beyond where: modeling spatial relationships and …...modeling relationships when variables are...

Lauren Bennett, Flora Vale, Alberto Nieto

Beyond Where: Modeling Spatial Relationships and Making Predictions

esriurl.com/spatialstats

Modeling Relationships

when variables are related, information can be learned about one variable by observing the values of the related variable(s)

Explore correlations

Predict unknown values

Understand key factors

Correlation: 0.992558

Divorce Rate in Maine vs Per Capita Consumption of Margarine

tylervigen.com

Correlation: 0.666004

Number People Who Drowned by Falling into a Swimming-Pool vs Number of Nicolas Cage Films

tylervigen.com

http://xkcd.com/552/

GeneralizedLinearRegression

Modeling linear relationships

a statistical process for estimating

linear relationships between variables

Dependent Variable

What are you trying to predict or understand?

Explanatory Variables

Variables you believe to cause or explain the dependent variable

Coefficients

Represent the strength and type of relationship that X has to y

Coefficients

Positive relationship- as obesity rates rise, diabetes rates also rise

Coefficients

Negative relationship- as foreclosure rates rise, home values drop

Coefficients

No relationship- the value for X is not correlated with the value for y

Residual

Model over and under predictions

observed value

predicted value

Residual

} over prediction

Residual

}

Residual

difference between the observed value

and the predicted value = ε

Evaluating our model

Every variable should be statistically significant *

Each variable should tell a different part of the story

Residuals should not be clustered in location or in value

Model should have a strong R-Squared

When comparing models, lower AICc is better

Exploratory Regression

GeographicallyWeightedRegression

Exploring spatial variation

each feature gets a

separate equation

Defining local

Defining local

Neighborhood Type

• Number of neighbors

Defining local

Neighborhood Type


• Distance band

Defining local

Neighborhood Type


• Distance band

Neighborhood Selection Method

• Golden search

• Manual intervals

• User defined

Local R-Squared

Condition NumberResiduals Predictions

Coefficients

Three model types

• Gaussian – continuous

• Logistic – binary

• Poisson – count

Gaussian – model a continuous variable

Sales profits

Healthcare spending

Mortality rateTemperature

Poisson – model a count variable

0 1 2 3 4 5

Number of people with cancer per 10,000

Traffic accidents

Crime counts

Sales per month

Logistic – model a binary variable

0 1

Fire damage

Pass/fail inspection

Disease presence

Insurance fraud

LocalBivariateRelationships

Examining relationships across space

“… the measurement of a relationship

the measurement

is taken.”

depends on where

Y

two variables

X

determine relationship significance and type

relationships across geography

… what does it mean for two variables to be related to

each other?

“good” relationships


null relationships

low entropy

higher entropy

null relationships


variable Y

low entropy

variable Y

high entropy

variable Y variable X

variable Y variable Xmutual

information

minimum spanning trees

low entropy

higher entropy

…how do we know if the

relationship is SIGNIFICANT???

Permutation-based distribution estimation

feature

each feature has two values

Y X

dependent explanatory

relationshipbetween and x

is evaluated for a neighborhood

Y X

neighborhood distribution of variables minimum spanning tree

0.56

entropy value

what is the probability that the observed entropyexists while and are

actually independent from each other?

Y X

observed local entropy

0.56

Y

X

0.56

start a permutation

Y

X

Y

X

0.56

the dependent values are shuffled, while explanatory values are kept the same

Y

X

Y

X

XX

Y

0.56

minimum spanning trees and entropy are calculated for the permutation

Y

X

Y

X

Y

X

0.56

Y

X

Y

X

Y

X

0.64

minimum spanning trees and entropy are calculated for the permutation

permutations

Y

X

Y

X

0.61

Y

X

Y

X

0.64

Y

X

Y

X

0.57

0.56

Y

X

Y

X

………

neighborhood

permutations

permutation entropy distribution

neighborhoodY

X

Y

X

0.61

Y

X

Y

X

0.64

Y

X

Y

X

0.57

0.56

Y

X

Y

X

………

observed entropy is converted to a p-value

permutation entropies

0.55 0.65

p-value = 0.0012

observed entropy0.56

entropy helps determine if relationships are significant, but

it does not tell uswhat type of relationship exists

classifying the local relationships

Positive Linear Negative Linear Convex Concave

Undefined Complex Not Significant

X

Y

start with the distribution of the significant feature’s

neighborhood

X

Y

estimate an ordinary linear regression model and

calculate AICc

square of X

Y

estimate a second linear regression model

using the square of the explanatory variable

AICc values are compared, and a model is selected

X

Y Y

square of X

linear model quadratic model

AICc: -42.3 AICc: -51.4

is the R2 value greater

than 0.05?

Yes

No

which model was chosen?

linear

quadratic

sign of coefficient?

sign of coefficient?

positive

negative

positive

negative

Positive Linear

Negative Linear

Convex

ConcaveUndefined Complex

entropy

classify type

Positive Linear

Convex

Negative Linear

Positive Linear

Negative Linear

Concave

Convex

Undefined Complex

Not Significant

"Essentially, all models are wrong, but some are useful."

- George E. P. Box

Want to learn more???


TUESDAY_________________________________________

1:45p Data Visualization for Spatial Analysis 146C

3:00p Machine Learning in ArcGIS 146C

4:15p From Means and Medians to Machine Learning: Spatial Statistics Basics and Innovations 146C

WEDNESDAY______________________________________

8:30a Machine Learning in ArcGIS 146C

11a Data Visualization for Spatial Analysis 146C

1:30p From Means and Medians to Machine Learning: Spatial Statistics Basics and Innovations 146C

2:45p Spatial Data Mining: Cluster Analysis and Space Time Analysis 146C

4:00p Beyond Where: Modeling Spatial Relationships and Making Predictions 146C

5:15p The Forest for the Trees: Making Predictions Using Forest-Based Classification and Regression 146C

Please fill out a course survey!!!

[email protected]@[email protected]

beyond where: modeling spatial relationships and …...modeling relationships when variables are...

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