loosmore course pointpaterrnspatialstatistics

8/3/2019 Loosmore Course PointPaterrnSpatialStatistics

1/116

Inference for Point Pattern Spatial

StatisticsN. Bert Loosmore

[email protected]

QERM 550

University of Washington

May 11 & 13, 2005

Inference for Point Pattern Spatial Statistics p.1/4


2/116

Outline

Use of Point Pattern Statistics in Ecology



3/116

Outline


The Failure of the Simulation Envelope



4/116

Outline



Diggles (1983, 2003) Goodness of Fit Test



5/116

Outline




Unresolved Implementation Issues



6/116

Outline





Parameterization Based on the Ecological ResearchQuestion



7/116

Outline





Parameterization Based on the Ecological ResearchQuestion

Characterizing Type I, II Error Rate Performance



8/116

Point Pattern Statistics in Ecology

Spatial processes

Ecological processes

0 50 100 150 200

0

50

100

150

200

Easting(m)

Northing(m)



9/116


Spatial processes


0 50 100 150 200

0

50

100

150

200

Easting(m)

Northing(m)

What pattern for thegreen points?



10/116


Spatial processes


0 50 100 150 200

0

50

100

150

200

Easting(m)

Northing(m)

What pattern for thered points?



11/116


Spatial processes


0 50 100 150 200

0

50

100

150

200

Easting(m)

Northing(m)

Do we see (or expect)stationarity?



12/116

Point Pattern Spatial Stats: How?

Evaluate observed pattern against ideas of



13/116



aggregation,

rMatClust() with 105 points, radius = 0.1



14/116



aggregation,

CSR,CSR pattern with 100 points


P i P S i l S H ?


15/116



aggregation,

CSR,

inhibition rSSI() with 100 points, radius = 0.05


P i P S i l S H ?


16/116



aggregation,

CSR,

inhibition

Analyze distances between events:


P i t P tt S ti l St t H ?


17/116



aggregation,

CSR,

inhibition

Analyze distances between events:G (nearest neighbor),




18/116



aggregation,

CSR,

inhibition


F(grid to nearest point),




19/116



aggregation,

CSR,

inhibition



K/L (all neighbors)




20/116



aggregation,

CSR,

inhibition



K/L (all neighbors)

Typically perform analysis using Simulation Envelope


D fi iti f th G d F St ti ti


21/116

Definition of the G and F Statistics

G statistic uses the nearest neighbor distances ( ) for eachof sample points as:

F statistic uses the distances ( ) from each of sample

points (typically located on a grid) to their nearest event as:

Under CSR, both the G and F statistic is approximated as

!

#

$


D fi iti f th K d L St ti ti


22/116

Definition of the K and L Statistics

K statistic uses the distances between all neighbors ( ) as:

#

#

Under CSR, K statistic can be approximated by

#

!

#

$

L statistic used to set mean

and (supposedly) stabilizevariance as:

#

!

#


Building the Simulation Envelope


23/116


A CSR pattern with

0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

1.0

Distance




24/116


99 CSR patterns with

0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

1.0

Distance


Using the Simulation Envelope


25/116

Using the Simulation Envelope

Plot after subtracting

#

0.00 0.05 0.10 0.15 0.20

0.3

0.2

0.1

0.0

0.1

0.2

0.3

rSSI(r=0.03, n=100)

Distance


Perceived Level Performance


26/116

Perceived Level Performance

Using all results from 19 simulations yields

, or

Throwing out upper and lower 2 simulations at eachdistance (

) from 99 simulations also yields


Kenkel (1988) Methods


27/116


Evaluated spatial locations of all live trees, all (live +standing dead) trees in a jack pine Pinus Bansiana forest.




28/116



Map of live + standing dead represents distributionfollowing early sapling mortality, but prior to the onset ofdensity-depending mortality.




29/116



Map of live + standing dead represents distributionfollowing early sapling mortality, but prior to the onset ofdensity-depending mortality.

Methods: Used MC techniques for the G and L statisticsto evaluate observed results against

of i) randomlocations (CSR) and ii) random mortality.


Kenkel (1988) Conclusions


30/116


G: live + dead shows no departure from randomnesswhereas live trees only shows significant regularity




31/116



L: live + dead shows no departure from CSR at smallscales, live trees show regularity at smaller scales




32/116



L: live + dead shows no departure from CSR at smallscales, live trees show regularity at smaller scales

But is this interpretation correct?


