TY - JOUR

T1 - Gradient modeling for multivariate quantitative data

AU - Sei, Tomonari

N1 - Funding Information:
Acknowledgments The author thanks professors M. Miyakawa, S. Aoki, A. Takemura and F. Komaki for their helpful comments and encouragement to write this paper. This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

PY - 2011/8

Y1 - 2011/8

N2 - We propose a new parametric model for continuous data, a "g-model", on the basis of gradient maps of convex functions. It is known that any multivariate probability density on the Euclidean space is uniquely transformed to any other density by using the gradient map of a convex function. Therefore the statistical modeling for quantitative data is equivalent to design of the gradient maps. The explicit expression for the gradient map enables us the exact sampling from the corresponding probability distribution. We define the g-model as a convex subset of the space of all gradient maps. It is shown that the g-model has many desirable properties such as the concavity of the log-likelihood function. An application to detect the three-dimensional interaction of data is investigated.

AB - We propose a new parametric model for continuous data, a "g-model", on the basis of gradient maps of convex functions. It is known that any multivariate probability density on the Euclidean space is uniquely transformed to any other density by using the gradient map of a convex function. Therefore the statistical modeling for quantitative data is equivalent to design of the gradient maps. The explicit expression for the gradient map enables us the exact sampling from the corresponding probability distribution. We define the g-model as a convex subset of the space of all gradient maps. It is shown that the g-model has many desirable properties such as the concavity of the log-likelihood function. An application to detect the three-dimensional interaction of data is investigated.

KW - Convex function

KW - Exact sampling

KW - Gradient representation

KW - Three-dimensional interaction

KW - g-Model

UR - http://www.scopus.com/inward/record.url?scp=79960072380&partnerID=8YFLogxK

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U2 - 10.1007/s10463-009-0261-1

DO - 10.1007/s10463-009-0261-1

M3 - Article

AN - SCOPUS:79960072380

SN - 0020-3157

VL - 63

SP - 675

EP - 688

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

IS - 4

ER -