Non-linearity and spatial correlation in landslide susceptibility mapping

Transcription

1 Non-linearity and spatial correlation in landslide susceptibility mapping C. Ballabio, J. Blahut, S. Sterlacchini University of Milano-Bicocca GIT /09/2009 1

2 Summary Landslide susceptibility modeling Non-linearity issues Few examples Application to a case study Modeling the residual spatial correlation 15/09/2009 2

3 Introduction Landslide susceptibility modeling Usually defined as a classification problem: if y=1 is an observed occurrence and y=0 is a point with no occurrence, and x is a series of variables, then we want to know: P(y=1 x)=f(x, θ) 15/09/2009 3

4 Introduction 6 Linearly separable classes Just find a separating line/plane/hyper -plane X X2 15/09/2009 4

5 Exactly what LDA, QDA and LR does 15/09/2009 5

6 Even for linearly separable classes the best function could be not linear 15/09/2009 6

7 What if the separation can not be performed by linear functions? How can we separate the two classes by using only X1 and X2? X /09/ X2

8 LDA does not work 15/09/2009 8

9 LDA does not work 15/09/2009 9

10 Neither does QDA 15/09/

11 Even far more flexible models fail to separate the classes 15/09/

12 ANNs get close to do the job, but require a lot of tuning 15/09/

13 Support Vector Machines Based on the Statistical Learning Theory (Vapnik, 1995) Very good performance in classification tasks Intrinsic Occam s razor logic: the simplest model is preferred Easy to avoid overfitting Not so Black-box 15/09/

14 Support Vector Machines Widely used in machine learning Bioinformatics / genetic classification Spatial mapping Robotics Digital soil mapping 15/09/

15 Support vector classification Use the best hyperplane Use the kernel trick 15/09/

16 6 Which is the best hyperplane? We need a way to define what optimal separation is X1 4 2? X2 15/09/

17 6 Find the widest gap between classes Fit a plane in the middle of the gap X X2 15/09/

18 Kernel Function The kernel linearize the data in an high dimensional space Makes possible to find a flat separating hyperplane 15/09/

19 Kernel Function The kernel linearize the data in an high dimensional space Makes possible to find a flat separating hyperplane 15/09/

20 Kernel Function Based on the dot product: x, x i i ' = l i= 1 [ x ] [ x Simple to elaborate i '] i But very powerful, can project data in high dimensional spaces: Reproducing Kernel Hilbert Spaces (RKHS) But it is not known beforehand which kernel is appropriate 15/09/

21 Kernels Linear: Polynomial: K ( x, x ) = x, i x i K ( x, x ) = x, i x i d Radial basis function: K( x, x i ) = exp x x 2σ i 2 2 x x Exponential RBF: K( x, x ) = exp i i 2 2σ 15/09/

22 SVM with Single Gaussian kernel Separates the classes almost perfectly Reproduces the general trend of the data 15/09/

23 The Staffora basin study area Triggering patterns for flows DEM derived covariates + geology and landuse 15/09/

24 Legend NB Probability Value High : Naïve Bayes ( WoE) prediction Not bad, but we got a lot of high probability areas Low : Kilometers 15/09/

25 Legend LDA Probability Value High : LDA (a.k.a. Maximum Likelihood) prediction Better than NB, but we still get a lot of high probabilities Low : Kilometers /09/

26 Legend SVM Probability Value High : SVM Just better Low : Kilometers /09/

27 Use cross-validation and ROC curves to compare the models Far less false positives Predicted in the cross-validation sample 15/09/

28 Success curves (Fabbri and Chung, 2003) It s a ROC with only true positives rate 15/09/

29 Why use Machine Learning? Increasing availability of low cost / high information topographic surveys i.e. LiDAR, Hyper-spectral data A lot raw derived information A ML system can automatically interpreter the data without the need of refinements (automatic mapping systems). 15/09/

30 What happened if we use only DEM derived data? We still get a decent prediction from SVM, but not from LR/LDA/NB 15/09/

31 0.15 Once we predict with SVM can we derive useful information from the data? There is still a lot of autocorrelation al low distances semivariance Residual spatial correlation distance 15/09/

32 detrended Residual correlation We can implement a Kriging system to model the residual information Or, we can use MK-SVM to model spatial variation semivariance detrended.svm.pred detrended.occurrence Model correlation Original trend svm.pred svm.pred.occurrence occurrence distance 15/09/

33 A monodimensional example Combination of cosine functions with different λ Plus some Additive Gaussian Noise Sample 30% of the data points y y y x /09/ x x

34 Multi-kernel analysis Two gaussian RBF kernels MK-SVR is able to separate the two signals, even in presence of noise. y2 y pred3 pred x /09/ x

35 Spatial SVM performance Within slope predicted probability Average probability close to max probability 15/09/

36 Spatial SVM performance Crossvalidation performance Average prob. still close to max probability 15/09/

37 Spatial SVM rules extraction 15/09/

38 Conclusions SVM clearly outperform most of the statistical techniques commonly applied for landslide susceptibility perdiction It IS a black box technique, but not so much several algorithms for feature selection and ranking are available Very good for automatic and real time mapping Can easily update the model if new data is provided Good for automatic mapping systems 15/09/