Ecosystems are: Hierarchically structured, Metastable, Far from equilibrium

Spatial RelationshipsTheoretical Framework:

“An Introduction to Applied Geostatistics“, E. Isaaks and R. Srivastava, (1989).

“Factorial Analysis”, C. J. Adcock, (1954)

“Spatial Analysis: A guide for ecologists”, M. Fortin and M. Dale, (2005)

Time

Basic paradigm:

Ecosystem processes (change) are constrained and controlled by the pattern of hierarchical scales

“Things” closer together (in both space and time) are more alike then things far apart – “Tobler’s Law” (1970, Economic Geography) “Everything is related to everything, but near things are more related then distant things”

Ecological “scale” is the space and time “distance” apart (lag) at which significant variation is NO LONGER correlated with “distance”

Distance

Scale

Variation

Applied GeostatisticsApplied Geostatistics

Spatial StructureSpatial StructureRegionalized VariableRegionalized Variable

Spatial AutocorrelationSpatial Autocorrelation

Moran I (1950)Moran I (1950) GEARY C (1954)GEARY C (1954)

SemivarianceSemivariance

StationarityStationarityStationarityStationarity AnistotropyAnistotropyAnistotropyAnistotropy

Applied GeostatisticsApplied Geostatistics

Notes on Introduction to Spatial Autocorrelation

Geostatistical methods were developed for interpreting data that varies continuously over a predefined, fixed spatial region. The study of geostatistics

assumes that at least some of the spatial variation observed for natural phenomena can be modeled by random processes with spatial autocorrelation.

D}i:{z(i)

Geostatistics is based on the theory of regionalized variables, variable distributed in space (or time). Geostatiscal theory supports that any

measurement of regionalozed variables can be viewed as a realization of a random function (or random process, or random field, or stochastic process)

Spatial StructureSpatial Structure

Geostatistical techniques are designed to evaluate the spatial structure of a variable, or the relationship between a value measured at a point in one place, versus a value from another

point measured a certain distance away.

Describing spatial structure is useful for:

Indicating intensity of pattern and the scale at which that pattern is exposed

Interpolating to predict values at unmeasured points across the domain (e.g. kriging)

Assessing independence of variables before applying parametric tests of significance

Regionalized Variables take on values according to spatial location.

Given a variable z, measured at a location i , the variability in z can be broken down into three components:

Where:

)()()( iii sfz

)(if A “structural” coarse scale forcing or trend

)(is A random” Local spatial dependency

error variance (considered normally distributed)

Usually removed by detrending

What we are interested in

Coarse scale forcing or trends can be removed by fitting a surface to the trend using regression and then working with regression residuals

Regionalized VariableRegionalized Variable

+Z1

+Z3

+Zn

+Z4 +Zi

+Z2

Regionalized Variable Zi

Variables are spatially correlated,Therefore:Z(x+h) can be estimated from Z(x) by using a regression model. ** This assumption holds true with a recognized increased in error, from other lest square models.

Function Z in domain D = a set of space dependent values

Histogram of samples zi

Z(x)

Z(x+h)

Cov(Z(x),Z(x+h))

D}i:{z(i)

X1321::µ

Y3253::µ

x y x·y x2 y2

b

aTan

θa

b

As θ decreases, a/b goes to 0

) θ1a

b

)

A Statement of the extent to which two data sets agree.

θ2

)

One distributionTwo distributions

21 TanTan

data deviates

Dev

iatio

ns

Pro

duct

of D

evia

tions

Sum

of

squa

res If you were to calculate correlation

by hand …. You would produce theseTerms.

Determined by the extent to which the two regressions lines depart from the horizontal and vertical.

Correlation Coefficient:

Correlation:

n

)x(x

n

)y(y

)/nx)(xy1(y

n

1i

2i

n

1i

2i

n

1iii

n

)x(x

n

)x(x

w/)x)(xx(xw

n

1i

2i

n

1i

2i

n

1i

n

1i

n

1jij

n

1jjiij

Spatial auto-correlation

CorrelationCoefficient

n

1i

2i

n

1i

n

1jij

n

1i

n

1jjiij

)x(x)w(

)x)(xx(xwN

=

Briggs UT-Dallas GISC 6382 Spring 2007

Spatial StructureSpatial Structure

Autocorrelation: := Degree of correlation to self

SpatialAutocorrelation:

