Інтерполяція випадкового процесу

Інтерполя́ція випадко́вого проце́су — одна із задач прогнозу теорії випадкових процесів. Лінійна інтерполяція випадкового процесу полягає в побудові оцінки $\xi (t)$ значення процесу ${\bar {\xi }}(t)$ у момент часу $\tau \ (0<\tau <T)$ , яка лінійно виражається через спостереження $\xi (t)$ при $t\leq 0$ і $t\geq T$ . При цьому звичайно шукають оцінки ${\tilde {\xi }}(t)$ , для яких квадрат середньоквадратичної похибки $\sigma ^{2}(t)=M\left|\xi (t)-{\tilde {\xi }}(t)\right|^{2}$ є мінімальним. Явні формули для вирішення задачі інтерполяції випадкового процесу отримані для стаціонарних випадкових процесів з дробово-раціональною спектральною густиною. Наприклад, якщо спектральна густина процесу $\xi (t)$ дорівнює $f(\lambda )={\frac {c}{\lambda ^{2}+\alpha ^{2}}}$ то ${\tilde {\xi }}(t)={\frac {1}{sh\ \alpha T}}(\xi (0)\ sh\ \alpha (T-\tau )+\xi (T)\ sh\ \ \alpha \tau )$ .

Вперше задачу лінійної інтерполяції випадкового процесу для стаціонарної послідовності $\xi _{n}$ спектральною густиною $f(\lambda )$ , що спостерігається при всіх $n$ , окрім $n=0$ , розглянув радянський математик О. М. Колмогоров. Виявилося, що квадрат середньоквадратичної похибки інтерполяції $\xi _{0}$ дорівнює

$\sigma ^{2}=2\pi \left[\int _{-\pi }^{\pi }{\frac {d\lambda }{f(\lambda )}}\right]^{-1}$ .

Інтерполяція практично безпомилкова, якщо

$\int _{-\pi }^{\pi }{\frac {d\lambda }{f(\lambda )}}=+\infty$ .

Література

Енциклопедія кібернетики : у 2 т. / за ред. В. М. Глушкова. — Київ : Гол. ред. Української радянської енциклопедії, 1973. стаття М. І. Ядренко

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