Spatial Statistics or Why Spatial is Special?

Transcription

1 Spatial Statistics or Why Spatial is Special? Curdin Derungs, GISLab Seite 1

2 Spatial is special

3 Spatial is special Longley et al s (2011) spatial is special -list: Seite 3

4 Spatial is special Longley et al s (2011) spatial is special -list: Space is multidimensional XKCD Seite 4

5 Spatial is special Longley et al s (2011) spatial is special -list: Space is multidimensional Ecological fallacy / spurious correlations / causality?

6 Spatial is special Longley et al s (2011) spatial is special -list: Space is multidimensional Ecological fallacy / spurious correlations / causality? Modifiable areal unit problem (MAUP) Spatial autocorrelation Seite 6

7 Modifiable areal unit problem - MAUP

8 Modifiable areal unit problem - MAUP p stands for problem, not solution i.e. there is no analytical solution to MAUP MAUP is a spatial data aggregation problem: different aggregation different results shape/form of aggregation units size/scale of aggregation units Seite 8

9 Why aggregating linguistic data? If you are more interested in distribution(s), than in individual answers Statistics is easier with continuous than with categorical data Typical examples: Aggregation into bins: admin. unit / country / continent Regular grid aggregation Density maps

10 A famous MAUP example: Gerrymandering How to win an election: cracking & packing Wikipedia

11 brilliantmaps.com/2016-county-election-map/ URPP Language and Space US presidential election: a good MAUP test-bed Underlying votes: 46% gop (red), 48% democrats (blue)

12 brilliantmaps.com/2016-county-election-map/ URPP Language and Space US presidential election: a good MAUP test-bed 1 st experiment: randomizing counties

13 US presidential election: a good MAUP test-bed 2 nd experiment: random aggregation units

14 US presidential election: a good MAUP test-bed 1 election, 1000 county randomisations election, 1000 random grid aggregations into 50 spatial units 456 into 10 spatial units 447 into 100 spatial units

15 What can we do? No analytical solution, but awareness Aggregation units should be as meaningless in shape and size as possible (e.g. grid cells) randomisation Sensitivity analysis: identifying MAUP hotspots

16 Spatial autocorrelation

17 Spatial autocorrelation is good and bad good bad Seite 17

18 What is spatial autocorrelation? Eiger Mönch Jungfrau Ulrich 1983, Seite 18

19 What is spatial autocorrelation? Die Zeit Appleyard 1970 Seite 19

20 Good spatial autocorrelation because it facilitates every-day live (and we seem to like it) it helps us identifying / describing / understanding / comparing different phenomena it makes phenomena predictable Seite 20

21 Describing spatial autocorrelation Seite 21

22 Describing spatial autocorrelation the distance similarity relation (e.g. Tobler 1970) Séguy Seite 22 Nerbonne 2010

23 Describing spatial autocorrelation the distance similarity relation (e.g. Tobler 1970) m r 2 m 1 F 12 m 1 m 2 r Seite 23

24 Semivariogram linguistic analysis usually stops with distance-similarity plots a next step would be to calculate a semivariogram quite similar to distance-similarity plots but only one value is calculated for distance bins a function is fitted through these variance values Seite 24

25 Semivariogram: variance per distance bin Seite 25

26 Semivariogram: fitting a curve (Gaussian, exp., spherical, etc.) Seite 26

27 variance URPP Language and Space Semivariogram: parameters Sill: maximum impact of space on phenomena Range: maximum distance of spatial impact Nugget: non-spatial variance, variance at distance=0 distance Seite 27

28 variance URPP Language and Space Semivariogram: Comparing phenomena distance Seite 28

29 variance URPP Language and Space How does the Semivariogram look like if the phenomena has no spatial autocorrelation? Pure Nugget No Spatial Impact distance Seite 29

30 Prediction with spatial autocorrelation Assume you know the values at three locations, how do you predict an unknown value? 1. You can assume spatial autocorrelation and thus give higher weight to more proximate locations (IDW)? d 3 d 1 d Seite 30

31 variance URPP Language and Space Prediction with spatial autocorrelation Assume you know the values at three locations, how do you predict an unknown value? 1. You can assume spatial autocorrelation and thus give higher weight to more proximate locations (IDW) 2. You can use the Semivariogram Kriging? d 3 d 1 d 2 d 1 d 2 d 3 distance Seite 31

32 Spatial Autocorrelation is good because 1. we can describe, characterise, compare phenomena 2. we can predict values at unseen locations but why is it sometimes bad? Seite 32

33 Bad Spatial Autocorrelation The US election again: Does income, unemployment and population count explain the US voting behaviour (on county level)? democrats(%) ~ income(pc) + unempl. % + population(abs)

34 Spatial Autocorrelation in residuals The US election again: Does income, unemployment and population count explain the US voting behaviour (on county level)? Spatial distribution of residuals: red = underblue = over-prediction the residuals are not randomly distributed spatial non-stationarity

35 Spatial Autocorrelation in residuals Spatial autocorrelation in residuals (Variogram):

36 What can we do against spatial non-stationarity? Depends on what you are most interested in: best possible model for prediction exploring model variation in space democrats(%) ~ income(pc) + unempl. % + population(abs)

37 Models for spatial non-stationarity Best prediction: Spatial Autoregressive Models y ~Ν Χβ, σ 2 y ~Ν Χβ, C, with C = spatially weighted covariance matrix y Χβ y Χβ σ y j y i C i C j C i < C j measures of y proximate to y i contribute more to the error than distant y j x (1) x (1)

38 Models for spatial non-stationarity Best prediction: Spatial Autoregressive Models y ~Ν Χβ, σ 2 y ~Ν Χβ, C, with C = spatially weighted covariance matrix Fortheringham 2006 Exploring spatial non-stationarity: Geographically Weighted Regression (GWR) one (independent) model is fit for each y i, incorporating its neighbours and a distance weighting function

39 GWR applied to US election Same model as before: democ. ~ inc. +unemp. +pop. but GWR explains 70% of the variation in democrat-votes and allows spatial analysis of model parameters

40 GWR model parameters democ. ~ inc. +unemp. +pop. unemployment blue = negative - red = positive relation

41 Summary spatial autocorrelation Good & Bad spatial autocorrelation allows spatial prediction but needs to be incorporated in statistical models we focused on the US elections. I hope it is straight forward to translate these insights to linguistics

42 thank you very much