Spatial Analysis IISpatial data analysisSpatial analysis and inference

Roadmap Spatial Analysis IOutline:

What is spatial analysis?Spatial JoinsStep 1: Analysis of attributesStep 2: Preparing for analyses: working with distanceStep 3: Spatial patterns analysisStep 4: Kernel density analysisSummary

Roadmap Spatial Analysis IIOutline:

What is spatial analysis?TransformationsIntroduction to spatial interpolationClassification of spatial interpolation methodsInterpolation methodsAreal interpolationKrigingSummary

Spatial analysisA method of analysis is spatial if the results depend on the locations of the objects being analyzed

move the objects and the results changeresults are not invariant under relocation

Spatial analysis requires both the attributes and locations of objects

Spatial analysis is the crux of GIS Attribute linkages

Spatial data

P,L,A,Px

Attribute data

NOIR

Spatial analysisMuch of what we do in spatial analysis is about transformations—transforming the data from one form of presentation (e.g., points, lines, areas) into another form of presentation (e.g., density [such as a density of crimes per CT], surfaces [such as creating a TIN or a DEM from mass points], buffers, lines [such as creating contour lines from a points, a TIN or a DEM]). (Also, normalizing vars. in MCE.)

Transformations are used for simple cartographic purposes (e.g., overlays), to aid in inductive reasoning, and for more complex purposes (e.g., hot spot analyses), as part of deductive reasoning.

Transforming spatial data from one form (point, line or area) into another form (area, line or point) is a

common and important precursor to much of spatial analysis.

Why and how are some of the ways in which we transform spatial data?

TransformationsCreate a contour map

Filled contours Or a

3-D surface

Given a set of points

ArcMap’s Hot Spot Analysis

Also look at Spatial Analyst: Kernel or Point Density

TransformationsOne of the most important means of transforming data is through spatial interpolation.

Can you describe / define interpolation?

What does interpolation provide for us (as geospatial scientists)? That is, why do we interpolate (field) data?

Uses of spatial interpolationTo create isolines (or other graphics) for visualization

To calculate some property of the surface at a location ( ) where measurements weren’t taken

To change the unit of comparison when using different data structures (e.g., census tracts to planning districts)

Both physical and social applications

Field representationslattice random points regular grid

areas TIN (Voronoi) contours

What are the ways we can represent field data in a GIS?

Hexagons

Basic forms of interpolationPoint to points (e.g., random points to a regular lattice)

Points to lines (e.g., random points to contour lines)

Lines to points (e.g., digitized contours to a regular grid)

Areas to areas (e.g., census tracts to planning districts)

Points

LinesAreas

Transformations

Point to Continuous

Point to Continuous

Elevation

Rainfall

Point to Continuous

RoadmapOutline:

What is spatial analysis?TransformationsIntroduction to spatial interpolationClassification of spatial interpolation methodsInterpolation methodsAreal interpolationKrigingSummary

ConsiderationsGiven the wide variety of spatial interpolation methods, you need to consider:

Which method best fits the data you have available.Which method best fits the process associated with the data.Which method will produce the result you need.

Know your data

A classification of interpolation methods

Global versus Local

Exact versus Approximate

Stochastic versus Deterministic

Abrupt versus Smooth

Global LocalAspatial

Could you describe what each of these dichotomies might encompass?

Windows

Exact versus Approximate

Exact interpolators honour the input data points (which doesn't mean that the surface is an exact replicate of the ‘true’ surface from which the points were collected, just that the surface falls exactly on the points)

Approximate interpolators allow for uncertainty in the input data points, which allows for smoothing

Stochastic versus DeterministicStochastic methods incorporate the concept of randomness (similar to a linear regression model—a surface of ‘best fit’)

Deterministic methods do not use probability theory (they exclude randomness).

Abrupt versus Smooth

Abrupt interpolators allow for barriers (e.g., faults, fronts)

Smooth interpolators produce a smooth surface

Issues to consider

What happens to trends?

What about minima / maxima?

