A New Wavelet-based Expansion of a Random Process

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1 Columbia International Publishing Journal of Applied Mathematics and Statistics doi: /jams esearch Article A New Wavelet-based Expansion of a andom Process Ievgen Turchyn 1* eceived: 28 July 2016; Published online: 1 October 2016 The author Published with open access at Abstract There has been obtained an expansion of a second-order stochastic process into a system of wavelet-based functions. This expansion is used for simulation of a sub-gaussian stochastic process with given accuracy and reliability. Keywords: Wavelets; Sub-Gaussian stochastic processes; Expansion 1. Introduction One noteworthy class of expansions of random processes is formed by wavelet-based expansions. Wavelet-based expansions with uncorrelated terms are especially important since they are convenient for simulation and approximation of random processes. Different wavelet-based expansions of stochastic processes were studied by Walter and Zhang (1994), Meyer et al. (1999), Pipiras (2004), Zhao et al. (2004), Didier and Pipiras (2008). Kozachenko and Turchyn (2008) obtained a theorem about a wavelet-based expansion for a wide class of stochastic process. Namely, a centered second-order process X(t) with the correlation function (u(t, ) L 2 ()) can be represented as the series which converges in L 2 (Ω) for any fixed t, where (t, s) = u(t, y)u(s, y)dy X(t) = k Z ξ 0k a 0k (t) + j=0 k Z η jk b jk (t) (1) *Corresponding evgturchyn@gmail.com 1 Oles Honchar Dnipropetrovsk National University 137

2 a 0k (t) = 1 2π u(t, y)φ 0k (y)dy,b jk (t) = 1 2π u(t, y)ψ jk (y)dy, φ(y) is an f-wavelet, ψ(y) is the corresponding m-wavelet, φ 0k (y) and ψ jk (y) are the Fourier transforms of φ 0k (y) = φ(y k) and ψ jk (y) = 2 j/2 ψ(2 j y k) correspondingly, ξ 0k and η jk are uncorrelated random variables. The rate of convergence of this expansion in different functional spaces was studied by Turchyn (2006), Kozachenko and Turchyn (2008), Turchyn (2011a), Turchyn (2011b). Conditions for uniform convergence with probability 1 and in probability were obtained by Kozachenko and Turchin (2009). A more general expansion of a random process (based on an arbitrary orthonormal basis in L 2 (T)) was considered by Kozachenko et al. (2011). We propose a new wavelet-type expansion of a second-order stochastic process based on a basis in L 2 ( 2 ) which is built using wavelets. This expansion is similar to (1) but is more flexible since we use two wavelets for this expansion instead of one wavelet in (1). We find the rate of convergence of our expansion in L p ([0, T]) for a sub-gaussian process in terms of simulation with given accuracy and reliability. A model based on our expansion can be used for simulation of a Gaussian process. 2. Auxiliary Facts Let φ L 2 () be such a function that the following assumptions hold: i) φ (y + 2πk) 2 = 1a. e., k Z where φ (y) is the Fourier transform of φ, φ (y) = exp{ iyx}φ(x)dx; ii) there exists a function m 0 L 2 ([0,2π]) such that m 0 (x) has period 2π and almost everywhere φ (y) = m 0 ( y 2 ) φ ( y 2 ) ; iii) φ (0) 0 and the function φ (y) is continuos at 0. The function φ(x) is called a f-wavelet. Let ψ(x) be the inverse Fourier transform of the function The function ψ(x) is called a m-wavelet. Let ψ (y) = m 0 ( y 2 + π) exp{ i y 2 }φ ( y 2 ). φ jk (x) = 2 j/2 φ(2 j x k), ψ jk (x) = 2 j/2 ψ(2 j x k),k Z, j = 0,1, 138

