Abstract Generated abstract
This note develops constructive methods for generalized smoothing and extrapolation problems for random vector functions, using the Hilbert space framework and notation established in an earlier work. It relates optimization with respect to general forecast quality criteria to auxiliary minimum variance problems, deriving formulas that connect their error correlation matrices through a nonnegative definite artificial scattering matrix. The paper also gives explicit relations between the corresponding weight functions, including parameterizations via arbitrary admissible functions and matrix transformations. These results reduce the original variational problems to simpler variance minimization problems and to finite-dimensional extremum investigations of the remaining quality criteria.
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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).