Determining Stochastic Dependence for Normally Distributed Vectors Using the Chi-Squared Metric. Valiveti, R. S. & Oommen, B. J. 26(6):975–987. ![Determining Stochastic Dependence for Normally Distributed Vectors Using the Chi-Squared Metric [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex A fundamental problem in information theory and pattern recognition involves computing and estimating the probability density function associated with a set of random variables. In estimating this density function, one can either assume that the form of the density function is known, and that we are merely estimating parameters that characterize the distribution or that no information about the density function is available. This problem has been extensively studied if the random variables are independent. If the random variables are dependent and are of the discrete sort, the problem of capturing this dependence between variables has been studied in Chow and Liu (IEEE Trans. Inf. Theory14, 462–467 (May 1968)). The analogous problem for normally distributed continuous random variables has been tackled in Chow et al. (Comput. Biomed. Res.12, 589–613 (1979)). In both these instances, the determination of the best dependence tree hinges on the well-known Expected Mutual Information Measure (EMIM) Metric. Recently Valiveti and Oommen studied the suitability of the chi-squared based metric in-lieu of the EMIM metric, for the discrete variable case (Pattern Recognition25, 1389–1400 (1992)). In this paper, we generalize the latter result and study the use of the chi-squared metric for determining dependence trees for normally distributed random vectors. We show that for such vectors, the chi-squared metric yields the optimal tree and that it is identical to the one obtained using the EMIM metric. The computation of the maximum likelihood estimate of the dependence tree is also discussed.
@Article{ valiveti_determining_1993,
title = {Determining Stochastic Dependence for Normally Distributed
Vectors Using the Chi-Squared Metric},
volume = {26},
issn = {0031-3203},
url = {http://www.sciencedirect.com/science/article/pii/0031320393900622},
doi = {https://doi.org/10.1016/0031-3203(93)90062-2},
abstract = {A fundamental problem in information theory and pattern
recognition involves computing and estimating the
probability density function associated with a set of
random variables. In estimating this density function, one
can either assume that the form of the density function is
known, and that we are merely estimating parameters that
characterize the distribution or that no information about
the density function is available. This problem has been
extensively studied if the random variables are
independent. If the random variables are dependent and are
of the discrete sort, the problem of capturing this
dependence between variables has been studied in Chow and
Liu ({IEEE} Trans. Inf. Theory14, 462–467 (May 1968)).
The analogous problem for normally distributed continuous
random variables has been tackled in Chow et al. (Comput.
Biomed. Res.12, 589–613 (1979)). In both these instances,
the determination of the best dependence tree hinges on the
well-known Expected Mutual Information Measure ({EMIM})
Metric. Recently Valiveti and Oommen studied the
suitability of the chi-squared based metric in-lieu of the
{EMIM} metric, for the discrete variable case (Pattern
Recognition25, 1389–1400 (1992)). In this paper, we
generalize the latter result and study the use of the
chi-squared metric for determining dependence trees for
normally distributed random vectors. We show that for such
vectors, the chi-squared metric yields the optimal tree and
that it is identical to the one obtained using the {EMIM}
metric. The computation of the maximum likelihood estimate
of the dependence tree is also discussed.},
pages = {975--987},
number = {6},
journaltitle = {Pattern Recognition},
author = {Valiveti, R. S. and Oommen, B. J.},
date = {1993}
}
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This problem has been extensively studied if the random variables are independent. If the random variables are dependent and are of the discrete sort, the problem of capturing this dependence between variables has been studied in Chow and Liu (IEEE Trans. Inf. Theory14, 462–467 (May 1968)). The analogous problem for normally distributed continuous random variables has been tackled in Chow et al. (Comput. Biomed. Res.12, 589–613 (1979)). In both these instances, the determination of the best dependence tree hinges on the well-known Expected Mutual Information Measure (EMIM) Metric. Recently Valiveti and Oommen studied the suitability of the chi-squared based metric in-lieu of the EMIM metric, for the discrete variable case (Pattern Recognition25, 1389–1400 (1992)). In this paper, we generalize the latter result and study the use of the chi-squared metric for determining dependence trees for normally distributed random vectors. We show that for such vectors, the chi-squared metric yields the optimal tree and that it is identical to the one obtained using the EMIM metric. 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In estimating this density function, one\n\t\t can either assume that the form of the density function is\n\t\t known, and that we are merely estimating parameters that\n\t\t characterize the distribution or that no information about\n\t\t the density function is available. This problem has been\n\t\t extensively studied if the random variables are\n\t\t independent. If the random variables are dependent and are\n\t\t of the discrete sort, the problem of capturing this\n\t\t dependence between variables has been studied in Chow and\n\t\t Liu ({IEEE} Trans. Inf. Theory14, 462–467 (May 1968)).\n\t\t The analogous problem for normally distributed continuous\n\t\t random variables has been tackled in Chow et al. (Comput.\n\t\t Biomed. Res.12, 589–613 (1979)). In both these instances,\n\t\t the determination of the best dependence tree hinges on the\n\t\t well-known Expected Mutual Information Measure ({EMIM})\n\t\t Metric. Recently Valiveti and Oommen studied the\n\t\t suitability of the chi-squared based metric in-lieu of the\n\t\t {EMIM} metric, for the discrete variable case (Pattern\n\t\t Recognition25, 1389–1400 (1992)). In this paper, we\n\t\t generalize the latter result and study the use of the\n\t\t chi-squared metric for determining dependence trees for\n\t\t normally distributed random vectors. We show that for such\n\t\t vectors, the chi-squared metric yields the optimal tree and\n\t\t that it is identical to the one obtained using the {EMIM}\n\t\t metric. The computation of the maximum likelihood estimate\n\t\t of the dependence tree is also discussed.},\n pages\t\t= {975--987},\n number\t= {6},\n journaltitle\t= {Pattern Recognition},\n author\t= {Valiveti, R. S. and Oommen, B. J.},\n date\t\t= {1993}\n}\n\n","author_short":["Valiveti, R. S.","Oommen, B. 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