Examples in Ecological Research


33/116

Examples in Ecological Research

Author (Year) Statistics Patterns in CI (%) Marginal

Used Sim Env (s) Results (y/n)

Batista and Maguire (1998) G, K 19 95% n

Dolezalet al.

(2004) K 99 95% yFreeman and Ford (2002) G, K 99 99% n

Grassi et al. (2004) K 99 95% n

Hirayama and Sakimoto (2003) K 19,99 95%, 99% n

Martens et al. (1997) L 99 95% n

Moeur (1997) G, K 200 90% n

Parish et al. (1999) G, K 19 95% n

Salvador-

Van Eysenrode et al. (2000) G, K 1000 95% y

Srutek et al. (2002) L 99 95% y

Tirado and Pugnaire (2003) K 1000 99% n


Outline


34/116

Outline





Parameterization Based on the Ecological Research

Question



Sim Env Level Performance


35/116


Simulation study with independent trials of a CSRpattern against a CSR envelope.

Designate failure if pattern exceeds envelope at anydistance. (Type I error)

Expected type I error rate 0.05 ...




36/116


Simulation study with independent trials of a CSRpattern against a CSR envelope.

Designate failure if pattern exceeds envelope at anydistance. (Type I error)

Expected type I error rate 0.05 ...

... actual type I error rate 0.5-0.7


Monte Carlo Simulation Theory


37/116

y

For a univariate continuous distribution,


Monte Carlo Simulation Theory


38/116

y

For a univariate continuous distribution,

But does the simulation envelope comprise a univariatedistribution?


How the Envelope is Really Made


39/116

p y

Simulation envelope built from 100 patterns:

0.00 0.05 0.10 0.15 0.20 0.25

0.3

0.2

0.1

0.0

0.1

0.2

0.3

55 patterns comprising

the simulation envelope

Distance


Failure of the Simulation Envelope


40/116

p

Although built from

patterns, complexity of both

1. G, F, and/or K statistics, and

2. spatial patterns

yields a multivariate result.

Since evaluation of the observed pattern occurs at manydistances we are performing simultaneous inference andthus is increased.

Further, if the simulation envelope is invalid, then how canwe use it to determine scale?


Outline


41/116






Question



Proper Statistical Methods


42/116

From Diggle (1983, 2003), for a given

:

1. At a single a priori distance - use upper and lowersimulated values

2. Across a range of distances - use Goodness of Fit test


The Goodness of Fit Test - 1


43/116

1. Represent the empirical results as:

#

observed pattern, and

#

for

simulated patterns




44/116

2. Calculate:

#

#

$

#

for

Summary statistic indicative of the total deviation of thegiven pattern from the theoretical result




45/116

2. Calculate:

#

#

$

#

for

but use

#

#

#

to reduce bias




46/116

3. Reject (fail to)

based on the rank of

using thep-value, calculated as

for

. So, if

(the largest), then

.

Now we have quantitative results to evaluate a patternssignificance based on an exact level test because of

proper MC methodsInference for Point Pattern Spatial Statistics p.22/4

Outline


47/116






Question





48/116

What is the optimal method to calculate

?

#

#

$

#

How to:

replace integration with summationincorporate edge correction methods

choose limits

#

, distance list

#

simulate patterns from null process


Replacing Integration with Summation


49/116

We can rewrite Eqn (1) as

#

#

$

#

#

#

$

#

But how accurate is this approximation?


Edge Correction


50/116

Used to eliminate bias from edge interfering with detectinga points neighbor

Reduced Sample edge correction approach:

Let

be the distance for point

to the closestboundary

Remove point

from calculation at distance#

where#

Other approaches (toroidal, isotropic, etc.)


Choice of Limits (

), Distance List (

)


51/116

Recommended default for#

, but applicationdependent!


Choice of Limits (

), Distance List (

)


52/116



#

,

#

are discrete, change where


Choice of Limits (

), Distance List (

)


53/116



#

,

#

are discrete, change wherenew neighbor detected, or


Choice of Limits (

), Distance List (

)


54/116



#

,

#


point removed from sample


Choice of Limits (

), Distance List (

)


55/116



#

,

#



Use empirical distance list for exact results from a singlepattern


Choice of Limits (

), Distance List (

)


56/116



#

,

#



Use empirical distance list for exact results from a singlepattern

Because of calculation, especially

#

, for exactsolution, need to use complete empirical distance list (i.e.from all patterns) for evaluation of each pattern