3 It is known that the family of functions {φ 0k, ψ jk, j = 0,1, ; k Z} is an orthonormal basis in L 2 (). Obviously the system {φ 0k / 2π, ψ jk / 2π, j = 0,1, ; k Z} also is an orthonormal basis in L 2 () and therefore any function f L 2 () can be represented as where f(x) = 1 2π k Z a 0k φ 0k (x) + 1 2π j=0 a 0k = 1 2π f(x)φ 0k (x)dx, b jk = 1 2π f(x)ψ jk (x)dx, series (2) converges in the norm of the space L 2 (). k Z b jk ψ jk (x), (2) More detailed information about wavelets is contained, for instance, in Härdle et al. (1998) or Walter and Shen (2000). Definition 2.1. A centered random variable ξ is called sub-gaussian if there exists such a constant a 0 that Eexp{λξ} exp{λ 2 a 2 /2},λ. The class of all sub-gaussian random variables on a standard probability space {Ω, B, P} is a Banach space with respect to the norm τ(ξ) = inf{a 0: Eexp{λξ} exp{λ 2 a 2 /2}, λ }. Definition 2.2 A sub-gaussian random variable ξ is called strictly sub-gaussian if τ(ξ) = (Eξ 2 ) 1/2. Example. A centered Gaussian random variable is strictly sub-gaussian. Definition 2.3. A stochastic process X = {X(t), t T} is called sub-gaussian if all the random variables X(t), t T, are sub-gaussian. Example. Any centered Gaussian process is sub-gaussian. Details about sub-gaussian random variables and processes can be found in Buldygin and Kozachenko (2000). 3. A wavelet-based Expansion Theorem 3.1. (Kozachenko et al. (2011)) Suppose that {X(t), t T} is a centered second-order random process with the correlation function (t, s) = EX(t)X(s), (Λ, B Λ, μ) is a measure space, {g k (λ), k Z} is an orthonormal basis in L 2 (Λ, μ) and u(t, ) L 2 (Λ, μ), t T. The correlation function (t, s) of X(t) can be represented as 139

4 if and only if the process X(t) can be represented as where (4) converges in L 2 (Ω), t T, (t, s) = u(t, λ)u(s, λ)dμ(λ), (3) Λ X(t) = k Z a k (t)ξ k,t T, (4) a k (t) = Λ u(t, λ)g k(λ)dμ(λ), ξ k are centered random variables such that Eξ k ξ l = δ kl. The following result enables us to expand a second-order process in a random series with uncorrelated terms which is built using two wavelets. Theorem 3.2. Let X = {X(t), t [0, T ]} be a centered random process such that E X(t) 2 < for all t [0, T ]. Let (t, s) = EX(t)X(s) and there exists such a Borel function u(t, y 1, y 2 ), y i, t [0, T ], that u(t, y 2 1, y 2 ) 2 dy 1 dy 2 < (i = 1,2)for all t [0, T ] and (t, s) = 2 u(t, y 1, y 2 )u(s, y 1, y 2 )dy 1 dy 2.(5) Let φ (1) (x), ψ (1) (x) and φ (2) (x), ψ (2) (x) be two pairs of a f-wavelet and the corresponding m-wavelet. Then the process X(t) can be represented as the following series which converges for any t [0, T ] in L 2 (Ω): where X(t) = a 0,k1,k 2 (t)ξ 0,k1,k 2 + b k;j,l (t)η k;j,l k 1 Z k 2 Z k Z j=0 l Z + d k;j,l (t)ζ k;j,l + e j1,l 1 ;j 2,l 2 (t)χ j1,l 1 ;j 2,l 2,(6) k Z j=0 l Z j 1 =0 l 1 Z j 2 =0 l 2 Z a 0,k1,k 2 (t) = 1 2π u(t, y 1, y 2 )φ 0,k1 (y1 )φ 0,k2 (y2 )dy 1 dy 2,(7) 2 b k;j,l (t) = 1 2π u(t, y 1, y 2 )φ 0,k (y1 )ψ j,l (y2 )dy 1 dy 2,(8) 2 d k;j,l (t) = 1 2π u(t, y 1, y 2 )ψ j,l (y1 )φ 0,k (y2 )dy 1 dy 2,(9) 2 e j1,l 1 ;j 2,l 2 (t) = 1 2π u(t, y 1, y 2 )ψ j1 2,l (y1 1 )ψ j2,l (y2 )dy 2 1 dy 2,(10) all the random variables ξ 0,k1,k 2, η k;j,l, ζ k;j,l, χ j1,l 1 ;j 2,l 2 are centered and uncorrelated, E ξ 0,k1,k 2 2 = E η k;j,l 2 = E ζ k;j,l 2 = E χ j1,l 1 ;j 2,l 2 2 =

