ENGRG Introduction to GIS

Transcription

1 ENGRG Introduction to GIS Michael Piasecki October 13, 2017 Lecture 06: Spatial Analysis

2 Outline Today Concepts What is spatial interpolation Why is necessary Sample of interpolation (size and pattern) Interpolation methods Thiessen polygon Local average Inverse Distance Weighted Spline Kriging 10/11/2017 ENGRG Intro to GIS 2

3 Spatial Interpolation Spatial interpolation is the process of using points with known values to estimate values at other points. These points with known values are called known points, control points, sampled points, or observations. Interpolation only works where values are spatially dependant, or spatially auto correlated, that is, where nearby location tend to have similar Z values. Examples of spatially auto correlated features: elevation, property value, crime levels, precipitation Cannot use interpolation where values are not spatially auto correlated 10/11/2017 ENGRG Intro to GIS 3

4 Why Spatial Interpolation? GIS require data at all points (subject to spatial resolution of question). Impossible to measure continuous variables at all points. Infinite number of points Finite amount of time and other resources Inaccessible points It is necessary to evaluate new values Evaluate new value if cell size is changed Replace missing or erroneous data (noise data) 10/11/2017 ENGRG Intro to GIS 4

5 Principles of Interpolation Problem: Impossible ever to measure any continuous variable at all points Objective: Estimate unknown points by similarity and proximity with known points Basic Principle: Points close together in space are more likely to have similar values than points far apart (Tobler's First Law of Geography, 1970, A computer model simulating urban growth in the Detroit region, Economic Geography 46 (2): ) Basis: Sample of Known Points Unknown points 10/11/2017 ENGRG Intro to GIS 5

6 First Element of Interpolation: The Known Points Sampling Sample size How many? Controllable (perhaps) Limits: resources, diminishing returns Sampling pattern Where? Know the study area Choice may introduce errors, artifacts, outliers, 10/11/2017 ENGRG Intro to GIS 6

7 Sample Size Calculation One of the most common questions any statistician gets asked is How large a sample size do I need?" Researchers are often surprised to find out that the answer depends on a number of factors and they have to give the statistician some information before they can get an answer! The answers are essentially different depending on whether the study is a survey designed to find out the proportion of something, or is designed to find a sample mean. We consider these cases separately. 10/11/2017 ENGRG Intro to GIS 7

8 Sample Size Calculation: Example How large a sample size do I need?" Example: A professor in the Sociology department is trying to determine the proportion of students who support gay marriage. She asks, How large a sample size do I need?" To answer a question like this we need to ask the researcher certain questions, such as: 1. How accurately do you need the answer? 2. What level of confidence do you intend to use? 3. (possibly) What is your current estimate of the proportion of UNC students who support gay marriage?. 10/11/2017 ENGRG Intro to GIS 8

9 Sample Size Calculation 10/11/2017 ENGRG Intro to GIS 9

10 Sample Size Calculation value Standard Error formula. Largest for p = Solve for 1 where p = prior judgment of correct value of p if not known: use 0.5 > most conservative n = population size z = critical value (e.g for 95% confidence interval, from Normal distribution) ME= Margin of Error 10/11/2017 ENGRG Intro to GIS 10

11 Critical Z values 10/11/2017 ENGRG Intro to GIS 11

12 Sample Size Calculation 10/11/2017 ENGRG Intro to GIS 12

13 Sample Size Calculation 10/11/2017 ENGRG Intro to GIS 13

14 Sample Size Calculation 10/11/2017 ENGRG Intro to GIS 14

15 Sample Size Calculation 10/11/2017 ENGRG Intro to GIS 15

16 Online Sampling Size Calculators 10/11/2017 ENGRG Intro to GIS 16

17 Sample Patterns Systematic pattern Random pattern Cluster pattern Adaptive pattern 10/11/2017 ENGRG Intro to GIS 17

18 Systematic Sampling Spaced uniformly in x and y Advantages: Easy to conceive No subjective judgment Easy to lay out field sample pattern Disadvantages: Not statistically representative: all areas are represented the same, but some areas have more variation. Travel, visiting each site may be difficult or impossible Bias: coincidence of sample interval with data patterns?? 10/11/2017 ENGRG Intro to GIS 18

19 Random Sample Random Sampling Random X (easting), Y (northing) location (calculate in spreadsheet ) Advantages Statistically robust no bias in sampling Unlikely to match pattern on landscape no aliasing. Easy to conceive and lay out Not subjective Disadvantages Does not distribute samples in areas of high variation Travel, visiting each site may be difficult or impossible 10/11/2017 ENGRG Intro to GIS 19

20 Cluster Sampling Cluster Sampling Group samples around centers Centers can be located systematically or randomly (or for convenience, if statistically valid) Samples within a cluster can be located randomly, systematically, or otherwise. Advantage Reduced travel time Take advantage of existing access Disadvantages: For cluster center placement, same as for systematic and random point placement 10/11/2017 ENGRG Intro to GIS 20

