Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 =...

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© 2003 Luc Anselin, All Rights Reserved Variogram Analysis Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign http://sal.agecon.uiuc.edu

Transcript of Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 =...

Page 1: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Variogram Analysis

Luc AnselinSpatial Analysis Laboratory

Dept. Agricultural and Consumer EconomicsUniversity of Illinois, Urbana-Champaign

http://sal.agecon.uiuc.edu

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© 2003 Luc Anselin, All Rights Reserved

Outline

�Geostatistical Perspective�Variogram and Correlogram�Modeling Variograms

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© 2003 Luc Anselin, All Rights Reserved

Geostatistical Perspective

Page 4: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Geostatistical Perspective

�Continuous Spatial Index� { Z(s): s ∈ D }

• s is spatial index• continuous in Rd

� sample of spatial locations• { s1, s2, ..., sN } = locations• { Z(s1), Z(s2), ..., Z(sN) } = random variable at

locations

�Spatial Random Field� model for a continuous spatial process� field data model

Page 5: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

Residential Sales Price, Baltimore MD (1980)sample points (darker is higher) and contours

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Air Quality (Ozone) in Los Angeles BasinContours from Spherical Variogram Interpolation

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© 2003 Luc Anselin, All Rights Reserved

Conceptual Framework

�Equilibrium = Stationarity� stochastic process

• not multiple realizations, but singlerealization

• the map is a single data point

� notion of stability needed to relatesample (= single observation) topopulation• even though there is only one data point,

act as if there are multiple observations

Page 8: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

Implications of Spatial Stationarity(source Y. Pannatier, Variowin)

Page 9: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Moment Conditions

�Constraints on Variability� ergodicity

• average over single realization same as overall possible

• whether you see one or many maps, informationis same

� moments must exist• no infinite variance

� moments must be “regular” over space• restrictions on heterogeneity, on range of

dependence

Page 10: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Spatial Stationarity

�Strict Stationarity� invariance of joint probability density

function under spatial shift (“translation”)• { z(x1), … , z(xk) } and { z(x1+h), … , z(xk + h) }• information about process the same no matter

where it is obtained

�Moment Stationarity� moments invariant under shift

• constant mean and constant variance• covariance only function of spatial separation h

Page 11: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

Baltimore House Sales PricesSpatial Trend Analysis

Page 12: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Linear Spatial TrendBaltimore House Prices

P = -166.02 - 0.148 X + 0.634 Y R2 = 0.27

Page 13: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Variogram and Correlogram

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© 2003 Luc Anselin, All Rights Reserved

Variogram

� Intrinsic Hypothesis� no spatial trend

• if there is a trend, take it out• residuals have no trend by construction

(mean = 0)

� variance constant� variance of first differences only a function

of displacement• Var { Z[s+h] – Z[s] }• how does the variability of the difference change

with h

Page 15: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Semi-Variogram

�General Variogram Function� 2γ(h) = Var [Z(s+h) - Z(s)]

• note the factor 2, hence γ(h) is 1/2 of variogramor semivariogram

�Regular Case� for intrinsic hypothesis, E[Z(s+h)]=E[Z(s)]

• constant mean, hence E[Z(s+h) – Z(s)] = 0• Var[Z(s+h) – Z(s)] = E{[Z(s+h) – Z(s)] -

E[Z(s+h) – Z(s)] }2

� 2γ(h) = E [Z(s+h) - Z(s)]2

• average of squared differences

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© 2003 Luc Anselin, All Rights Reserved

(Semi)Variogram Cloud Plot(residuals of Baltimore trend surface)

Page 17: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

Brushing a Variogram Cloud Plotidentification of potential outliers

Page 18: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Estimating a Variogram

�Method of Moments� 2γ(h) = { 1 / | N(h) | } x ΣN(h) [Z(s+h)-Z(s)]2

� average of squared differences by distancebin• h: distance bin• N(h): number of pairs in distance bin for h

�Rules of Thumb� at least 30 pairs in each bin� h < D / 2 (D is max distance)

• distance of reliability

Page 19: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

Variogram for residuals from second ordertrend surface

Page 20: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Covariogram

�Second Order Stationarity� covariance “regular” over space� assumptions stronger than for (semi)

variogram� define regularity for second order moments

�Covariogram� C(h) = Cov [ Z(s+h),Z(s) ]

• covariance as a function of “distance” h

� C(0) = Var [ Z(s) ]• at zero distance, covariance = variance of process

Page 21: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Correlogram

�Correlogram� ρ(h) = C(h) / C(0)� notion of an “autocorrelation” function� covariogram C(h) standardized by variance

C(0)

�Semi-Variogram and Correlogram� 2γ(h) = E { [Z(s+h) - Z(s)]2 }

= 2 E {Z(s)}2 - 2 E {Z(s+h). Z(s)}� γ(h) = C(0) - C(h)� correlogram decreases with distance� semivariogram increases with distance

Page 22: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Correlogram(Baltimore trend surface residuals)

Page 23: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Modeling Variograms

Page 24: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Range and Sill

� Limit Behavior� γ(h) = C(0) - C(h)� as h → ∞ , C(h) → 0 or γ(h) → C(0)

• limit on range of dependence

�Sill� C(0) is sill = Var[ Z(s) ]

�Range� h s.t. γ(h) = C(0) is range

• use range in spatial sampling• range is distance beyond which there is no spatial

autocorrelation

Page 25: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

range = 7

sill = 1

Theoretical Spherical Variogram

Page 26: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Nugget Effect

�Behavior Near h = 0� γ(h) → c0 > 0 as h → 0

� not possible mathematically• γ(0) = 0 by definition

�Interpretation of Nugget Effect� measurement error for h < hmin

� scale problems

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© 2003 Luc Anselin, All Rights Reserved

Valid Variogram Models

�Moment Conditions� C(•) positive definite ∀ h� γ(•) negative definite ∀ h

� Isotropic Variogram Models• γ (h, θ) = 0 for h = 0• γ (h, θ) same in all directions

�Non-Isotropic Models� directional

• different variogram for different directions

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© 2003 Luc Anselin, All Rights Reserved

Examples of Variogram Models

�Spherical� γ (h, θ) = c0 + cs{1.5h/a - 0.5(h/a)3}

for 0 < h ≤ a = c0 + cs for h ≥ a

� c0 = nugget effect, c0 + cs = sill, a = range

�Exponential� γ (h, θ) = c0 + cs{ 1 - e-(3h/a) }

� a is “practical range” 95% of asymptoticrange

Page 29: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Spherical Variogram

Source Pannatier 96

Page 30: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Exponential Variogram

Source Pannatier 96

Page 31: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

© 2003 Luc Anselin, All Rights Reserved

Fitting a Variogram Model

�Parameter Estimates� from empirical variogram� methods

• nonlinear least squares• weighted least squares• “eyeball”

�Parameters must satisfyconstraints� valid parameter space

Page 32: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

Fit of Spherical Variogram Model to Baltimore Residuals

Page 33: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

Fit of Exponential Variogram Model to Baltimore Residuals

Page 34: Variogram Analysis - INPE...s{1.5h/a - 0.5(h/a)3} for 0 < h ≤ a = c 0 + c s for h ≥ a c 0 = nugget effect, c 0 + c s = sill, a = range Exponential γ (h, θ) = c 0 + c s{ 1 - e-(3h/a)}

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