Karhunen-Loève decomposition of Gaussian measures on Banach spaces

Transcription

1 Karhunen-Loève decomposition of Gaussian measures on Banach spaces Jean-Charles Croix Génie Mathématique et Industriel (GMI) First workshop on Gaussian processes at Saint-Etienne - November 2017, 7th. 1 / 42

2 Contents 1 Motivation 2 Karhunen-Loève decomposition in Hilbert spaces Gaussian vectors in R n Gaussian elements in Hilbert spaces 3 Karhunen-Loève decomposition in Banach spaces 4 Particular case of (R n,. ) 5 Particular case of C(K), K metric and compact. Continuous Gaussian processes Illustration on Brownian motion Summability of λ 6 Conclusion and open questions 7 References 2 / 42

3 Motivation 3 / 42

4 Motivation The problem of representation of Gaussian elements in linear series is used in: 1 Simulation (e.g. truncated Karhunen-Loève series), 2 Approximation and Dimension reduction (e.g. PCA or POD), 3 Optimal quantization, 4 Bayesian inverse problems. If the existence of an optimal basis is well known in Hilbert spaces it s not always explicit (eigenvalue problem),... it s not the case in Banach spaces. What if we are interested in non-hilbertian norms? 4 / 42

5 Motivation In the case of a continuous Gaussian process on [0, 1] we may inject it in L 2 ([0, 1], dx) and use Hilbertian geometry Basis functions f0 f1 f2 f3 f Figure: Karhunen-Loève basis of Brownian motion in L 2 ([0, 1], dx). 5 / 42

6 Motivation or tackle the problem directly in C([0, 1]) f0 f1 f2 f3 f4 f5 f6 f7 Basis functions Figure: Brownian motion basis functions in C([0, 1]) (Paul Levy s construction)....in the hope of better approximation w.r.t. supremum norm! 6 / 42

7 Karhunen-Loève decomposition in Hilbert spaces 7 / 42

8 Gaussian vector in Euclidian spaces Consider the euclidian space R n with canonical inner product.,.. Gaussian random vector Let (Ω, F, P) a probability space and X : (Ω, F, P) (R n, B(R n )) a measurable mapping, then X is a Gaussian vector if and only if y R n, X, y is a Gaussian random variable. 8 / 42

9 Gaussian vector in Euclidian spaces Consider the euclidian space R n with canonical inner product.,.. Gaussian random vector Let (Ω, F, P) a probability space and X : (Ω, F, P) (R n, B(R n )) a measurable mapping, then X is a Gaussian vector if and only if y R n, X, y is a Gaussian random variable. Covariance Given a (centered) Gaussian random vector X, define the bilinear form: x, y R n, Cov(x, y) = E[ X, x X, y ], (1) which uniquely defines Σ : R n R n such that: x, y R n, Σx, y = Cov(x, y). (2) 8 / 42

10 Cameron-Martin and Gaussian spaces Considering the Gaussian vector X gives a fundamental structure: Gaussian space: Vect( X, e k, k [1, n]) L 2 (P) Cameron-Martin space H X = Range(Σ) equipped with.,. X = Σ 1.,. Loève isometry The following application: is an isometry. X, x L 2 (P) Σx H X (3) 9 / 42

11 Representation of Gaussian vectors in R n For any orthonormal basis (e k ) in R n, we can write: n X (ω) = X (ω), e k e k a.s. (4) k=1 10 / 42

12 Representation of Gaussian vectors in R n For any orthonormal basis (e k ) in R n, we can write: n X (ω) = X (ω), e k e k a.s. (4) k=1 X (ω), e i, X (ω), e j are independent if and only if Cov(e i, e j ) = Σe i, e j = Σe i, Σe j X = / 42

13 Representation of Gaussian vectors in R n For any orthonormal basis (e k ) in R n, we can write: n X (ω) = X (ω), e k e k a.s. (4) k=1 X (ω), e i, X (ω), e j are independent if and only if Cov(e i, e j ) = Σe i, e j = Σe i, Σe j X = 0. For any orthonormal basis (x k ) in H X : X (ω) = dim(h X ) k=1 X (ω), x k X x k (5) 10 / 42

