Spatial analysis. 0 move the objects and the results change

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2 0 Outline: Roadmap 0 What is spatial analysis? 0 Transformations 0 Introduction to spatial interpolation 0 Classification of spatial interpolation methods 0 Interpolation methods 0 Areal interpolation 0 Kriging 0 Summary

3 Spatial analysis 0 A method of analysis is spatial if the results depend on the locations of the objects being analyzed 0 move the objects and the results change 0 results are not invariant under relocation 0 Spatial analysis requires both the attributes and locations of objects 0 Spatial analysis is the crux of GIS Attribute linkages Spatial data Attribute data P,L,A,P x NOIR

4 Spatial analysis 0 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 line or area) into another form (area, line or another form of presentation (e.g., density [such as a density point) of is crimes a common per CT], and surfaces important [such as element creating of a TIN or a DEM from spatial mass points], analysis. lines [such as creating contour lines from a points, a TIN or a DEM]). Transforming spatial data from one form (point, Why and how are some of the ways in which we 0 Transformations are used for simple cartographic purposes (e.g., transform overlays) and spatial for more data? complex purposes (e.g., hot spot analyses).

5 Transformations Given a set of points Create a contour map ArcMap s Hot Spot Analysis Filled contours Or a 3-D surface

6 Transformations 0 One of the most important means of transforming data is through spatial interpolation. 0 Can you describe / define interpolation? 0 What does interpolation provide for us (as geospatial scientists)? That is, why do we interpolate data?

7 Uses of spatial interpolation 0 To create isolines (or other graphics) for visualization 0 To calculate some property of the surface at a location ( ) where measurements weren t taken 0 To change the unit of comparison when using different data structures (e.g., census tracts to planning districts) 0 Both physical and social applications

8 Six field representations lattice random points regular grid What are the ways we can represent field data in a GIS? areas TIN contours

9 Basic forms of interpolation 0 Point to points (e.g., random points to a regular grid) 0 Points to lines (e.g., random points to contour lines) 0 Lines to points (e.g., digitized contours to a regular grid) 0 Areas to areas (e.g., census tracts to planning districts) Points Transformations Areas Lines

10 0 Outline: Roadmap 0 What is spatial analysis? 0 Transformations 0 Introduction to spatial interpolation 0 Classification of spatial interpolation methods 0 Interpolation methods 0 Areal interpolation 0 Kriging 0 Summary

11 Considerations 0 Given the wide variety of spatial interpolation methods, you need to consider: 0 Which method best fits the data you have available. 0 Which method best fits the process associated with the data. 0 Which method will produce the result you need. Know your data

12 0 Global versus Local 0 Exact versus Approximate 0 Stochastic versus Deterministic 0 Abrupt versus Smooth Could you describe what each of these dichotomies might encompass? Global Aspatial Local A classification of interpolation methods

13 0 Exact interpolators honour the input data points (which doesn't mean that the surface is 'exact') 0 Approximate interpolators allow for uncertainty in the input data points, which allows for smoothing Exact versus Approximate

14 Stochastic versus Deterministic 0 Stochastic methods incorporate the concept of randomness (similar to a linear regression model a surface of best fit ) 0 Deterministic methods do not use probability theory (they exclude randomness).

15 0 Abrupt interpolators allow for barriers (e.g., faults, fronts) 0 Smooth interpolators produce a smooth surface Abrupt versus Smooth

16 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. How are the parameters selected? How does the data distribution affect the results? Issues to consider

17 Roadmap 0 Outline: 0 What is spatial analysis? 0 Transformations 0 Introduction to spatial interpolation 0 Classification of spatial interpolation methods 0 Interpolation methods 0 Areal interpolation 0 Kriging 0 Summary

18 0 Exact methods of point-based interpolation 0 Proximal: local, exact, abrupt, deterministic 0 best for nominal data (aka Thiessen polygons) Interpolation methods Thiessen polygons are the dual of a Delaunay triangulation