:= The relationship is a function of distance

SpatialStructure which is:

Exogenous (induced) … induced external spatial dependenceEndogenous (inherent) … inherent spatial autocorrelation

SpatialDependence:

Compare values at given distance apart -- LAGS

ABCD

Point – Point Autocorrelation A - B Positive A - C None A - D Negative

Direction ofAutocorrelation:

Anisotropic := varies in intensity and range with orientationIsotropic := varies similarly in all directions

Spatial StructureSpatial Structure

Given: Spatial Pattern is an outcome of the synthesis of dynamic processes operating at various spatial and temporal scales

Therefore: Structure at any given time is but one realization of several potential outcomes

Assuming: All processes are Stationary (homogeneous)

Where: Properties are independent of absolute location and direction in space

Therefore: Observations are independent which := they are homoscedastic and form a known distribution

That is: ijjZiZ jiXX ,,,, 22

Stationarity is a property of the process NOT the data allowing spatial inferences

And:

Stationarity is scale dependentFurthermore:

Inference (spatial statistics) apply over regions of assumed stationarity

Thus:

SpaceSpace

A B C D E F G H I J

First Order Neighbors TopologyBinary Connectivity Matrix

Distance ClassConnectivity Matrix

1

1 1

1 0 1

1 0 0 1

0 0 0 1 1

0 0 1 1 0 1

0 0 0 0 0 1 1

0 1 1 0 0 0 1 1

0 1 0 0 0 0 0 1 1

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

1

1 2

1 2 1

1 2 2 1

2 3 2 1 1

2 2 1 1 2 1

3 2 2 2 2 1 1

2 1 1 2 3 2 1 1

2 1 2 3 3 2 2 1 1

A

B

C

D

E

F

G

H

I

J

J I H

B G C F D A E

Topological v’s Euclidean

1= connected, 0=not connected

Spatial AutocorrelationSpatial Autocorrelation

Positive autocorrelation:Negative autocorrelation:

No autocorrelation:

A variable is thought to be autocorrelated if it is possible to predict its value at a given

location, by knowing its value at other nearby locations.

Autocorrelation is evaluated using structure functions that assess the spatial structure or dependency of the variable.

Two of these functions are autocorrelation and semivariance which are graphed as a correlogram and semivariogram, respectively.

Both functions plot the spatial dependence of the variable against the spatial separation or lag distance.

SpaceSpace

A B C D …. JA 0.0B 2.00 0.00C 1.41 3.16 0.00::J

A B C D …. JA 0B 2 0C 1 3 0::J

A B C D …. JA 0B 0 0C 1 0 0::J

A B C D …. J

Euclidean Distance Matrix

Euclidean Distance Matrix

Connectivity Matrix

Weighted Matrix

A B C D E F G H I J K L

A 0B 0 0C 0.7 0 0D 0.7 0.7 0:J

Moran I (1950)Moran I (1950)

The numerator is a covariance (cross-product) term; the denominator is a variance term.

Where:

n is the number of pairsZi is the deviation from the mean for value at location i (i.e., Z i = xi – x for variable x) Zj is the deviation from the mean for value at location j (i.e., Z j = xj – x for variable x) wij is an indicator function or weight at distance d (e.g. wij = 1, if j is in distance class d from point i, otherwise = 0) Wij is the sum of all weights (number of pairs in distance class)

2

iiij

i jjiij

(d)ZW

ZZwn

I

• A cross-product statistic that is used to describe autocorrelation

• Compares value of a variable at one location with values at all other locations

Values range from [-1, 1] Value = 1 : Perfect positive correlation Value = -1: Perfect negative correlation

Moran I (1950)Moran I (1950)

Again; where for variable x:

n is the number of pairswij(d) is the distance class connectivity matrix (e.g. wij = 1, if j is in distance class d from point i, otherwise = 0) W(d) is the sum of all weights (number of pairs in distance class)

2

1

_

_

1 1

_

)()(

1

1

n

ii

j

n

jii

n

ijj

iij

dd

xxn

xxxxdw

WI

i

2iij

i j

2jiij

(d)Z2W

])y(yw1)[[(NC

• A squared difference statistic for assessing spatial autocorrelation

• Considers differences in values between pairs of observations, rather than the covariation between the pairs (Moran I)

GEARY C (1954)GEARY C (1954)

The numerator in this equation is a defference term that gets squared.

