Full Text
UDC 519.3+519.212.3
MATHEMATICS
D. B. YUDIN
METHODS FOR SOLVING GENERALIZED PROBLEMS OF SMOOTHING AND EXTRAPOLATION OF RANDOM FUNCTIONS
(Presented by Academician A. A. Dorodnitsyn, 4 IV 1967)
In (¹) the conditions for the existence of a solution of generalized problems of smoothing and extrapolation of random functions were investigated, and a formal apparatus for the analysis of these problems was constructed. In the present note constructive methods are set forth for solving the problems formulated in (¹). The article uses the concepts and preserves the notation and terminology adopted in (¹).
Sec. 1. Let us formulate problem I of smoothing and extrapolation of random functions in terms of the Hilbert space \(H_r^n\).
Problem I. It is required to choose a system of elements \(h^\alpha \in G^\alpha\), \(\alpha = 1, 2, \ldots, n\), on which the quality index of the forecast \(R(k^{\alpha\beta}) = \bar R(h^\alpha)\) of the random vector-function \(\eta(t)=\{\eta^\alpha(t)\}\) at the point \(t_0+t_y\) attains its upper bound.
We associate with problem I problem \(I^\sigma\) of smoothing and extrapolation by the minimum of variance. Let us formulate it in terms of the space \(H_r^n\).
Problem \(I^\sigma\). It is required in each of the \(G^\alpha\) to select an element \(h_\sigma^\alpha\) on which the minimum \(k^{\alpha\alpha}\) is attained.
Let us denote the correlation matrices of forecast errors corresponding to the forecast quality indices \(R(k^{\alpha\beta})\) and \(\sigma_\alpha^2 = k^{\alpha\alpha}\), \(\alpha = 1, 2, \ldots, n\), respectively, by \(\|k_R^{\alpha\beta}\|\) and \(\|k_\sigma^{\alpha\beta}\|\).
Theorem 1. The correlation matrices of the errors of smoothing and extrapolation corresponding to the solutions of problems I and \(I^\sigma\) are related by the relation
\[ \|k_R^{\alpha\beta}\|=\|k_\sigma^{\alpha\beta}\|+\|k_P^{\alpha\beta}\|, \tag{1} \]
where \(\|k_P^{\alpha\beta}\|\) is some nonnegative definite symmetric matrix—the matrix of artificial scattering.
Introduce the following notation. \(\mathcal P_{\alpha\beta}^R(\tau)\), \(\mathcal P_{\alpha\beta}^\sigma(\tau)\), \(\alpha,\beta=1,2,\ldots,n\), are the weight functions corresponding to the solutions of problems I and \(I^\sigma\), respectively; \(w(t_0,\tau)=w(\tau)\) is an arbitrary function belonging to \(H^{(K)}(t_0,T)\), distinct from the identically zero function and satisfying the conditions
\[ \int_0^T \psi_j^\alpha(t_0-\tau)w(\tau)\,d\tau=0, \qquad \alpha=1,2,\ldots,n;\quad j=1,2,\ldots,r. \]
Let, moreover,
\[ a_{\alpha\beta} = \int_0^T\int_0^T k_{\xi^\alpha,\xi^\beta}(t_0-\tau_1,t_0-\tau_2) w(\tau_1)w(\tau_2)\,d\tau_1\,d\tau_2 . \tag{2} \]
Define the matrix of numbers \(\|\chi_{\alpha\beta}\|\), \(\alpha,\beta=1,2,\ldots,n\), by the equation
\[ \|\chi_{\beta\mu}\|\,\|a_{\mu\nu}\|\,\|\chi_{\alpha\nu}\|^T = \|k_R^{\alpha\beta}\|-\|k_\sigma^{\alpha\beta}\| = \|k_P^{\alpha\beta}\|. \tag{3} \]
System (3) has a solution (more precisely, an infinite set of solutions), since there always exists a transformation with matrix \(\|\chi_{\beta\mu}\|\) which transforms the positive definite quadratic form with matrix \(\|a_{\mu\nu}\|\) into the nonnegative definite quadratic form with matrix \(\|k_P^{\alpha\beta}\|\).