14 Representation of Gaussian vectors in R n For any orthonormal basis (e k ) in R n, we can write: n X (ω) = X (ω), e k e k a.s. (4) k=1 X (ω), e i, X (ω), e j are independent if and only if Cov(e i, e j ) = Σe i, e j = Σe i, Σe j X = 0. For any orthonormal basis (x k ) in H X : X (ω) = dim(h X ) k=1 X (ω), x k X x k (5) The Spectral theorem (on covariance operator) exhibits a particular (Karhunen-Loève) basis (bi-orthogonal and Σh k = λ k h k ): X (ω) = dim(h X ) k=1 λk X (ω), h k h k (6) 10 / 42

15 Representation of Gaussian vectors in R n For any orthonormal basis (e k ) in R n, we can write: n X (ω) = X (ω), e k e k a.s. (4) k=1 X (ω), e i, X (ω), e j are independent if and only if Cov(e i, e j ) = Σe i, e j = Σe i, Σe j X = 0. For any orthonormal basis (x k ) in H X : X (ω) = dim(h X ) k=1 X (ω), x k X x k (5) The Spectral theorem (on covariance operator) exhibits a particular (Karhunen-Loève) basis (bi-orthogonal and Σh k = λ k h k ): X (ω) = dim(h X ) k=1 λk X (ω), h k h k (6) Note P k the linear projector on the k-th biggest eigenvalues, then: k n, min rank(p)=k E[ X PX 2 ] = E[ X P k X 2 ] = λ k λ dim(hx ) 10 / 42

16 Representation of Gaussian vectors in R n e 2 λ1 λ2 B HX (0, 1) e 1 B E (0, 1) Figure: Karhunen-Loève basis in dimension / 42

17 Gaussian element in Hilbert spaces Consider the (real) Hilbert space H with inner product.,.. Gaussian random element Let (Ω, F, P) a probability space and X : (Ω, F, P) (H, B(H)) a measurable mapping, then X is a Gaussian element if and only if y H, X, y is a Gaussian random variable. Covariance Given a Gaussian random element X, define the bilinear form: x, y H, Cov(x, y) = E[ X, x X, y ], (8) and the associated covariance operator C : H H such that: x, y H, Cx, y = Cov(x, y). (9) 12 / 42

18 Representation of Gaussian elements in Hilbert spaces The covariance operator C : H H is positive, symmetric and trace-class (see Vakhania 1987). 13 / 42

19 Representation of Gaussian elements in Hilbert spaces The covariance operator C : H H is positive, symmetric and trace-class (see Vakhania 1987). The Cameron-Martin space H X is the completion of Range(C) w.r.t. x, y C = C 1 x, y (it s a proper subspace of H, H X H, see Vakhania 1987). 13 / 42

20 Representation of Gaussian elements in Hilbert spaces The covariance operator C : H H is positive, symmetric and trace-class (see Vakhania 1987). The Cameron-Martin space H X is the completion of Range(C) w.r.t. x, y C = C 1 x, y (it s a proper subspace of H, H X H, see Vakhania 1987). For any Hilbert basis (e k ) in H we can write: X (ω) = k 0 X (ω), e k e k a.s. (10) 13 / 42

21 Representation of Gaussian elements in Hilbert spaces The covariance operator C : H H is positive, symmetric and trace-class (see Vakhania 1987). The Cameron-Martin space H X is the completion of Range(C) w.r.t. x, y C = C 1 x, y (it s a proper subspace of H, H X H, see Vakhania 1987). For any Hilbert basis (e k ) in H we can write: X (ω) = k 0 X (ω), e k e k a.s. (10) For any basis (x k ) in H X and (ξ k ) i.i.d. N (0, 1): X (ω) d = k 0 ξ k (ω)x k (11) 13 / 42

22 Representation of Gaussian elements in Hilbert spaces The covariance operator C : H H is positive, symmetric and trace-class (see Vakhania 1987). The Cameron-Martin space H X is the completion of Range(C) w.r.t. x, y C = C 1 x, y (it s a proper subspace of H, H X H, see Vakhania 1987). For any Hilbert basis (e k ) in H we can write: X (ω) = k 0 X (ω), e k e k a.s. (10) For any basis (x k ) in H X and (ξ k ) i.i.d. N (0, 1): X (ω) d = k 0 ξ k (ω)x k (11) Spectral theorem applies and exhibits a (Karhunen-Loève) bi-orthogonal basis (h k ): X (ω) = k 0 λk ξ k (ω)h k a.s. 13 / 42

