Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Interpretable Sparse Sliced Inverse Regression for digitized functional data

410 views

Published on

Séminaire de l'Institut de Mathématique de Bordeaux
8 avril 2016

Published in: Science
  • Be the first to comment

  • Be the first to like this

Interpretable Sparse Sliced Inverse Regression for digitized functional data

  1. 1. Interpretable Sparse Sliced Inverse Regression for digitized functional data Victor Picheny, Rémi Servien & Nathalie Villa-Vialaneix nathalie.villa@toulouse.inra.fr http://www.nathalievilla.org Séminaire Institut de Mathématiques de Bordeaux 8 avril 2016 Nathalie Villa-Vialaneix | IS-SIR 1/26
  2. 2. Sommaire 1 Background and motivation 2 Presentation of SIR 3 Our proposal 4 Simulations Nathalie Villa-Vialaneix | IS-SIR 2/26
  3. 3. Sommaire 1 Background and motivation 2 Presentation of SIR 3 Our proposal 4 Simulations Nathalie Villa-Vialaneix | IS-SIR 3/26
  4. 4. A typical case study: meta-model in agronomy climate (daily time series: rain, temperature...) plant phenotypes predictions (yield, N leaching...) Agronomic model Nathalie Villa-Vialaneix | IS-SIR 4/26
  5. 5. A typical case study: meta-model in agronomy climate (daily time series: rain, temperature...) plant phenotypes predictions (yield, N leaching...) Agronomic model Agronomic model: based on biological and chemical knowledge; Nathalie Villa-Vialaneix | IS-SIR 4/26
  6. 6. A typical case study: meta-model in agronomy climate (daily time series: rain, temperature...) plant phenotypes predictions (yield, N leaching...) Agronomic model Agronomic model: based on biological and chemical knowledge; computationaly expensive to use; Nathalie Villa-Vialaneix | IS-SIR 4/26
  7. 7. A typical case study: meta-model in agronomy climate (daily time series: rain, temperature...) plant phenotypes predictions (yield, N leaching...) Agronomic model Agronomic model: based on biological and chemical knowledge; computationaly expensive to use; useful for realistic predictions but not to understand the link between the inputs and the outputs. Nathalie Villa-Vialaneix | IS-SIR 4/26
  8. 8. A typical case study: meta-model in agronomy climate (daily time series: rain, temperature...) plant phenotypes predictions (yield, N leaching...) Agronomic model Agronomic model: based on biological and chemical knowledge; computationaly expensive to use; useful for realistic predictions but not to understand the link between the inputs and the outputs. Metamodeling: train a simplified, fast and interpretable model which can be used as a proxy for the agronomic model. Nathalie Villa-Vialaneix | IS-SIR 4/26
  9. 9. A first case study: SUNFLO [Casadebaig et al., 2011] Inputs: 5 daily time series (length: one year) and 8 phenotypes for different sunflower types Output: sunflower yield Data: 1000 sunflower types × 190 climatic series (different places and years) (n = 190 000) of variables in R5×183 × R8 Nathalie Villa-Vialaneix | IS-SIR 5/26
  10. 10. Main facts obtained from a preliminary study R. Kpekou internship The study focused on the influence of the climate on the yield: 5 functional variables digitized at 183 points. Nathalie Villa-Vialaneix | IS-SIR 6/26
  11. 11. Main facts obtained from a preliminary study R. Kpekou internship The study focused on the influence of the climate on the yield: 5 functional variables digitized at 183 points. Main result: Using summary of the variables (mean, sd...) on several weeks and an automatic aggregating procedure in a random forest method, led to obtain good accuracy in prediction. Nathalie Villa-Vialaneix | IS-SIR 6/26
