PCA of the yield, WUE, PFP and fruit quality of greenhouse tomatoes

PCA is the general name for a technique that uses sophisticated underlying mathematical principles to transform a number of possibly correlated variables into a smaller number of variables called principal components^63–65. The process for the analysis is as follows:

I) Select sample parameters. Normalization seeks to obtain comparable scales, which allow for attribute comparisons. The dimensionality reduction approach involves minimizing the squared errors via a vector coordinate transformation, and the measurement data are defined based on the following equation:

where n is the measured value of the sample number (i.e., yield of tomato fruit, WUE, PFP and fruit quality in this study), and p is the variable number.

II) Sample parameters are converted to standardized values. It is convenient to standardize the sample with the following equation:

where x¯j=1n∑i=1nxij, sj2=1n−1∑i=1n(xij−x¯j)2, and n is the measured value of the sample number.

III) The correlation matrix is calculated for the different irrigation and fertilization treatments and is defined based on the following equation:

where r _ij is the correlation coefficient of the original variable, r _ij = r _ji, and r _ij is given by the following equation:

IV) The eigenvalues of the R values and the eigenvectors for each sample number are calculated. A Jacobi iteration is used to determine the eigenvalues, as defined in the following equation:

where λ is the eigenvalue, E is the identity matrix and R is the correlation matrix. Next, these eigenvalues are sized down as λ ₁ ≥ λ ₂ ≥ … ≥λ _p ≥ 0, and the respective eigenvector e _i (i = 1, 2, ……) solved for:

where e _ij is the j-th component of e _i.

V) The contribution rate (C _r) and accumulative contribution rate (AC _r), with eigenvalues, are calculated using the following equations:

From the calculation results, the principal components corresponding to the characteristic value were greater than 1. The sample number of the principal components was selected as t; then, the factor of the former t was used as the corresponding data object associated with the component matrix S ₁, S ₂, ···, S _t.

VI) The mathematical model is established based on the PCA, as defined in the following equation:

where S _1i, S _2i, …, S _ti (i = 1, 2, …, t) are the eigenvectors corresponding to the principal components, and X ₁, X ₂, …, X _p are the standardized values, the value of which is converted based on the sample parameters.

VII) The evaluation process is determined based on the comprehensive evaluation index (F). The F value is defined by the following equation:

where λ ₁, λ ₂ … λ _t are the characteristic values corresponding to the principal components, and Q ₁, Q ₂, …, Q _t are the evaluation values of the different irrigation and fertilization treatments. A higher comprehensive evaluation index indicates a better treatment.