23 Representation of Gaussian elements in Hilbert spaces The covariance operator C : H H is positive, symmetric and trace-class (see Vakhania 1987). The Cameron-Martin space H X is the completion of Range(C) w.r.t. x, y C = C 1 x, y (it s a proper subspace of H, H X H, see Vakhania 1987). For any Hilbert basis (e k ) in H we can write: X (ω) = k 0 X (ω), e k e k a.s. (10) For any basis (x k ) in H X and (ξ k ) i.i.d. N (0, 1): X (ω) d = k 0 ξ k (ω)x k (11) Spectral theorem applies and exhibits a (Karhunen-Loève) bi-orthogonal basis (h k ): X (ω) = k 0 λk ξ k (ω)h k a.s. Eckart-Young theorem is still valid (functional PCA,...) k > 0, min E[ X PX rank(p)=k 2 H] = E[ X P k X 2 H] = λ i (12) i>k 13 / 42

24 Karhunen-Loève decomposition in Banach spaces 14 / 42

25 Gaussian element in Banach spaces Consider the (real) Banach space E with duality pairing.,. E,E. Gaussian random element Let (Ω, F, P) a probability space and X : (Ω, F, P) (E, B(E)) a measurable mapping, then X is a Gaussian element if and only if f E, X, f E,E is a Gaussian random variable. Covariance Given a Gaussian random element X, define the bilinear form: f, g E, Cov(f, g) = E[ X, f E,E X, g E,E ], (13) and the associated covariance operator C : E E (see Vakhania 1987 or Bogachev 1998) such that: f, g E, Cf, g E,E = Cov(f, g). (14) 15 / 42

26 Representation of Gaussian elements in Banach spaces The covariance operator C : E E is positive, symmetric and nuclear (see Vakhania 1987). 16 / 42

27 Representation of Gaussian elements in Banach spaces The covariance operator C : E E is positive, symmetric and nuclear (see Vakhania 1987). The space H X is the completion of Range(C) w.r.t. x, y X = y, C 1 x E,E (it s a proper subspace of E, H X E, see Vakhania 1987, Bogachev 1998). 16 / 42

28 Representation of Gaussian elements in Banach spaces The covariance operator C : E E is positive, symmetric and nuclear (see Vakhania 1987). The space H X is the completion of Range(C) w.r.t. x, y X = y, C 1 x E,E (it s a proper subspace of E, H X E, see Vakhania 1987, Bogachev 1998). For any Hilbert basis (x k ) in H X and (ξ k ) i.i.d. N (0, 1), we have: X d = k 1 ξ k x k (15) 16 / 42

29 Representation of Gaussian elements in Banach spaces The covariance operator C : E E is positive, symmetric and nuclear (see Vakhania 1987). The space H X is the completion of Range(C) w.r.t. x, y X = y, C 1 x E,E (it s a proper subspace of E, H X E, see Vakhania 1987, Bogachev 1998). For any Hilbert basis (x k ) in H X and (ξ k ) i.i.d. N (0, 1), we have: X d = k 1 ξ k x k (15) C : E E No Spectral theorem. 16 / 42

30 Representation of Gaussian elements in Banach spaces The covariance operator C : E E is positive, symmetric and nuclear (see Vakhania 1987). The space H X is the completion of Range(C) w.r.t. x, y X = y, C 1 x E,E (it s a proper subspace of E, H X E, see Vakhania 1987, Bogachev 1998). For any Hilbert basis (x k ) in H X and (ξ k ) i.i.d. N (0, 1), we have: X d = k 1 ξ k x k (15) C : E E No Spectral theorem. We know that (Bogachev 1998, Vakahnia 1991): (x k ) Range(C) Hilbert basis in H X, (x k ) H X such that k 0 h k 2 E < +, 16 / 42

31 Representation of Gaussian elements in Banach spaces The covariance operator C : E E is positive, symmetric and nuclear (see Vakhania 1987). The space H X is the completion of Range(C) w.r.t. x, y X = y, C 1 x E,E (it s a proper subspace of E, H X E, see Vakhania 1987, Bogachev 1998). For any Hilbert basis (x k ) in H X and (ξ k ) i.i.d. N (0, 1), we have: X d = k 1 ξ k x k (15) C : E E No Spectral theorem. We know that (Bogachev 1998, Vakahnia 1991): (x k ) Range(C) Hilbert basis in H X, (x k ) H X such that k 0 h k 2 E < +, If the basis (x k ) is in Range(C) with x k = Cx k, then: X (ω) = k 0 X (ω), x k E,E x k (16) 16 / 42