How does the data distribution affect the results?How are the parameters selected?

In previous lectures we’ve talked about a few of the problematic issues

that might arise in spatial analysis, although at the time spatial interpolation

wasn’t specifically mentioned.

Rule uncertainty

Roadmap

Outline:What is spatial analysis?TransformationsIntroduction to spatial interpolationClassification of spatial interpolation methodsInterpolation methodsAreal interpolationKrigingSummary

Know your data?

20 30

13

?

What is the expected value?

Process?

Interpolation methods

Exact methods of point-based interpolation

Proximal: local, exact, abrupt, deterministicbest for nominal data (aka Thiessen polygons)

Thiessen polygons are the ‘dual’ of a Delaunay triangulation

Note: The point patterns are identical.

Splines

B-Splines: “french curves” piecewise polynomials, local, exact, can be smooth, not min/max bound

Manual methods

Manual interpolation: knowledge-based, local, abrupt, tend to be exact, subjective

Often associated with geological mapping

Interpolation methods

Approximate methods of point-based interpolation

Trend surface analysis: similar to regression, global, smooth, deterministic

The simplest surface:z = a + bx + cy

What is this called?

Trend surface analysis

The graph illustrates a quadratic or second-order surface: z = a + bx + cy + dx2 + exy + fy2

Fourier methods

Fourier Series: assumes that the surface can be approximated by overlaying a series of sine and cosine waves

global, smooth, deterministic

The first four partial sums of the Fourier series

for a square wave

Distance-weighted methods

Moving Average / Distance-Weighted Average:can be exact or, more typically, approximate, local to global, smooth or abrupt the most widely used spatial interpolation method in Geography

an almost unlimited number of modifications or variations are available, including:

variations on the distance function imposing constraints on the point selection process (e.g., by direction, limiting the number of points, limiting the distance).

Moving averages

Moving averages are widely used in time series analysesThe smoother curves (dark blue on left, red on right) represent the moving average of the original time series. Note that the smoothed curve can never be higher nor lower than any of the data points.

Moving averages cannot extendacross the entire range of the data.

Why?

point iknown value zilocation xiweight wi distance di

unknown value (z to be interpolated)location x

∑∑=i

ii

ii wzwz )(x

βii dw 1=

The estimate is a weighted average

Weights decline with distance,

β is usually given a value of 2, but it can be any value > 1

Distance-weighted formulaSpatial Moving Averages (SMA)

Effects of changing parametersIDW: Changing the exponent from 2 to 4

21 ii dw = 41 ii dw =

TINs

Triangulated Irregular Networks (TINs)Not really a form of interpolation, per se, although using TINs one can create contours or regions.

Exact, local, abrupt, deterministic.

Contour placement

Although the process of 'stringing' contours between the vertices of the TIN triangles (or between two grid points) may seem unproblematic, there are some interesting issues (issues which should be addressed in any contouring exercise, but typically are never explicitly addressed).

Basically: what assumption should be made with respect to how the surface should be modeled between the known points?

Contour placement

The default for most programs, although this is likely more realistic.

RoadmapOutline:

What is spatial analysis?TransformationsIntroduction to spatial interpolationClassification of spatial interpolation methodsInterpolation methodsAreal interpolationKrigingSummary

What of areal interpolation?If the areas can be represented by a single point, and the data can be considered to be a field, you can use a point-based interpolation method (e.g., pop density of CTs).

Areal interpolation is actually a complex process. Polygon overlays , and using the proportional areas as weights, is a typical

approach (but one that is not reversible nor volume preserving—total pop after ≠ total pop before).Pycnophylactic interpolation is a reversible, volume preserving method (see

lecture notes for a link to a video that describes this method in detail).

Polygon overlays

The problem with (nonreversible) polygon overlays:

The fundamental assumption is that the field has a homogeneous distribution throughout the zones.

Note that the totals assigned to the two polygons in the bottom row

do not equal the values within each polygon in the top row.

Each area is split into 2 equal parts,

assuming equal pop in each part.