The matrices satisfying equation (3) are
\[ \|x_{\alpha\beta}\|=\|c_{\alpha\mu}\|\|d_{\mu\nu}\|\|h_{\nu\lambda}\|\|c_{\lambda\beta}\|^{-1}. \tag{4} \]
Here \(\|c_{\alpha\beta}\|\) is a nonsingular matrix which simultaneously reduces the matrix \(\|a_{\alpha\beta}\|\) to normal form and the matrix \(\|k_P^{\alpha\beta}\|\) to canonical form:
\[ \|c_{\alpha\mu}\|\|a_{\mu\nu}\|\|c_{\alpha\nu}\|^{T}=E_n, \]
\[ \|c_{\alpha\mu}\|\|k_P^{\mu\nu}\|\|c_{\alpha\nu}\|= \begin{Vmatrix} g_1 & 0 & \cdots & 0\\ 0 & g_2 & \cdots & 0\\ . & . & \cdots & .\\ 0 & 0 & \cdots & g_n \end{Vmatrix}, \]
\[ \|h_{\nu\lambda}\|= \begin{Vmatrix} \sqrt{g_1} & 0 & \cdots & 0\\ 0 & \sqrt{g_2} & \cdots & 0\\ . & . & \cdots & .\\ 0 & 0 & \cdots & \sqrt{g_n} \end{Vmatrix}, \]
\(\|d_{\mu\nu}\|\) is an arbitrary orthogonal matrix.
There is the following relation between the weight functions determining the solutions of Problems I and \(I^\sigma\).
Theorem 2. The weight functions on which the solutions of Problems I and \(I^\sigma\) are attained are connected by the formulas
\[ \mathscr{P}_{\alpha\beta}^{R} = \mathscr{P}_{\alpha\beta}^{\sigma} + x_{\alpha\beta}w(\tau), \qquad \alpha,\beta=1,2,\ldots,n. \tag{5} \]
2.
Let us formulate Problem II of smoothing and extrapolation in terms of the spaces \(H^n\) and \(H^{(K)}(t_0,T)\).
Problem II. It is required to choose a random vector \(\zeta=\{\zeta^\alpha\}\subset L=L^n(t_0,T)\subset H^n\) (or, equivalently, a system of weight functions \(\mathscr{P}_{\alpha\beta}(t_0,\tau)\in H^{(K)}(t_0,T)\), \(\alpha,\beta=1,2,\ldots,n\)), for which the quality criterion of the prediction
\[ R(m^\alpha,k^{\alpha\beta})=\bar R(\zeta^\alpha)=R(\mathscr{P}_{\alpha\beta}) \]
of the random vector-function \(\eta(t)=\{\eta^\alpha(t)\}\) at the point \(t_0+t_y\) attains its upper bound.
Let us select in \(L=L^n(t_0,T)\) the set of elements \(G_c\) of the form \(\xi=\xi_0+c\) \((c=\bar{\xi})\) and associate with Problem II the auxiliary Problems \(II^{\sigma_1}\) and \(II^{\sigma_2}\).
Problem \(II^{\sigma_1}\). It is required, for each \(\alpha\), \(\alpha=1,2,\ldots,n\), to find a random variable \(\xi^\alpha\in G_0\) on which the minimum of
\[ \mathrm{M}[\eta^\alpha(t_0+t_y)-\xi^\alpha]^2 \]
is attained.
Problem \(II^{\sigma_2}\). It is required to find a random vector \(\xi\in G_1\) on which the minimum of \(\mathrm{M}[\xi]^2\) is attained.
