Sliced inverse regression
Introduction '''Sliced inverse regression (or''' '''SIR''') is a tool for dimensionality reduction in the field of multivariate statistics.In statistics, regression analysis is a method of studying the relationship between a response variable ''y'' and its input variable , which is a ''p''-dimensional vector. There are several approaches in the category of regression. For example, parametric methods include multiple linear regression, and non-parametric methods include local smoothing.As the number of observations needed to use local smoothing methods scales exponentially with high-dimensional data (as ''p'' grows), reducing the number of dimensions can make the operation computable. Dimensionality reduction aims to achieve this by showing only the most important dimension of the data. SIR uses the inverse regression curve, , to perform a weighted principal component analysis.
Model Given a response variable  and a (random) vector  of explanatory variables, '''SIR''' is based on the model:where  are unknown projection vectors,  is an unknown number smaller than ,  is an unknown function on as it only depends on arguments, and  is a random variable representing error with  and a finite variance of . The model describes an ideal solution, where  depends on  only through a dimensional subspace; i.e., one can reduce the dimension of the explanatory variables from to a smaller number without losing any information.An equivalent version of  is: the conditional distribution of  given  depends on  only through the  dimensional random vector . It is assumed that this reduced vector is as informative as the original  in explaining .The unknown  are called the ''effective dimension reducing directions'' (EDR-directions). The space that is spanned by these vectors is denoted by the ''effective dimension reducing space'' (EDR-space).
Relevant linear algebra background Given , then , the set of all linear combinations of these vectors is called a linear subspace and is therefore a vector space. The equation says that vectors  span , but the vectors that span space  are not unique.The dimension of  is equal to the maximum number of linearly independent vectors in . A set of  linear independent vectors of  makes up a basis of . The dimension of a vector space is unique, but the basis itself is not. Several bases can span the same space. Dependent vectors can still span a space, but the linear combinations of the latter are only suitable to a set of vectors lying on a straight line.
Inverse regression Computing the inverse regression curve (IR) means instead of looking for *, which is a curve in it is actually*, which is also a curve in , but consisting of  one-dimensional regressions.The center of the inverse regression curve is located at . Therefore, the centered inverse regression curve is*which is a  dimensional curve in .
Inverse regression versus dimension reduction The centered inverse regression curve lies on a -dimensional subspace spanned by . This is a connection between the model and inverse regression.Given this condition and , the centered inverse regression curve  is contained in the linear subspace spanned by , where .
Estimation of the EDR-directions After having had a look at all the theoretical properties, the aim now is to estimate the EDR-directions. For that purpose, weighted principal component analyses are needed. If the sample means ,  would have been standardized to . Corresponding to the theorem above, the IR-curve  lies in the space spanned by , where . As a consequence, the covariance matrix  is degenerate in any direction orthogonal to the . Therefore, the eigenvectors  associated with the largest eigenvalues are the standardized EDR-directions.
Algorithm The algorithm to estimate the EDR-directions via SIR is as follows.'''1.''' Let  be the covariance matrix of . Standardize  to: ( can also be rewritten as: where . )'''2.''' Divide the range of  into  non-overlapping slices  is the number of observations within each slice and  is the indicator function for the slice:: '''3.''' Compute the mean of  over all slices, which is a crude estimate  of the inverse regression curve :: '''4.''' Calculate the estimate for :: '''5.''' Identify the eigenvalues  and the eigenvectors  of , which are the standardized EDR-directions.'''6.''' Transform the standardized EDR-directions back to the original scale. The estimates for the EDR-directions are given by:: 	(which are not necessarily orthogonal)
References *Li, K-C. (1991) "Sliced Inverse Regression for Dimension Reduction", Journal of the American Statistical Association, 86, 316–327  Jstor*Cook, R.D. and Sanford Weisberg, S. (1991) "Sliced Inverse Regression for Dimension Reduction: Comment", Journal of the American Statistical Association, 86, 328–332  Jstor*Härdle, W. and Simar, L. (2003) ''Applied Multivariate Statistical Analysis'',  Springer Verlag. *Kurzfassung zur Vorlesung Mathematik II im Sommersemester 2005, A. Brandt