Sliced inverse regression

Sliced inverse regression (SIR) is a tool for dimensionality reduction in the field of multivariate statistics.[1]

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

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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

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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

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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

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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

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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

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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

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  1. ^ Li, Ker-Chau (1991). "Sliced Inverse Regression for Dimension Reduction". Journal of the American Statistical Association. 86 (414): 316–327. doi:10.2307/2290563. ISSN 0162-1459.