  12. 12. Question and mathematical framework A functional regression problem: X: random variable (functional) & Y: random real variable E(Y|X)? Nathalie Villa-Vialaneix | IS-SIR 7/26
  13. 13. Question and mathematical framework A functional regression problem: X: random variable (functional) & Y: random real variable E(Y|X)? Data: n i.i.d. observations (xi, yi)i=1,...,n. xi is not perfectly known but sampled at (fixed) points xi = (xi(t1), . . . , xi(tp))T ∈ Rp . We denote: X =   xT 1 ... xT n   . Nathalie Villa-Vialaneix | IS-SIR 7/26
  14. 14. Question and mathematical framework A functional regression problem: X: random variable (functional) & Y: random real variable E(Y|X)? Data: n i.i.d. observations (xi, yi)i=1,...,n. xi is not perfectly known but sampled at (fixed) points xi = (xi(t1), . . . , xi(tp))T ∈ Rp . We denote: X =   xT 1 ... xT n   . Question: Find a model which is easily interpretable and points out relevant intervals for the prediction within the range of X. Nathalie Villa-Vialaneix | IS-SIR 7/26
  15. 15. Related works (variable selection in FDA) LASSO / L1 regularization in linear models [Ferraty et al., 2010, Aneiros and Vieu, 2014] (isolated evaluation points), [Matsui and Konishi, 2011] (selects elements of an expansion basis), [James et al., 2009] (sparsity on derivatives: piecewise constant predictors) [Fraiman et al., 2015] (blinding approach useable for various problems: PCA, regression...) [Gregorutti et al., 2015] adaptation of the importance of variables in random forest for groups of variables Nathalie Villa-Vialaneix | IS-SIR 8/26
  16. 16. Related works (variable selection in FDA) LASSO / L1 regularization in linear models [Ferraty et al., 2010, Aneiros and Vieu, 2014] (isolated evaluation points), [Matsui and Konishi, 2011] (selects elements of an expansion basis), [James et al., 2009] (sparsity on derivatives: piecewise constant predictors) [Fraiman et al., 2015] (blinding approach useable for various problems: PCA, regression...) [Gregorutti et al., 2015] adaptation of the importance of variables in random forest for groups of variables Our proposal: a semi-parametric (not entirely linear) model which selects relevant intervals combined with an automatic procedure to define the intervals. Nathalie Villa-Vialaneix | IS-SIR 8/26
  17. 17. Sommaire 1 Background and motivation 2 Presentation of SIR 3 Our proposal 4 Simulations Nathalie Villa-Vialaneix | IS-SIR 9/26
  18. 18. SIR in multidimensional framework SIR: a semi-parametric regression model for X ∈ Rp Y = F(aT 1 X, . . . , aT d X, ) for a1, . . . , ad ∈ Rp (to be estimated), F : Rd+1 → R, unknown, and , an error, independant from X. Standard assumption for SIR Y X | PA (X) in which A is the so-called EDR space, spanned by (ak )k=1,...,d. Nathalie Villa-Vialaneix | IS-SIR 10/26
  19. 19. Estimation Equivalence between SIR and eigendecomposition Nathalie Villa-Vialaneix | IS-SIR 11/26
  20. 20. Estimation Equivalence between SIR and eigendecomposition A is included in the space spanned by the first d Σ-orthogonal eigenvectors of the generalized eigendecomposition problem: Γa = λΣa, with Σ = E (X − E(X|Y)))T E(X|Y) and Γ = E E(X|Y)T E(X|Y) Nathalie Villa-Vialaneix | IS-SIR 11/26
