Introduction

We will discuss What Is Principal Component Analysis? Using a smaller number of “summary indices” that are simpler to display and understand, principal component analysis, or PCA, is a statistical technique that lets you condense the information contained in huge data tables. Measurements reflecting the characteristics of production samples, chemical substances or processes (continuous or batch), time points in a process, batches from a process, biological people, or DOE-protocol trials can all be the basis for the underlying data. One of the multivariate statistical methods that is most often use the principal component analysis. It is a statistical technique that goes by the general name of factor analysis and has been extensively employed in the fields of pattern recognition and signal processing.

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The MVDA’s mother approach is PCA.

PCA serves as the foundation for multivariate data analysis using projection techniques. In order to identify trends, jumps, clusters, and outliers, it is crucial to describe a multivariate data table as a smaller number of variables (summary indices). The links between observations and variables as well as those within the variables may be revealed by this overview.

Although PCA dates back to Cauchy, Pearson is credited with developing it in statistics. Pearson defined PCA as the process of locating “lines and planes of closest fit to systems of points in space” [Jackson, 1991].

With its great flexibility, PCA can analyse datasets with various characteristics, such as multicollinearity, missing values, categorical data, and erroneous measurements. The objective is to identify the key information in the data and express it as a collection of summary indices known as primary components.

How PCA operates

Think of a matrix called X that has N rows, or “observations,” and K columns (aka “variables”). We create a variable space with the same number of dimensions as the variables for this matrix (see figure below). One coordinate axis is represented by each variable. The length has been standardised for each variable in accordance with a scaling criterion, typically by scaling to unit variance.

The K-dimensional variable space is then filled with each observation (row) of the X-matrix in the following step. Therefore, the rows of the data table create a cluster of points in this area.

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Mean centering

The next step in mean-centering is to subtract the data from the variable averages. In the K-space, the averages vector refers to a particular point.

The coordinate system is moved so that the average point is now the origin when the averages are subtracted from the data.

The first essential element

The data set is prepared for computation of the first summary index, the first principal component, after mean-centering and scaling to unit variance (PC1). The line in the K-dimensional variable space that most closely resembles the data in a least squares sense is this component. The average point is intersected by this line. To obtain a coordinate value along the PC-line, each observation (yellow dot) may now be projected onto this line. The score is another name for this new coordinate value.

The second essential element

Usually, only one summary index or primary component is needed to adequately represent a data set’s systematic variance. As a result, the second principal component (PC2), a second summary index, is computed. A line orthogonal to the first PC and representing the second PC in the K-dimensional variable space. Additionally traversing the average point, this line maximises the approximation of the X-data.

Conclusion

So far we have discussed What Is Principal Component Analysis and How It Is Used?

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