
• High power gives more weight to closer points.
• Radius type.
• Considers how far away to look.
• Barrier.
• Search can be limited by other polygons or
  polyline.

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IDW Characteristics

• Inverse Distance Weighting (IDW) is a quick
  deterministic interpolator that is exact. There
  are very few decisions to make regarding model
  parameters. It can be a good way to take a first
  look at an interpolated surface. However, there
  is no assessment of prediction errors, and IDW
  can produce "bulls eyes" around data locations.
  There are no assumptions required of the data.

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IDW a local interpolator
We usually uses power functions greater than 1. A
r 2 is known as the inverse distance squared
weighted interpolation.
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Adv. and disadv.
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3. Global Polynomial interpolator

• Global Polynomial (GP) is a quick deterministic
  global interpolator that is smooth (inexact).
  There are very few decisions to make regarding
  model parameters. It is best used for surfaces
  that change slowly and gradually. However, there
  is no assessment of prediction errors and it may
  be too smooth. Locations at the edge of the data
  can have a large effect on the surface. There are
  no assumptions required of the data.

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GP interpolation fits a polynomial regression to
x,y coordinates
First order
Second order
Third order
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4. Local Polynomial interpolator

• Global Polynomial interpolation is the only
  method in Geostatistical Analyst that does not
  use a search neighborhood. If you add the idea of
  a search neighborhood to Global Polynomial
  interpolation, you get Local Polynomial
  interpolation
• Local Polynomial (LP) is a moderately quick
  deterministic interpolator that is smooth
  (inexact). It is more flexible than the global
  polynomial method, but there are more parameter
  decisions. There is no assessment of prediction
  errors. The method provides prediction surfaces
  that are comparable to kriging with measurement
  errors. Local polynomial methods do not allow you
  to investigate the autocorrelation of the data,
  making it less flexible and more automatic than
  kriging. There are no assumptions required of the
  data.

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• Local polynomial interpolation creates a surface
  from many different formulas, each of which is
  optimized for a neighborhood
• The neighborhood shape, maximum and minimum
  number of points, and sector configuration can be
  specified. In addition, as with IDW, the sample
  points in a neighborhood can be weighted by their
  distance from the prediction location. Thus,
  local polynomial interpolation produces surfaces
  that better account for local variation.

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5. Radial Basis Functions

• Radial Basis Functions (RBF) are moderately quick
  deterministic interpolators that are exact. They
  are much more flexible than IDW, but there are
  more parameter decisions. There is no assessment
  of prediction errors. The method provides
  prediction surfaces that are comparable to the
  exact form of kriging. Radial Basis Functions do
  not allow you to investigate the autocorrelation
  of the data, making it less flexible and more
  automatic than kriging. Radial Basis Functions
  make no assumptions about the data.
• RBF seems like a rubber membrane that is fitted
  to each of the measured data points while
  minimizing the total curvature of the surface.
  Because the surface must pass through each
  sampled point, radial basis functions are exact
  interpolators

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Radial basis functions (Spline)

• Radial basis functions (RBF) methods are a series
  of exact interpolation techniques, that is, the
  surface must go through each measured sample
  value.
• There are five different basis functions
  thin-plate spline, spline with tension,
  completely regularized spline, multiquadric
  function, and inverse multiquadric spline
• RBFs are conceptually similar to fitting a rubber
  membrane through the measured sample values while
  minimizing the total curvature of the surface.

IDW will never predict values above the maximum
measured value or below the minimum measured
value. Spline can
http//www.math.ucla.edu/baker/java/hoefer/Spline
.htm
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Radial basis functions
Wi
Bias or error
Wi
When running spline in Spatial Analyst weight
- regularized 0, 0.001, 0.01, 0.1, 0.5
the higher the weight,
the smoother the surface - tension 0, 1,
5, 10 the higher, the
coarser number of points - used in the
calculation. the more points. the
smoother the surface
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6. Kriging

• Kriging is a moderately quick interpolator that
  can be exact or smoothed depending on the
  measurement error model. It is very flexible and
  allows you to investigate graphs of spatial
  autocorrelation. Kriging uses statistical models
  that allow a variety of map outputs including
  predictions, prediction standard errors,
  probability, etc. The flexibility of kriging can
  require a lot of decision-making. Kriging assumes
  the data come from a stationary stochastic
  process (1. constant mean throughout the region,
  or 2. variance of differences between any two
  samples is independent of position, but depends
  on separated distance), , and some methods assume
  normally-distributed data.

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Trend and error

• In Kriging, a predicted value depends on two
  factors a trend and an additional element of
  variability. This is an intuitive idea with
  plenty of analogies in the real world. For
  instance, if you go from the ocean to the top of
  a mountain, you have an upward trend in
  elevation. However, there is likely to be
  variation on the wayyou will go both up and down
  when crossing valleys, streams, knobs and other
  features.
• In Kriging, the trend part of a prediction is
  called the trend. The fluctuation part is called
  spatially-autocorrelated random error. "Error"
  doesn't mean a mistakeit just means a
  fluctuation from the trend. Z(s) µ(s) e (s)
• Assumption one e (s) is zero, (positive errors
  and negative errors)
• Assumption two the autocorrelation of the error
  is purely spatial it depends only on distance
  and not on any other property (such as position)
  of a location.

