Principal component analysis, sometimes known as principal components analysis, is a powerful statistical tool for condensing large data sets into more easily readable tables while preserving key information. It is a versatile tool widely adopted by businesses to turn complex data into manageable formats, simplify the decision-making process, and highlight profitable strategies.

To help you master this strategic technique, our experts at Business2Community have curated this detailed guide. From performing the analysis to understanding its limitations, we’ve covered everything you need to know.

## Principal Component Analysis – Key Takeaways

• Principal component analysis reduces the dimensionality of large data sets with many variables by identifying principal components.
• This statistical technique concerns linear algebra and can investigate the correlations among variables.
• Due to its complex nature, small businesses should adopt other analytical tools to cross-examine results to enhance accuracy.

## What is a Principal Component Analysis?

Principal component analysis summarizes a large quantity of data points into a smaller data set while retaining the most vital information from the original data. It is a dimensionality reduction method that examines the linear combinations of variables, including dependent variables and independent variables.

In principal component analysis, the first principal component refers to data with the maximum variance, i.e. the most statistical information, and the second principal component explains data with the second-biggest variance, and so on.

This data science technique is crucial in the research process as it highlights statistical structures like covariance matrix, eigenvalues, the mean, and standard deviation. It allows you to process and visualize the data matrix more easily, which is handy when analyzing intricate information like stock returns.

### Principal Component Analysis vs. Factor Analysis

Beginners often confuse principal component analysis with factor analysis. Although they both work wonderfully in condensing massive data sets, they are different in their purposes.

Factor analysis is used to identify latent factors and the underlying structure that can’t be observed directly, whereas principal component analysis reduces the dimensionality of your data to find the principal components within.

Factor analysis is a great tool to supplement your research but it can’t replace principal component analysis.

### Who Needs to Do a Principal Component Analysis?

Being able to interpret the covariance matrix is useful in a wide range of business settings. As a form of cluster analysis for visualizing the correlation matrix, this data analysis technique is fundamental to strategic planning.

To illustrate how different professionals can make good use of principal component analysis, here are a few examples:

• AI programmers/experts: Data compression facilitates computer vision development and makes machine learning algorithms easier. AI professionals can better control the statistical learning process of AI and improve performance.
• Stock traders/analysts: If you’re a stock trader, the principal component analysis method enables you to evaluate the covariance matrix of different stocks and financial products so you can learn about the risks and opportunities involved.
• Business owners: As a business owner, studying the linear combinations and covariance matrix helps you pinpoint all the variables crucial to your decision-making process. You can make more informed business decisions with the relative importance of different factors in mind.

## How to Perform a Principal Component Analysis

Using principal component analysis to find the principal components and covariance matrix can be a complicated process. There are statistical software programs you can use to run the calculations to eliminate human error and generate quicker results.

That said, you should still learn about the process and assumptions of principal component analysis to be able to manually adjust the analysis or correct errors when necessary.

### Assumptions of Principal Component Analysis

To obtain valid results from your data set, these are the principal component analysis assumptions:

• All the variables must have a linear relationship and be able to form a linear regression line.
• Sampling adequacy must be fulfilled. The original data points need to be large enough to produce a meaningful final data set. Inadequate sample size may fail to pick out the right principal components.
• There must be at least two variables in the study and they can be measured at a continuous level to find the cumulative proportion of total variance.
• Whether you’re working with categorical data or numerical values, the data you work with should be suitable for dimension reduction, meaning they have to be highly correlated.
• There should not be a significant amount of outliers in the study.

Now that we’ve gone through the various assumptions, it’s time to look at the principal component analysis process.

### Step 1: Standardization/Normalization of the Data Set

An unscaled data set can contain uneven weight from the initial variables, which will obscure your final results. Initial variables with larger ranges will overpower those with smaller ranges, leading to biased data analysis results. You need to make sure there are no normalization constraints in your original data.

The first step to creating a valid principal component analysis is to standardize your data set. You can do so by calculating the z-score of each variable.

The transformed data will contribute equally to the analysis, which is essential in obtaining valid principal component scores from your original variables.

### Step 2: Conduct the Covariance Matrix Computation

After normalizing your data set, you need to calculate the covariance matrix. Some variables may be closely related in such a way that there is redundant information. A covariance matrix computation identifies these relationships in your data set and to which extent the variables are related.

If the covariance coefficient is positive, the two variables are positively correlated. If it is negative, the variables are inversely correlated.

Regardless of which data set you work with, the covariance matrix will always be a p x p symmetric matrix, where p denotes the number of dimensions you are working with.

### Step 3: Calculate the Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors demonstrate the orthogonal direction in the plot, which shows you the variability of your data set and where the maximum variance is. They are used to find the principal components in your data matrix.