The Geary C statistic is more sensitive to extreme values & clustering than the Moran I, and behaves like a distance measure:

Values range from [0,3]

Value = 0 : Positive autocorrelation Value = 1 : No autocorrelation Value > 1 : Negative autocorrelation

Ripley’s K (1976)The L (d) transformationsRipley’s K (1976)

The L (d) transformations

1

,1 1

NN

jikA

Li

ijj

(d) Where:A = areaN = nuber of pointsD = distanceK(i,j) = the weight, which is 1 when |i-j| < d, 0 when |i-j| > d

Determines if features are clustered at multiple different distance. Sensitive to study area boundary. Conceptualized as “number of points” within a set of radius sets.

If events follow complete spatial randomness, the number of points in a circle follows a Poisson distribution (mean less then 1) and defines the “expected”.

ji

jiij

(d) xx

xxdw

G

)(

Where:d = distance classWij = weight matrix, which is 1 when |i-j| < d, 0 when |i-j| > d

General GGeneral G

Effectively Distinguishes between “hot and cold” spots. G is relatively large if high values cluster, low if low values cluster.

Numerator are “within” a distance bound (d), expressed relative to the entire study area.

SemivarianceSemivariance

2)(1

)(2 jii j

ijd

yywn

d

Where :j is a point at distance d from ind is the number of points in that distance class (i.e., the sum of the weights wij for that distance class)wij is an indicator function set to 1 if the pair of points is within the distance class.

2)()(2

1)( idi

dn

i

yydn

d

The geostatistical measure that describes the rate of change of the regionalized variable is known as the semivariance.

Semivariance is used for descriptive analysis where the spatial structure of the data is investigated using the semivariogram and for predictive applications where the

semivariogram is fitted to a theoretical model, parameterized, and used to predict the regionalized variable at other non-measured points (kriging).

The sill is the value at which the semivariogram levels off (its asymptotic value)

The range is the distance at which the semivariogram levels off (the spatial extent of structure in the data)

The nugget is the semivariance at a distance 0.0, (the y –intercept)

A semivariogram is a plot of the structure function that, like autocorrelation, describes the relationship between measurements taken some distance apart.

Semivariograms define the range or distance over which spatial dependence exists.

Autocorrelation assumes stationarity, meaning that the spatialstructure of the variable is consistent over the entire domain of the dataset.

The stationarity of interest is second-order (weak) stationarity, requiring that:

(a) the mean is constant over the region(b) variance is constant and finite; and (c) covariance depends only on between-sample spacing

In many cases this is not true because of larger trends in the data In these cases, the data are often detrended before analysis. One way to detrend data is to fit a regression to the trend, and use only the residuals for autocorrelation analysis

StationarityStationarityStationarityStationarity

Autocorrelation also assumes isotropy, meaning that the spatial structure of the variable is consistent in all directions.

Often this is not the case, and the variable exhibits anisotropy, meaning that there

is a direction-dependent trend in the data.

AnistotropyAnistotropyAnistotropyAnistotropy

If a variable exhibits different ranges in different directions, then there is a geometric anisotropy. For example, in a dune deposit, larger range in the wind direction

compared to the range perpendicular to the wind direction.

bdco γ(d)

)]/exp(1[γ(d) 22 adcco

)]/exp(1[γ(d) adcco

Gaussian:

Linear:

Spherical:

Exponential:

For predictions, the empirical semivariogram is converted to a theoretic one by fitting a statistical model (curve) to describe its range, sill, & nugget.

adcc

adadadcc

o

o

,

)],2/()2/3[γ(d)

33

There are four common models used to fit semivariograms:

Where:

c0 = nugget

b = regression slope

a = range

c0+ c = sill

Assumes no sill or range

• Check for enough number of pairs at each lag distance (from 30 to 50). • Removal of outliers

• Truncate at half the maximum lag distance to ensure enough pairs

• Use a larger lag tolerance to get more pairs and a smoother variogram

• Start with an omnidirectional variogram before trying directional variograms

• Use other variogram measures to take into account lag means and variances (e.g., inverted covariance, correlogram, or relative variograms)

• Use transforms of the data for skewed distributions (e.g. logarithmic transforms).

• Use the mean absolute difference or median absolute difference to derive the range

Variogram Modeling SuggestionsVariogram Modeling Suggestions