# Difference between revisions of "Faq difference between a loading and a weighting"

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===Possible Solutions:=== | ===Possible Solutions:=== | ||

− | When performing Principal Components Analysis (PCA), you get loadings, P, which are an orthonormal basis which can be used to calculate scores: <code>T = X*P</code> or to estimate data <code>X = T*P'</code> | + | When performing Principal Components Analysis (PCA), you get loadings, <code>P</code>, which are an orthonormal basis which can be used to calculate scores: <code>T = X*P</code> or to estimate data <code>X = T*P'</code> |

These operations are invertible (repeating them gives the same result) because the loadings are the eigenvectors of <code>X'X</code>. | These operations are invertible (repeating them gives the same result) because the loadings are the eigenvectors of <code>X'X</code>. | ||

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When using Partial Least Squares (PLS), you get loadings, <code>P</code>, but also weights, <code>W</code>, because the decomposition is based on <code>X'Y</code>. The weights and loadings must be used together to calculate scores: <code>T = X*W*pinv(P'*W)</code> From a phenomenological point of view, the weights represent features in <code>X</code> which are related to the original <code>Y</code> values. The loadings represent the features in <code>X</code> which are related to the scores, <code>T</code>, which are the given factor's estimate of <code>Y</code>. | When using Partial Least Squares (PLS), you get loadings, <code>P</code>, but also weights, <code>W</code>, because the decomposition is based on <code>X'Y</code>. The weights and loadings must be used together to calculate scores: <code>T = X*W*pinv(P'*W)</code> From a phenomenological point of view, the weights represent features in <code>X</code> which are related to the original <code>Y</code> values. The loadings represent the features in <code>X</code> which are related to the scores, <code>T</code>, which are the given factor's estimate of <code>Y</code>. | ||

− | Note, by the way, that the weights are the ones used to calculate the regression vector (that which is used to make a prediction). Loadings are only used when calculating scores and, of course, Hotelling's | + | Note, by the way, that the weights are the ones used to calculate the regression vector (that which is used to make a prediction). Loadings are only used when calculating scores and, of course, Hotelling's T<sup>2</sup>. |

## Latest revision as of 13:57, 8 January 2019

### Issue:

What is the difference between a loading and a weighting?

### Possible Solutions:

When performing Principal Components Analysis (PCA), you get loadings, `P`

, which are an orthonormal basis which can be used to calculate scores: `T = X*P`

or to estimate data `X = T*P'`

These operations are invertible (repeating them gives the same result) because the loadings are the eigenvectors of `X'X`

.

When using Partial Least Squares (PLS), you get loadings, `P`

, but also weights, `W`

, because the decomposition is based on `X'Y`

. The weights and loadings must be used together to calculate scores: `T = X*W*pinv(P'*W)`

From a phenomenological point of view, the weights represent features in `X`

which are related to the original `Y`

values. The loadings represent the features in `X`

which are related to the scores, `T`

, which are the given factor's estimate of `Y`

.

Note, by the way, that the weights are the ones used to calculate the regression vector (that which is used to make a prediction). Loadings are only used when calculating scores and, of course, Hotelling's T^{2}.

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