What Is Principal Component Analysis (PCA)? Meaning, Working, and Applications

• Principal component analysis (PCA) is defined as a statistical technique to reduce the dimensionality of complex, high-volume datasets by extracting the principal components containing the most information and rejecting noise or less important data while preserving all the crucial details.
• PCA is applied in various fields, from neuroscience to financial services and, most importantly, machine learning algorithms, to help machines quickly learn from large datasets.
• This article explains the meaning of PCA, its working and steps, and its business applications.

What Is Principal Component Analysis?

Principal component analysis (PCA) is a statistical technique to reduce the dimensionality of complex, high-volume datasets by extracting the principal components that contain the most information and rejecting noise or less important data while preserving all the crucial details.

Extracting Two Principal Components (PCs) From a Broad Dataset of Genetic Data

Source: BioTuringOpens a new window

PCA is a method to reduce the dimensionality of enormous data collections. This approach transforms an extensive collection of variables into a smaller group that retains nearly all the data in the larger set.

Lowering the number of variables in a data set inevitably reduces its precision, just as shrinking an image will leave out several details you could see in its larger counterpart. The purpose of dimension reduction is to forgo a certain level of precision for simplification. Smaller data collections are easier to investigate and visualize. This facilitates and accelerates the analysis of data points by machine learning (ML) algorithms.

PCA is designed to limit the total quantity of variables in a data set while maintaining as many details as possible.

The altered new features or PCA’s results are known as principal components (PCs) once PCA has been performed. The number of PCs is the same as or lesser than the number of original features in the dataset. The following are the characteristics of these principal components:

• The principal component must correspond to the linear arrangement of the initial features.
• The character of these components is orthogonal. This indicates there is no relationship within a pair of variables.
• From 1 to n, the significance of each component diminishes. This indicates that the number one PC is the most important, while the number “n” PC is the least significant.

PCA is a highly adaptable approach to analyzing datasets that may include multicollinearity, absent numbers, categorized data, and imprecise metrics, among others. The objective is to extract essential details from the data and convey it as an array of overarching indices known as principal components.

How was the PCA algorithm developed?

Karl Pearson, a British mathematician credited with establishing the field of mathematical statistics, invented PCA in 1901. He created PCA as a mechanical analog to the principal axis theorem. Harold Hotelling, an American statistician and economist, separately devised and named PCA in the 1930s.

 Depending on the field of application, PCA is differently named in different fields:

• The Hotelling transform in multivariate quality control
• The proper orthogonal decomposition (POD) in mechanical engineering
• The discrete Karhunen–Loève transform (KLT) in signal processing
• Empirical modal analysis in structural dynamics
• Empirical orthogonal functions (EOF) in meteorological science

In general, when referring to PCA, we are referring to calculating the principal component and applying those components to the data to induce a change. This makes the algorithm widely applicable across fields of study, including artificial intelligence and machine learning.

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Why is PCA important?

The quantity of data mandatory for obtaining a statistically significant outcome rises exponentially as the total number of features or parameters in a dataset increases. This may result in problems like overfitting, a spike in computation time, and a decline in the accuracy of machine learning structures. This is called the “curse of dimensionality” and happens when interacting with high-dimensional data.

The primary goal of PCA is to tackle the dimensionality problem, so let’s delve deeper into this idea.

High-dimensional data in machine learning relates to data with many distinctive characteristics or changing elements. As the number of dimensions grows, the range of conceivable feature combinations rises exponentially. This renders it computationally challenging to obtain a sample of the indicative data. In addition, it becomes difficult to carry out clustering and classification tasks.

In addition, some machine learning algorithms are highly cognizant of the number of dimensions, mandating more data to reach identical levels of accuracy, as achievable with lower-dimensional data.

To address this, advanced feature engineering techniques, including feature selection or feature extraction, are employed. Principal component analysis is a form of feature extraction that attempts to minimize the total amount of input features while preserving as much of the original data as practicable.

Pros and cons of principal component analysis

PCA, in some form or another, is essential in data operations. It helps convert huge and complex real-world datasets into representative data pools that a computer can understand and work with. However, it does have a few disadvantages. Before we look at the cons of PCA, here are its pros:

• PCA algorithms are relatively simple for machines to calculate and compute.
• The application of PCA accelerates machine learning operations and algorithms.
• It prohibits predictive algorithms from having problems with data overfitting.
• By eliminating superfluous correlated variables, you may boost the efficacy of ML algorithms.
• PCA frequently results in enhanced data visualization.
• You can reduce disruptions in data that cannot be instantaneously disregarded.