19 0 B-Splines: french curves 0 piecewise polynomials, local, exact, can be smooth, not min/max bound Splines

20 0 Manual interpolation: knowledge-based, local, abrupt, tend to be exact, subjective 0 Often associated with geological mapping Manual methods

21 0 Approximate methods of point-based interpolation 0 Trend surface analysis: similar to regression, global, smooth, deterministic 0 The simplest surface: z = a + bx + cy Interpolation methods

22 0The graph illustrates a quadratic or second-order surface: z = a + bx + cy + dx 2 + exy + fy 2 A simple review of how to use TSA Trend surface analysis

23 0 Fourier Series: assumes that the surface can be approximated by overlaying a series of sine and cosine waves 0 global, smooth, deterministic Fourier methods The first four partial sums of the Fourier series for a square wave

24 0 Moving Average / Distance-Weighted Average: 0 can be exact or, more typically, approximate, local to global, smooth or abrupt 0 the most widely used spatial interpolation method in Geography 0 an almost unlimited number of modifications or variations are available, including: 0 variations on the distance function 0 imposing constraints on the point selection process (e.g., by direction, limiting the number of points, limiting the distance). Distance-weighted methods

25 0 Moving averages are widely used in time series analyses 0 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

26 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) wi z i w 1 i 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

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

28 0 Triangulated Irregular Networks (TINs) 0 Not really a form of interpolation, per se, although using TINs one can create contours or regions. 0 Exact, local, abrupt, deterministic. TINs

29 0 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). 0 Basically: what assumption should be made with respect to how the surface should be modeled between the known points? Contour placement

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

31 0 Outline: Roadmap 0 What is spatial analysis? 0 Transformations 0 Introduction to spatial interpolation 0 Classification of spatial interpolation methods 0 Interpolation methods 0 Areal interpolation 0 Kriging 0 Summary

32 What of areal interpolation? 0 If the areas can be represented by a single point, and the data can be considered to be a field, you can use a pointbased interpolation method (e.g., pop density of CTs). 0 Areal interpolation is actually a complex process. 0 Polygon overlays, and using the proportional areas as weights, is a typical approach (but one that is not reversible nor volume preserving). 0 Pycnophylactic interpolation is a reversible, volume preserving method (see lecture notes for a link to a video that describes this method in detail). (An ArcGIS script.)

33 0 The problem with (nonreversible) polygon overlays: 0 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.

34 Pycnophylactic interpolation

35 0 Outline: Roadmap 0 What is spatial analysis? 0 Transformations 0 Introduction to spatial interpolation 0 Classification of spatial interpolation methods 0 Interpolation methods 0 Areal interpolation 0 Kriging 0 Summary

36 Geostatistical methods: Kriging 0 Kriging 0 stochastic, exact, smooth or abrupt, global or local 0 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. 0 Kriging was developed to model those stochastic concepts. 0 It is based on the concept of a regionalized variable that has three components:

37 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. Components of a Regionalized Variable

38 The structural component (e.g., a linear trend) The spatially correlated component The random noise component (non-fitted) Components of a Regionalized Variable

39 0 Kriging is implemented using a semi-variogram 0 There are many different varieties of kriging, and selecting the appropriate one requires careful consideration of the data. 0 ArcGIS's help file--look up the term kriging provides a lot of information on the various types of kriging that are commonly used in spatial analysis. 0 ArcGIS s tutorial for the Geostatistical Analyst is also very informative (in particular consider the Geostatistical Wizard) Kriging

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

41 0 This is an example of a semivariogram produced using ArcGIS's Geostatistical Analyst. Kriging

42 Kriging 0One 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?" 0Using 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).

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

44 IDW versus kriging

45 Summary 0 Interpolation is a very important process in GIS, and in particular in spatial analysis. ArcGIS's help files provide a lot of useful information. 0 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