Introduce the following notation: \(\xi_R^\alpha\), \(\xi_{\sigma_1}^\alpha\), \(\xi_{\sigma_2}^\alpha\) are random variables determining the solutions of Problems II, \(II^{\sigma_1}\), and \(II^{\sigma_2}\), respectively; \(m_R^\alpha\), \(k_R^{\alpha\beta}\) are the first and second moments of the smoothing and extrapolation errors corresponding to the solution of Problem II;
\[ c_R^\alpha=\mathrm{M}\xi_R^\alpha=\mathrm{M}\eta^\alpha-m_R^\alpha;\quad k_{\sigma_1}^{\alpha\beta} = \mathrm{M}\{[\eta^\alpha(t_0+t_y)-\xi_{\sigma_1}^\alpha][\eta^\beta(t_0+t_y)-\xi_{\sigma_1}^\beta]\}; \]
\[ k_{\sigma_2}=\mathrm{M}(\xi_{\sigma_2}-1)^2;\quad k_{\sigma_1\sigma_2}^{\alpha} = \mathrm{M}\{\xi_{\sigma_1}^{\alpha}(\xi_{\sigma_2}-1)\}. \]
Theorem 3. The statistical characteristics of the solutions of Problems II, \(II^{\sigma_1}\), and \(II^{\sigma_2}\) are connected by the formulas
\[ \|k_R^{\alpha\beta}\| = \|k_{\sigma_1}^{\alpha\beta}\| - \|c_R^\alpha k_{\sigma_1\sigma_2}^{\beta} + c_R^\beta k_{\sigma_1\sigma_2}^{\alpha}\| + \|c_R^\alpha c_R^\beta k_{\sigma_2}\| + \|k_P^{\alpha\beta}\|, \tag{6} \]
\[ m_R^\alpha=\bar{\eta}^{\alpha}-c_R^\alpha, \qquad \alpha,\beta=1,2,\ldots,n, \tag{7} \]
where \(\|k_P^{\alpha\beta}\|\) is a symmetric nonnegative definite matrix—the matrix of artificial scattering.
Let \(\mathscr{P}_{\alpha\beta}^{R}(\tau)\), \(\mathscr{P}_{\alpha\beta}^{\sigma_1}(\tau)\), and \(\mathscr{P}_{\beta}^{\sigma_2}(\tau)\) be the weight functions corresponding to the solutions of problems II, \(\mathrm{II}^{\sigma_1}\), and \(\mathrm{II}^{\sigma_2}\), and let \(w(\tau)\) be an arbitrary function, not identically zero, satisfying the conditions
\[ \int_{0}^{T} M\xi^{\alpha}(t_0-\tau) w(\tau)\, d\tau = 0, \qquad \alpha = 1,2,\ldots,n. \]
Suppose, in addition, that the parameters \(\varkappa_{\alpha\beta}\) are computed from the equations
\[ \sum_{\mu=1}^{n}\sum_{\nu=1}^{n} \varkappa_{\alpha\mu}\varkappa_{\beta\nu} a_{\mu\nu} = k_{P}^{\alpha\beta}, \]
where \(\|k_{P}^{\alpha\beta}\|\) satisfies equation (6), and the constants \(a_{\mu\nu}\) are computed from \(k_{\xi^\alpha,\xi^\beta}(t_1,t_2)\) and \(w(\tau)\) by formulas (2).
Theorem 4. The weight functions for which the solutions of problems II, \(\mathrm{II}^{\sigma_1}\), and \(\mathrm{II}^{\sigma_2}\) are attained are connected by the relations
\[ \mathscr{P}_{\alpha\beta}^{R}(\tau) = \mathscr{P}_{\alpha\beta}^{\sigma_1}(\tau) + c_{R}^{\alpha}\mathscr{P}_{\beta}^{\sigma_2}(\tau) + \varkappa_{\alpha\beta} W(\tau), \qquad \alpha,\beta=1,2,\ldots,n. \tag{8} \]
Theorems 1–4 reduce the solution of the complicated variational problems I and II to the analysis of the substantially simpler variational problems \(\mathrm{I}^{\sigma}\), \(\mathrm{II}^{\sigma_1}\), and \(\mathrm{II}^{\sigma_2}\), considered in the literature, and to the investigation for an extremum of the functions
\(R(k^{\alpha\beta}) = R^0(k_{P}^{\alpha\beta})\) and
\(R(m^{\alpha}, k^{\alpha\beta}) = R^0(c^{\alpha}, k_{P}^{\alpha\beta})\), respectively.
Received
30 III 1967
CITED LITERATURE
- D. B. Yudin, DAN, 177, No. 3 (1967).