5 Proof. It is enough to apply Theorem 3.1 to the process X(t) and the orthonormal basis F defined as F = F 1 F 2 F 3 F 4, where F 1 = φ 0,k1 (y1 )φ 0,k2 (y2 )/(2π), k 1, k 2 Z, F 2 = φ 0,k1 (y1 )ψ j2,k 2 (y 2 )/(2π), k 1, k 2 Z, j 2 = 0,1,, F 3 = ψ j1,k 1 (y 1 )φ 0,k2 (y2 )/(2π), k 1, k 2 Z, j 1 = 0,1,, (1) F 4 = ψ j1,k 1 (2) (y 1 )ψ j2,k 2 (y 2 )/(2π), k 1, k 2 Z, j 1, j 2 = 0,1, emark. We can generalize expansion (6) if we consider a centered random field X = {X(t), t T} (where T d ) which correlation function can be represented as (t, s) = n u(t, y 1, y 2,, y n )u(s, y 1, y 2,, y n )dy 1 dy 2 dy n,(11) u(t,,,, ) L 2 ( n ), and apply Theorem 3.1 to the orthonormal basis in L 2 ( n ) which is the (i) (i) tensor product of orthonormal bases {φ 0,k (x)/ 2π, k Z; ψ j,l (x)/ 2π, j = 0,1, ; k, l Z}, i = 1,2,, n (φ (i) (x) and ψ (i) (x) are pairs of a f-wavelet and the corresponding m-wavelet). Let us consider the case d = n. If X = {X(t), t T d } is a centered stationary random field which has the spectral density f(y), y d, then its correlation fucntion (t, s) can be represented as (11) if we set u(t, y) = f(y)exp{ i(t, y)}, where t = (t 1, t 2,, t d ), y = (y 1, y 2,, y d ). So X(t) can be expanded into a series of types (4)and(6). 4. Inequalities for the Coefficients Lemma 4.1. Let X(t) be a stochastic process which satisfies the conditions of Theorem 3.2 together with the f-wavelets φ (1) (x), φ (2) (x) and the corresponding m-wavelets ψ (1) (x), ψ (2) (x), the function u(t, y 1, y 2 ) from (5) is such that u(t, y 1, y 2 ) v 1 (t, y 1 )v 2 (t, y 2 ), (12) where v i (t, ) L 2 ()(i = 1,2). Assume that φ (i) (y) are absolutely continuous, v i (t, y) are absolutely continuous with respect to y for any fixed t, there exist v i (t, y)/ y, dφ (i) (y)/dy, dψ (i) (y)/dy and dψ (i) (y) C dy i, v i (t, y) H i (t)v i (y), v i(t, y) G y i (t)v i (y), 141

6 sup H i (t) <, t [0,T] sup G i (t) <, t [0,T] v i (y) y dy <, v i (y)dy <, v dφ (i) (y) i (y) dy <, dy v (y) φ (i) i (y) dy <, v i (y) y dy <, i = 1,2. Define v (y) φ (i) i (y) dy <, lim v i(t, y) φ (i) (y) = 0,t, y lim v i(t, y) ψ (i) (y/2 j ) = 0,t, y (i) 1 S 1 = 2π v dφ (i) (y) i (y) dy, dy (i) 1 S 2 = 2π v (y) φ (i) i (y) dy, j = 0,1,, Q 1 (i) = C i 2π v i (y)dy, C i (i) Q 2 = 2π v i (y) y dy, L (i) = 1 2π v (y) φ (i) i (y) dy, W (i) = i = 1,2. Then the following inequalities hold: C i 2π v i (y) y dy, a 0,k1,k 2 (t) A a,a(t) k 1 k 2,k 1 0, k 2 0,(13) 142

