Pollution Sources Detection via Principal Component Analysis and Rotation

Transcription

1 Pollution Sources Detection via Principal Component Analysis and Rotation Vanessa Kuentz 1 in collaboration with : Marie Chavent 1 Hervé Guégan 2 Brigitte Patouille 1 Jérôme Saracco 1,3 1 IMB, Université Bordeaux 1, Talence, France ( vanessa.kuentz@math.u-bordeaux1.fr) 2 ARCANE-CENBG, Gradignan, France 3 GREThA, Université Bordeaux 4, Pessac, France 1/22 Applied Stochastic Models and Data Analysis, Crete, 2007

2 Outline Factor Analysis 1 Factor Analysis Notations Recall about PCA Factor Analysis model Link between FA and PCA 2 Motivation Criterion Remark on rotation in PCA 3 A three steps process Statistical treatment Some confirmation of the results 2/22 Applied Stochastic Models and Data Analysis, Crete, 2007

3 Issue Factor Analysis Air pollution = small particles + liquid droplets High concentrations of particles = danger to human health, especially fine particles Development of air quality control strategies identification of pollution sources Receptor modeling, commonly used = Principal Component Analysis Part of scientific project initiated by French Ministry of Ecology 3/22 Applied Stochastic Models and Data Analysis, Crete, 2007

4 Notations Recall about PCA Factor Analysis model Link between FA and PCA X = (x j i ) n,p numerical data matrix where n objects are described on p < n variables x 1,..., x p X = ( x j i ) n,p standardized data matrix: x j s j ( x j,s j sample mean and standard deviation of x j ) R sample correlation matrix of x 1,..., x p : i = x j i x j R = X M X where M = 1 m I n (with m = n or n 1) We have : R = Z Z with Z = M 1/2 X 4/22 Applied Stochastic Models and Data Analysis, Crete, 2007

5 Notations Recall about PCA Factor Analysis model Link between FA and PCA Replace p variables by q p uncorrelated components The SVD of Z (of rank r p) is Z = UΛ 1/2 V with : - Λ : diagonal matrix (r, r) of nonnull eigenvalues λ 1,..., λ r of Z Z (in decreasing order) - U : orthonormal matrix (n, r) of eigenvectors of ZZ associated with first r eigenvalues - V : orthonormal matrix (p, r) of eigenvectors of Z Z associated with first r eigenvalues Decomposition of X: X = M 1/2 UΛ 1/2 V 5/22 Applied Stochastic Models and Data Analysis, Crete, 2007

6 Notations Recall about PCA Factor Analysis model Link between FA and PCA q = r : In PCA, equation X = M 1/2 UΛ 1/2 V is written : X = ΨV with Ψ = M 1/2 UΛ 1/2 We have Ψ = XV (VV = I p ) Columns of Ψ : principal components ψ k, k = 1,..., q q < r : In practice, user retains only first q < r eigenvalues of Λ Approximation of X : ˆ Xq = Ψ q V q (Ψ q, V q : matrices Ψ and V reduced to first q columns) 6/22 Applied Stochastic Models and Data Analysis, Crete, 2007

7 Notations Recall about PCA Factor Analysis model Link between FA and PCA FA : underlying common factors + dimension reduction Model is written : x = Af + e (p 1) (p q)(q 1) (p 1) with : - x = ( x 1,..., x p ) random vector of R p - A : loading (or pattern) matrix - f = (f 1,..., f q ) random vector of q common factors - e = (e 1,..., e p ) random vector of p unique factors x j : linear combination of common factors + unique factor 7/22 Applied Stochastic Models and Data Analysis, Crete, 2007

8 Notations Recall about PCA Factor Analysis model Link between FA and PCA FA model : various estimation methods - Principal factor method - Maximum likelihood method - Principal component method -... Commonly used method : PCA - Easy implementation - Reasonable results - Default method in statistical softwares 8/22 Applied Stochastic Models and Data Analysis, Crete, 2007

9 Notations Recall about PCA Factor Analysis model Link between FA and PCA q = r : In FA (with PCA estimation), equation X = M 1/2 UΛ 1/2 V is written : X = FA with : -F = M 1/2 U : factor scores matrix -A = V Λ 1/2 : loading (or pattern) matrix We have F = XVΛ 1/2 (V V = I q ) Columns of F : common factors f k, k = 1,..., q q < r : In practice, user retains only first q < r eigenvalues of Λ Approximation of X : ˆ X q = F q A q (F q, A q : matrices F and A reduced to first q columns) 9/22 Applied Stochastic Models and Data Analysis, Crete, 2007

10 Notations Recall about PCA Factor Analysis model Link between FA and PCA We have : PCA : Ψ = XV } F = ΨΛ 1/2 1/2 FA : F = XVΛ factors f k correspond to standardized principal components : f k = ψk λk, k = 1,..., q 10/22 Applied Stochastic Models and Data Analysis, Crete, 2007

