Sufficient dimension reduction

In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency.In a regression setting, it is often useful to summarize the distribution of y ∣ x {displaystyle ymid { extbf {x}}}   graphically. For instance, one may consider a scatter plot of y {displaystyle y}   versus one or more of the predictors. A scatter plot that contains all available regression information is called a sufficient summary plot.Suppose R ( x ) = A T x {displaystyle R({ extbf {x}})=A^{T}{ extbf {x}}}   is a sufficient dimension reduction, where A {displaystyle A}   is a p × k {displaystyle p imes k}   matrix with rank k ≤ p {displaystyle kleq p}  . Then the regression information for y ∣ x {displaystyle ymid { extbf {x}}}   can be inferred by studying the distribution of y ∣ A T x {displaystyle ymid A^{T}{ extbf {x}}}  , and the plot of y {displaystyle y}   versus A T x {displaystyle A^{T}{ extbf {x}}}   is a sufficient summary plot.If a subspace S {displaystyle {mathcal {S}}}   is a DRS for y ∣ x {displaystyle ymid { extbf {x}}}  , and if S ⊂ S d r s {displaystyle {mathcal {S}}subset {mathcal {S}}_{drs}}   for all other DRSs S d r s {displaystyle {mathcal {S}}_{drs}}  , then it is a central dimension reduction subspace, or simply a central subspace, and it is denoted by S y ∣ x {displaystyle {mathcal {S}}_{ymid x}}  . In other words, a central subspace for y ∣ x {displaystyle ymid { extbf {x}}}   exists if and only if the intersection ⋂ S d r s { extstyle igcap {mathcal {S}}_{drs}}   of all dimension reduction subspaces is also a dimension reduction subspace, and that intersection is the central subspace S y ∣ x {displaystyle {mathcal {S}}_{ymid x}}  .There are many existing methods for dimension reduction, both graphical and numeric. For example, sliced inverse regression (SIR) and sliced average variance estimation (SAVE) were introduced in the 1990s and continue to be widely used. Although SIR was originally designed to estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different.