Methods and formulas for x- and y-statistics in Partial Least Squares Regression

Select the method or formula of your choice.

X-scores

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.

Formula

Notation

TermDescription
nthe number of observations
mthe number of components
i the observations from 1 to n
jthe terms from 1 to p
X the design matrix
Wthe x-weight matrix

X-loadings

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 mth component. The x-loadings form a p × m matrix.

Formula

Notation

TermDescription
pthe number of terms
mthe number of components
ithe observations from 1 to n
jthe terms from 1 to p
t the x-scores
X the predictors

X-weights

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.

Formula

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

Notation

TermDescription
pthe number of terms
mthe number of components
ithe observations from 1 to n
jthe terms from 1 to p
X the x-residual matrix
u the y-scores

X-residuals

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.

Formula

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

Notation

TermDescription
nthe number of observations
pthe number of terms
ithe observations from 1 to n
jthe terms from 1 to p
t the x-scores
l the x-loadings

X-calculated values

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.

Formula

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.

Notation

TermDescription
nthe number of observations
pthe number of predictors
ithe number of observations from 1 to n
jthe number of predictors from 1 to p
t the x-scores
l the x-loadings

Y-scores

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.

Formula

Notation

TermDescription
nthe number of observations
mthe number of components
kthe number of responses from 1 to r
Y the y matrix
c the y-loadings

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.

Formula

Notation

TermDescription
rthe number of responses
mthe number of components
ithe observations from 1 to n
kthe responses from 1 to r
Y the responses
t the x-scores

Y-residuals

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.

Formula

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

Notation

TermDescription
nthe number of observations
rthe 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

Y-calculated values

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.

Formula

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.

Notation

TermDescription
nthe number of observations
rthe 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
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