As for the disadvantages, PCA can be hard to interpret to begin with.

After calculating the principal components by employing the algorithm, determining key characteristics can sometimes be challenging. Calculating covariances or covariance matrices without a firm grasp of statistics is also daunting. Occasionally, the PCA results can be harder to comprehend than the initial array of components, regardless of whether they are vast or intricate.

That is why, today, most users rely on software applications for principal component analysis to reduce the chance of errors. Popular options include Matlab, XLSTAT, SPSS, and the R programming language.

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How Does Principal Component Analysis Work?

To understand how PCA works, we must first understand a principal component’s meaning. Let us consider the diagram below — it has many data points but only two PCs sufficient to understand the big picture.

How PCA Works

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In the figure above, multiple coordinates are drawn on a 2-D plane. There are two primary elements. PC1 is the principal component that describes the greatest amount of data variance. PC2 is an additional primary element that is orthogonal to PC1 (i.e., uncorrelated with it). Consequently, a PC is an uninterrupted line that accounts for most of the data’s variation. It has a magnitude and a direction.

Key concepts defining PCA’s working

Next, let us look at a few concepts central to the working of PCA:

1. Dimensionality

As mentioned, “dimensionality” is the number of characteristics or parameters utilized in the data. When dealing with high-dimensional data, like datasets and numerous variables (hence the curse), it can be tricky to visualize and analyze the relationships between variables.

While minimizing the number of variables in a dataset, dimensionality reduction techniques like principal component analysis (PCA) are implemented to preserve the most essential data. The original variables are transformed into the principal components, which are linear amalgamations of the initial variables.

2. Correlation

Correlation is a statistical measure indicating the direction and intensity of the linear relationship between the two variables. The covariance matrix is a square matrix that depicts the correlations between all variables in the dataset. It is determined by correlation. Using the correlation coefficient, the intensity and direction of the linear relationship between the two variables can be determined. It is through this calculation that we eventually arrive at the principal component.

3. Orthogonality

This term refers to the arrangement of the principal components as orthogonal to each other. In a nutshell, there is no duplication of information among the primary components, and they are independent. Each PC in the principal component analysis is constructed so as to maximize the variance it clarifies while conforming to the requirement that it be perpendicular to all other PCs. In our diagram, therefore, PC1 is orthogonal to PC2.

4. Eigenvectors

In algebra, an eigenvector is a vector that varies by no more than a scalar factor when subjected to a linear transformation. The equivalent eigenvalue, often represented by lambda, is the scaling factor for the eigenvector. In simple terms, these two concepts help calculate the actual and maximum variance from the original data possible in a dataset so that its dimensionality can be reduced accurately.

5. Covariance matrix

Covariance matrices are indispensable to the PCA algorithm’s calculation of the data’s principal components. It is an indicator of the correlation between two random variables. In principal component analysis, the covariance matrix will help you calculate the importance of values by computing their potential changes together as they are correlated.

Now that these fundamental concepts have been clarified, let us progress to the PCA process and its various elements.

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Principal component analysis steps

It is crucial to note that advanced statistical knowledge is a prerequisite to running PCA manually. Modern data modeling software simplifies the computations, even if you are a data science beginner. However, knowing how this extremely important machine learning algorithm works is always useful. Here are the key steps.

How Principal Component Analysis Works

1. Perform standardization on the initial set of continuous variables

Standardization must happen before principal component analysis (PCA) because PCA is extremely susceptible to variations in the initial variables. If there are significant differences within the spectrum of original variables, these parameters with the larger ranges will prevail. This will result in biased outcomes.

Mathematically, standardization can be performed by deducting the mean and dividing each variable value by its standard deviation. Once this is achieved, all variables will be adjusted to the same level.

2. Find out the covariance matrix

This phase aims to comprehend how the variables of the original data set deviate from the mean in relation to one another or to determine if there’s any relationship among them. This is because variables can become so highly correlated that they hold redundant data. To be able to identify such correlations, the covariance matrix is computed. Remember that the covariance matrix is merely a tabulation that summarizes the relationships between all possible variable combinations.

