1. Modeling Spatial Phenomena
Lecture 4
SIE 512 Spatial Analysis
Fall 2017

2. Modeling Spatial Phenomena
We want to develop models for random variables distributed in space, and possibly evolving in time.
A stochastic process operating in a region of space
Referred to as a spatial stochastic process or a spatial random field.
Key Objective: investigate statistical models for spatial stochastic processes.

3. Modeling Spatial Phenomena
A statistical model for a stochastic phenomena consists of specifying a probability distribution for a random variable.
For a simple phenomena like the toss of dice we know the complete behavior
y=(1,..6)
For complex environmental processes operating in space and evolving in time, the model specification becomes more difficult.

4. Modeling Geographical Phenomena
Data records in a spatial data set are identified as having a location specified by an (x,y) coordinate pair
Points from Goodchild
Positional uncertainty
It is impossible to measure positions on the earth surface exactly.
Geometry underlying all geographic information is approximate.
Object centered approach: limited accuracy assumes that the true position lies somewhere within a given distance of a points apparent position.

5. Modeling Geographical Phenomena
Points from Goodchild
Spatial dependence is endemic:
A variable has spatial dependence if correlations exist between values at distinct points
Similarity between values is a function of distance
Tobler’s first law of geography: “All things are similar but nearby things are more similar than distant things”.
Spatial autocorrelation: the coincidence of value similarity with location similarity. (Moran’s I, Geary’s C, variogram, covariogram)

6. Modeling Geographical Phenomena
Points from Goodchild
Geographic space is heterogeneous:
How to balance nomothetic view with idiographic view:
search for principles that are true everywhere – versus study of places that are unique
Nomothetic view assumes some level of spatial homogeneity
Earth’s surface is highly variable
Any particular study area represents a small portion of variation
Changing the bounds of a study area can produce different results

7. Modeling Geographical Phenomena
Points from Goodchild
Geographic space is heterogeneous:
Any general laws relevant to the geographic world are likely to be of limited predictive power.
Unexplained variation in a law is likely to be a geographic variable.
The law might apply everywhere but a and b vary geographically
A law
As a linear function

8. Modeling Geographical Phenomena
Points from Goodchild
Geographic information as derivative and uncertain:
Geographic information is based on some set of measurements but processed or derived through various operations to construct “maps” e.g a soils map
Through this derivation the potential exists for multiple maps or realizations.
Uncertainty concept – multiple databases or maps could arise through the derivation process and each may be equally likely

9. Modeling Geographical Phenomena
Points from Goodchild
Geographic attributes are scale specific:
Examples:
Computation of slope is scale specific
Land use classification is scale specific
Modifiable areal unit problem (MAUP)

10. Modeling Spatial Phenomena
A random variable measured at a set of locations s is a spatial random field or spatial stochastic process
Any particular set of measurements of a spatial random field are called a realization of the spatial random field.
Spatial random field
A set of possibly non-independent random variables

11. A multi-dimensional sample of one

12. Modeling Spatial Phenomena
It is often convenient to model a spatial random field as the sum of a collection of components.
Data = Structure + Error
A structural or deterministic component – consisting of a trend or large scale variation
A spatially autocorrelated random process
An uncorrelated random variable representing uncorrelated random variation and measurement error

13. Modeling Spatial Phenomena
The behavior of spatial phenomena modeled as a combination of first and second order effects
First Order effects
Describes large scale variation in the mean due to location or other explanatory variables.
Second order effects
Spatial dependence in a process - correlation in the deviations of values of the process from the mean

14. Modeling Spatial Phenomena
First Order effects
The mean function (the stochastic expectation) characterizes trends and systematic structures in space/time.
The covariance function on the residuals
C(si, sj) = E((Z(si)-m(si))(Z(sj)-m(sj)))
expresses correlations and dependencies – small scale interactions in space.
Second order effects

15. Spatial Phenomena
Because of combined first and second order effects – the assumption of independently distributed random spatial variables is usually violated
We typically need to replace the independence assumption with an alternative that incorporates covariance structure to accommodate the second order effects.
The second order component is typically modeled as a stationary spatial process

16. Stationarity Stationarity is a form of location invariance
Stationarity is the quality of a process in which the statistical parameters (e.g. mean and standard deviation) of the process do not change with space or time.
Some degree of stationarity is typically assumed to make inferences about the data

17. Stationarity Second order stationarity
A spatial process is said to be stationary if its statistical properties are independent of absolute location in R.
Implies the mean E(Z(s)), and Variance Var(Z(s)) are constant in R and do not depend on location.
Second order stationarity
The covariance COV(Z(si), Z(sj)) between any two locations depends only on the relative location of the sites, the distance and direction between them, and not the absolute location in R.

18. Non Stationarity
If the process drifts over R, i.e the mean, variance, or covariance structure changes the process is said to be non-stationary. R
Increasing west to east
Non-stationarity in the mean N

19. Intrinsic Stationarity
Increments are stationary
A process
is intrinsically stationary if
and

20. Second order stationarity
The covariance COV(Y(si), Y(sj)) between any two locations depends only on the relative location of the sites, the distance and direction between them, and not the absolute location in R. s1 s2 s7 s8 s3 s4 s5 s6 s9
s10

21. Isotropic Anisotropic
Refers to a spatial process that evolves the same in all directions – direction independent
We say a process is isotropic if in addition to being stationary, the covariance depends only on distance between locations and not the direction in which they are separated.
Anisotropic
A spatial process in which the correlation and covariance differs with direction
Most methods assume spatial correlation is isotropic

22. Stationary and Isotropic
Second order stationarity and isotropy s1 s2 s7 s8 s3 s4 s5 s6 s9
s10
Depends only on relative distance not direction between observations