32 Representation of Gaussian elements in Banach spaces Factorization of covariance operators (Vakhania 1987, Bogachev 1998). Let C be the covariance operator of a Gaussian element, then: C = SS, (17) where S : H E is a bounded operator and H a Hilbert space. 17 / 42

33 Representation of Gaussian elements in Banach spaces Factorization of covariance operators (Vakhania 1987, Bogachev 1998). Let C be the covariance operator of a Gaussian element, then: C = SS, (17) where S : H E is a bounded operator and H a Hilbert space. Examples: In the Hilbert case, S = C 1 2, S : H X E the inclusion map, S : f E L2 (P) E[f (X )X ] E (S is the injection from E to L 2 ) 17 / 42

34 Representation of Gaussian elements in Banach spaces Factorization of covariance operators (Vakhania 1987, Bogachev 1998). Let C be the covariance operator of a Gaussian element, then: C = SS, (17) where S : H E is a bounded operator and H a Hilbert space. Examples: In the Hilbert case, S = C 1 2, S : H X E the inclusion map, S : f E L2 (P) E[f (X )X ] E (S is the injection from E to L 2 ) (Luschgy and Pagès 2009) Let C = SS with S : H E. Then for any basis (h n ) in H: X d = k 0 ξ k Sh k, (18) where (ξ k ) are i.i.d. N (0, 1). 17 / 42

35 Representation of Gaussian elements in Banach spaces This methodology has been widely used: When E = C([0, 1]), S : f E L 2 ([0, 1], dx)... We will now give a different methodology, mimicking the Hilbert case, to construct a Hilbert basis in H X. We will proceed as follows: 1 Find directions of maximum variance, 2 Choose a right notion of orthogonality to iterate, 3 Study the asymptotics. 18 / 42

36 Decomposition of the Cameron-Martin space 1 For any (Gaussian) covariance operator, f E Cov(f, f ) R + may be interpreted as a Rayleigh quotient, is weakly sequentially continuous, is quadratic. 19 / 42

37 Decomposition of the Cameron-Martin space 1 For any (Gaussian) covariance operator, f E Cov(f, f ) R + may be interpreted as a Rayleigh quotient, is weakly sequentially continuous, is quadratic. 2 f 0 B E (0, 1) (non-unique) such that Cov(f 0, f 0 ) = max f E 1 Cov(f, f ) = λ 0 (Banach-Alaoglu), 19 / 42

38 Decomposition of the Cameron-Martin space 1 For any (Gaussian) covariance operator, f E Cov(f, f ) R + may be interpreted as a Rayleigh quotient, is weakly sequentially continuous, is quadratic. 2 f 0 B E (0, 1) (non-unique) such that Cov(f 0, f 0 ) = max f E 1 Cov(f, f ) = λ 0 (Banach-Alaoglu), 3 If λ 0 > 0, let x 0 E such that Cf 0 = λ 0 x 0, then P 0 : x E x, f 0 E,E x 0 is a projector of unit norm. 19 / 42

39 Decomposition of the Cameron-Martin space 1 For any (Gaussian) covariance operator, f E Cov(f, f ) R + may be interpreted as a Rayleigh quotient, is weakly sequentially continuous, is quadratic. 2 f 0 B E (0, 1) (non-unique) such that Cov(f 0, f 0 ) = max f E 1 Cov(f, f ) = λ 0 (Banach-Alaoglu), 3 If λ 0 > 0, let x 0 E such that Cf 0 = λ 0 x 0, then P 0 : x E x, f 0 E,E x 0 is a projector of unit norm. 4 X = (P 0 X, (I P 0 )X ). 19 / 42