  21. 21. Estimation Equivalence between SIR and eigendecomposition A is included in the space spanned by the first d Σ-orthogonal eigenvectors of the generalized eigendecomposition problem: Γa = λΣa, with Σ = E (X − E(X|Y)))T E(X|Y) and Γ = E E(X|Y)T E(X|Y) Estimation (when n > p) compute X = 1 n n i=1 xi and ˆΣ = 1 n XT (X − X) Nathalie Villa-Vialaneix | IS-SIR 11/26
  22. 22. Estimation Equivalence between SIR and eigendecomposition A is included in the space spanned by the first d Σ-orthogonal eigenvectors of the generalized eigendecomposition problem: Γa = λΣa, with Σ = E (X − E(X|Y)))T E(X|Y) and Γ = E E(X|Y)T E(X|Y) Estimation (when n > p) compute X = 1 n n i=1 xi and ˆΣ = 1 n XT (X − X) split the range of Y into H different slices: τ1, ... τH and estimate ˆE(X|Y) = 1 nh i: yi∈τh xi h=1,...,H , with nh = |{i : yi ∈ τh}|, ˆΓ = ˆE(X|Y)T DˆE(X|Y) with D = Diag n1 n , . . . , nH n Nathalie Villa-Vialaneix | IS-SIR 11/26
  23. 23. Estimation Equivalence between SIR and eigendecomposition A is included in the space spanned by the first d Σ-orthogonal eigenvectors of the generalized eigendecomposition problem: Γa = λΣa, with Σ = E (X − E(X|Y)))T E(X|Y) and Γ = E E(X|Y)T E(X|Y) Estimation (when n > p) compute X = 1 n n i=1 xi and ˆΣ = 1 n XT (X − X) split the range of Y into H different slices: τ1, ... τH and estimate ˆE(X|Y) = 1 nh i: yi∈τh xi h=1,...,H , with nh = |{i : yi ∈ τh}|, ˆΓ = ˆE(X|Y)T DˆE(X|Y) with D = Diag n1 n , . . . , nH n solving the eigendecomposition problem ˆΓa = λˆΣa gives the eigenvectors a1, . . . , ad ⇒ ˆA = (a1, . . . , ad), p × d Nathalie Villa-Vialaneix | IS-SIR 11/26
  24. 24. Equivalent formulations SIR as a regression problem [Li and Yin, 2008] shows that SIR is equivalent to the (double) minimization of E(A, C) = H h=1 ˆph Xh − X − ˆΣACh 2 for Xh = 1 nh i: yi∈τh , A a (p × d)-matrix and C a vector in Rd . Nathalie Villa-Vialaneix | IS-SIR 12/26
  25. 25. Equivalent formulations SIR as a regression problem [Li and Yin, 2008] shows that SIR is equivalent to the (double) minimization of E(A, C) = H h=1 ˆph Xh − X − ˆΣACh 2 for Xh = 1 nh i: yi∈τh , A a (p × d)-matrix and C a vector in Rd . Rk: Given A, C is obtained as the solution of an ordinary least square problem... Nathalie Villa-Vialaneix | IS-SIR 12/26
  26. 26. Equivalent formulations SIR as a regression problem [Li and Yin, 2008] shows that SIR is equivalent to the (double) minimization of E(A, C) = H h=1 ˆph Xh − X − ˆΣACh 2 for Xh = 1 nh i: yi∈τh , A a (p × d)-matrix and C a vector in Rd . Rk: Given A, C is obtained as the solution of an ordinary least square problem... SIR as a Canonical Correlation problem [Li and Nachtsheim, 2008] shows that SIR rewrites as the double optimisation problem maxaj,φ Cor(φ(Y), aT j X), where φ is any function R → R and (aj)j are Σ-orthonormal. Nathalie Villa-Vialaneix | IS-SIR 12/26
  27. 27. Equivalent formulations SIR as a regression problem [Li and Yin, 2008] shows that SIR is equivalent to the (double) minimization of E(A, C) = H h=1 ˆph Xh − X − ˆΣACh 2 for Xh = 1 nh i: yi∈τh , A a (p × d)-matrix and C a vector in Rd . Rk: Given A, C is obtained as the solution of an ordinary least square problem... SIR as a Canonical Correlation problem [Li and Nachtsheim, 2008] shows that SIR rewrites as the double optimisation problem maxaj,φ Cor(φ(Y), aT j X), where φ is any function R → R and (aj)j are Σ-orthonormal. Rk: The solution is shown to satisfy φ(y) = aT j E(X|Y = y) and aj is also obtained as the solution of the mean square error problem: min aj E φ(Y) − aT j X 2 Nathalie Villa-Vialaneix | IS-SIR 12/26
