What Is Principal Component Analysis?

Principal component analysis (PCA) is an unsupervised machine learning algorithm that attempts to reduce the dimensionality (number of features) within a dataset while retaining as much information as possible. It simplifies complex, high-volume datasets to accelerate further analysis by other machine learning algorithms. Data sets with too many dimensions, like images, require significant time and computing effort when used in ML training. Principle component analysis lowers the variables in the data set by extracting critical information and cutting out unnecessary details and noise. Removing information lowers precision but increases processing speed for machine learning use cases.

Principal components are summarized representations of the original datasets. These representations consist of equal or fewer variables than the number of original variables. While PCA can produce multiple principal components, the first and second are the most important. Both form a model plane called principal component loadings. It helps engineers understand how the original variables correspond to the principal components.

Without them, the machine learning algorithm lacks data points to model the variable's distribution.

First principal component

The algorithm extracts the most important variable through PCA and stores it in the first principal component. The first component is a representation in dimensional space that retains the most information or maximum variance of the dataset. You can imagine it as a line that passes through the average value of all variables and is close to most.

Second principal component

PCA stores the second most important information in the second principal component. The second principal also passes through the average value but perpendicularly. It retains the second-highest information but is not correlated with the first principal component.

Other principal components

The process repeats until the algorithm constructs all principal components. The final principal components follow a diminishing order based on their feature importance. The first two principal components comprise the most essential ML training data representations. Machine learning engineers can discard the remaining components to simplify model learning and produce reasonably accurate results.

It's important to note that every principal component is unrelated. For example, variations in the first principal component don't affect the second.

The training datasets that machine learning models use to learn have become increasingly complex. With more features to learn from, training an ML model requires more time and computing power. Principal component analysis (PCA) simplifies the dataset and helps reduce specific computing challenges that ML engineers face.

Prevents overfitting

Overfitting is a condition in which the machine learning model fails to predict accurately in real life despite performing well during training. One cause of overfitting is the vast number of variables present in training data. PCA greatly reduces the number of variables while preserving the critical information, reducing the risk of overfitting.

Reduces model training time

Training a model with a complex training dataset requires more time. Because there are so many variables involved, machine learning algorithms must analyze high data volume before performing accurately. PCA reduces the dimensionality of the training dataset, resulting in simpler computing and faster training completion.

Overcome multi-collinearity issues

Multi-collinearity occurs when two or more variables depend on each other in a training dataset. When training specific regressive models, multi-collinearity can impact the training outcome and result in inaccurate predictions. PCA detects and removes collinear variables from the dataset by compressing them into independent features.

PCA's ability to simplify complex datasets is helpful in these applications.

Image compression

You can compress high-resolution image files into smaller sizes with PCA. The algorithm recreates linear combinations of existing image features without losing important information.

Noise filtering

PCA reconstructs a dataset by compressing critical information into the initial variables. Engineers can discard redundant data that doesn't facilitate model learning during training.

Data visualization

PCA makes mapping complex data points into charts easier, as its dimensional reduction of the original data set allows for a more straightforward interpretation and plotting of the data points in charts and graphs.

Financial prediction

Financial applications process large volumes of multi-dimensional data to analyze and predict probable outcomes. They use PCA to extract principal components representing key distributions to simplify calculations.

Facial recognition

Facial recognition systems compare a facial scan with large databases. Scientists developed a method called eigenface based on PCA to reduce process complexity. The technique represents human faces by unique variations, enabling detection without relying on distinct features like eyes, mouth, and nose.

When applying principal components analysis, it's important to understand how the concepts below apply to reconstructing dataset variables.

Orthogonality

Orthogonality describes the non-correlation relationship between principal components. The first and second principal components are perpendicular when plotted on a two-dimensional plane. Both intersect at the average point but represent entirely different variances of data point distributions.

Dimensionality

Dimensionality is the number of features in a dataset. Machine learning models face difficulties when learning from high-dimensional data. Therefore, engineers apply dimensionality reduction techniques like PCA to compress features into a lower dimensional space. Low-dimensional data contains fewer features and is easier to process.

Correlation

Correlation is a statistical analysis of two or more variables in a dataset. It measures the strength and direction of correlation between different variables. Variables are correlated if changes in one variable affect another. In PCA, there shouldn't be any correlation between principal components to ensure their independence from external influence.

Covariance matrix

Like correlation, covariance is an indicator of the relationship between multiple variables. A covariance matrix calculates and compares the relationship of variables in the dataset. When computing the covariance matrix, the PCA algorithm evaluates every possible combination of the initial features in a symmetrical data matrix.

Eigenvector and eigenvalue

Both the eigenvector and eigenvalue are calculated from the covariance matrix. The eigenvector tells you which principal component has the most information. Meanwhile, the eigenvalue indicates the importance of each eigenvector. Arranging eigenvectors according to their eigenvalue produces an ordered series of principal components.

Feature vector

The feature vector is a matrix that consists of retained eigenvectors after the less important ones are discarded. It converts the original dataset into a lower-dimensional representation on a new data plane.

Principal component analysis compresses high-dimensional datasets with linear transformation during unsupervised learning. Several computing nodes calculate the summarized variables, later combined as a single set of principal components. Below, we explain the steps that each node takes.

Standardization

Initially, each node standardizes the variables to prevent bias. This is important because PCA's accuracy is highly sensitive to variances of the original variables. Mathematically, standardization requires transforming the variables to a smaller scale by dividing the differences of their mean and value by the standard deviation.

Covariance

Next, the node calculates the covariance matrix to determine if any variables are related. It determines the principal components to retain by calculating the eigenvectors and eigenvalues from the covariance matrix.

A high correlation between variables might indicate redundant information, which increases computing time during training. The node removes unhelpful or redundant ones by arranging the principal components in order of eigenvalue.

Transformation

So far, data has only been analyzed, and the original dataset still needs to be updated. All the above steps lead to a feature vector representing the most important principal component. Often, the feature vector contains fewer variables than the original number. The node then transforms the original dataset by combining it with the feature vector to produce a simpler and more compact final dataset.

Besides principal component analysis, there are other methods that machine learning engineers use for dimensional reduction.

Multivariate data analysis

Multivariate data analysis consists of different techniques to analyze and understand datasets consisting of multiple variables. Principal component analysis is one of the techniques within several multivariate statistical methods.

Factor analysis

Factor analysis works similarly to principal component analysis by reconstructing variables into smaller parts. However, their purposes differ. Factor analysis measures the relationship of various variables, while principal component analysis produces linear combinations of weighted variables.

Linear discriminant analysis

Both principal component analysis and linear discriminant analysis are methods to reduce dimensionality. PCA seeks to represent the most variance in a linear direction in unsupervised learning. Meanwhile, linear discriminant analysis is a supervised learning method that analyzes labels to improve the accuracy of classification tasks.

Amazon Sagemaker provides ready-to-deploy foundation models and tools for principal component analysis. You can distribute PCA workloads to multiple nodes and receive the summarized component upon completion. It performs singular value decomposition on the summarized values to derive the principal components.

Sagemaker supports regular and randomized PCA.

• Use regular PCA if your dataset contains sparse or moderate features.

• Randomized PCA is better for datasets with large numbers of variables.

With Sagemaker, you can effortlessly train, deploy, and scale machine learning models with the reconstructed datasets.

Get started with principal component analysis on AWS by signing up for a free account today.