But before seeing that this holds true, we will have a look at how the inverse regression curve is computed within the SIR-Algorithm, which will be introduced in detail later. What comes is the "sliced" part of SIR. We estimate the inverse regression curve by dividing the range of formula_3 into formula_65 nonoverlapping intervals (slices), to afterwards compute the sample means formula_66 of each slice. These sample means are used as a crude estimate of the IR-curve, denoted as formula_67. There are several ways to define the slices, either in a way that in each slice are equally many observations, or we define a fixed range for each slice, so that we then get different proportions of the formula_68 that fall into each slice. As just mentioned, the centered inverse regression curve lies on a formula_7-dimensional subspace spanned by formula_63 (and therefore also the crude estimate we compute). This is the connection between our Model and Inverse Regression. We shall see that this is true, with only one condition on the design distribution that must hold. This condition is, that:  I.e. the conditional expectation is linear in formula_72, that is, for some constants formula_73. This condition is satisfied when the distribution of formula_74 is elliptically symmetric (e.g. the normal distribution). This seems to be a pretty strong requirement. It could help, for example, to closer examine the distribution of the data, so that outliers can be removed or clusters can be separated before analysis. Given this condition and formula_20, it is indeed true that the centered inverse regression curve formula_59 is contained in the linear subspace spanned by formula_77, where formula_78. The proof is provided by Duan and Li in "Journal of the American Statistical Association" (1991). After having had a look at all the theoretical properties, our aim is now to estimate the EDR-directions. For that purpose, we conduct a (weighted) principal component analysis for the sample means formula_79, after having standardized formula_74 to formula_81. Corresponding to the theorem above, the IR-curve formula_82 lies in the space spanned by formula_83, where formula_84. (Due to the terminology introduced before, the formula_85 are called the "standardized effective dimension reducing directions".) As a consequence, the covariance matrix formula_86 is degenerate in any direction orthogonal to the formula_85.