40 Decomposition of the Cameron-Martin space 1 For any (Gaussian) covariance operator, f E Cov(f, f ) R + may be interpreted as a Rayleigh quotient, is weakly sequentially continuous, is quadratic. 2 f 0 B E (0, 1) (non-unique) such that Cov(f 0, f 0 ) = max f E 1 Cov(f, f ) = λ 0 (Banach-Alaoglu), 3 If λ 0 > 0, let x 0 E such that Cf 0 = λ 0 x 0, then P 0 : x E x, f 0 E,E x 0 is a projector of unit norm. 4 X = (P 0 X, (I P 0 )X ). 5 Iterate on X 1 = (I P 0 )X. 19 / 42

41 Decomposition of the Cameron-Martin space 1 For any (Gaussian) covariance operator, f E Cov(f, f ) R + may be interpreted as a Rayleigh quotient, is weakly sequentially continuous, is quadratic. 2 f 0 B E (0, 1) (non-unique) such that Cov(f 0, f 0 ) = max f E 1 Cov(f, f ) = λ 0 (Banach-Alaoglu), 3 If λ 0 > 0, let x 0 E such that Cf 0 = λ 0 x 0, then P 0 : x E x, f 0 E,E x 0 is a projector of unit norm. 4 X = (P 0 X, (I P 0 )X ). 5 Iterate on X 1 = (I P 0 )X. Bay & Croix 2017 (λ n ) is non-increasing, λ n 0 and ( λ k x k ) is a Hilbert basis in H X. 19 / 42

42 Decomposition of the Cameron-Martin space 1 For any (Gaussian) covariance operator, f E Cov(f, f ) R + may be interpreted as a Rayleigh quotient, is weakly sequentially continuous, is quadratic. 2 f 0 B E (0, 1) (non-unique) such that Cov(f 0, f 0 ) = max f E 1 Cov(f, f ) = λ 0 (Banach-Alaoglu), 3 If λ 0 > 0, let x 0 E such that Cf 0 = λ 0 x 0, then P 0 : x E x, f 0 E,E x 0 is a projector of unit norm. 4 X = (P 0 X, (I P 0 )X ). 5 Iterate on X 1 = (I P 0 )X. Bay & Croix 2017 (λ n ) is non-increasing, λ n 0 and ( λ k x k ) is a Hilbert basis in H X. We will note x0 = f 0 and xn = f n n 1 k=0 x k, f n E,E xk P n = n k=0 x k x k (similar to Gram-Schmidt). such that 19 / 42

43 Properties of the decomposition Few comments about the previous construction... Properties This construction gives the following properties: X (ω) = k 0 X (ω), x k E,E x k a.s. C = SS with S : f E k 0 λ k x k, f E,E x k Note C n = n k=0 λ kx k x k then C C n L(E,E) = λ n+1. Recovers Karhunen-Loève basis in the Hilbert case. (x k, x k ) E E is bi-orthogonal. Vect(x k, k 0) E = H X E, k N, f k E = x k E = 1. Remarks k 0 λ k need not be finite (not a nuclear representation of C). Optimality seems out of reach (Rate optimality?). 20 / 42

44 Particular case of (R n,. ) 21 / 42

45 Application in R n Here, E = (R n,. 1 ) and (x, f ) E E, x, f E,E Suppose a centered dataset (y i ) i [1,p] (R n ) p, Form the usual covariance matrix Σ M n (R), Find the direction of maximal variance f 0 R n : = n i=1 x if i. λ 0 = f T 0 Σf 0 = max i [1,n] Σ (i,i) with f 0 = (0,..., 0, 1, 0,..., 0) (19) If λ 0 > 0 then x 0 = Σf0 λ 0. Σ 1 = Σ λ 0 x 0 x T 0 Iterate. 22 / 42

46 Application in R n Residuals norm as r k = 1 n n i=1 x i k j=0 x i, xj E,E x j, Projections norm as p k = 1 n n i=1 k j=0 x i, xj E,E x j. Figure: Residuals (decreasing) and projections (increasing) norm (blue for L2-PCA, red for Banach PCA) 23 / 42

47 Particular case of C(K), K metric and compact. 24 / 42

48 E = C(K) with K metric compact. Let X be a Gaussian element such that X C(K) almost-surely and suppose that k : (s, t) K 2 Cδ s, δ t E,E = Cov(X s, X t ) R (20) is continuous. The Cameron-Martin space H X coincides here with the Reproducing Kernel Hilbert space (RKHS). The previous decomposition becomes: 1 Set n = 0 and k n = k, 2 Find x n K such that k n (x n, x n ) = max x K k n (x, x), 3 If λ n = k n (x n, x n ) > 0 then k n+1 (s, t) = k n (s, t) kn(s,xn)kn(xn,t) λ n, 4 n n / 42