23. Modeling Spatial Processes
For spatial modeling, the general approach is to assume non-stationarity (heterogenity) in the mean but stationarity in the second order effects.
Without an assumption of stationarity in the covariance, it becomes difficult to fit a spatial model – too many parameters to estimate and not enough data
Assumptions
Heterogeneity in the mean
Stationary covariance in the deviations from the mean

24. Modeling Spatial Phenomena
In order to separate data into components we first estimate the trend term and subtract it from the data.
The trend is usually estimated using a linear regression model or the median polish technique.
To use a linear regression model we use a variation called a trend model in which predictor variables are functions of the coordinates x, and y

25. Specifying the Spatial Model
For complex spatial phenomena we rarely have all the theoretical knowledge to develop an appropriate statistical model.
So we rely on observation data to help specify a model
Assume a set of observations { z1, z2, z3, … zn } taken at a set of locations {s1, s2, s3, … sn } in Rd
The set is a realization of a spatial process
It is simply one observation from the joint probability distribution of our spatial random variables
How to construct a model from a sample of one?

26. Specifying the Spatial Model
Approach to model specification involves a combination of data plus some assumptions on the nature of the phenomena
Using some general knowledge on a phenomena we can specify a general form of a probability distribution with certain parameters left unspecified.
This general form of the model is then refined by fitting it to the observed data

27. Specifying the Spatial Model
Suppose we have some reason to propose a linear regression model for a spatial random field
By proposing this model we make certain assumptions:
1. The random variables are independent.
2. Their probability distributions only differ in their mean value
3. The mean value is a simple linear function of location
4. Each Z(s) has a normal distribution about this mean and a constant variance
We can say the probability distribution is thus

28. Specifying the Spatial Model
The assumption of independence in the model removes the need to specify a joint probability distribution for every subset of the random variables Z(s).
The distributions just differ in their means as a function of the parameters
We now use data to estimate these parameters for our proposed model
How do we estimate the parameters?

29. Parameter Estimation Method of Moments Estimators
Equates the sample moments of the data with the moments of the probability distribution.
The rth moment of the distribution of a random variable X is denoted as:
First moment - mean

30. Maximum Likelihood Estimation
Method of maximum likelihood estimation is generally considered to be the most robust of the parameter estimators
The aim of maximum likelihood estimation is to find the parameter value(s) that make the joint probability of the observed data most probable.
If the probability of a event X dependent on a model with parameter p is written
P ( X | p )
then we can write the likelihood as
L ( p | X )
that is, the likelihood of the parameter given the data.

31. Maximum Likelihood Estimation
Say we toss a coin 100 times and observe 56 heads and 44 tails. Instead of assuming that p is 0.5, we want to find the MLE for p based on our observed data. Then we could ask whether or not this value differs significantly from 0.50.
How do we do this?
We find the value for p that makes the observed data most likely.
The observed data are considered fixed
They will be constants that are plugged into our proposed model - a binomial probability model :
N = 100 (total number of tosses)
n = 56 (total number of heads)

32. p L
Assume p takes some values between [0,1]

33. If we graph these data across the full range of possible values for p [0,1] we see the following likelihood surface.

34. Maximum Likelihood Estimation
Suppose we have a random sample of m forest plots.
In each forest plot we count the number of trees above a certain size diameter. The number of trees observed in a given plot is a discrete random variable, X.
In our random sample then we observe the values of m random variables, X1, X2, ... , Xm, one for each plot.
We observe x1 trees in plot 1, x2 trees on plot 2, etc.
We can describe a probability for this as

35. Maximum Likelihood Estimation
This expresses the joint probability
Which we can also write as
If events A and B are independent

36. Maximum Likelihood Estimation
We assume a Poisson distribution is a sensible model for the counts of trees in plots. So each of the probability terms in the product is a Poisson probability.
Plugging these in and regrouping terms yields the following.

37. Maximum Likelihood Estimation
Using notation for probability functions with parameters we can write
This is the probability of our data. If we knew λ we could calculate the probability of obtaining any set of values x1, x2, ... , xm.
For fixed λ if we summed this expression over all possible values of x1, x2, ... , xm we would get 1.

38. Maximum Likelihood Estimation
When we look at the function from the perspective of considering the observed values x1, x2, ..., xn to be fixed "parameters" and θ as the function's variables allowed to vary freely, the distribution function is the likelihood.
The likelihood function is still the joint probability function for our data under the assumed probability model but by another name.

39. Maximum Likelihood Estimation
If X is a random variable with pdf:
where θ1, θ2,..., θk are k unknown parameters which need to be estimated, with N independent observations, x1, x2,... xN
The likelihood function is given by:
i = 1, 2, ..., N
The log likelihood function is given by:

40. Maximum Likelihood Estimation
Starting with a generic probability model and proceeding to our independent Poisson model, the loglikelihood takes the following form.

41. Maximum Likelihood Estimation
Going back to the example of a proposed linear regression model for observed spatial data ( z1, z2 , z3 , … zn ) the joint probability distribution or density is
Where q represents the parameters b of the proposed model
We want to compute the likelihood: probability of model parameters given the data

42. Maximum Likelihood Estimation
The probability distribution of our proposed model was

43. Maximum Likelihood Estimation
For models with assumed independently distributed random variables that are normally distributed with constant variance, maximum likelihood reduces to ordinary least squares.
Maximum likelihood estimation provides standard errors for the estimated parameters.
Related to how peaked the likelihood function is at its maximum. Sharp peaks provide more reliable estimates – smaller standard errors.

44. Modeling Spatial Phenomena
Recall a model for a spatial random field as:
A structural or deterministic component – consisting of a trend or large scale variation
A spatially autocorrelated random process
An uncorrelated random variable representing uncorrelated random variation and measurement error
ML and OLS do not account for this term