Abstract

The problem of assessing tightness of the interdependence between random vectors of different dimensionality is considered. These random vectors can obey arbitrary multidimensional continuous distribution laws. An analytical expression is derived for the coefficient of tightness of the interdependence between random vectors. It is expressed in terms of the coefficients of determination of conditional regressions between the components of random vectors. For the case of Gaussian random vectors, a simpler formula is obtained, expressed through the determinants of each of the random vectors and determinant of their association. It is shown that the introduced coefficient meets all the basic requirements imposed on the degree of tightness of the interdependence between random vectors. This approach is more preferable compared to the method of canonical correlations providing determination of the actual tightness of the interdependence between random vectors. Moreover, it can also be used in case of non-linear correlation dependence between the components of random vectors. The measure thus introduced is rather simply interpretable and can be applied in practice to real data samplings. Examples of calculating the tightness of the interdependence between Gaussian random vectors of different dimensionality are given.

Highlights

  • Ðàññìîòðåíà ïðîáëåìà îöåíèâàíèÿ òåñíîòû âçàèìîçàâèñèìîñòè ìåæäó ñëó÷àéíûìè âåêòîðàìè ðàçíîé ðàçìåðíîñòè

  • An analytical expression is derived for the coefficient of tightness of the interdependence between random vectors

  • It is expressed in terms of the coefficients of determination of conditional regressions between the components of random vectors

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Summary

SCALAR MEASURE OF THE INTERDEPENDENCE BETWEEN RANDOM VECTORS

 íàñòîÿùåå âðåìÿ äàííàÿ ïðîáëåìà ðåøåíà ëèøü ÷àñòè÷íî — äëÿ ñëó÷àÿ òåñíîòû âçàèìîñâÿçè ìåæäó êîìïîíåíòàìè ñëó÷àéíîãî âåêòîðà. Äëÿ ãàóññîâñêèõ ñëó÷àéíûõ âåêòîðîâ ïðåäëîæåíà [5] ñêàëÿðíàÿ ìåðà — êîýôôèöèåíò òåñíîòû ñîâìåñòíîé ëèíåéíîé êîððåëÿöèîííîé ñâÿçè, ðàâíûé. Êðîìå îöåíêè òåñíîòû âçàèìîñâÿçè ìåæäó êîìïîíåíòàìè ñëó÷àéíîãî âåêòîðà, íåîáõîäèìî òàêæå îöåíèâàòü òåñíîòó âçàèìîçàâèñèìîñòè ìåæäó ñëó÷àéíûìè âåêòîðàìè. Ýòîò ìåòîä ÿâëÿåòñÿ îáîáùåíèåì ïàðíîé ëèíåéíîé êîððåëÿöèè è ïîçâîëÿåò íàõîäèòü ìàêñèìàëüíûå êîððåëÿöèîííûå ñâÿçè ìåæäó äâóìÿ ãðóïïàìè ñëó÷àéíûõ âåëè÷èí. Âî-âòîðûõ, íàõîäèòñÿ ìàêñèìàëüíàÿ âåëè÷èíà êîýôôèöèåíòà êîððåëÿöèè ìåæäó êàíîíè÷åñêèìè ïåðåìåííûìè, â òî âðåìÿ êàê òðåáóåòñÿ îöåíèòü òåñíîòó ôàêòè÷åñêîé âçàèìîñâÿçè, êîòîðàÿ ìîæåò çíà÷èòåëüíî îòëè÷àòüñÿ îò ìàêñèìàëüíî âîçìîæíîé. Öåëü äàííîé ðàáîòû — ïîïûòêà íà îñíîâå ýíòðîïèéíîãî ïîäõîäà îáîáùèòü ìåðû (1) – (5) íà ñëó÷àé îöåíêè òåñíîòû âçàèìîçàâèñèìîñòè ìåæäó äâóìÿ ãðóïïàìè ïåðåìåííûõ, óñòðàíèâ íåäîñòàòêè è îãðàíè÷åíèÿ, ïðèñóùèå ìåòîäó êàíîíè-.

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