7 a 0,k1,0(t) A a,0(t),k k 1 1 0,(14) a 0,0,k2 (t) A 0,a(t),k k 2 2 0,(15) a 0,0,0 (t) A 0,0 (t),(16) where b k;j,l (t) B a,b(t) 2 j/2,k 0, l 0,(17) k l b 0;j,l (t) B 0,b(t) 2 j/2,l 0,(18) l b k;j,0 (t) B a,0(t),k 0,(19) k 23j/2 b 0;j,0 (t) B 0,0(t),(20) 23j/2 d k;j,l (t) D a,b(t) 2 j/2,k 0, l 0,(21) k l d 0;j,l (t) D 0,b(t) 2 j/2,l 0,(22) l d k;j,0 (t) D a,0(t),k 0,(23) k 23j/2 d 0;j,0 (t) D 0,0(t),(24) 23j/2 E b,b (t) e j1,l 1 ;j 2,l 2 (t) 2 j1/2 2 j2/2 l 1 l 2,l 1 0, l 2 0,(25) e j1,0;j 2,l 2 (t) e j1,l 1 ;j 2,0(t) E 0,b (t) 2 3j 1/2 2 j 2/2 l 2,l 2 0,(26) E b,0 (t) 2 j 1/2 2 3j 2/2 l 1,l 1 0,(27) e j1,0;j 2,0(t) E 0,0(t) 2 3j 1/2 2 3j 2/2,(28) (1) (2) A a,a (t) = (S 1 H1 (t) + S 2 G1 (t))(s 1 H2 (t) + S 2 G2 (t)), (2) (2) A 0,a (t) = (S 1 H2 (t) + S 2 G2 (t))l (1) H 1 (t), (1) (1) A a,0 (t) = (S 1 H1 (t) + S 2 G1 (t))l (2) H 2 (t), A 0,0 (t) = L (1) L (2) H 1 (t)h 2 (t), (1) (2) B a,b (t) = (S 1 H1 (t) + S 2 G1 (t))(q 1 H2 (t) + Q 2 G2 (t)), 143

8 B 0,b (t) = L (1) H 1 (t)(q 1 (2) H2 (t) + Q 2 (2) G2 (t)), B a,0 (t) = W (2) H 2 (t)(s 1 (1) H1 (t) + S 2 (1) G1 (t)), B 0,0 (t) = L (1) W (2) H 1 (t)h 2 (t), D a,b (t) = (S 1 (2) H2 (t) + S 2 (2) G2 (t))(q 1 (1) H1 (t) + Q 2 (1) G1 (t)), D 0,b (t) = L (2) H 2 (t)(q 1 (1) H1 (t) + Q 2 (1) G1 (t)), D a,0 (t) = W (1) H 1 (t)(s 1 (2) H2 (t) + S 2 (2) G2 (t)), D 0,0 (t) = L (2) W (1) H 1 (t)h 2 (t), E b,b (t) = (Q 1 (1) H1 (t) + Q 2 (1) G1 (t))(q 1 (2) H2 (t) + Q 2 (2) G2 (t)), E 0,b (t) = W (1) H 1 (t)(q 1 (2) H2 (t) + Q 2 (2) G2 (t)), E b,0 (t) = W (2) H 2 (t)(q 1 (1) H1 (t) + Q 2 (1) G1 (t)), E 0,0 (t) = W (1) W (2) H 1 (t)h 2 (t). Proof. Let us prove, for instance, inequality (13). We have a 0,k1,k 2 (t) 1 2π v 1 (t, y 1 )φ 0,k1 (y1 )dy 1 v 2 (t, y 2 )φ 0,k2 (y2 )dy 2. Estimating the integrals in the right-hand side by means of Lemma 1 from Turchyn (2011a), we obtain (13). Inequalities (14) (28) are proved in a similar way. 5. Simulation Expansion (6) may be used for simulation of stochastic processes. If a process X(t) satisfies the conditions of Theorem 3.2, then we can consider as a model of X(t) the process k 1 a 1 k 2 a 1 k b 1 X (t) = k 1 = (k a 1 1) k 2 = (k a 2 1) a 0,k 1,k 2 (t)ξ 0,k1,k 2 + k= (k b 1) j=0 k= (k d 1) j d 1 j=0 l d 1 l= (l d 1) d k;j,l(t)ζ k;j,l j e 1 1 j 1 =0 l e 1 1 l 1 = (l e 1 1) j e 2 1 j 2 =0 l e 2 1 k + d 1 + j b 1 l b 1 l= (l b 1) b k;j,l(t)η k;j,l l 2 = (l 2 e 1) e j 1,l 1 ;j 2,l 2 (t)χ j1,l 1 ;j 2,l 2, (30) where ξ 0,k1,k 2, η k;j,l, ζ k;j,l, χ j1,l 1 ;j 2,l 2 are the random variables from expansion (6), the functions a 0,k1,k 2 (t), b k;j,l (t), d k;j,l (t), e j1,l 1 ;j 2,l 2 (t) are calculated using (7) (10), the parameters k 1 a, k 2 a, k b, j b, l b, k d, j d, l d, j 1 e, l 1 e, j 2 e, l 2 e are strictly greater than