3. Calculate the eigenvectors and eigenvalues of the covariance matrix to arrive at the PCs

Eigenvectors and eigenvalues are the linear algebraic concepts required to calculate the principal components of data obtained from the covariance matrix.

As previously mentioned, PCs are new variables derived from linear combinations or composites of the original variables. These combinations are performed so that the newly created variables do not correlate, and most of the initial variables’ data is condensed or compacted into the initial components.

So, the idea is that 10-dimensional datasets provide users with ten principal components. But PCA attempts to place as much information as possible in the primary component, then as much remaining data as possible in the second, and so on until the resulting figure resembles the one shown below.

Percentage of Variance (Information) for Each by PC

Source: BuiltInOpens a new window

In PCA, there are the same number of principal components as variables in the data. Consequently, PCs are put together so that the first PC accounts for the most probable variation in the data set.

See the diagram below.

How PCA Constructs the Principal Components

Source: BuiltInOpens a new window

The first principal component will be the diagonal purple line since it captures the highest variance possible in the data (i.e., the blue dots). If you keep (visually) rotating the line with the red dots, you will eventually touch upon all the possible maximum variances and, therefore, all the PCs. That is precisely what is achieved by eigenvectors and eigenvalues, but at a statistical level, not a visual level.

The eigenvectors in a matrix are the orientations of the axes wherein the variance (information) is the highest. Eigenvalues are merely the coefficients associated with eigenvectors, which reflect the variance presented by every PC.

4. Figure out which PCs to retain

By calculating the eigenvectors and sorting these based on their eigenvalues in descending sequence, the order of significance of the PCs can be determined. In step 4, you must decide whether to retain all these components or remove those with less significance (low eigenvalues). This is accomplished using a feature vector.

The feature vector is an array in which the eigenvectors of the parts we choose to retain are organized in a column. This represents the initial stage toward dimension reduction. If we retain just k eigenvectors (or components) from the initial n, the resulting dataset will consist of only k variables in contrast to the original n.

5. Replot the data on your original axes

Other than standardization, we don’t change the data in the previous stages. We just choose the PCs and generate the feature vector, but the original axes of the source data set remain unchanged.

In this final phase, the feature vector is utilized to reposition the data from their original axes to those embodied in the principal components. This can be accomplished by multiplying the initial data set’s transposition by the feature vector’s transposition.

Note that in mathematics, the transpose of a dataset is one where the rows and columns are swapped for each other.

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Applications of Principal Component Analysis

Now that you know the meaning of PCA and the overall steps in its functioning, let us consider its key applications. Due to the ubiquity of data modeling operations, dimensionality reduction is used in many fields. This includes:

1. Biology and medicine

The discipline of neuroscience employs spike-triggered covariance analysis, a type of principal components analysis. PCA assists in identifying the stimulus properties that increase the probability of a neuron causing an “action” response.

2. Financial services

PCA reduces the number of dimensions in a complicated financial problem. Let us assume that an investment banker’s portfolio comprises 150 securities. To quantitatively analyze these equities, they will need a 150-by-150 correlation matrix, which renders the issue extremely complex. Nevertheless, PCA may assist in extracting 15 principal components that best define the stock variance. This would simplify the problem while additionally detailing the fluctuations of each of the 150 equities.

3. Facial recognition technology

An array of eigenvectors employed for the computer vision challenge of detecting human faces is called an eigenface. PCA is central to the eigenfaces method, as it generates the collection of possible faces that are likely to occur. Principal component analysis decreases the statistical complexity of face image depiction while maintaining its essential characteristics. This is crucial for facial recognition technology.

4. Image compression

Consider that we are given an extensive collection of 64×64 images of human features. Now, we wish to portray and retain pictures with significantly lower dimensions. Using the PCA concept, photographs can be compressed and stored in smaller, similarly precise files. However, it should be noted that reconstructing an image requires further computations.

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Takeaway

PCA is a foundational concept in large-scale data operations, fueling advanced AI and ML algorithms and scientific research. Massive sets of information are necessary to make algorithms more accurate, yet the complexity of the data can overwhelm algorithmic models. Principal component analysis (PCA) applies a relatively simple mathematical method to extract the most important dimensions with too much dimensionality so you know precisely which data to focus on.

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