21 Adaptive Patterns Adaptive Sampling higher sampling density where important variable(s) have greatest variation. Advantages Sampling efficiency optimized Logistics can be planned in many cases Disadvantages Preliminary sampling may be necessary, e.g. soil chemistry Significant prep time Subjective 10/11/2017 ENGRG Intro to GIS 21

22 The Second Element of Interpolation: The Unknown Points All interpolation methods use sampled values and positions to model unsampled points Deterministic interpolation techniques create surfaces from measured points, based on either the extent of similarity (e.g., Inverse Distance Weighted) or a degree of smoothing (e.g., Trend Surface Analysis). Geostatistical interpolation techniques (e.g., Kriging) are based on statistics and are used for more advanced prediction surface modeling, which also includes error or uncertainty of predictions. NOTE: Different methods will (almost always) produce different results. 10/11/2017 ENGRG Intro to GIS 22

23 Classification of Interpolation Local or global Exact or approximate (inexact) Deterministic or stochastic Gradual or abrupt 10/11/2017 ENGRG Intro to GIS 23

24 Global vs. Local A global interpolation method uses every known point available to estimate an unknown value. A local interpolation method, on the other hand, uses a sample of known points to estimate an unknown value. 10/11/2017 ENGRG Intro to GIS 24

25 Exact vs. Inexact Exact interpolation predicts a value at the point location that is the same as its known value. In other words, exact interpolation generates a surface that passes through the control points. Inexact interpolation or approximate interpolation predicts a value at the point location that differs from its known value. 10/11/2017 ENGRG Intro to GIS 25

26 Deterministic vs. Stochastic A deterministic interpolation method provides no assessment of errors with predicted values. Estimated value +- 2 A stochastic interpolation method, on the other hand, offers assessment of prediction errors with estimated variances. 10/11/2017 ENGRG Intro to GIS 26

27 Gradual or Abrupt Gradual methods: produce smooth surface between data points appropriate for interpolating data of low local variability Abrupt methods: produce surfaces with a stepped appearance appropriate for interpolating data of high local variability or data with discontinuities 10/11/2017 ENGRG Intro to GIS 27

28 Interpolation Methods Different methods will produce different results with same input data. No single method is more accurate than others for all situations. Accuracy may be determined by comparison with a second set of withheld samples for accuracy checking. The researcher should select the method based on knowledge of the study area, phenomena of interest, and available resources. 10/11/2017 ENGRG Intro to GIS 28

29 Choose Interpolation Methods The best interpolation method for any given application depends on The characteristics of the variable to be estimated the cost of sampling available resources accuracy requirements of the user Relative performance of interpolators has been determined for some variables in some locations This is truly a matter of trying out different methods, once you have some data. 10/11/2017 ENGRG Intro to GIS 29

30 The Basic Equation for Spatial Interpolation z o ' n i 1 w i z Where: Z 0 is the attribute value to be predicted at the unvisited site Z i is the attribute value at the i point of the nearby locations w i is the weight assigned to the attribute at point i, w i should sum up to 1 (to be unbiased) n is the total number of nearby locations involved Key issues: 1) How many nearby points should we include for a given un visited site? 2) How to select these nearby points? 3) How to allocate the weight for each nearby point? i 10/11/2017 ENGRG Intro to GIS 30

31 Interpolation methods Most GIS packages offer a number of methods Typical methods are: Thiessen polygons Spatial moving average Trend Surfaces Inverse Distance Weighted Kriging 10/11/2017 ENGRG Intro to GIS 31

32 Nearest Neighbor Interpolation (Thiessen polygons) Also called Thiessen Polygons, Voronoi polygons, Dirichlet cells Value assigned from nearest sample Value assigned across unusually large cells may be questionable; watch for them Edge effects 10/11/2017 ENGRG Intro to GIS 32

33 Thiessen polygon construction 10/11/2017 ENGRG Intro to GIS 33

34 Polygon Models: Thiessen (Voronoi polygons, Dirichlet cells) Polygons ò ò ò ò ò ò Thiessen Polygons: Areas of Influence An Exact Interpolator : Original Values are Preserved 10/11/2017 ENGRG Intro to GIS 34

35 Question What categories does the Thiessen polygon method fall into: exact or approximate? deterministic or stochastic? gradual or abrupt? local or global? For what could or could NOT it be used for? 10/11/2017 ENGRG Intro to GIS 35

36 Local Average Method (Spatial Moving Average) Calculates new value of each location based on range of values associated with neighbouring points Neighbourhood determined by a filter size, shape and character of filter Avoids shaky assignments at extreme distances Potential problem: Assigns zero to sites with no samples within radius Not exact interpolator 10/11/2017 ENGRG Intro to GIS 36

37 Spatial Moving Average 10/11/2017 ENGRG Intro to GIS 37

38 Local Average: Fixed Radius Raster grid specified Sample points within a fixed, search radius with center in each cell are averaged. 10/11/2017 ENGRG Intro to GIS 38