9 emark. If X(t) is a Gaussian process then ξ 0,k1,k 2, η k;j,l, ζ k;j,l, χ j1,l 1 ;j 2,l 2 in (30) are independent random variables with distribution N(0,1) and the model X (t) can be used for computer simulation of X(t). Definition 5.1. We say that the model X (t) approximates a process X(t) with given reliability 1 δ (0 < δ < 1) and accuracy ε > 0 in L p ([0, T]) if T 1/p P {( X(t) X (t) p dt) > ε} δ. 0 emark. We will consider only real-valued f-wavelets and m-wavelets below. Definition 5.2. We say that the condition 1 holds for a stochastic process X(t) if it satisfies the conditions of Theorem 3.2, u(t,, ) L 1 ( 2 ) L 2 ( 2 ) and inverse Fourier transform u (t, y 1, y 2 ) of the function u(t, y 1, y 2 ) with respect to (y 1, y 2 ) is a real-valued function. Theorem 5.1. (Turchyn (2011a)) Suppose that X(t) is a sub-gaussian random process that satisfies condition 1, X (t) is the model of the process defined by (30), random variables ξ 0,k1,k 2, η k;j,l, ζ k;j,l, χ j1,l 1 ;j 2,l 2 in expansion (6) of X(t) are independent and strictly sub-gaussian, p 1, δ (0; 1), ε > 0, T (0, T ). If sup t [0,T] E X(t) X (t) 2 ε 2 min { 2T 2/p ln(2/δ), ε 2 pt 2/p}, then the model X (t) approximates the process X(t) with reliability 1 δ and accuracy ε in L p ([0, T]). The following theorem is the main result of the article. Theorem 5.2 Suppose that a sub-gaussian random process X = {X(t), t [0, T ]} satisfies the condition 1 and the conditions of Lemma 4.1 together with f-wavelets φ (1) (x), φ (2) (x) and the corresponding m -wavelets ψ (1) (x), ψ (2) (x), the random variables ξ 0,k1,k 2, η k;j,l, ζ k;j,l, χ j1,l 1 ;j 2,l 2 in expansion (6) of X(t) are independent and strictly sub-gaussian, p 1, T (0, T ), δ (0; 1), ε > 0. Denote by A a,0 (T),, E 0,b (T) the suprema of the functions A a,0 (t),, E 0,b (t) correspondingly on [0, T] (where A a,0 (t),, E 0,b (t) are defined in Lemma 4.1), If ε 2 = min { 2T 2/p ln(2/δ), ε 2 pt 2/p}. k 1 a 1 + (16(A a,0 (T)) (A a,a (T)) 2 )/, k 2 a 1 + (16(A 0,a (T)) (A a,a (T)) 2 )/, k b 1 + ((256/7)(B a,0 (T)) (B a,b (T)) 2 )/, 145

