Linear predictive functional model on environmental data: case of chlorophyll-a oceanographic profiles Séverine Bayle 1 , Pascal Monestiez 1 , David Nerini 2 1 INRA, UR 546 Biostatistics and Spatial Processes (BioSP), F-84914 AVIGNON. 2 Mediterranean Institute of Oceanography (MIO) - UMR 7294, Pytheas Institute (OSU), Aix-Marseille University, Campus de Luminy, Case 901, 13288 MARSEILLE Cedex 09. 7 th Days of functional statistics, Montpellier, June 28-29, 2012 Séverine Bayle (INRA) 1 / 26
Plan of the talk Introduction 1 Methodology 2 Results 3 Conclusion 4 Séverine Bayle (INRA) 2 / 26
Introduction Context and purpose of the study Physical data (profiles) collected within the framework of ANR project IPSOS-SEAL between October 2009 and January 2010 in Southern Ocean around Kerguelen islands : Chlorophyll-a (Chl-a) : CTD-Fluo and Argos devices Brightness : TDR + GPS devices Séverine Bayle (INRA) 3 / 26
Introduction Capturing elephant seals for installing devices Séverine Bayle (INRA) 4 / 26
Introduction Capturing elephant seals for installing devices Séverine Bayle (INRA) 5 / 26
Introduction Capturing elephant seals for installing devices Séverine Bayle (INRA) 6 / 26
Introduction Capturing elephant seals for installing devices Séverine Bayle (INRA) 7 / 26
Introduction Capturing elephant seals for installing devices Séverine Bayle (INRA) 8 / 26
Introduction Capturing elephant seals for installing devices Séverine Bayle (INRA) 9 / 26
Introduction Capturing elephant seals for installing devices Séverine Bayle (INRA) 10 / 26
Introduction Elephant seal dataset 0 0 -50 -50 Depth (m) Depth (m) -100 -100 -150 -150 60 80 100 120 140 160 0.0 0.5 1.0 1.5 2.0 Light (W/m²) Chl-A (mg/l) Séverine Bayle (INRA) 11 / 26
Introduction Context and purpose of the study Primary productivity : production of vegetal matter Photosynthesis : permitted through the oceanic phytoplankton content in Chl-a → Vital link between living and inorganic stocks of carbon Séverine Bayle (INRA) 12 / 26
Introduction Context and purpose of the study Primary productivity : production of vegetal matter Photosynthesis : permitted through the oceanic phytoplankton content in Chl-a → Vital link between living and inorganic stocks of carbon Measurement of Chl-a concentration throughout the water column in Southern Ocean is used as an indicator of the amount of phytoplankton and allows to know the distribution of primary productivity Séverine Bayle (INRA) 12 / 26
Introduction Context and purpose of the study Primary productivity : production of vegetal matter Photosynthesis : permitted through the oceanic phytoplankton content in Chl-a → Vital link between living and inorganic stocks of carbon Measurement of Chl-a concentration throughout the water column in Southern Ocean is used as an indicator of the amount of phytoplankton and allows to know the distribution of primary productivity Few Chl-a data profiles recorded : devices which record fluorescence are energy-intensive Séverine Bayle (INRA) 12 / 26
Introduction Context and purpose of the study Primary productivity : production of vegetal matter Photosynthesis : permitted through the oceanic phytoplankton content in Chl-a → Vital link between living and inorganic stocks of carbon Measurement of Chl-a concentration throughout the water column in Southern Ocean is used as an indicator of the amount of phytoplankton and allows to know the distribution of primary productivity Few Chl-a data profiles recorded : devices which record fluorescence are energy-intensive But a lot of brightness data profiles Séverine Bayle (INRA) 12 / 26
Introduction Context and purpose of the study Primary productivity : production of vegetal matter Photosynthesis : permitted through the oceanic phytoplankton content in Chl-a → Vital link between living and inorganic stocks of carbon Measurement of Chl-a concentration throughout the water column in Southern Ocean is used as an indicator of the amount of phytoplankton and allows to know the distribution of primary productivity Few Chl-a data profiles recorded : devices which record fluorescence are energy-intensive But a lot of brightness data profiles Idea : reconstruct Chl-a profiles from brigthness profiles Séverine Bayle (INRA) 12 / 26
Introduction Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles, only data profiles collected during day were kept Séverine Bayle (INRA) 13 / 26