  28. 28. SIR in large dimensions: problem In large dimension (or in Functional Data Analysis), n < p and ˆΣ is ill-conditionned and does not have an inverse ⇒ Z = (X − InX T )ˆΣ−1/2 can not be computed. Nathalie Villa-Vialaneix | IS-SIR 13/26
  29. 29. SIR in large dimensions: problem In large dimension (or in Functional Data Analysis), n < p and ˆΣ is ill-conditionned and does not have an inverse ⇒ Z = (X − InX T )ˆΣ−1/2 can not be computed. Different solutions have been proposed in the litterature based on: prior dimension reduction (e.g., PCA) [Ferré and Yao, 2003] (in the framework of FDA) regularization (ridge...) [Li and Yin, 2008, Bernard-Michel et al., 2008] sparse SIR [Li and Yin, 2008, Li and Nachtsheim, 2008, Ni et al., 2005] Nathalie Villa-Vialaneix | IS-SIR 13/26
  30. 30. SIR in large dimensions: ridge penalty / L2-regularization of ˆΣ Following [Li and Yin, 2008] which shows that SIR is equivalent to the minimization of E2(A, C) = H h=1 ˆph Xh − X − ˆΣACh 2 , Nathalie Villa-Vialaneix | IS-SIR 14/26
  31. 31. SIR in large dimensions: ridge penalty / L2-regularization of ˆΣ Following [Li and Yin, 2008] which shows that SIR is equivalent to the minimization of E2(A, C) = H h=1 ˆph Xh − X − ˆΣACh 2 +µ2 H h=1 ˆph ACh 2 , [Bernard-Michel et al., 2008] propose to penalize by a ridge penalty in a high dimensional setting. Nathalie Villa-Vialaneix | IS-SIR 14/26
  32. 32. SIR in large dimensions: ridge penalty / L2-regularization of ˆΣ Following [Li and Yin, 2008] which shows that SIR is equivalent to the minimization of E2(A, C) = H h=1 ˆph Xh − X − ˆΣACh 2 +µ2 H h=1 ˆph ACh 2 , [Bernard-Michel et al., 2008] propose to penalize by a ridge penalty in a high dimensional setting. They also show that this problem is equivalent to finding the eigenvectors of the generalized eigenvalue problem ˆΓa = λ ˆΣ + µ2Ip a. Nathalie Villa-Vialaneix | IS-SIR 14/26
  33. 33. SIR in large dimensions: sparse versions Specific issue to introduce sparsity in SIR sparsity on a multiple-index model. Most authors use shrinkage approaches. First version: sparse penalization of the ridge solution If (ˆA, ˆC) are the solutions of the ridge SIR as described in the previous slide, [Ni et al., 2005, Li and Yin, 2008] propose to shrink this solution by minimizing Es,1(α) = H h=1 ˆph Xh − X − ˆΣDiag(α)ˆA ˆCh 2 + µ1 α L1 (regression formulation of SIR) Nathalie Villa-Vialaneix | IS-SIR 15/26
  34. 34. SIR in large dimensions: sparse versions Specific issue to introduce sparsity in SIR sparsity on a multiple-index model. Most authors use shrinkage approaches. Second version: [Li and Nachtsheim, 2008] derive the sparse optimization problem from the correlation formulation of SIR: min as j n i=1 Pˆaj (X|yi) − (as j )T xi 2 + µ1,j as j L1 , in which Pˆaj is the projection of ˆE(X|Y = yi) = Xh onto the space spanned by the solution of the ridge problem. Nathalie Villa-Vialaneix | IS-SIR 15/26
  35. 35. Characteristics of the different approaches and possible extensions [Li and Yin, 2008] [Li and Nachtsheim, 2008] sparsity on shrinkage coefficients estimates nb optimization pb 1 d sparsity common to all dims specific to each dim Nathalie Villa-Vialaneix | IS-SIR 16/26
  36. 36. Characteristics of the different approaches and possible extensions [Li and Yin, 2008] [Li and Nachtsheim, 2008] sparsity on shrinkage coefficients estimates nb optimization pb 1 d sparsity common to all dims specific to each dim Extension to block-sparse SIR (like in PCA)? Nathalie Villa-Vialaneix | IS-SIR 16/26
