Spatial Analysis II. Spatial data analysis Spatial analysis and inference

Transcription

1 Spatial Analysis II Spatial data analysis Spatial analysis and inference

2 Roadmap Spatial Analysis I Outline: What is spatial analysis? Spatial Joins Step 1: Analysis of attributes Step 2: Preparing for analyses: working with distance Step 3: Spatial patterns analysis Step 4: Kernel density analysis Summary

3 Roadmap Spatial Analysis II Outline: What is spatial analysis? Transformations Introduction to spatial interpolation Classification of spatial interpolation methods Interpolation methods Areal interpolation Kriging Summary

4 Spatial analysis A method of analysis is spatial if the results depend on the locations of the objects being analyzed move the objects and the results change results 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 Attribute data P,L,A,P x NOIR

5 Spatial analysis Much 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 common a points, and a TIN important a DEM]). precursor (Also, normalizing to much vars. of in MCE.) Transforming spatial data from one form (point, line or area) into another form (area, line or point) is a spatial analysis. Transformations are used for simple cartographic purposes (e.g., overlays), to aid in inductive reasoning, and for more complex Why and purposes how are (e.g., some hot spot of the analyses), ways in as which part of we deductive reasoning. transform spatial data?

6 Transformations Given a set of points Create a contour map ArcMap s Hot Spot Analysis Filled contours Or a 3-D surface Also look at Spatial Analyst: Kernel or Point Density

7 Transformations One 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?

8 Uses of spatial interpolation To 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

9 Field representations lattice random points regular grid What are the ways we can represent field data in a GIS? areas TIN (Voronoi) contours Hexagons

10 Basic forms of interpolation Point 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 Transformations Areas Lines

11 Point to Continuous

12 Point to Continuous Elevation Rainfall

13 Point to Continuous

14 Roadmap Outline: What is spatial analysis? Transformations Introduction to spatial interpolation Classification of spatial interpolation methods Interpolation methods Areal interpolation Kriging Summary

15 Considerations Given 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

16 Global versus Local Exact versus Approximate Stochastic versus Deterministic Abrupt versus Smooth Could you describe what each of these dichotomies might encompass? Global Aspatial Local Windows A classification of interpolation methods

17 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 Exact versus Approximate

18 Stochastic versus Deterministic Stochastic 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).

19 Abrupt interpolators allow for barriers (e.g., faults, fronts) Smooth interpolators produce a smooth surface Abrupt versus Smooth

20 What happens to trends? What about minima / maxima? In previous lectures we ve talked about a few of the problematic issues that might arise, although at the time spatial interpolation wasn t specifically mentioned. Rule uncertainty How are the parameters selected? How does the data distribution affect the results? Issues to consider

21 Roadmap Outline: What is spatial analysis? Transformations Introduction to spatial interpolation Classification of spatial interpolation methods Interpolation methods Areal interpolation Kriging Summary

22 Know your data? 20 30? Process? 13 What is the expected value?

23 Exact methods of point-based interpolation Note: The point patterns are identical. Proximal: local, exact, abrupt, deterministic best for nominal data (aka Thiessen polygons) Interpolation methods Thiessen polygons are the dual of a Delaunay triangulation

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

25 Manual interpolation: knowledge-based, local, abrupt, tend to be exact, subjective Often associated with geological mapping Manual methods

26 Approximate methods of point-based interpolation Trend surface analysis: similar to regression, global, smooth, deterministic The simplest surface: z = a + bx + cy Interpolation methods

27 The graph illustrates a quadratic or second-order surface: z = a + bx + cy + dx 2 + exy + fy 2 Trend surface analysis

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29 Fourier Series: assumes that the surface can be approximated by overlaying a series of sine and cosine waves global, smooth, deterministic Fourier methods The first four partial sums of the Fourier series for a square wave

30 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). Distance-weighted methods

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

32 point i known value z i location x i weight w i distance d i unknown value location x (z to be interpolated) z (x) = i w = 1 w z i i β i d i i w i Weights decline with distance, β is usually given a value of 2 The estimate is a weighted average Distance-weighted formula Spatial Moving Averages (SMA)

33 IDW: Changing the exponent from 2 to 4 Effects of changing parameters w = i 24 1 d i

34 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. TINs

35 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

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

37 Roadmap Outline: What is spatial analysis? Transformations Introduction to spatial interpolation Classification of spatial interpolation methods Interpolation methods Areal interpolation Kriging Summary

38 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).

39 The problem with (nonreversible) polygon overlays: The fundamental assumption is that the field has a homogeneous distribution throughout the zones. Each area is split into 2 equal parts, assuming equal pop in each part. Polygon overlays 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.

40 Pycnophylactic interpolation

41 ArcGIS s Areal Interpolation Method

42 Areal interpolation

43 Roadmap Outline: What is spatial analysis? Transformations Introduction to spatial interpolation Classification of spatial interpolation methods Interpolation methods Areal interpolation Kriging Summary

44 Geostatistical methods: Kriging Kriging 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:

45 STRUCTURAL This may be represented by the mean or a constant trend. SPATIALLY CORRELATED Data often exhibit positive spatial correlations. data RANDOM NOISE Measurement errors, other errors, random fluctuations. Topography is a reflection of many processes operating at different scales; with Kriging we hope to develop models of some of those processes. Components of a Regionalized Variable

46 This is what we are attempting to model. The structural component (e.g., a linear trend) The spatially correlated component The random noise component (non-fitted) Components of a Regionalized Variable

47 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

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

49 Kriging The 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.

50 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

51 N= number of pairs f1 = Head f2 = Tail

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

53 Semivariogram Nugget 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. The lag represents the distance at which the spatial autocorrelation is at its global value.

54 Semivariogram Sill is the value of the semivariance 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)).

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

56 Kriging One 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).

57 TIN Ordinary kriged prediction map Kriging Prediction standard error map showing data points

58 IDW versus kriging

59 Summary Interpolation 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

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