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The x-scores are linear combinations of the terms; similar to principal component scores. The x-scores form an *n* × *m* matrix of uncorrelated columns. Providing a window to the x-space, the x-scores are projections of the observations on the PLS components. PLS fits the x-scores, which replace the original terms in the data, using least squares estimation.

Term | Description |
---|---|

n | the number of observations |

m | the number of components |

i | the observations from 1 to n |

j | the terms from 1 to p |

X | the design matrix |

W | the x-weight matrix |

The x-loadings are linear coefficients that link the terms to the x-scores; similar to eigenvectors in principal components analysis. The loading values indicate the importance of the corresponding term to the *m*^{th} component. The x-loadings form a *p* × *m* matrix.

Term | Description |
---|---|

p | the number of terms |

m | the number of components |

i | the observations from 1 to n |

j | the terms from 1 to p |

t | the x-scores |

X | the predictors |

The x-weights describe the covariance between the terms and the responses. In the algorithm, the weights ensure the x-scores are orthogonal, or unrelated to one another, and are used to calculate the x-scores. The x-weights form a *p * ×* m* matrix.

Minitab scales the vector of weights so that the length of the vector is 1.

Term | Description |
---|---|

p | the number of terms |

m | the number of components |

i | the observations from 1 to n |

j | the terms from 1 to p |

X | the x-residual matrix |

u | the y-scores |

The x-residuals contain the variance in the predictors not explained by the PLS model. Observations with relatively large x-residuals are outliers in the x-space, indicating that they are not well explained by the model.

The x-residuals are the differences between the actual predictor values and the x-calculated values and are on the same scale as the original predictors. The x-residual matrix, similar to the original x-matrix, is an *n* × *p* matrix.

The x-residual matrix is initialized to the standardized x-matrix. After calculating the *m* ^{th} component and obtaining the x-score vector and the x-loading vector, Minitab calculates the x-residuals.

Minitab then calculates the unstandardized x-residuals by multiplying the standardized x-residuals by the standard deviation of the predictor values.

Term | Description |
---|---|

n | the number of observations |

p | the number of terms |

i | the observations from 1 to n |

j | the terms from 1 to p |

t | the x-scores |

l | the x-loadings |

The x-calculated values are linear combinations of the x-scores; contain the variance in the predictors explained by the PLS model. Observations with relatively small x-calculated values are outliers in the x-space and are not well explained by the model.

The x-calculated matrix, similar to the original x-matrix, is an *n* × *p* matrix, where *n* equals the number of observations and *p* equals the number of predictors. The x-calculated values are on the same scale as the predictors.

The x-calculated matrix is initialized to the zero matrix. After calculating the *m* ^{th} component and obtaining the x-score vector and the x-loading vector, Minitab calculates the x-calculated values. If the number of components equals the number of predictors, then the x-calculated value equals the original x-value.

Minitab then calculates the unstandardized x-calculated values by multiplying the standardized x-calculated values by the standard deviation of the predictor values and adding the mean.

Term | Description |
---|---|

n | the number of observations |

p | the number of predictors |

i | the number of observations from 1 to n |

j | the number of predictors from 1 to p |

t | the x-scores |

l | the x-loadings |

The y-scores are linear combinations of the response variables. The y-scores form an *n* × *m* matrix. Providing a window to the y-space, the y-scores are projections of the observations on the PLS components.

Term | Description |
---|---|

n | the number of observations |

m | the number of components |

k | the number of responses from 1 to r |

Y | the y matrix |

c | the y-loadings |

The y-loadings are linear coefficients that link the responses to the y-scores. The loading values indicate the importance of the corresponding response to the *m* ^{th} component. The y-loadings form an *r* ×*m* matrix.

Term | Description |
---|---|

r | the number of responses |

m | the number of components |

i | the observations from 1 to n |

k | the responses from 1 to r |

Y | the responses |

t | the x-scores |

The y-residuals contain the remaining variance in the responses not explained by the PLS model. Observations with relatively large y-residuals are outliers in the y-space, indicating that they are not well explained.

The y-residuals are the differences between the actual response values and the y-calculated values, and are on the same scale as the original responses. The y-residual matrix, similar to the original y-matrix, is an *n* × *r* matrix.

The y-residual matrix is initially set to the standardized Y matrix. After Minitab calculates the *m* ^{th} component and obtains the x-score and y-loading vectors, Minitab determines the standardized y-residuals.

Minitab then calculates the unstandardized y-residuals by multiplying the standardized y-residuals by the standard deviation of the corresponding response values.

Term | Description |
---|---|

n | the number of observations |

r | the number of responses |

i | the observations from 1 to n |

k | the responses from 1 to r |

t | the x-scores |

c | the y-loadings |

The y-calculated values are linear combinations of the x-scores; contain the variance in the responses explained by the PLS model. Observations with relatively small y-calculated values are outliers in the y-space and are not well explained.

The y-calculated matrix, like the original y-matrix, is an *n* x *r* matrix. The y-calculated matrix is initially set to the zero matrix. After Minitab calculates the *m* ^{th} component and obtains the x-score and y-loading vectors, Minitab determines the standardized y-calculated values.

Minitab then calculates the unstandardized y-calculated values by multiplying the standardized y-calculated values by the standard deviation of the corresponding response and adding the mean.

Term | Description |
---|---|

n | the number of observations |

r | the number of responses |

i | the observations from 1 to n |

k | the responses from 1 to r |

t | the x-scores |

c | the y-loadings |