Autocorrelation between e(s1) and e(s1h) is the
same as the one between e(s2) and e(s2h)
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Ordinary Kriging

• In many cases, however, there is no trend in the
  dataor, if there is one, it is weak enough that
  your predictions are just as good when you ignore
  it. Assuming that there is no trend in the data
  is mathematically equivalent to assuming that the
  data have a constant mean value.
• If the mean is a simple constant, such as µ(s)
  µ (i.e., no trend) for all locations s, and if µ
  is unknown (you do not have prior knowledge of
  the mean value), then this is the model on which
  Ordinary Kriging is based.

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Universal Kriging

• Sometimes, there is a trend where the data values
  change consistently with the spatial coordinates.
  Mathematically, this is represented as a linear
  regression equation on the spatial x- and
  y-coordinates. Trends that vary (do not have a
  constant mean), and for which the regression
  coefficients are unknown, form models for
  Universal Kriging.

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Simple Kriging and indicator Kriging

• A known constant mean for the entire dataset,
  then you have the model for Simple Kriging.
• look at the left side of the equation, Z(s)
  µ(s) e(s). You can do some useful math
  operations on Z(s). For example, suppose you want
  to predict the probability that Z(s) is above or
  below some threshold value, such as 0.12 ppm for
  ozone concentration. You can transform Z(s) to an
  indicator variable, where it gets the value 0 if
  Z(s) is below the threshold and 1 if it is above
  it. This is called Indicator Kriging.

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Ordinary Kriging
Simple Kriging
Universal Kriging
Indicator Kriging
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7. Cokriging

• Cokriging is a moderately quick interpolator that
  can be exact or smoothed depending on the
  measurement error model. Cokriging uses multiple
  datasets and is very flexible, allowing you to
  investigate graphs of cross-correlation and
  autocorrelation. Cokriging uses statistical
  models that allow a variety of map outputs
  including predictions, prediction standard
  errors, probability, etc. The flexibility of
  cokriging requires the most decision-making.
  Cokriging assumes the data come from a stationary
  stochastic process, and some methods assume
  normally-distributed data.

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8. definitions of output maps (for Kriging
methods)

• Prediction maps (interpolated maps) estimate
  values at locations where measurements have not
  been taken.
• Standard error maps (the square root of the
  variance of a prediction) show the distribution
  of prediction error for a surface. Error tends to
  be highest in places where there is little or no
  sample data.
• Quantile maps show the values with 100 percent
  probability that the true values are less than
  the quantile map values.
• Probability maps show the odds that the true
  value at a location is greater than a threshold
  value. The probability of exceeding a threshold
  is determined from predicted values, the error
  distribution, and the specified threshold value

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Kriging family and output maps

• Transformation and trend removal can help justify
  assumptions of normality and stationarity

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creating a prediction map

• while applying a transformation
• while using detrending
• while considering error for nugget

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9. Performing diagnostics

• before produce a final surface, you should know
  how well the model predicts the values at unknown
  locations. cross-validation and validation help
  you make an informed decision as to which model
  provides the best predictions.
• cross-validation uses all of the data to estimate
  the autocorrelation model. Then it removes each
  data location, one at a time, and predicts the
  associated data value. compare the predicted
  value with measured value
• validation first remove part of the data (test
  dataset) using Create Subset tool, and then uses
  the rest of the data (training dataset) to
  develop the trend and autocorrection models to be
  used for prediction
• in both methods, graphs and summary statistics
  used for diagnostics are the same predicted,
  prediction error (predicted-measured),
  standardized error (error/estimated kriging
  standard error), normal QQPlot (standardized
  error and standard normal distribution)

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create subset for validation
demo
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basic rules for good predicts

• Mean error should be close to 0,
• RMS (root-mean-square) error, average standard
  error, and mean standardized error should be as
  small as possible,
• Root mean square stardardized error should be
  close to 1
• uncertainty of prediction standard errors
  average estimated standard error versus RMS
  prediction error.
• - equal, good
• - larger than RMS, overestimate
• - less than RMS, underestimate
• or RMS standardized error
• - 1, good,
• - lt1, overestimate
• - gt1, underestimate

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10. Compare model results

• model surfaces can be compared using
  cross-validation statistics
• previous basic rules for good predict still use
  in here

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11. Displaying geostatistical layers
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12. Comparison of different mathods
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here stochastic is geostatistic
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Summary

• Kriging performs statistical analysis of the
  error in its predictions. This allows it to
  create four kinds of surfaces prediction,
  standard error, quantile, and probability.
• Prediction maps estimate values at locations
  where measurements have not been taken. (All
  interpolators make prediction maps.)
• Standard error maps show the distribution of
  prediction error for a surface. Error tends to be
  highest in places where there is little or no
  sample data.
• Quantile maps show the values that the true
  values are unlikely to exceed.
• Probability maps show the odds that the true
  value at a location is greater than a threshold
  value.
• The various interpolation methods (Inverse
  Distance Weighting, Global Polynomial, Local
  Polynomial, Radial Basis Functions, and Kriging)
  offer trade-offs in speed, flexibility, and their
  advantages and disadvantages.
• Fast interpolators produce output surfaces
  quickly, but are not as good at capturing subtle
  surface variations.
• Exact interpolators predict values equal to the
  observed value at all sampled locations. Smooth
  interpolators do not.
• Flexible interpolators allow users to fine-tune
  the output, while inflexible interpolators allow
  users to avoid making lots of choices

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Main references

• ESRI book using ArcGIS Geostatistical Analyst
• ESRI visual campus at campus.esri.com