Resolution of Simulated Patterns


57/116

Complexity? - Number of distances grows with

,




58/116


,

Resolution (i.e.

vs

) of simulatedpatterns should be equivalent to that of observed

pattern




59/116


,

Resolution (i.e.

vs


patternLimiting resolution helps constrain complexity




60/116


,

Resolution (i.e.

vs



is highly accurate for ecological

data (Freeman and Ford, 2002)




61/116


,

Resolution (i.e.

vs



is highly accurate for ecological

data (Freeman and Ford, 2002)

Combining resolution and default#

leads to at most

25,000 distances in

#

, regardless of

,

or test statistic, andprovides an exact solution


Outline


62/116






Question



Parameterization - 1


63/116

How to run any given test based on the ecologicalresearch question

Number of simulations ( )


Parameterization - 1


64/116

How to run any given test based on the ecologicalresearch question

Number of simulations ( )

Choice of

, including choice of#


versus


65/116

Uncertainly in realized p-value (

) results from the use ofMC simulations

Ramifications of ? Affects precision of

through

actual simulated patterns against which observedpattern tested, and

number of those patterns

Note about exact level performance (across many testsvs. variation of p-value for single test)


Distribution of


66/116

Let

and

for

. The p-value forthe test is then:


Distribution of


67/116

Let

and

for

. The p-value forthe test is then:

The expected value of P is:

Assuming Y comes from

, then

. So,

each of the


Variance of P (

)


68/116

Looking at the variance of

we have

$

$

$


Variance of P (

)


69/116

Looking at the variance of

we have

$

$

$

Hence we can model the theoretical distribution of

as

from a binomial(p,s) distribution.Inference for Point Pattern Spatial Statistics p.33/4

Managing Uncertainty in


70/116

Rem that binomial quickly converges to Normal




71/116

Rem that binomial quickly converges to NormalCreate 95% CI on (true p-value) near

as

$




72/116


as

$

95% of CI created this way should contain the truevalue of , and so set decision rule: e.g. reject

if

CI contains or fully below 0.05




73/116


as

$


if


Choose acceptable range of uncertainty for .




74/116


as

$


if


Choose acceptable range of uncertainty for . For

example if

is ok, use $




75/116


as

$


if

CI contains or fully below 0.05Choose acceptable range of uncertainty for . For

example if

is ok, use $

Use relationship between $

and to find value of


as a function of


76/116

0 500 1000 1500 2000

0.

01

0.

02

0

.03

0.

04

0.0

5

0.

06

0.

07

# of Simulations


Choice of


77/116

Use all available ecological knowledge for a moreinformative test


Choice of


78/116


Null point process just needs to be able to be simulated,

many models available (e.g. spatstat) or write yourown!


Choice of


79/116




At the very least, choose simple inhibition model basedon physical separation


Choice of


80/116





EDA vs. confirmatory analysis, results in iterative

nature of research, with (hopefully) tests onindependent data sets


Choice of


81/116





EDA vs. confirmatory analysis, results in iterative

nature of research, with (hopefully) tests onindependent data sets

Use the model to determine information on scale!


Example of model fitting


82/116

Attempt to fit a clustered model, representingestablishment processes to the lower SW quadrant ofthe WRCCRF data, for all trees

in height.




83/116


in height.

Used Poisson Clustered model, with

represents thenumber of parents and represents the expectednumber of children per parent, and where clustering ofchildren around each parent are described as

!

$

$

$

$




84/116


in height.



!

$

$

$

$

How to choose values for

and

? (

)




85/116


in height.



!

$

$

$

$

How to choose values for

and

? (

)

Note that my null model here describes not only theprocess, but also the parameter values.


Example of model fitting - 2


86/116

This is Exploratory Data Analysis!



Thi i E l D A l i !


87/116

This is Exploratory Data Analysis!If we knew the theoretical value of G, K for this model,use Diggles Least Squares Estimation method



Thi i E l t D t A l i !


88/116


Otherwise, use GoF test to estimate parameter space





89/116


Otherwise, use GoF test to estimate parameter spaceFind

for different combinations of and acceptmodel where





90/116


Otherwise, use GoF test to estimate parameter space

0 20 40 60 80 100

0.

1

0.

2

0.

3

0.

4



Inference? For the observed data if this model fits then


91/116

Inference? For the observed data, if this model fits, thenlarger suggests lower (i.e. few parents) and so morechildren/parent.





92/116


Conversely a smaller clustering radius requires higher

and so fewer children per parent.