39 Question What categories does the Spatial Moving Average method fall into: exact or approximate? deterministic or stochastic? gradual or abrupt? local or global? What could it be used for? 10/11/2017 ENGRG Intro to GIS 39

40 Inverse Distance Weighting (IDW) z j s i 1 s i 1 zi d 1 d n ij n ij Exact Interpolator: Value calculated directly from three (usually) nearest samples Near one sample point, its influence greatly predominates. Must know! 10/11/2017 ENGRG Intro to GIS 40

41 Exponent Value: Degree of Influence 10/11/2017 ENGRG Intro to GIS 41

42 Question What categories does the Inverse Distance Weighted (IDW) method fall into: exact or approximate? deterministic or stochastic? gradual or abrupt? local or global? What could it be used for? 10/11/2017 ENGRG Intro to GIS 42

43 Trend Surface Analysis First order (linear) trend surface Second order (quadratic) trend surface 10/11/2017 ENGRG Intro to GIS 43

44 Surface Models: Trend Surface Analysis Y X Z = b o + b 1 X + b 2 Y + [polynomial terms + interaction terms] + e Inexact interpolator: Surface usually not through sample points. 10/11/2017 ENGRG Intro to GIS 44

45 Surface Models: Trend Surface Analysis Regression SUMMARY OUTPUT Regression Statistics Multiple R 0.83 R Square 0.69 Adjusted R 0.58 Standard E 3.16 Observatio 20 ANOVA df SS MS F Significance Regression Residual Total Coefficien Standard Error t Stat P-value Intercept X Y X^ Y^ XY Only independent variables are the X,Y coordinates. Z = * X * Y * X * Y * XY 10/11/2017 ENGRG Intro to GIS 45

46 Trend Surface Interpolation Statistical approach Not exact, but can be life like, i.e., similar to natural phenomena, which usually vary smoothly. 10/11/2017 ENGRG Intro to GIS 46

47 Question What categories does the Trend Surface Analysis method fall into: exact or approximate? deterministic or stochastic? gradual or abrupt? local or global? What could it be used for? 10/11/2017 ENGRG Intro to GIS 47

48 Interpolation with Splines Force line to pass through known points. Joined polynomial functions Can be 1 3 order Segments meet at join points Iterative process 10/11/2017 ENGRG Intro to GIS 48

49 Spatial Interpolation in ArcGIS Geostatistical Analyst The ArcGIS GeoStatistical Analyst extension provides a variety of interpolation methods for the creation of an optimal interpolated surface from your data. A friendly wizard helps you through the interpolation process. 10/11/2017 ENGRG Intro to GIS 49

50 Interpolation with Geostatistics Several options; best known is: Kriging Developed by D.G. Krige and G. Matheron. Statistically based estimator Depends on Variogram analysis: defined as a fct describing the degree of spatial dependence of a spatial random field or stochastic process the variance of the difference between field values at two locations x and y across realizations in the field Weighting to interpolate variable value similar to IDW, but weights are based on spatial autocorrelation in the sample, not merely distance. 10/11/2017 ENGRG Intro to GIS 50

51 Kriging Steps 1. Calculate semivariance: spatial autocorrelation 2. Identify a model to estimate the weights 3. Calculate covariance within the data 4. Make the estimation 5. Check for error 10/11/2017 ENGRG Intro to GIS 51

52 Geostatistics with the ArcGIS Geostatistical Analyst Extension 10/11/2017 ENGRG Intro to GIS 52

53 Geostatistical Analyst Again, use of this methodology deserves an entire course 10/11/2017 ENGRG Intro to GIS 53

54 Cross Validation The validation technique compares the interpolation methods by first dividing known points into two samples: one sample for developing the models for each interpolation method to be compared and the other sample for testing the accuracy of the models Steps Remove a known point from the data set. Use the remaining points to estimate the value at the point previously removed. Calculate the predicted error of the estimation by comparing the estimated with the known value. 10/11/2017 ENGRG Intro to GIS 54

55 Root mean square (RMS) error RMS 1 n n (z i 1 2 i,act zi,est) Standardized RMS = 1 n n i 1 (z i,act s 2 z i,est ) 2 10/11/2017 ENGRG Intro to GIS 55

56 Summary Global Method Local Method Deterministic Stochastic Deterministic Stochastic Trend surface (inexact)* Regression (inexact) Thiessen (exact) Inverse Distance Weighted (exact) Kriging (exact) Splines (exact) Local Average (inexact) *Given some required assumptions, trend surface analysis can be treated as a special case of regression analysis and thus a stochastic method. 10/11/2017 ENGRG Intro to GIS 56

57 Summary Concepts What is spatial interpolation and why is necessary Sample pattern: systematic; random; cluster; adaptive Classification of interpolation methods Global vs Local Exact vs Inexact Graduate vs abrupt Deterministic vs Stochastic Interpolation methods Thiessen polygon Local average Inverse Distance Weighted Spline Kriging 10/11/2017 ENGRG Intro to GIS 57