Linear predictive functional model on environmental data: case of - - PowerPoint PPT Presentation

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Linear predictive functional model on environmental data: case of chlorophyll-a oceanographic profiles Sverine Bayle 1 , Pascal Monestiez 1 , David Nerini 2 1 INRA, UR 546 Biostatistics and Spatial Processes (BioSP), F-84914 AVIGNON. 2

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Linear predictive functional model on environmental data: case of chlorophyll-a

  • ceanographic profiles

Séverine Bayle1, Pascal Monestiez1, David Nerini2

1INRA, UR 546 Biostatistics and Spatial Processes (BioSP), F-84914 AVIGNON. 2Mediterranean Institute of Oceanography (MIO) - UMR 7294, Pytheas Institute (OSU),

Aix-Marseille University, Campus de Luminy, Case 901, 13288 MARSEILLE Cedex 09.

7th Days of functional statistics, Montpellier, June 28-29, 2012

Séverine Bayle (INRA) 1 / 26

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Plan of the talk

1

Introduction

2

Methodology

3

Results

4

Conclusion

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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

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Introduction

Capturing elephant seals for installing devices

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Introduction

Capturing elephant seals for installing devices

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Introduction

Capturing elephant seals for installing devices

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Introduction

Capturing elephant seals for installing devices

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Introduction

Capturing elephant seals for installing devices

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Introduction

Capturing elephant seals for installing devices

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Introduction

Capturing elephant seals for installing devices

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Introduction

Elephant seal dataset

60 80 100 120 140 160

  • 150
  • 100
  • 50

Light (W/m²) Depth (m) 0.0 0.5 1.0 1.5 2.0

  • 150
  • 100
  • 50

Chl-A (mg/l) Depth (m)

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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

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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

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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

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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

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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

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Introduction

Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles,

  • nly data profiles collected during day were kept

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Introduction

Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles,

  • nly 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)

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Introduction

Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles,

  • nly 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

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Introduction

Context and purpose of the study In order to calibrate relationships between 2 kinds of data profiles,

  • nly 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

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Methodology

Functional data analysis Chl-a and brightness functional profiles can be considered as curves zci(t) = yi(t) + ǫi(t), zbi(s) = xi(s) + ǫi(s)

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Methodology

Functional data analysis Chl-a and brightness functional profiles can be considered as curves zci(t) = yi(t) + ǫi(t), zbi(s) = xi(s) + ǫi(s) Modeling these functional profiles needs definition of basis functions φk, k = 1, . . . , K

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Methodology

Functional data analysis Chl-a and brightness functional profiles can be considered as curves zci(t) = yi(t) + ǫi(t), zbi(s) = xi(s) + ǫi(s) Modeling these functional profiles needs definition of basis functions φk, k = 1, . . . , K Fonctional profiles are defined as linear combinations of these basis functions : yi(t) =

K

  • k=1

cikφk(t), xi(s) =

K

  • k=1

dikφk(s)

c1, . . . , cK and d1, . . . , dK : expansion coefficients φ1, φ2, . . . , φK : basis functions

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Methodology

Functional data analysis Reconstruct functional profiles y and x using data (t, zci) and (s, zbi), i = 1, . . . , n

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Methodology

Functional data analysis Reconstruct functional profiles y and x using data (t, zci) and (s, zbi), i = 1, . . . , n Utilisation of 10 splines of order 4 1/n

n

  • i=1

(x(ti) − y(ti))2 + λ

  • (x′′(u))2du

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Methodology

Functional data analysis Reconstruct functional profiles y and x using data (t, zci) and (s, zbi), i = 1, . . . , n Utilisation of 10 splines of order 4 1/n

n

  • i=1

(x(ti) − y(ti))2 + λ

  • (x′′(u))2du

λ : Trade-off between smoothness of the curve and sum of squared deviations between model and data

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Methodology

Functional data analysis Reconstruct functional profiles y and x using data (t, zci) and (s, zbi), i = 1, . . . , n Utilisation of 10 splines of order 4 1/n

n

  • i=1

(x(ti) − y(ti))2 + λ

  • (x′′(u))2du

λ : Trade-off between smoothness of the curve and sum of squared deviations between model and data We work now with splines coefficients ck and dk

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Methodology

  • 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 −150 −100 −50 Chl−a (µg/l) Depth (m)

Number of basis functions = number of knots + order of splines

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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

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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

FDA Package on R

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Results

Chl-a functional profiles well predicted...