93/116




Is this model a good fit? What might the physiologicaland/or ecological implications be?





94/116




Is this model a good fit? What might the physiologicaland/or ecological implications be?

gives us hints about scale.


, Variance stabilization

#

should be chosen before the test and based on f


95/116

should be chosen before the test, and based onresearch question. (i.e. what is the interaction distance ofinterest?)


, Variance stabilization

#

should be chosen before the test and based onh i (i h i h i i di f


96/116

should be chosen before the test, and based onresearch question. (i.e. what is the interaction distance ofinterest?)

0.00 0.05 0.10 0.15 0.20

0.1

0

0.0

5

0

.00

0.0

5

Distance

Variance stabilization - to make variance independent of#

.


Outline



97/116






Question



Type I Error Rate ( ) - 1

Simulation study of Type I error rate performance


98/116

Simulation study of Type I error rate performance

Evaluated different

levels, for different point patternintensities (

)

Results within LRT boundaries


Type I Error Rate ( ) - 2

Simulations of 1000 independent trials using


99/116

Simulations of 1000 independent trials using

0 50 100 150 200 250

0.

00

0.

05

0.

10

0

.15

a) Type I error rates for G

0 50 100 150 200 250

0.

00

0.

05

0.

10

0

.15

b) Type I error rates for K

# points (

) # points (

)Inference for Point Pattern Spatial Statistics p.43/4

Type II Error Rate (1-Power)

Type II error rate is the prob of accepting

given that is really true


100/116

Type II error rate is the prob of accepting given that is really true.






101/116

Type II error rate is the prob of accepting given that is really true.

Requires definition of

.






102/116

ype e o ate s t e p ob o accept g g e t atis really true.


.

Power will be a function of how far

is from

.(Easy to think of this distance when using Normaldistribution, but more difficult to conceptualize here.)






103/116

yp p p g gis really true.


.

Power will be a function of how far

is from

.(Easy to think of this distance when using Normaldistribution, but more difficult to conceptualize here.)

Often overlooked for spatial point process analysis, butcan be simulated.


Analysis of Type II Error Rate

Analysis of power against

of CSR for WRCCRFexample for different parameterizations of

.


104/116

example for different parameterizations of .

Type II error rate tells us the ability to distinguish thepattern from CSR.

As increases, larger clusters are more like CSR.

0.05 0.15 0.25 0.35

0.0

0.2

0

.4

0.6

0.8

1.0

a)=20

Power

0.05 0.15 0.25 0.35

0.0

0.2

0

.4

0.6

0.8

1.0

b)=40

Power


Power of the G Statistic

Large deviation at small distances may be swamped out


105/116

g y p

0.00 0.05 0.10 0.15 0.20

0.3

0.2

0.1

0.0

0.1

0.2

0.3

rSSI(r=0.02)

rSSI(r=0.03)

Distance


Parameters that may improve Power

Rewriting Equation (2) in its full form (Diggle, 2003):


106/116

g q ( ) ( gg )

#

#

#

$

#





107/116

#

#

#

$

#

#

, as parameters to improve Power against certain





108/116

#

#

#

$

#

Use of

#

not well explored, but could be used toemphasize certain distances.

For my calculations,

#





109/116

#

#

#

$

#

For

#

,

use

for L statistic.

use

for power against clustered patterns(Diggle, 2003)

other?


Conclusions

Simulation envelope does not result in expected Type Ierror rates. Limits are not confidence intervals.


110/116


Conclusions



111/116

For more precise, reliable results, implement Digglesgoodness of fit test


Conclusions



112/116


Previous marginal results should be re-examined


Conclusions



113/116



Choice of

,#

based on research question and

previous knowledge


Conclusions



114/116



Choice of

,#


previous knowledgeEvaluate the Power of your test


Conclusions



115/116



Choice of

,#


previous knowledgeEvaluate the Power of your test

R software availability:

http://students.washington.edu/nhl/masters.html


R software resources

CRAN (Comprehensive R Archive Network) sitehttp://cran.r-project.org/


116/116

p p j g

A. Baddeleys spatstat packagehttp://www.maths.uwa.edu.au/ adrian/spatstat.html

P. Diggles splancs packagehttp://www.maths.lancs.ac.uk/ rowlings/Splancs/

UW R and S-plus user support grouphttp://mailman1.u.washington.edu/mailman/listinfo/s plus


loosmore course pointpaterrnspatialstatistics

Documents