Introduction
•Statistical options tend to be limited in most GIS
applications.
•This is likely to be redressed in the future.
•We will look at spatial statistics in general terms, and
conclude with a review of the software available.
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Professor has explained the following concepts in these Lecture Slides : Basic Concepts, Introduction, Basic Concepts, Statistical Options, Application, General Terms, Spatial Statistics, Spatially Continuous Data, Areal Data, Interaction Data
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2012/2013
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Introduction
- Statistical options tend to be limited in most GIS
applications.
- This is likely to be redressed in the future.
- We will look at spatial statistics in general terms, and
conclude with a review of the software available.
Basic Concepts
- Spatial statistics differ from ‘ordinary’ statistics by the
inclusion of locational properties.
- This makes spatial statistics more complex.
- The book by Bailey and Gatrell (1995) provides an
accessible introduction. They identify four categories:
- Point pattern data;
- Spatially continuous data;
- Areal data; and
- Interaction data.
- Obvious correspondence with conceptual models.
Random Variables
- Statistical models deals with phenomena that are
stochastic (i.e. are subject to uncertainty).
- A random variable Y has values that are subject to
uncertainty (but may not necessarily be random).
- The distribution of possible values is referred to as the
probability distribution.
- Represented by a function fY(y)
- Random variables may be discrete or continuous.
Probabilities
- Probability that y is between a and b is given by:
if Y is discrete
if Y is continuous (probability density)
- Cumulative probability ( or distribution function ) FY is
given by:
if Y is discrete if Y is continuous
∑^ ( )
=
b y a
fY y
∫^ (^ )
b a fY^ y dy
( ) (^) ∑ ( ) =−∞
=
y Y (^) u Y F y f u
FY^ (^ y )^ =∫− y ∞ fY ( ) udu
Joint Probability
- Can generalise to situations where there is more than one
random variable.
- Joint probability distribution (or density): fXY(x,y)
- Covariance : COV(X,Y) = Σ ((X - E(X)).(Y - E(Y)))
- Correlation : ρ X,Y = COV(X,Y) / σ X. σ y
- Independence : Neither variable affects the other. Joint
probability is product of individual probabilities:
fXY(x,y)=f X (x).f Y (y)
Statistical Models
- A statistical model specifies the probability distribution for the phenomenon being modelled.
- If modelling ozone levels in a region R we would have a probability distribution for each location s (where s is a 2x1 vector of x,y coordinate pairs). Individual points can be referred to as s 1 , s 2 etc.
- The complete set of random variables may be referred to as a spatial stochastic process.
- The probability distribution for near points will probably be more similar than for distant points, so our random variables will probably not be independent.
Specifying Models(2)
- Assumptions may be expressed in general terms (e.g. a
Normal distribution, a regression model) with unspecified parameters.
- The model can be fitted using observed data to estimate
the parameters.
- After evaluating the model we may decide to change its
general form.
A Regression Model
- To illustrate, to model our ozone data we might make the following assumptions: - The random variables {Y( s ), s ∈ R} are independent; - They have the same distribution, but different means; - Their means are a simple linear function of location,
say E(Y( s )) = β 0 + β 1 s 1 + β 2 s 2 ;
- Each Y( s ) has a normal distribution about this mean
with the same variance σ^2.
- These assumptions would enable us to estimate the
parameters from the available data.