10 l b 1 + (256(B a,b (T)) (B 0,b (T)) 2 )/, j b max {1 + log 2 64(B 0,b (T)) (B a,b (T)) 2, 1 + log 8 64(B a,0 (T)) (B 0,0 (T)) 2 7 }, k d 1 + ((256/7)(D a,0 (T)) (D a,b (T)) 2 )/, l d 1 + (256(D a,b (T)) (D 0,b (T)) 2 )/, j d max {1 + log 2 64(D 0,b (T)) (D a,b (T)) 2, 1 + log 8 64(D a,0 (T)) (D 0,0 (T)) 2 7 }, j 1 e max{1 + log 2 768(E b,b (T)) (E b,0 (T)) 2 /7, 2 + log 8 24(E 0,b (T)) 2 /7 + 24(E 0,0 (T)) 2 /49 }, j 2 e max{1 + log 2 768(E b,b (T)) (E 0,b (T)) 2 /7, 2 + log 8 24(E b,0 (T)) 2 /7 + 24(E 0,0 (T)) 2 /49 }, l 1 e (E b,b (T)) (E b,0 (T)) 2 /7, l 2 e (E b,b (T)) (E 0,b (T)) 2 /7, then the model X (t) defined by (30) approximates the process X(t) with reliability 1 δ and accuracy ε in L p ([0, T]). Proof. It is easy to see using Lemma 4.1 that under the assumptions of the theorem E X(t) X (t) 2 = k 1 : k 1 ka 1 k 2 Z a 0,k1,k 2 (t) 2 + k 1 = (k a 1 1) k 2 : k 2 ka 2 a 0,k1,k 2 (t) 2 + k: k kb k + b 1 j=0 k= (k b 1) j b 1 k + d 1 k= (k d 1) j=j d + j 1 =je 1 j + e 1 1 j 1 =0 j e 2 1 k b 1 k 1 a 1 l Z b k;j,l (t) 2 + k= (k b 1) j=j b l Z b k;j,l (t) 2 j=0 l: l l b b k;j,l(t) 2 + k: k k k d 1 d j=0 j d 1 l Z d k;j,l (t) 2 l Z d k;j,l (t) 2 + k= (k d 1) j=0 l: l l d d k;j,l(t) 2 j 1 e 1 j 2 =0 l 1 Z l 2 Z e j1,l 1 ;j 2,l 2 (t) 2 + j 1 =0 j 2 =je 2 l 1 Z l 2 Z e j1,l 1 ;j 2,l 2 (t) 2 j 1 e 1 j 2 =0 l 1 : l 1 le 1 l 2 Z e j1,l 1 ;j 2,l 2 (t) 2 + j 1 =0 j 2 =0 l 1 = (l e 1 1) l 2 : l 2 le 2 e j1,l 1 ;j 2,l 2 (t) 2 for all t [0, T]. Therefore and it remains to apply Theorem 5.1. j 2 e 1 sup E X(t) X (t) 2 t [0,T] Example. It is easy to see that a centered Gaussian process X = {X(t), t [0, T ]} with the l 1 e 1 146

11 correlation function where u(t, y 1, y 2 ) = (t, s) = u(t, y 1, y 2 )u(s, y 1, y 2 )dy 1 dy 2, t 4l (1 + t 2l ) 2 + B(1 + t 2l )(y 1 2m 1 + y 2 2m 2 ) + Ay 1 2m 1 y 2 2m 2, l, m 1, m 2 N, m 1 2, m 2 2, A 1, B 1, together with two Daubechies wavelets of arbitrary order satisfies the conditions of Theorem Conclusion We consider an expansion of a second-order stochastic process based on two wavelets. This expansion may be regarded as a generalization of an expansion from Kozachenko and Turchyn (2008). We use our expansion to build a model of a stochastic process and obtain a theorem about simulation of a sub-gaussian stochastic process with given accuracy and reliability in L p ([0, T]) by this model. Acknowledgements The author would like to thank professor Yuriy V. Kozachenko for valuable discussions which helped to substantially improve the quality of the paper and an anonymous referee for the constructive comments. eferences Buldygin, V.V., & Kozachenko, Yu.V. (2000). Metric Characterization of andom Variables and andom Processes. Providence, I: American Mathematical Society. Didier, G., & Pipiras, V. (2008). Gaussian stationary processes: adaptive wavelet decompositions, discrete approximations, and their convergence, J. Fourier Anal. Appl. 14, Härdle, W., Kerkyacharian, G., Picard, D., & Tsybakov, A. (1998). Wavelets, Approximation and Statistical Applications. N.Y.: Springer. Kozachenko, Y., & Turchyn, Y. (2008). On Karhunen-Loève-like expansion for a class of random processes. Int. J. Stat. Manag. Syst. 3, Kozachenko, Yu.V., & Turchin, E.V. (2009). Conditions for the uniform convergence of expansions of φ-sub- Gaus- sian stochastic processes in function systems generated by wavelets. Theory Probab. Math. Stat. 78, Kozachenko, Yu.V., ozora, I.V., & Turchyn, Ye.V. (2011). Properties of some random series. Comm. Statist. Theory Methods 40, Meyer, Y., Sellan, F., & Taqqu, M. S. (1999). Wavelets, generalized white noise and fractional integration: the syn- thesis of fractional Brownian motion. J. Fourier Anal. Appl. 5(5),

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