Introduction Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles, only data profiles collected during day were kept To be more accurate in estimation and smoothing of profiles, only Chl-a data profiles which have 18 observations recorded every 10 meters between -5 et -175 meters were kept (407 profiles selected) Séverine Bayle (INRA) 13 / 26
Introduction Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles, only data profiles collected during day were kept To be more accurate in estimation and smoothing of profiles, only Chl-a data profiles which have 18 observations recorded every 10 meters between -5 et -175 meters were kept (407 profiles selected) Selection of Chl-a and brightness data profiles collected at the same time : 208 profiles altogether Séverine Bayle (INRA) 13 / 26
Introduction Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles, only data profiles collected during day were kept To be more accurate in estimation and smoothing of profiles, only Chl-a data profiles which have 18 observations recorded every 10 meters between -5 et -175 meters were kept (407 profiles selected) Selection of Chl-a and brightness data profiles collected at the same time : 208 profiles altogether Reconstruction of one Chl-a data profile is made for each 208 pairs Séverine Bayle (INRA) 13 / 26
Methodology Functional data analysis Chl-a and brightness functional profiles can be considered as curves z ci ( t ) = y i ( t ) + ǫ i ( t ) , z bi ( s ) = x i ( s ) + ǫ i ( s ) Séverine Bayle (INRA) 14 / 26
Methodology Functional data analysis Chl-a and brightness functional profiles can be considered as curves z ci ( t ) = y i ( t ) + ǫ i ( t ) , z bi ( s ) = x i ( s ) + ǫ i ( s ) Modeling these functional profiles needs definition of basis functions φ k , k = 1 , . . . , K Séverine Bayle (INRA) 14 / 26
Methodology Functional data analysis Chl-a and brightness functional profiles can be considered as curves z ci ( t ) = y i ( t ) + ǫ i ( t ) , z bi ( s ) = x i ( s ) + ǫ i ( s ) Modeling these functional profiles needs definition of basis functions φ k , k = 1 , . . . , K Fonctional profiles are defined as linear combinations of these basis functions : K K � � y i ( t ) = c ik φ k ( t ) , x i ( s ) = d ik φ k ( s ) k = 1 k = 1 c 1 , . . . , c K and d 1 , . . . , d K : expansion coefficients φ 1 , φ 2 , . . . , φ K : basis functions Séverine Bayle (INRA) 14 / 26
Methodology Functional data analysis Reconstruct functional profiles y and x using data ( t , z ci ) and ( s , z bi ) , i = 1 , . . . , n Séverine Bayle (INRA) 15 / 26
Methodology Functional data analysis Reconstruct functional profiles y and x using data ( t , z ci ) and ( s , z bi ) , i = 1 , . . . , n Utilisation of 10 splines of order 4 n � ( x ( t i ) − y ( t i )) 2 + λ ( x ′′ ( u )) 2 du � 1 / n i = 1 Séverine Bayle (INRA) 15 / 26
Methodology Functional data analysis Reconstruct functional profiles y and x using data ( t , z ci ) and ( s , z bi ) , i = 1 , . . . , n Utilisation of 10 splines of order 4 n � ( x ( t i ) − y ( t i )) 2 + λ ( x ′′ ( u )) 2 du � 1 / n i = 1 λ : Trade-off between smoothness of the curve and sum of squared deviations between model and data Séverine Bayle (INRA) 15 / 26
Methodology Functional data analysis Reconstruct functional profiles y and x using data ( t , z ci ) and ( s , z bi ) , i = 1 , . . . , n Utilisation of 10 splines of order 4 n � ( x ( t i ) − y ( t i )) 2 + λ ( x ′′ ( u )) 2 du � 1 / n i = 1 λ : Trade-off between smoothness of the curve and sum of squared deviations between model and data We work now with splines coefficients c k and d k Séverine Bayle (INRA) 15 / 26
Methodology 0 ● ● ● ● −50 ● ● ● ● Depth (m) ● −100 ● ● ● ● ● −150 ● ● ● ● 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Chl−a (µg/l) Number of basis functions = number of knots + order of splines Séverine Bayle (INRA) 16 / 26
Methodology Functional linear model We consider a fully functional linear model Assumption : relationship between derivative of brightness function and Chl-a function � y ( t ) = α ( t ) + β ( s , t ) x ( s ) ds + ǫ ( t ) y(t) : Chl-a profile reconstructed (or predicted) t and s : Depths x(s) : Derivative of brightness function α ( t ) : Univariate coefficient (functional intercept) β ( s , t ) : Bivariate coefficient ǫ ( t ) : Functional error Séverine Bayle (INRA) 17 / 26
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