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General Calculation For Principal Component Analysis
PCA or Principal component analysis is a statistical process that simplifies the complexity of sample spaces along multiple dimensions while preserving their information. Suppose that there is a sample with each individual with p variables (X1, X2 ... XP), that is, the sample space p dimensions. PCA makes it possible to find several underlying factors (z <p) that are nearly the same as the original p variable. Where previously p-values were required to characterize each individual, they are now sufficient z-values. Each of these new variables is called a principal component.
Principal component analysis refers to a family of technologies known as untrained education. The purpose of guided learning methods described in previous chapters is to predict from a set of response variables and predictors. For this, it has p characteristics (x 1, x 2"¦ exp), and response variables and n measured in observations. In the case of unsupervised learning, feedback is not taken into consideration and variables, because the goal is not to predict and, but for example, to identify subgroups, extract information using predictors. The main problem facing unavailable learning methods is the difficulty of validating the results, as there are no response variables available to them.
The PCA method, therefore, makes it possible to 'condense' the information provided by multiple components into only a few components. This is a very useful way to implement it after using other statistical techniques such as regression, clustering ... Nevertheless, we must remember that it is still necessary to have the value of the original variable to compute the components.
Analysis In Principal Components
The principal components method aims to convert a set of variables, to the entitled originals, in a different set of variables called main workings. The latter are characterized by being unrelated to each other and, also, they can be ordered according to the information they have incorporated. As a measure of the amount of data fused in a component, its variance is used.
That is, the better its variance, the better the amount of information it has incorporated said component. For this reason, the first component that has a greater variance, while the last component has the lowest variance.
In general, the extraction of principal components is carried out on variables typified to avoid problems derived from the scale, although it can also be applied to variables expressed in deviations from the mean. If p variables are typified, the calculation of the variances is p, since the variance of a variable Typified is by definition 1.
The new set of variables that is obtained by the principal components method is equal in number to the original variables. It is important to note that the sum of its variances is equal to the calculation of the variances of the original variables.
The variance between the 2 sets of variables is that the components Principal are calculated in such a way that they are unrelated to each other. When the variables originals are highly correlated with each other, most of their variability can be explained with very few components.
If the original variables were completely unrelated to each other, the analysis of principal components would be completely uninteresting, since in that case, the principal components would match the original variables.
The principal components are expressed as a linear combination of the variable's original. From an application point of view, the principal components method is considered as a reduction method, that is, a method that allows reducing the dimension of the number of inventive variables that have been measured in the analysis.
It does not tie in with all the main components, note that the largest number corresponds to the total number of variables. Having all of them will not simplify the problem, so the researcher will have to choose from several options, as they are infrequent and explanatory, suggesting an acceptable ratio of global variation or point cloud inertia that would indicate a reasonable loss of information.
The application can be simplified for some components, the latter for other multivariate techniques (regression, clusters, etc.) by reducing many variables.
The principal component method can be considered as a data reduction method, addressing other problems such as factor rotation, contrasts, this is done within the factor analysis which implies more formalism. In this sense, the method of principal components is part of descriptive statistics.
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