Applications of Spatial Data Analysis

Applications of Spatial Data Analysis

Earvin Balderama

Department of Mathematics and StatisticsLoyola University Chicago

April 23, 2015

Applications of Spatial Data Analysisc© 2015 by Earvin Balderama <[email protected]>

A little philosophy...

“Nothing puzzles me more than time and space; and yet nothing troubles meless...”

Charles Lamb (1810)


A little philosophy...

“Nothing puzzles me more than time and space; and yet nothing troubles meless...as I never think of them.”

Charles Lamb (1810)


Spatial Statistics

1 Studies dependencies in data due to their proximity.2 Answers the questions of “When?” and “Where?”


Spatial Statistics

1 Studies dependencies in data due to their proximity.2 Answers the questions of “When?” and “Where?”

Fact“There would be no History without Geography.”


What’s so spatial about spatial data?

We usually think of Y1,Y2, . . . ,Yn as independent observations.But if the Yi’s are from locations in Rd, they may be spatially correlated.

Three main types of spatial data1 Point-referenced data (geostatistics)2 Point pattern data3 Areal data



1 Point-referenced data (geostatistics)Data: The observed values from fixed locations.Goal: Interpolation over continuous space.e.g., predicting the values between locations.



1 Point-referenced data (geostatistics)2 Point pattern data

Data: The random locations of an observed point process.Goal: Model the underlying data-generating process.e.g., predicting the location of certain values.



1 Point-referenced data (geostatistics)Data: The observed values from fixed locations.Goal: Interpolation over continuous space.e.g., predicting the values between locations.

2 Point pattern dataData: The random locations of an observed point process.Goal: Model the underlying data-generating process.e.g., predicting the location of certain values.

3 Areal dataData: Observations on a (regular or irregular) lattice.equal-sized square grid cells, or by county or state lines.disease mapping, etc.


Where are all the old people? (Areal data example)


Where are all the old people? (Areal data example)


Spatial data not restricted to locations on Earth


Red Banana

Red Bananas


Red Banana Data

Red bananas found in Costa Rica

1 Species Musa velutina1 Native to India and parts of

southeast Asia.2 Usually grown as an ornamental

plant.2 788 observed plants

1 Dec 2006 – Jan 20082 La Selva Biological Station3 Measured height, GPS location

3 318 selected for weekly heightmeasurements; the rest were dug out.

1 Empirical growth rate2 Estimate birth times

10.43

10.44

−84.02 −84.01 −84.00Longitude

Latit

ude

Height (m) 1 2 3 4 5 6


Red Banana Data

Spread of red banana beginning in 2002

6 months

10.43

10.44

−84.02 −84.01 −84.00Longitude

Latit

ude


Red Banana Data


1 year

10.43

10.44

−84.02 −84.01 −84.00Longitude

Latit

ude


Red Banana Data


2 years

10.43

10.44

−84.02 −84.01 −84.00Longitude

Latit

ude


Red Banana Data


3 years

10.43

10.44

−84.02 −84.01 −84.00Longitude

Latit

ude


Red Banana Data


4 years

10.43

10.44

−84.02 −84.01 −84.00Longitude

Latit

ude


Red Banana Data

Finding models for spread

Goal: Find a model to characterize the spread of an invasive species inspace-time.

Epidemic-type aftershock sequence (ETAS)1

Models a sequence of earthquakes via a conditional intensity.

λ(t, x, y|Ht) = µ(x, y) +∑{i:ti<t}

g(t − ti, x− xi, y− yi;Mi)

1 λ is the infinitesimal rate at which events are expected to occurconditioned on the prior history of process,

2 µ is a non-homogeneous background rate,3 g explains how aftershocks are “triggered."

1Ogata, Y., 1998. Space-Time Point-Process Models for Earthquake Occurrences. The Inst. of Statistical Mathematics.


Red Banana Data

Estimate background rate using a kernel smooth

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10.43

10.44

−84.02 −84.01 −84.00Longitude

Latit

ude


Red Banana Data

Estimate background rate using a kernel smooth


Red Banana Data


Space-time ETAS triggering function

g(t, x, y;M) =K0

(t + c)p ·e−α(M−M0)

(x2 + y2 + d)q


Red Banana Data


Space-time ETAS triggering function

g(t, x, y;M) =K0

(t + c)p ·e−α(M−M0)

(x2 + y2 + d)q

Omori’s law2

Frequency of aftershocks at time t follow a power-law distribution.

n(t) =K

(t + c)p

Magnitude frequency law3

Magnitudes of earthquakes follow an exponential distribution.