  37. 37. Sommaire 1 Background and motivation 2 Presentation of SIR 3 Our proposal 4 Simulations Nathalie Villa-Vialaneix | IS-SIR 17/26
  38. 38. IS-SIR: a two step approach Background: Back to the functional setting, we suppose that t1, ..., tp are split into D intervals I1, ..., ID. Nathalie Villa-Vialaneix | IS-SIR 18/26
  39. 39. IS-SIR: a two step approach Background: Back to the functional setting, we suppose that t1, ..., tp are split into D intervals I1, ..., ID. First step: Solve the ridge problem on the digitized functions (viewed as high dimensional vectors) to obtain ˆA and ˆC: min A,C H h=1 ˆph Xh − X − ˆΣACh 2 + µ2 H h=1 ˆph ACh 2 Nathalie Villa-Vialaneix | IS-SIR 18/26
  40. 40. IS-SIR: a two step approach Background: Back to the functional setting, we suppose that t1, ..., tp are split into D intervals I1, ..., ID. First step: Solve the ridge problem on the digitized functions (viewed as high dimensional vectors) to obtain ˆA and ˆC: min A,C H h=1 ˆph Xh − X − ˆΣACh 2 + µ2 H h=1 ˆph ACh 2 Second step: Sparse shrinkage using the intervals. If PˆA (E(X|Y = yi)) = (Xh − X)T ˆA for h st yi ∈ τh and if Pi = (P1 i , . . . , Pd i )T and Pj = (Pj 1 , . . . , Pj n)T , we solve: arg min α∈RD d j=1 Pj − (X∆(ˆaj)) α 2 + µ1 α L1 with ∆(ˆaj) the (p × D)-matrix such that ∆kl(ˆaj) = ˆajl if tl ∈ Ik and 0 otherwise. Nathalie Villa-Vialaneix | IS-SIR 18/26
  41. 41. IS-SIR: Characteristics uses the approach based on the correlation formulation (because the dimensionality of the optimization problem is smaller); uses a shrinkage approach and optimizes shrinkage coefficients in a single optimization problem; handles functional setting by penalizing entire intervals and not just isolated points. Nathalie Villa-Vialaneix | IS-SIR 19/26
  42. 42. Parameter estimation H (number of slices): usually, SIR is known to be not very sensitive to the number of slices (> d + 1). We took H = 10 (i.e., 10/30 observations per slice); Nathalie Villa-Vialaneix | IS-SIR 20/26
  43. 43. Parameter estimation H (number of slices): usually, SIR is known to be not very sensitive to the number of slices (> d + 1). We took H = 10 (i.e., 10/30 observations per slice); µ2 and d (ridge estimate ˆA): L-fold CV for µ2 (for a d0 large enough) Note that GCV as described in [Li and Yin, 2008] can not be used since the current version of the L2 penalty involves the use of an estimate of Σ−1 . Nathalie Villa-Vialaneix | IS-SIR 20/26
  44. 44. Parameter estimation H (number of slices): usually, SIR is known to be not very sensitive to the number of slices (> d + 1). We took H = 10 (i.e., 10/30 observations per slice); µ2 and d (ridge estimate ˆA): L-fold CV for µ2 (for a d0 large enough) using again L-fold CV, ∀ d = 1, . . . , d0, an estimate of R(d) = d − E Tr Πd ˆΠd , in which Πd and ˆΠd are the projector onto the first d dimensions of the EDR space and its estimate, is derived similarly as in [Liquet and Saracco, 2012]. The evolution of ˆR(d) versus d is studied to select a relevant d. Nathalie Villa-Vialaneix | IS-SIR 20/26
