Context. Sardine Anchovy Sprat

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2 Context Sardine Anchovy Sprat

3 Reflexion strength % change in biomass Impact of a 1cm change ~ 11-12% size size 3

4 Criteria (no hierarchy): Distance Bathymetry Similarities in the echograms 4

5 25 0.5cm size classes discrete sampling High nb of 0, some correlation between bars Dimension reduction by PCA 5

6 A histogram should not be reduced to a set of bars. It is rather an organised sequence of bars. Probability distributions The georeferenced target is a function, i.e. no longer a variable Functional kriging Nerini, D., Monestiez, P. and C., Mante Cokriging for functional data, J. Mult. Anal. 101, Petitgas, P., Doray, M., Massé, J.,, and Grellier P., 2011.Spatially explicit estimation of fish length histograms, with application to anchovy habitats in the Bay of Biscay. ICES Journal of Marine Science, 68(10), Caballero, W., Giraldo, R., and Mateu, J A universal kriging approach for spatial functional data. Stochastic Environmental Research and Risk Assessment, 27(7), A Menafoglio, P Secchi, M Dalla Rosa, A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space. Electronic Journal of Statistics. 6

7 Principles and existence of a solution (Nérini et al., 2010) Each observation is a realization of a functional random variable Y i at position x i (2D), that take its values in an Hilbert space equipped with standard inner product and the associated norm. Under stationary hypothesis, the mean function is E Y( x) μ The estimation of the function at an unknown location is: Y n * 0 B i ( Y i ) i 1 where B i refers to linear operators. These operators can be defined by their kernels so that: Bi ( Yi )( t) βi( s, t) Yi ( t) ds 7

8 Principles and existence of a solution * Unbiasedness E[ Y Y ] implies that n Bi ( μ) μ i 1 μ being unknown, one must choose operators that verify that constraint whatever μ One must choose the kriging operators so that n Bi K i 1 K( f) f, f Such an operator K exists if we restrict ourselves to Reproducing Kernel Hilbert Space RKHS. Regularity constraints are added to the more general HS case. Optimality is also tractable in this framework. 8

9 What about real case? Computers do not like infinite dimensional spaces. use of a finite number of basis functions Yi p k 1 α k ( x ) S i k ( t) where the S k s form an orthogonal basis. In that case, the problem reduces to the co-kriging of the α k 9

10 Using Legendre polynomials Legendre polynomials are orthogonal. S k ( t) Legendre ( t) k Linear model of Coregionalisation of the α k Empirical Distributions (N) Decomposition on N Legendre polynomials cokriging of the α cok k 10

11 Using B-Splines B-Spline functions are not-orthogonal. Functional PCA: Dimension reduction + Orthogonal basis of eigen functions S k ( t) PCk Bspline1,..., B( t) Linear model of coregionalisation Empirical distributions Decomposition on numerous non-orthogonal B-Splines Princ. Comp. cokriging of the princ. comp. 11

12 3 methods N proportions by size classes Series of functions B>>N B-Splines N Legendre polynomials Dimension reduction PCA ~few Princ. components Dimension reduction Functionnal PCA ~ few Princ. components Dimension reduction ~ few polynomials Principal curve One curvilinear variable 12

13 Cross validation to compare between methods Use yearly data from 1993 to 2013 (n = 448 samples) Loop on the number of principal components or Legendre polynomials Conditional on this number, define a linear model of CoRegionalization (mean annual strucures, modeld with nugget + 2 sphericals) Leave One sample Out and re-estimate it using the data of the running year Compute GOF, Kolmogorov distance, negative probabilities (quantity and frequence) to evaluate the quality of the estimation ( pi * pi )² * GOF K P p 1 dist max Fi Fi i p i p ² i 0 13

14 GOF between known and re-estimated histograms Legendre polynomials PCA PCA B-Splines Optimality for ~10 Legendre polynomials and 3-4 p.components 14

15 GOF between known and re-estimated histograms PCA_BSplines - PCA Optimality for ~10 Legendre polynomials and 4 p.components Legendre < B-Splines = ACP 15

16 Negative probabilities of known and re-estimated histograms Legendre polynomials PCA PCA_BSplines - PCA Up to 20% of probabilities < 0 with Legendre polynomials significantly less for PCA and functional PCA Legendre << B-Splines = ACP 16

17 Conclusions For all criteria (GOF, K dist et P - ): Legendre < B-Splines = PCA The winner is thus. PCA No functional kriging required 17

18 PC2 PC1 PC1 PC2 PC3 73% of the variability between distributions is made of : Translation (PC1) Some bimodality (PC2) Some dissymmetry (PC3) 18

19 . Cokriging of the 5 first PCA PC1 PC1 PC2 Mean annual variogram Linear model of coregionalisation PC1 PC2 PC3 PC3 PC2 RGeostats PC3

20 Section of interpolated size distributions (for latitude = 43 ) 20

21 PC2 Points sorted along a principal curve PC1 Curvilinear distance variogram Geographical distance 21

22 GOF Space dimension in which the principal curve is built 22

23 Honouring the constraints is not straightforward in cokriging Kriging of the curvilinear index (monovariate kriging) Space dimensions from 2 to 5. GOF for ACP GOF for principal curve Nota: Rebuilding true histo from 3 PCs produces negative proportions 23

24 ACP generate less negative probabilities than princ. curve Space dimensions from 2 to 5. ACP generate more negative probabilities than princ. curve Proba negative ACP proba negative Principal curve 24

25 Take home message Functional kriging works better with Bsplines than Legendre polynomials. However, in the particular case presented here, PCA remains the best method. Particularities rely on the discrete nature of the input data. The problem can be reduced down to a monovariate kriging of the curvilinear index of the PCA coordinates with an optimum for a space dimension equals to 4. Positiveness or monotony constraints (e.g. pdf or cdf) are not insured by the cokriging step. Principal curve approach reduces the problem as the space dimension increases. 25

26 Some perspectives Sensitivity to the size window used for determining the principal curve MAF and their potential principal curves Aitchison metric to solve positivity External variable(s) 26

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28 PCA of the size classes 73% of explained variance by first 3 PC >97% with 10 PC Very good reconstruction with 10 PCs 28

29 Spatial model for the first 10 functional PCs Mean annual variogram Linear model of coregionalisation RGeostats

30 Estimation of the ponctual full size distributions From the cok of the first 10 functional PCs Frequency Section of interpolated size distributions (for latitude = 43 ) Longitude ( ) Size (cm) 30