In the world of Data Science and Machine Learning which are growing exponentially, dimensionality reduction plays an important role. For the purpose of handling large and complex datasets, one of the most popularly used technique is the Principal Component Analysis.
Whether you’re an experienced professional or just a beginner in Data Science, Principal Component Analysis in Machine Learning is important. It has various applications including data compression, feature extraction, visualisation, etc. The following blog will take you on a journey to understand PCA in Machine Learning with components and types.
What is Principal Component Analysis in Machine Learning?
Principal Component Analysis (PCA) is a popular technique in Machine Learning and Statistics used for dimensionality reduction and data compression. It allows you to transform high-dimensional data into a lower-dimensional space while retaining the most important information or patterns present in the original data.
The primary objective of PCA is to identify the principal components (also known as eigenvectors) that capture the maximum variance in the data. These principal components are orthogonal to each other, meaning they are uncorrelated, and they are sorted in descending order of the variance they explain. The first principal component explains the most variance, the second one explains the second most variance, and so on.
Process of Principal Component Analysis:
The PCA process involves the following steps:
- Data Preprocessing: Standardize or normalize the data to ensure that all features have the same scale, as PCA is sensitive to the scale of the features.
- Covariance Matrix: Calculate the covariance matrix of the standardized data. The covariance between two variables measures how they vary together.
- Eigenvalue Decomposition: Perform eigenvalue decomposition on the covariance matrix to obtain its eigenvectors and eigenvalues. The eigenvectors are the principal components, and the eigenvalues represent the amount of variance explained by each corresponding eigenvector.
- Selecting Principal Components: Sort the eigenvectors in descending order of their eigenvalues and select the top k eigenvectors to retain. Typically, you choose the number of principal components such that they explain a significant amount of the total variance, e.g., 95% or 99%.
- Projection: Use the selected eigenvectors to transform the original data onto the lower-dimensional space. This is achieved by projecting the data points onto the new axis defined by the principal components.
- The reduced-dimensional data can be used for visualization, data compression, and noise reduction. It also helps improve the performance of Machine Learning models by reducing the number of features and removing multicollinearity. Additionally, PCA can be helpful in identifying the most important features that contribute most to the variance in the data.
Keep in mind that PCA is a linear transformation technique, and it might not be appropriate for some nonlinear data distributions. In such cases, nonlinear dimensionality reduction techniques like t-SNE (t-Distributed Stochastic Neighbor Embedding) or autoencoders may be more suitable.
Principal Component Analysis in Machine Learning Example:
Let’s walk through a simple example of Principal Component Analysis (PCA) using Python and the popular Machine Learning library, scikit-learn. In this example, we’ll use the well-known Iris dataset, which contains measurements of iris flowers along with their species. We’ll perform PCA to reduce the data to two dimensions and visualize the results.
- Import the Libraries
- Load the Iris Dataset and preprocess the data
- Perform PCA and select the number of principal components
- Visualise the reduced data
The resulting scatter plot will show the data points projected onto the two principal components. Each color corresponds to a different species of iris flowers (Setosa, Versicolor, Virginica). PCA has transformed the high-dimensional data into a 2D space while retaining the most important information (variance) present in the original data.
Keep in mind that the principal component analysis example above uses a small dataset for illustrative purposes. In practice, PCA is most valuable when dealing with high-dimensional datasets where visualizing and understanding the data becomes challenging without dimensionality reduction. Additionally, the number of principal components (here, 2) can be adjusted based on the specific use case and the desired amount of variance to be retained.
Types of Principal Component Analysis:
Principal Component Analysis (PCA) is a powerful technique used for dimensionality reduction and data compression. It helps in transforming high-dimensional data into a lower-dimensional space while preserving the essential information. There are various types or variants of PCA, each with its specific use cases and advantages. In this explanation, we’ll cover four main types of PCA:
- Standard PCA: Standard PCA is the basic form of PCA and is widely used for dimensionality reduction. It involves finding the principal components by performing eigenvalue decomposition on the covariance matrix of the standardized data. The principal components are orthogonal to each other and sorted in descending order of variance explained. Standard PCA is effective when the data is linear and the variance is well-distributed across the dimensions. However, it may not be suitable for highly nonlinear datasets.
- Incremental PCA: Incremental PCA is an efficient variant of PCA that is particularly useful for handling large datasets that do not fit into memory. In standard PCA, the whole dataset is required to compute the covariance matrix, making it computationally expensive for large datasets. Incremental PCA, on the other hand, processes data in batches or chunks, allowing you to perform PCA incrementally. This way, it’s possible to reduce memory requirements and speed up the computation for massive datasets.