P(Mag > M) = e−βM

2Omori, F., 1894. On the aftershocks of earthquakes. Journal of the College of Science, Imperial University of Tokyo, 7,111-200.

3Gutenberg, B., Richter, C., 1944. Frequency of Earthquakes in California. Bulletin of the Seismological Society of America, 34,

185-188.


Red Banana Modified ETAS

Spatial and temporal clustering

Inter-event distanceDistance between pairs of plantsborn within 6 weeks of each other.

inter−event squared distance (km2)

Den

sity

0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0

Inter-event timeTime between pairs of plants that areless than 100 meters apart.

inter−event birth time (weeks)

Den

sity

0 50 100 150

0.00

00.

005

0.01

00.

015

0.02

00.

025


Red Banana Modified ETAS

Modified ETAS for red banana

Triggering density function

g(t, x, y) =αβ

πe−αt−β(x2+y2)

Conditional intensity

λ(t, x, y|Ht) = (1− p)µ(x, y) +pαβπ

∑{i:ti<t}

e−α(t−ti)−β{(x−xi)2+(y−yi)2}

1 α is the temporal clustering parameter,2 β is the spatial clustering parameter,3 p is the proportion of plants that were “triggered" by previous plants.


Red Banana Estimation

Estimate model parameters by MLE

log-Likelihood

l =n∑

i=1

logλ(ti, xi, yi)−∫∫

A

∫ ∞0

λ(t, x, y)dtdxdy

α̂ = 0.076 (0.005)

β̂ = 0.029 (0.002)

p̂ = 0.577 (0.019)

1 About 58% of plants are direct descendants of previous plants.2 Model fit evaluated using super-thinning4 with the R packagestppresid.

4Clements, R., Schoenberg, F., Veen, A., 2012. Evaluation of space-time point process models using super-thinning. Environmetrics, 23,

606-616.


Red Banana Estimation

Simulating 100 years

0 500 1000 1500 2000 2500

0500

1000

1500

2000

2500

After 1 year

xy

0 500 1000 1500 2000 2500

0500

1000

1500

2000

2500

After 2 years

x

y

0 500 1000 1500 2000 2500

0500

1000

1500

2000

2500

After 5 years

x

y

0 500 1000 1500 2000 2500

0500

1000

1500

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2500

After 10 years

xy

0 500 1000 1500 2000 2500

0500

1000

1500

2000

2500

After 50 years

x

y

0 500 1000 1500 2000 2500

0500

1000

1500

2000

2500

After 100 years

x

y


Sea birds

Sea Birds

←− Northern Gannet


Sea birds Motivation

Development of offshore wind energy facilities



Detrimental effects on seabird life



Statistical motivation

Northern Gannet data

Count

Fre

quen

cy

0 500 1000 1500

020

0040

0060

0080

00

Observed count Frequency0 9553

1− 10 177811− 100 184

101− 1000 111001+ 1


Sea birds Data

Data collection

1 Boat and aerial surveys2 1992− 20103 43,701 transects4 133,890 separate sightings5 > 2 million total birds6 ∼ 150 unique species


Sea birds Data

Data collection

1 Boat and aerial surveys2 1992− 20103 43,701 transects4 133,890 separate sightings5 > 2 million total birds6 ∼ 150 unique species

Goal: Model the space-timedistribution of seabirds in theAtlantic ocean, and create mapsthat assess risk.


Sea birds Data

Discretize spatial domain

15984 sites4 × 4 km each site35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude


Sea birds Data

Consider data from July 2002—November 2010

Amount of Effort35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

10+

5−9

1−4

0


Sea birds Model

Mixture models

1 Must account for excess zeros.1 Zero-inflated models.2 Hurdle models.

2 Must account for over-dispersion.1 Negative Binomial (NB) instead of Poisson for “typical" counts.


Sea birds Model

Mixture models


2 Must account for over-dispersion.1 Negative Binomial (NB) instead of Poisson for “typical" counts.2 Generalized Pareto distribution (GPD) for “extreme" values.


Sea birds Model

Mixture models


2 Must account for over-dispersion.1 Negative Binomial (NB) instead of Poisson for “typical" counts.2 Generalized Pareto distribution (GPD) for “extreme" values.

