Principle Component Analysis: A Complete Overview(2021)

Introduction 

Artificial intelligence is flourishing in today’s world. With the world advancing towards data science and digitalization this technology of principle component analysis helps to process data without the loss of any important info. Machine learning is quite useful for a huge amount of data, but it has its own disadvantage, i.e. the blasphemy of dimensionality. These large voluminous data sets are increasing more and more with time making data more complicated and convoluted. For the purpose of getting rid of this curse, the dimensionality reduction process was introduced. This process involves techniques that filter the datasets and produce only that information useful and necessary for guidance and decision-making. 

This blog will help you understand all there is to know about principle component analysis and how it works step by step so that every firm or individual can use it irrespective of their mathematical background. Principal component analysis originated in a paper by Hotelling and was published in the Journal of Educational Psychology. Since then, various fields of biological, physical, and engineering sciences have put it to use in their data analysis. Principle component analysis serves as a means of decreasing the dimensionality of voluminous data sets. 

In this article let us look at:

1. What is Principle component analysis?
2. Step by step process of principle component analysis
3. Why we need principle component analysis?
4. Principle component analysis with Python

1. What is Principle component analysis?

Principle component analysis is a technique to reduce dimensional data that helps you to recognize associations and correlations in a dataset to enable it to transform into another set of data that is much easily readable, easily processed, and of a much lower dimension than the original set without losing the initial significant data. The main idea behind PCA analysis is to discover associations and correlations in a data set. Next, the data set is reduced, and the significant and important data in the initial dataset is retained.

The process of principle component analysis is crucial to solving problems driven by data and the usage of data sets of higher dimensions. Smaller data sets are very easy to visualize and make the job of data analysis much easier, faster, and efficient. 

2. Step by step process of principle component analysis

Now that we have understood what principle component analysis is, let’s move on to the various steps which are to be followed in this procedure: 

Step 1:

Standardisation:

Standardisation is the first step in this procedure. It is all about filtering and allocating your dataset in such a way that all variables, along with their values, fall under a parallel range. For example, if you have a dataset containing variable values ranging from 10-100, 100-1000, and 1000-10000 the output which is going to be produced by these variables will be biased as the variables with a higher range will affect the data more than variables with a lower range. Hence, the standardisation of data is essential to bring down the data to an equivalent and comparable range. Standardisation involves subtracting each data value from the average and distributing it by the complete divergence in the data set. 

Step 2:

Computing the covariance matrix:

This next step in the process of principle component analysis helps to identify the dependencies and correlations between the different features of a dataset. The correlation between the different data sets is explained with the help of this covariance matrix. This step aims to understand the difference or relationship between the variables of data with respect to each other. In order to identify the correlation between variables, we compute the covariance matrix. The covariance matrix is a p x p symmetric matrix, p being the number of dimensions. 

For example, for a data set with three dimensions and variables, x, y, and z, the given matrix form is drawn. 

If the covariance sign is positive, then the two variables act together, i.e. increase or decrease together (positively correlated), and if the sign of covariance is negative, the variables are inversely correlated. 

Step 3:

Calculating the Eigenvalues and Eigenvectors: 

These two terms refer to mathematical constructs that must be computed from the covariance matrix to determine the principle components of a database. 

What are the principle components? Principle components are the new set of variables obtained after standardising the old and initial set of variables. The new set of variables are highly significant. They possess the most useful information brought out by a data set that was scattered among the initial variables. 

The two algebraic formulations, eigenvectors and eigenvalues, go hand in hand. Eigenvectors use the covariance matrix to discover that point of data which possesses the most amount of variances. The principal components are identified with the help of Eigenvectors while eigenvalues identify the scalars of the computed eigenvectors.

Step 4:

Computing the principle components 

Once the eigenvectors and eigenvalues are computed, the next step is to arrange them in a descending order where the highest eigenvalue of the eigenvector is the most significant one forming the principle component. The less significant principle component can then be removed in order to filter the data and decrease its size. The last step is to construct a feature matrix containing all important data featuring all the variables possessing the maximum information about the data. 

Step 5:

Reducing the dimensions of the data set

The final step of the principle component analysis is reorganising the original data representing the most significant information of the data set. 

3. Why we need principle component analysis?

Often data produced is highly dimensional and includes a ton of insignificant information which is not required by the data scientists. This is where principle components analysis comes into use as it enables the data scientists to allocate the data and set aside those variables which possess the most value and significance. Huge chunks of data is a curse as it wastes time, and decreases productivity. Principle component analysis enables to build a better predictive model than the initial one. 

4. Principle component analysis with Python

The following steps enable us to perform principle component analysis with the help of python programming language:

• Import required packages.
• Import data set.
• Format the data.
• Standardisation.
• Compute covariance matrix.
• Calculate eigenvectors and eigenvalues.
• Compute the feature vector.
• Use the principle component analysis function to reduce the dimensionality of the dataset. 
• Projecting the variance with respect to the principle components.

Conclusion 

To sum it up the idea of principle component analysis is very simple, i.e. to reduce and decrease the number of datasets and to preserve the most significant and important information for further use. The principle components analysis serves various advantages – removes correlated features, improves algorithm performance, reduces overfitting, and improves visualization. However, as anything has its own pros and cons, principle component analysis also has some. When you implement principle component analysis, independent variables become less interpretable. Principle components are not easily readable and interpretable as original features.

Data standardization is essential before principle component analysis. Although principle components try their best to cover all significant information and record all important variables, often important information is lost when compared to the original data. Principle components must be chosen with the utmost care and must cover maximum variance. This was an overview of principle component analysis, the steps to implement it, and its uses. After reading the blog, we hope you have understood how does principle component analysis work, and when to use PCA analysis. 

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