  45. 45. Parameter estimation H (number of slices): usually, SIR is known to be not very sensitive to the number of slices (> d + 1). We took H = 10 (i.e., 10/30 observations per slice); µ2 and d (ridge estimate ˆA): L-fold CV for µ2 (for a d0 large enough) using again L-fold CV, ∀ d = 1, . . . , d0, an estimate of R(d) = d − E Tr Πd ˆΠd , in which Πd and ˆΠd are the projector onto the first d dimensions of the EDR space and its estimate, is derived similarly as in [Liquet and Saracco, 2012]. The evolution of ˆR(d) versus d is studied to select a relevant d. µ1 (LASSO) glmnet is used, in which µ1 is selected by CV along the regularization path. Nathalie Villa-Vialaneix | IS-SIR 20/26
  46. 46. An automatic approach to define intervals 1 Initial state: ∀ k = 1, . . . , p, τk = {tk } Nathalie Villa-Vialaneix | IS-SIR 21/26
  47. 47. An automatic approach to define intervals 1 Initial state: ∀ k = 1, . . . , p, τk = {tk } 2 Iterate along the regularization path, select three values for µ1: Nathalie Villa-Vialaneix | IS-SIR 21/26
  48. 48. An automatic approach to define intervals 1 Initial state: ∀ k = 1, . . . , p, τk = {tk } 2 Iterate along the regularization path, select three values for µ1: P% of the coefficients are zero, P% of the coefficients are non zero, best GCV. define: D− (“strong zeros”) and D+ (“strong non zeros”) Nathalie Villa-Vialaneix | IS-SIR 21/26
  49. 49. An automatic approach to define intervals 1 Initial state: ∀ k = 1, . . . , p, τk = {tk } 2 Iterate define: D− (“strong zeros”) and D+ (“strong non zeros”) merge consecutive “strong zeros” (or “strong non zeros”) or “strong zeros” (resp. “strong non zeros”) separated by a few numbers of intervals which are of undetermined type. Until no more iterations can be performed. Nathalie Villa-Vialaneix | IS-SIR 21/26
  50. 50. An automatic approach to define intervals 1 Initial state: ∀ k = 1, . . . , p, τk = {tk } 2 Iterate define: D− (“strong zeros”) and D+ (“strong non zeros”) merge consecutive “strong zeros” (or “strong non zeros”) or “strong zeros” (resp. “strong non zeros”) separated by a few numbers of intervals which are of undetermined type. Until no more iterations can be performed. 3 Output: Collection of models (first with p intervals, last with 1), M∗ D (optimal for GCV) and corresponding GCVD versus D (number of intervals). Nathalie Villa-Vialaneix | IS-SIR 21/26
  51. 51. An automatic approach to define intervals 1 Initial state: ∀ k = 1, . . . , p, τk = {tk } 2 Iterate define: D− (“strong zeros”) and D+ (“strong non zeros”) merge consecutive “strong zeros” (or “strong non zeros”) or “strong zeros” (resp. “strong non zeros”) separated by a few numbers of intervals which are of undetermined type. Until no more iterations can be performed. 3 Output: Collection of models (first with p intervals, last with 1), M∗ D (optimal for GCV) and corresponding GCVD versus D (number of intervals). Final solution: Minimize GCVD over D. Nathalie Villa-Vialaneix | IS-SIR 21/26
  52. 52. Sommaire 1 Background and motivation 2 Presentation of SIR 3 Our proposal 4 Simulations Nathalie Villa-Vialaneix | IS-SIR 22/26
  53. 53. Simulation framework Data generated with: Y = d j=1 log X, aj with X(t) = Z(t) + in which Z is a Gaussian process with mean µ(t) = −5 + 4t − 4t2 and the Matern 3/2 covariance function with parameters σ = 0.1 and θ = 0.2/ √ 3, is a centered Gaussian variable independant of Z, with standard deviation 0.1.; aj = sin t(2+j)π 2 − (j−1)π 3 IIj (t) two models: (M1), d = 1, I1 = [0.2, 0.4]. For (M2), d = 3 and I1 = [0, 0.1], I2 = [0.5, 0.65] and I3 = [0.65, 0.78]. Nathalie Villa-Vialaneix | IS-SIR 23/26
  54. 54. Simulation framework Nathalie Villa-Vialaneix | IS-SIR 23/26
  55. 55. Simulation framework Nathalie Villa-Vialaneix | IS-SIR 23/26
