Sufficient dimension reduction

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of <a href="/facts/Dimension_reduction/XoRpcrB0">dimension reduction</a> with the concept of <a href="/facts/Sufficient_statistic/eClRXHpd">sufficiency</a>.
Dimension reduction has long been a primary goal of <a href="/facts/Regression_analysis/n6z5Tf7K">regression analysis</a>. Given a response variable y and a p-dimensional predictor vector 
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\textbf {x}}}
 
, regression analysis aims to study the distribution of 
 
 
 
 y
 ∣
 
 
 x
 
 
 
 
 {\displaystyle y\mid {\textbf {x}}}
 
, the <a href="/facts/Conditional_distribution/0eGm3P9W">conditional distribution</a> of 
 
 
 
 y
 
 
 {\displaystyle y}
 
 given 
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\textbf {x}}}
 
. A dimension reduction is a function 
 
 
 
 R
 (
 
 
 x
 
 
 )
 
 
 {\displaystyle R({\textbf {x}})}
 
 that maps 
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\textbf {x}}}
 
 to a subset of 
 
 
 
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle \mathbb {R} ^{k}}
 
, k < p, thereby reducing the <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of 
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\textbf {x}}}
 
. For example, 
 
 
 
 R
 (
 
 
 x
 
 
 )
 
 
 {\displaystyle R({\textbf {x}})}
 
 may be one or more <a href="/facts/Linear_combinations/CVjL6kuN">linear combinations</a> of 
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\textbf {x}}}
 
.
A dimension reduction 
 
 
 
 R
 (
 
 
 x
 
 
 )
 
 
 {\displaystyle R({\textbf {x}})}
 
 is said to be sufficient if the distribution of 
 
 
 
 y
 ∣
 R
 (
 
 
 x
 
 
 )
 
 
 {\displaystyle y\mid R({\textbf {x}})}
 
 is the same as that of 
 
 
 
 y
 ∣
 
 
 x
 
 
 
 
 {\displaystyle y\mid {\textbf {x}}}
 
. In other words, no information about the regression is lost in reducing the dimension of 
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\textbf {x}}}
 
 if the reduction is sufficient.

Sufficient dimension reduction open-in-new

Sufficient dimension reduction