Data Continuous Analysis
Analysis of Spatially Continuous Data
Bailey and Gatrell Chapter 5
Focus is on patterns in the attribute values not locations as in the analysis of point patterns The locations are simply sites at which attribute values have been recorded within a region Attributes are conceptually spatially continuous. Examples include observations on rainfall, temperature, salinity, air quality variables such as ozone, soil variables such as permeability, conductivity, pH,
Lecture 13 October 20, 2009
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The Data
Observations on a spatially stochastic process Y s , s R that varies continuously over a region R and has been sampled at fixed point locations si. Referred to as y ( si ) for random variable Y ( si ) Shortened to y or Y(si) for the vector Y = (Y(s1) ,...Y(sn)) for i the random variable Y Often referred to as geostatistical data
Analysis Objectives:
Infer the nature of spatial variation in an attribute over the whole of a region R based on sampled point values. Examine first order effects variations in the mean value of surface (large scale), and second order effects (spatial dependence between values at any 2 locations) locations). Model the pattern of variability of an attribute and determine factors that might relate to it Obtain predictions of a value at un-sampled locations
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Continuous Data Analysis
Develop descriptions that capture global trends as well as local variability Consider first and second order effects
Visualizing Spatially Continuous Data
Use symbols that will represent the information on the data values Proportional circles or rectangles are often used The size of the circle is proportional to the data value e.g. radius equal to the square root of data values Or height of the rectangle is proportional to the data value Colors can be used to reinforce the same data value or add a different variable
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E Y ( s ) s
COV Y ( si ), Y ( s j )
Spatial dependence between Y (si )and Y (s j ) (s (s
Propose models consisting of two components First order component representing large (coarse) scale variation Second order component representing fine scale spatial dependence
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Visualizing Spatially Continuous Data
When mapping symbols to classes the number of classes and the type of class interval "influence the message" Larger numbers of data values typically require more classes to cover the range Rule of thumb Number of classes equal to 1+ 3.3 log n where n is the number of observations Start by first examining the distribution of values before selecting class intervals For data with very skewed distributions it is useful to transform the data values first.
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Visualizing Spatially Continuous Data
Equal Intervals The equal interval method divides the range of attribute values into equal sized sub-ranges. Good if data values are uniformly distributed over their range If data are skewed there will be large number of values in a few classes, and classes with few, if any map features Trimmed Equal Intervals Assign top and bottom ten percent to separate classes and equally divide remainder
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Equal Intervals
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Trimmed Equal Interval
Visualizing Spatially Continuous Data
Percentiles of the distribution each class contains the same number of features, a good method of classification for evenly distributed data, features with greatly different values can be placed in a single class Standard deviates Class breaks are set above and below the mean at intervals of either 1/4, 1/2, or 1 standard deviations until all the data values are contained within the classes. Natural break intervals Natural breaks find groupings and patterns inherent in the data by minimizing the sum of the variance within each of the classes
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Quantiles
Standard Deviates
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Natural Breaks
Exploring Spatially Continuous Data
First consider approaches to investigate variation in the mean value E Y ( s ) s over the region
Spatial moving averages Interpolation based on tessellations Kernel estimation Next consider approaches to investigate second order effect or spatial dependence among values within the region Covariogram, variogram
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Spatial Moving Averages
Estimate mean by averaging values at nearby data points Unweighted average Average n data values nearest to location s Weighted W i ht d average
Methods Based on Tessellations
Estimate (s) from a tiling of the observed sample locations si Common tessellation is the Delauney Triangulation n locations in the plane can be assigned a territory closer to the point than to any other Dirchelet Tessellation, Voronoi, or Theissen polygons Lines joining contiguous locations form a triangulation
^ ( s) wi ( s ) yi
i 1
n
w ( s) 1
i
wi ( s) hi
wi ( s ) e hi
, parameter to change the 19 degree of smoothing
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hi is the distance from s to si
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Methods Based on Tessellations