49 Illustration on Brownian motion: Step 1 Figure: Left: Brownian motion samples and point-wise (symmetrized) variance. Right: First basis function x 0(t) = t associated to f 0 = δ / 42

50 Illustration on Brownian motion: Step 2 Figure: Left: Brownian bridge samples and point-wise (symmetrized) variance. Right: Second basis function x 1(t) = min(t, 0.5) 0.5t, associated to f 1 = δ / 42

51 Illustration on Brownian motion: Step 3 Figure: Left: Conditional Brownian motion samples and point-wise (symmetrized) variance. Right: Third and fourth basis functions. 28 / 42

52 A word on summability The proposed decomposition need not be a nuclear representation of C, that is k 0 λ k [0, + ]. Indeed, the Brownian case shows us: k 0 λ k = = + (21) 32 However, a simple transformation gives a nuclear representation in this case: f 2 1 = f 2 1 f 2 2 Cov( f 2 1, f ) = Cov(f 2 1, f 1 2 f 2 2 = f f 2 2 Cov( f 2 2, 2 f 2 2 ) = Cov(f 2 2, f 2 2 Doing similar transformations at each step gives: k 0 2 ) 2 ) = 1 8 = 1 8 (22) (23) λ k = = 2 (24) 256 Remark that E[ k n ξ kh k 2 ] k n λ k, thus the approximation error is exponentially decreasing! 29 / 42

53 Conclusion and open questions 30 / 42

54 Conclusion and open questions Conclusions New practical method to represent Gaussian elements, based on covariance operator. Numerical solution in the case of E = C(K) (K metric and compact). Rediscovers Paul Levy s construction of Brownian motion. Open question & future work Other properties of (x k, x k ) E E? Rate optimality of finite dimensional approximations in C(K)? Multiplicity of decomposition elements? Links with approximation numbers (ex: l(x ) = E[ X 2 ] 1 2 ) approximation theory (s-numbers)? Properties of Banach spaces? Optimality for projection norm? Dual problem? Interpretation of k 0 λ k when it is finite? Evolution of x k E? 31 / 42

55 References 32 / 42

56 References X. Bay, J.-C. Croix. Karhunen-Loève decomposition of Gaussian measures on Banach spaces. ArXiV, H. Luschgy, G. Pagès. Expansions For Gaussian Processes And Parseval Frames. Electronic Journal of Probability, 14, 2009, N. Vakhania. Canonical Factorization of Gaussian covariance operators and some of its applications. Theory of Probability and Its Applications, V. Bogachev. Gaussian Measures. Mathematical Surveys and Monographs vol. 62, AMS, N. Vakhania, V. Tarieladze, S. Chobanyan. Probability Distributions on Banach spaces. Mathematics and Its Applications, Springer Netherlands, / 42

57 Appendix 34 / 42

58 Examples with different kernels Basis functions 1.2 f0 f1 f2 f3 1.0 f4 f5 f6 f Figure: Wiener measure basis functions. 35 / 42

59 Examples with different kernels 10 1 Variances Variances Figure: Wiener measure basis variances. 36 / 42

60 Examples with different kernels 1.5 Basis functions f0 f1 0.5 f2 f3 f4 f5 f6 f Figure: k(s, t) = exp ) ( (s t)2 basis functions / 42

61 Examples with different kernels 10 1 Variances Variances Figure: k(s, t) = exp ) ( (s t)2 basis variances / 42

62 Examples with different kernels Basis functions x0 x1 1.0 x2 x3 x4 x5 x6 x7 x8 x9 x10 x Figure: Matern 3/2 kernel basis functions. 39 / 42

63 Examples with different kernels 10 1 Variances Variances Figure: Matern 3/2 kernel basis variances. 40 / 42

64 Examples with different kernels Basis functions 1.2 f0 f1 f2 f3 1.0 f4 f5 f6 f Figure: Fractional brownian motion (H = 75%) basis functions. 41 / 42

65 Examples with different kernels Basis functions 1.2 f0 f1 f2 f3 1.0 f4 f5 f6 f Figure: Fractional brownian motion (H = 25%) basis functions. 42 / 42