To find the eigenvalues and eigenvectors, you need to solve the equation:

### Step 4: Rank the Eigenvalues

Arrange the calculated eigenvalues on a chart in ascending or descending order. Each corresponding eigenvalue represents a dimension you’re working with. These corresponding eigenvalues form a feature vector that contains the condensed information of the data set.

The eigenvalues show the explained variance of variables in the linear combination. In essence, this technique puts as much information as possible in the first principal component and as much information left as possible in the second principal component.

### Step 5: Find the Principal Components

You can decide how many principal components you’re interested in by choosing the corresponding eigenvalues in your data set. These principal components become the new variables, represented by different linear combinations.

The first principal component represents data points with maximum variance explained by the correlation matrix and the second principal component denotes data with the second-most variance.

Most of the time, the first two principal components are enough to summarize your data set and offer valuable information. Still, when you conduct this dimension reduction method, you need to decide how many components you want to include in your conclusion based on your needs and research purposes.

## Examples of Principal Component Analysis

Now that you know the steps and assumptions of principal component analysis calculations, it’s time to look at two real-life examples of how finding principal components and the covariance matrix can boost your business performance.

### Example 1: Use Covariance to Forecast Stock Returns

You’re a stock trader interested in knowing how two stocks are correlated in such a way that you can foresee changes and gain an edge in curating profitable investment strategies.

Here are the 3-day returns of two stocks:

Using the covariance formula:

[(2% – 1.57%) x (5.3% – 4.77%) + (1.2% – 1.57%) x (4.2% – 4.77%) + (1.5% – 1.57%) x (4.8% – 4.77)%] / (3-1) = 0.218

A positive value indicates a positive correlation. When one variable increases, the other variable will increase as well. Covariance matrices assist you in identifying stock return fluctuations more effectively by revealing the linear combination in your data sets.

### Example 2: Adjust Production Levels With Principal Component Analysis

In this principal component analysis example, we’ll show you how to interpret the analysis results to adjust your production levels accordingly as a production manager.

Here is a principal component analysis score plot created by Sartorius to demonstrate the consumption of different food items:

Items on the same side, like frozen vegetables (Frozen Veg) and sweetener (Sweetner), of the plot origin are positively correlated whereas, items on the opposite side of the plot origin, such as Olive Oil and Frozen Veg, are inversely correlated.

This data analysis allows you to prepare for upcoming changes in demand. For example, when the consumption of frozen vegetables increases, sweeteners will as well. On the other hand, the demand for olive oil will fall. Knowing the dynamics among variables in the data set is vital to your role in optimizing production levels to ensure efficiency and profitability.

## How to Adjust a Principal Component Analysis

Sometimes, the covariance matrix and the principal component analysis results may not be satisfactory. Perhaps the principal components don’t align with expectations or the covariance matrix fails to represent data sets accurately.

These are some factors you can manipulate to influence the principal components:

• Increase sampling size: Ample samples ensure the accuracy of the results. Increasing your sampling size can possibly change the principal components if the previous analysis was made with an inadequate sample size.
• Update production methods: Using new technologies or updating the production process can lead to an increase in consumption for a particular product. You can adjust your production strategies to examine their impacts on the covariance matrix.
• Change the initial variables: Adjusting the initial variables allows you to spot missing values and hidden dynamics among factors. It can change the principal components and construct a more comprehensive view of your data.

## Limitations of Principal Component Analysis

Although principal component analysis is an insightful tool that breaks down the essence of massive data points, it still has its limitations.

First, principal component analysis is not a beginner-friendly tool, especially if you don’t have a business background. You may need to outsource the process to professionals, which can be a burdensome cost.

Second, dimensionality reduction can lead to a slight decrease in accuracy due to missing values. In most cases, the loss is minute but it can still damage the authenticity of the analysis.

Third, due to its difficult calculation process, finding principal components can be time-consuming. It isn’t suitable if you need to process a large amount of data quickly.

To compensate for its disadvantages, you should utilize other analytical tools like multivariate data analysis and reliability analysis. Multivariate data analysis investigates and explains the relationships among several variables while reliability analysis determines the trustworthiness of your test.

These additional tools can validate your research and allow you to present your findings with greater confidence.

## The Value of Principal Component Analysis

Principal component analysis is a powerful tool for simplifying complicated data sets, investigating the relationships among variables, and grouping principal components that contain the most important information.

From business owners to stock traders and financial analysts, being able to perform and interpret principal component analysis could be a major contributor to greater business success. You can curate smarter business strategies, adjust production levels, and understand consumer preferences with this tool.

When incorporating principal component analysis into your strategic planning progress, it is important to be aware of its limitations. The best way is to utilize multiple analysis techniques to cover all your business needs.