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Partial least squares regression
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<p>Partial least squares (PLS) regression is a <a href="/facts/Statistics/DNPsKGYU">statistical</a> method that bears some relation to <a href="/facts/Principal_component_regression/w3hCdpLa">principal components regression</a> and is a reduced rank regression; instead of finding <a href="/facts/Hyperplane/2s2ycYYd">hyperplanes</a> of maximum <a href="/facts/Variance/ULBJKXD1">variance</a> between the response and independent variables, it finds a <a href="/facts/Linear_regression/5n998IhK">linear regression</a> model by projecting the <a href="/facts/Predicted_variable/dINfzIF2">predicted variables</a> and the <a href="/facts/Observable_variable/IAfRc11t">observable variables</a> to a new space of maximum covariance (see below). Because both the <i>X</i> and <i>Y</i> data are projected to new spaces, the PLS family of methods are known as bilinear factor models. Partial least squares discriminant analysis (PLS-DA) is a variant used when the <i>Y</i> is categorical. </p><p>PLS is used to find the fundamental relations between two <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> (<i>X</i> and <i>Y</i>), i.e. a <a href="/facts/Latent_variable/ohg99BnW">latent variable</a> approach to modeling the <a href="/facts/Covariance/hILocGNH">covariance</a> structures in these two spaces. A PLS model will try to find the multidimensional direction in the <i>X</i> space that explains the maximum multidimensional variance direction in the <i>Y</i> space. PLS regression is particularly suited when the matrix of predictors has more variables than observations, and when there is <a href="/facts/Multicollinearity/bQXzB6sA">multicollinearity</a> among <i>X</i> values. By contrast, standard regression will fail in these cases (unless it is <a href="/facts/Tikhonov_regularization/eq6kH8PX">regularized</a>). </p><p>Partial least squares was introduced by the Swedish statistician <a href="/facts/Herman_Wold/ubWRKL81">Herman O. A. Wold</a>, who then developed it with his son, Svante Wold. An alternative term for PLS is <i>projection to latent structures</i>, but the term <i>partial least squares</i> is still dominant in many areas. Although the original applications were in the social sciences, PLS regression is today most widely used in <a href="/facts/Chemometrics/xEgJ6fYV">chemometrics</a> and related areas. It is also used in <a href="/facts/Bioinformatics/D5x2L8ee">bioinformatics</a>, <a href="/facts/Sensometrics/T4wSzDQm">sensometrics</a>, <a href="/facts/Neuroscience/o5RV8Cas">neuroscience</a>, and <a href="/facts/Anthropology/W6WK1C3a">anthropology</a>. </p>