  56. 56. Ridge step (model M1) Selection of µ2: µ2 = 1 Nathalie Villa-Vialaneix | IS-SIR 24/26
  57. 57. Ridge step (model M1) Selection of d: d = 1 Nathalie Villa-Vialaneix | IS-SIR 24/26
  58. 58. Definition of the intervals D = 200 (initial state) 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.8 a^ 1 Nathalie Villa-Vialaneix | IS-SIR 25/26
  59. 59. Definition of the intervals D = 147 (retained solution) 0.2 0.4 0.6 0.8 1.0 0.000.020.040.060.08 a^ 1 Nathalie Villa-Vialaneix | IS-SIR 25/26
  60. 60. Definition of the intervals D = 43 0.2 0.4 0.6 0.8 1.0 −0.050.000.05 a^ 1 Nathalie Villa-Vialaneix | IS-SIR 25/26
  61. 61. Definition of the intervals D = 5 0.2 0.4 0.6 0.8 1.0 −0.04−0.020.000.020.040.060.08 a^ 1 Nathalie Villa-Vialaneix | IS-SIR 25/26
  62. 62. Definition of the intervals q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q 0 50 100 150 200 0.0190.0200.0210.0220.023 Number of intervals CVerror Nathalie Villa-Vialaneix | IS-SIR 25/26
  63. 63. Conclusion IS-SIR: sparse dimension reduction model adapted to functional framework; fully automated definition of relevant intervals in the range of the predictors Nathalie Villa-Vialaneix | IS-SIR 26/26
  64. 64. Conclusion IS-SIR: sparse dimension reduction model adapted to functional framework; fully automated definition of relevant intervals in the range of the predictors Perspective: application to real data block-wise sparse SIR? Nathalie Villa-Vialaneix | IS-SIR 26/26
  65. 65. Aneiros, G. and Vieu, P. (2014). Variable in infinite-dimensional problems. Statistics and Probability Letters, 94:12–20. Bernard-Michel, C., Gardes, L., and Girard, S. (2008). A note on sliced inverse regression with regularizations. Biometrics, 64(3):982–986. Casadebaig, P., Guilioni, L., Lecoeur, J., Christophe, A., Champolivier, L., and Debaeke, P. (2011). SUNFLO, a model to simulate genotype-specific performance of the sunflower crop in contrasting environments. Agricultural and Forest Meteorology, 151(2):163–178. Ferraty, F., Hall, P., and Vieu, P. (2010). Most-predictive design points for functiona data predictors. Biometrika, 97(4):807–824. Ferré, L. and Yao, A. (2003). Functional sliced inverse regression analysis. Statistics, 37(6):475–488. Fraiman, R., Gimenez, Y., and Svarc, M. (2015). Feature selection for functional data. Journal of Multivariate Analysis. In Press. Gregorutti, B., Michel, B., and Saint-Pierre, P. (2015). Grouped variable importance with random forests and application to multiple functional data analysis. Computational Statistics and Data Analysis, 90:15–35. James, G., Wang, J., and Zhu, J. (2009). Functional linear regression that’s interpretable. Annals of Statistics, 37(5A):2083–2108. Li, L. and Nachtsheim, C. (2008). Nathalie Villa-Vialaneix | IS-SIR 26/26
  66. 66. Sparse sliced inverse regression. Technometrics, 48(4):503–510. Li, L. and Yin, X. (2008). Sliced inverse regression with regularizations. Biometrics, 64:124–131. Liquet, B. and Saracco, J. (2012). A graphical tool for selecting the number of slices and the dimension of the model in SIR and SAVE approches. Computational Statistics, 27(1):103–125. Matsui, H. and Konishi, S. (2011). Variable selection for functional regression models via the l1 regularization. Computational Statistics and Data Analysis, 55(12):3304–3310. Ni, L., Cook, D., and Tsai, C. (2005). A note on shrinkage sliced inverse regression. Biometrika, 92(1):242–247. Nathalie Villa-Vialaneix | IS-SIR 26/26

×