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

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Results

...But some problems remain !

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −150 −100 −50 Amount of chlorophyll−a (µg/l) Depth (m)

Reconstructed profile Predicted profile

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Results

Cross validation Is 10 basis functions the optimal number to use ?

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Results

Cross validation Is 10 basis functions the optimal number to use ? Check by cross validation → Computation for each marine mammal → One profile is withdrawn (validation set), and others profiles represent training set → Calculation of mean square error → Repetition choosing another validation set which has not yet been used for the validation of the model → Mean of all mean square errors is calculated to estimate prediction error

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Results

Cross validation Is 10 basis functions the optimal number to use ? Check by cross validation → Computation for each marine mammal → One profile is withdrawn (validation set), and others profiles represent training set → Calculation of mean square error → Repetition choosing another validation set which has not yet been used for the validation of the model → Mean of all mean square errors is calculated to estimate prediction error 5 basis functions are enough to minimize prediction error

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Results

Comparison of R2 between the use of 5 and 10 basis functions Calculation of R2 between measured Chl-a profiles and predicted Chl-a profiles : R2

i = ||yi − ¯

yi||2 − ||ˆ yi − yi||2 ||yi − ¯ yi||2

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Results

Comparison of R2 between the use of 5 and 10 basis functions Calculation of R2 between measured Chl-a profiles and predicted Chl-a profiles : R2

i = ||yi − ¯

yi||2 − ||ˆ yi − yi||2 ||yi − ¯ yi||2 Number of basis functions Elephant seal Mean R2 Median R2 5 1st 0.87 0.93 2nd 0.77 0.87 3rd 0.70 0.85 10 1st 0.87 0.93 2nd 0.78 0.86 3rd 0.70 0.84

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Results

Characterization of fine scale variations (one day) Only one Chl-a functional profile 21 profiles predicted from 21 brightness functional profiles → Highlighting of fine-scale structures

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Conclusion

Discussion Method well suited to predict Chl-a profiles

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Conclusion

Discussion Method well suited to predict Chl-a profiles Applicable under similar conditions. Make sure that brightness profiles are recorded during day

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Conclusion

Discussion Method well suited to predict Chl-a profiles Applicable under similar conditions. Make sure that brightness profiles are recorded during day Difficulty of choice of number of basis functions : cross validation seems to indicate a few number

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Conclusion

Discussion Method well suited to predict Chl-a profiles Applicable under similar conditions. Make sure that brightness profiles are recorded during day Difficulty of choice of number of basis functions : cross validation seems to indicate a few number Chl-a data required pre-treatment (data day), this has a significant influence on the adjustment

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Conclusion

Prospect Promotion of many historical records of brightness profiles over a large geographic coverage which will enable monitoring of phytoplankton production

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Conclusion

Prospect Promotion of many historical records of brightness profiles over a large geographic coverage which will enable monitoring of phytoplankton production Methodological development (function linmod on R) to integrate several explanatory variables in the model

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Conclusion

Prospect Promotion of many historical records of brightness profiles over a large geographic coverage which will enable monitoring of phytoplankton production Methodological development (function linmod on R) to integrate several explanatory variables in the model How to account for Chl-a profiles registered by night ? → Interpolation using kriging

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Conclusion

References Ramsay J. & Silverman B. (2005). Functional Data Analysis. Springer, New-York. Ramsay J., Hooker G. & Graves S. (2009). Functional Data Analysis with R and MATLAB. Springer, New-York. Nerini, D., Monestiez, P . & Manté, C. (2010). Cokriging for spatial functional data. Journal of Multivariate Analysis, 101 :409-418

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Thanks for your attention !

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