Each triangular face is associated a with function that is used to interpolate values at unknown points within the triangle Interpolated values are then used to construct isolines
TIN and constructed Isolines
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Methods Based on Tessellations
The resulting contours are an exploratory device to examine variation in the mean value Natural neighborhood interpolation
Kernel Estimation
^ ( s)
n 2 i 1
1 s si k
We are now interested in an estimate of the mean value for the attribute yi rather than intensity of events
^ ( s ) wi ( s ) yi
i 1
n
si s
i 1
n
1 s si k yi 2
Weights are now based on areas of Vornoi polygon around si "stolen" by the tile around s
Represents the amount of the attribute per unit area To find the average we need to divide by number of observations per unit area
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Kernel Estimation
s si k y i ^ ( s ) i 1n s s k i i 1
n
The effect of increasing the bandwidth is to increase the degree of smoothing
Kernel Estimate for mean August temperature
Variation on the weighted moving average
w ( s) y
i 1 i
n
i
s si k wi ( s ) n s s k j j 1
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Covariogram and Variogram
Used to explore the spatial dependence of deviations in attribute values from their mean The covariance function is analogous to the K function for analyzing second order properties in point patterns In the continuous data case we are interested in the way the deviations of observations from their mean values covary over the region In most cases we expect positive covariance or correlation at short distances for spatially continuous phenomena
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Covariogram and Variogram
Assume we have a spatial stochastic process
Y s , s R
E Y ( s ) as s
VAR(Y ( s )) as 2 ( s )
The covariance of the process at any two points si and sj
C si , s j E ((Y ( si ) ( si ))(Y ( s j ) ( s j )))
Correlation is: Variance
si , s j
C ( si , s j )
( si ) ( s j )
C ( s, s ) 2 ( s )
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Covariogram and Variogram
The process is stationary if: (s )
Covariogram and Variogram
(s)
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Mean and variance are independent of location and constant throughout the region and
The process is isotropic if the dependence is only a function of distance and not direction That is dependent only on the length of the vector h Then
C ( si , s j ) C ( si s j ) C (h)
Referred to as the covariogram or covariance function of the process
C ( si , s j ) C (h)
Covariance depends only on the vector difference h and
( si , s j ) (h)
C (h)
(h)
Referred to as the correlogram
C (0) 2
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Covariogram and Variogram
Intrinsic Stationarity A weaker assumption of stationarity There is a constant mean and constant variance in the differences between values at locations separated by a given distance and direction
Covariogram and Variogram
For stationary processes the covariogram, correlogram and variogram are related
( h)
C ( h)
2
E (Y ( s h) Y ( s )) 0 VAR(Y ( s h) Y ( s )) 2 (h)
(h) 2 C (h)
(h)
is strictly the semivariogram but commonly referred to as the variogram
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Covariogram and Variogram
2 C ( 0)
covariogram
correlogram
C (h)
(h)
2
h
h
(h)
( h)
variogram
C ( h)
2
Variogram stops increasing beyond a certain distance and becomes stable at a limit value () called Sill () Var{Y ( s )} C (0) Range distance at which this occurs - transition from state in which spatial correlation exists to absence of correlation Nugget value of variation
( h ) 2 C ( h)
h
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(h) at h = 0 due to measurement error or micro-scale 34
Exploratory Analysis Steps
Estimate an isotropic variogram or covariogram Estimate directional variograms in 2 or 3 broad directions to examine directional effects - anisotrophy A general exploratory technique is the variogram cloud
Estimation of the Variogram
2^ (h) 1 2 hyi y j n ( h ) si s j
Sum squared differences over all pairs less than or equal to a distance h Assuming isotropy, variogram is estimated over all directions for a given separation distance h
Estimation of the Covariogram 1 ^ C (h) yi y y j y n ( h ) si s j h
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Estimation of the Variogram
3000 2500 se emivariance 363677 2000 1500 1000 126191 500 0 0 5 10 15 q1b 0 Nug(0) + 3100 Sph(12) 20 25 distance 30 35 40 108521 1063443 1066870 74336777295 8 930802037260 554407 1
n(h) varies as h increases so the reliability of estimates vary
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Anisotrophic Variograms
Variogram Cloud
The variogram cloud is the distribution of the variance between all pairs of points at all possible distances (h). Plot of or Plot of
( yi y j ) 2
against h (si sj)
(| yi y j |) / 2
against h
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10
2*10^6
Variogram Cloud
Variogram Cloud
8000
1.5*10^6
ga amma
10^6
gam mma
5*10^5
0
0 0
2000
40 000
6000
1000
2000 distance
3000
0
20
40
60 distance
80
100
120
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140
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Radon Data
600 g gamma 0 0 100 200 3 300 400 500
50
100 distance
150
200
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