Pycnophylactic interpolation

ArcGIS’s Areal Interpolation Method

Arealinterpolation

RoadmapOutline:

What is spatial analysis?TransformationsIntroduction to spatial interpolationClassification of spatial interpolation methodsInterpolation methodsAreal interpolationKrigingSummary

Geostatistical methods: KrigingKriging

stochastic, exact, smooth or abrupt, global or local

Natural data are difficult to model using smooth functions because naturally-occurring random fluctuations and measurement error combine to cause irregularities in sampled data values.

Kriging was developed to model those stochastic concepts.

It is based on the concept of a regionalized variable that has three components:

Components of a Regionalized Variable

data

STRUCTURAL – This may be represented by the

mean or a constant trend.

SPATIALLY CORRELATED – Data often exhibit positive spatial

correlations.

RANDOM NOISE – Measurement errors, other errors,

random fluctuations.

Topography is a reflection of many processesoperating at different scales; with Kriging we hope to develop models of some of those processes.

Components of a Regionalized Variable

The random noise component (non-fitted)

The spatially correlated component

The structural component (e.g., a linear trend)

This is what we areattempting to model.

Kriging

Kriging is implemented using a semi-variogram

There are many different varieties of kriging (e.g, ordinary, universal, simple, indicator), and selecting the appropriate one requires careful consideration of the data.

ArcGIS's help file--look up the term kriging—provides a lot of information on the various types of kriging (and co-kriging) that are commonly used in spatial analysis.

ArcGIS’s tutorial for the Geostatistical Analyst is also very informative (in particular consider the Geostatistical Wizard)

Kriging

The semivariogram is based on modeling the (squared) differences in the z-values as a function of the distances between the known points.

h

KrigingThe first step in ordinary kriging is to construct a semivariogram from the points being interpolated.

A semivariogram consists of two parts: an experimental semivariogram (the data) a model semivariogram (the math).

The experimental semivariogram is found by calculating the variance (g) of each point in the set with respect to each of the other points and plotting the (semi)variances versus distance (h) between the points.

Semivariogram

Variance between points is represented over space. The above shows how variance between point values tends to increase as distance between points increases.

The values of the points could be rainfall, elevation, demographic continuous data, ozone levels, percentage of gold in a sample…

N= number of pairsf1 = Headf2 = Tail

h = distance (0 -> range)

SemivariogramThe semivariogram is used to explore a data set visually and to estimate at what distance the spatial autocorrelation becomes insignificant (i.e., the range).

SemivariogramNugget represents subgrid scale variation that cannot be estimated by reasons of the sampling grid spacing or measurement error.

Range is a measure of the spatial autocorrelation between data points, represented in terms of the lag distance. It represents the distance at which the spatial autocorrelation is at its global value.

SemivariogramSill is the value of the semi-variance as the lag tends towards infinity – in non-standardized data it is equal to the total variance of the dataset.

Lag refers to distances (e.g., 1250m) that typically have ‘bands’ associated with them (e.g., +/- 250m). Distance may be a direct measure or a transformation of distance (e.g., log(1000 m)).

N= number of pairsf1 = Headf2 = Tail

h = distance (0 -> range)

Kriging

This is an example of a semivariogram produced using ArcGIS's Geostatistical Analyst.

KrigingOne of the very useful outputs from a kriging

analysis is the uncertainty surface that can be generated--we can answer the question: "How good are the predictions?"

Using some of the data that we used in lab 2 (where you created a TIN), I created an ordinary kriged map and a map showing the standard error of the predictions (and a TIN for comparison).

Kriging Prediction standard error mapshowing data points

TINOrdinary krigedprediction map

IDW versus kriging

SummaryInterpolation is a very important process in GIS, and in particular in spatial analysis. ArcGIS's help files provide a lot of useful information.

It is important to know, first, about your data (how was it collected, the spatial distribution of the collection points, the process responsible for creating the 'field' you are mapping) and, second, about the spatial interpolation method (its assumptions, faults and fine points).

Know your data