11 Motivation Criterion Remark on rotation in PCA Loading matrix A q : a k j = corr(x j, f k ) identification of groups of correlated elements Problem : Intermediate values difficult association variables/factors Objective : - Column of A q : values close to 0 or 1 - Row of A q : only one value close to 1 Solution : - Non-uniqueness of FA solution : rotation of factors - Orthogonal transformation matrix T (TT = T T = I q ) - Ăq = A q T : correlation of variables to rotated factors f k - How to determine T? 11/22 Applied Stochastic Models and Data Analysis, Crete, 2007

12 Motivation Criterion Remark on rotation in PCA Several criteria : most used is varimax Maximizes empirical variance of squared elements of each column of Ăq = (ă α j ) p,q : 2 p p q (ăj α ) 4 (ăj α ) 2 j=1 j=1 p p α=1 with ă α j = a j t α and u.cs. t α (t α ) = 1, t l (t k ) = 0, k l. 12/22 Applied Stochastic Models and Data Analysis, Crete, 2007

13 Motivation Criterion Remark on rotation in PCA Possible rotation in PCA as in FA Must apply T to the good matrices keep property of non correlation of components One must not write : but : X = X = Ψ q {}}{ M 1/2 UΛ 1/2 TT V F q {}}{ M 1/2 UTT Λ 1/2 V rotation of standardized principal components = factors of FA 13/22 Applied Stochastic Models and Data Analysis, Crete, 2007

14 A three steps process Statistical treatment Some confirmation of the results Part of scientific project of French Ministry of Ecology Improve air quality : identification + quantification of pollution sources Three steps process : - Collecting of fine particles in French urban site (Anglet) : n = 61 samples (Dec. 2005) - Measurements of p = 16 chemical species : PIXE method (Particle Induced X-ray Emission) - Statistical treatment of X = (x j i ) n,p : concentration of jth chemical compound in ith sample 14/22 Applied Stochastic Models and Data Analysis, Crete, 2007

15 A three steps process Statistical treatment Some confirmation of the results FA model with PCA estimation Choice of number of factors : q = 5 Orthogonal rotation of factors with varimax criterion Detection of groups of correlated chemical compounds Identification of air pollution sources 15/22 Applied Stochastic Models and Data Analysis, Crete, 2007

16 A three steps process Statistical treatment Some confirmation of the results f 1 f 2 f 3 f 4 f 5 Al2O SiO P SO Cl K Ca Mn Fe2O Cu Zn Br Pb C-Org Table: Loading matrix before rotation A 5 Lots of intermediate values difficult association variables/factors 16/22 Applied Stochastic Models and Data Analysis, Crete, 2007

17 A three steps process Statistical treatment Some confirmation of the results f 1 f 2 f 3 f 4 f 5 Al2O SiO P SO Cl K Ca Mn Fe2O Cu Zn Br Pb C-Org Factor Possible source Factor1 Soil dust Factor2 Combustion Factor3 Industry Factor4 Vehicle Factor5 Sea Table: Pollution sources Table: Rotated loading matrix Ă5 Obvious usefulness of rotation : clear associations ex: Zn + Pb highly correlated to f 3 industrial pollution source 17/22 Applied Stochastic Models and Data Analysis, Crete, 2007

18 A three steps process Statistical treatment Some confirmation of the results Rotated factors : columns of F 5 = F 5 T = ( f i k ) n,5 Score f i k = relative contribution of kth source to ith sample Confrontation of rotated factors with external parameters : - Meteorological data (wind, temperature,...) - Periodicity day/night - Other chemical coumpounds 18/22 Applied Stochastic Models and Data Analysis, Crete, 2007

19 A three steps process Statistical treatment Some confirmation of the results Figure: Evolution of vehicle pollution source Stronger contribution during day than night confirmation of cars pollution source 19/22 Applied Stochastic Models and Data Analysis, Crete, 2007

20 A three steps process Statistical treatment Some confirmation of the results Figure: Evolution of heating pollution and temperatures Middle of period : } Increase in contribution confirmation of heating pollution Decrease in temperature 20/22 Applied Stochastic Models and Data Analysis, Crete, 2007

21 A three steps process Statistical treatment Some confirmation of the results Figure: Map of sampling site and correlations wind/sea pollution } Strong correlation to N-W wind validation of sea pollution Location toward Atlantic Ocean 21/22 Applied Stochastic Models and Data Analysis, Crete, 2007

22 Conclusion A three steps process Statistical treatment Some confirmation of the results FA (with PCA estimation) followed by rotation : identification of five sources of particulate emission No prior knowledge : number, composition? more complex sampling site Sources quantification : percentage of total fine dust mass of each source (JDS, Angers, 2007) 22/22 Applied Stochastic Models and Data Analysis, Crete, 2007