Generalized Pareto distribution (GPD) properties1 µ is the lower bound (threshold)2 σ > 0 is the scale3 ξ is the shape

1 if ξ < 0, the distribution is bounded above2 if ξ > 0.5, the variance is infinite3 if ξ > 1, the mean is infinite


Sea birds Model

Double-hurdle model

P(Yij|θ) =

pij if Yij = 0,(1− pij) · (1− qij) · NB(mij, r) if 1 ≤ Yij < µ,

(1− pij) · qij ·GPD(µ, σ, ξ) if Yij ≥ µ.


Sea birds Model

Double-hurdle model

P(Yij|θ) =



1 p = Pr(zero-count)


Sea birds Model

Double-hurdle model

P(Yij|θ) =




2 m = mean of typical-count distribution.


Sea birds Model

Double-hurdle model

P(Yij|θ) =





3 q = Pr(large-count | nonzero-count)


Sea birds Model

Double-hurdle model

P(Yij|θ) =



1 p = Pr(zero-count)logit(p) = Xβ(p) + S(p)




Sea birds Model

Double-hurdle model

P(Yij|θ) =




2 m = mean of typical-count distribution.log(m) = log(E) + Xβ(m) + S(m)



Sea birds Model

Double-hurdle model

P(Yij|θ) =




2 m = mean of typical-count distribution.log(m) = log(E) + Xβ(m) + S(m)

3 q = Pr(large-count | nonzero-count)logit(q) = Xβ(q)


Sea birds Spatial model

Modeling spatial random effects

1 Guassian Markov random field

π(S|τ) ∝ τ rank(Q)/2 exp(− τ

2S′QS

)






2S′QS

)2 Inverse covariance matrix Q = D− ρA

D is diagonal with entries the number of neighbors,A is the adjacency matrix,ρ = 1 specifies the intrinsic CAR prior.






2S′QS

)2 Inverse covariance matrix Q = D− ρA

D is diagonal with entries the number of neighbors,A is the adjacency matrix,ρ = 1 specifies the intrinsic CAR prior.

3 Q is a 15984×15984 matrixEigen-decompose Q = VΛV−1,V is a matrix whose columns are eigenvectors,Λ is diagonal whose entries are eigenvalues.



Hierarchical modeling

Spatial random effects, S = Vn×n· α

n×1≈ V

n×k· α

k×1

1 Choose the first k� n eigenvectors (k = 50 explains 67% of variance),

logit(p) = Xβ(p) + Vα(p)

log(m) = log(E) + Xβ(m) + Vα(m)

logit(q) = Xβ(q)

Biophysical covariates, X = [1, x1, x2, . . . , x7]

1 Bathymetry, Distance-to-shore,2 Sea surface temperature, Chlorophyll,3 Fourier basis sin(π6 month), cos(π6 month) for temporal variation.


Sea birds Results

Predictive maps for northern gannet

JanuaryP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

JanuaryP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


FebruaryP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

FebruaryP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


MarchP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

MarchP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


AprilP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

AprilP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


MayP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

MayP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


JuneP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

JuneP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


JulyP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

JulyP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


AugustP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

AugustP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


SeptemberP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

SeptemberP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


OctoberP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

OctoberP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


NovemberP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

NovemberP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


Sea birds Results


DecemberP(y ≥ 1)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.00

0.25

0.50

0.75

1.00

DecemberP(y ≥ 7)

35.0

37.5

40.0

42.5

−76 −72 −68 −64Longitude

Latit

ude

0.0

0.1

0.2

0.3


What was the most popular NYTimes.com article of 2013?


How Y’all, Youse, and You Guys Talk

Fun:http://spark.rstudio.com/jkatz/DialectMap/http://www.nytimes.com/interactive/2013/12/20/sunday-review/dialect-quiz-map.html


http://spark.rstudio.com/jkatz/DialectMap/

http://www.nytimes.com/interactive/2013/12/20/sunday-review/dialect-quiz-map.html

http://www.nytimes.com/interactive/2013/12/20/sunday-review/dialect-quiz-map.html

My Dialect Map


How ya’ll, youse, and you guys talk

1 Examine regional variation in English dialect in continental US.2 Important in linguistic research.3 Point-referenced data, coded by zip code.4 Estimate pt, the probability vector for any location t.5 k-nearest neighbor kernel smoothing.


Fall Course

Coming this Fall...STAT 388/488 - Applied Spatial Statistics

1 No required textbook2 Data analyses using R3 Project-based course4 Final poster presentation


Thank you!

Thank You!


Documents

Applications of Spatial Data Analysis