- Kernel PCA: Kernel PCA is an extension of PCA that can handle nonlinear data distributions. It uses the kernel trick to implicitly transform the original data into a higher-dimensional space, where linear PCA can be applied effectively. The kernel function computes the dot product between data points in the higher-dimensional space without explicitly mapping them. This allows Kernel PCA to capture nonlinear relationships among data points, making it suitable for a broader range of datasets.
- Sparse PCA: Sparse PCA is a variation of PCA that introduces sparsity in the principal components. In standard PCA, all components contribute to each data point in the transformed space. However, in sparse PCA, only a small subset of components is selected to represent each data point, leading to a sparse representation. This can be useful for feature selection or when the data is thought to have only a few dominant features. Sparse PCA can lead to more interpretable and compact representations of the data.
Each type of PCA has its strengths and weaknesses, and the choice of which variant to use depends on the specific characteristics of the dataset and the problem at hand.
In summary, PCA is a versatile tool that allows us to reduce the dimensionality of data while preserving essential information. Standard PCA is effective for linear data distributions, but if the data is nonlinear or too large to fit in memory, we can turn to Incremental PCA or Kernel PCA, respectively. Additionally, Sparse PCA can provide more interpretable and compact representations by introducing sparsity in the principal components.
Before applying PCA or its variants, it’s essential to preprocess the data properly, handle missing values, and consider the scale of the features. Additionally, the number of principal components to retain should be carefully chosen based on the amount of variance explained or the specific application requirements. Overall, PCA remains a fundamental technique in the field of Machine Learning and data analysis, offering valuable insights and simplification for complex datasets.
Difference Between Factor Analysis & Principal Component Analysis:
Factor Analysis (FA) and Principal Component Analysis (PCA) are both techniques used for dimensionality reduction and exploring underlying patterns in data, but they have different underlying assumptions and objectives. Let’s explore the main differences between Factor Analysis and Principal Component Analysis:
|Factor Analysis (FA)
Principal Component Analysis (PCA)
|Factor Analysis, on the other hand, is a statistical model that assumes that the observed variables are influenced by a smaller number of latent (unobservable) variables called factors. These latent factors are the underlying constructs that explain the correlations among the observed variables. FA assumes that there is an error component in the observed variables, which is not explained by the factors.
|PCA is a mathematical technique that focuses on finding the orthogonal axes (principal components) that capture the maximum variance in the data. It does not make any assumptions about the underlying structure of the data. The principal components are derived solely based on the variance-covariance matrix of the original data.
|The primary goal of the Factor Analysis is in idenifying the latent factors that explain the observed correlations among the variables. FA ensures ri uncover the underlying structure or common factors that generate the observed data. Accordingly, it focuses on providing a meaningful and interpretable representation of data by explaining the shared variance through different factors.
|Maximising the variance explained by each principle component is the basic objective of principal component analysis (PCA). Its goal is to find a low-dimensional representation of the data while retaining as much volatility as possible. Interpreting the various components or their relationships to the source variables is not the focus of PCA.
|The latent factors are allowed to be connected with one another in factor analysis. This method can identify shared information among the observed variables and accepts the possibility that the components may be related. Factor Analysis provides a more adaptable and nuanced depiction of the connected patterns in the data by allowing for correlations between components.
|The main components in main Component Analysis (PCA) are orthogonal to one another, demonstrating that they are uncorrelated. Although it makes component interpretation easier, the orthogonality attribute may not always accurately reflect the underlying structure of the data.
|However, when researchers want to understand the latent variables that affect the observed data, they use factor analysis (FA). The social sciences and psychology frequently use this method to pinpoint the underlying theories that underlie observed attitudes or behaviours.
|PCA is extensively used for noise reduction, data preprocessing, and visualisation. Without explicitly modelling the underlying structure, it helps discover the data’s most important dimensions (or “principal components)”
The above blog provides you with clear and detailed understanding on PCA in Machine Learning. Principal Component Analysis in Machine Learning helps you in reducing the dimensionality of the complex datasets. The step-by-step guide has covered all the essential requirements that will help you learn about PCA effectively.
What does principal component analysis measure?
PCA algorithm in Machine Learning is a statistical technique that is used for reducing the dimensions and extracting features in multivariate data. It measures the directions of the variance which is at maximum in the data and transforms the original variables ti a new set if uncorrelated variables.
Is PCA linear or nonlinear?
PCA is a linear technique that is based on linear algebra and assumes that the relationship betweeb the variables are also linear in a data.
What are the applications of Principal Component Analysis in remote sensing?
Principal Component Analysis application in remote sensing has several aspects which includes data compression, image enhancement, feature extraction and classification. It also includes data a visualisation, noise reduction, change detection, atmospheric correction and hyperspectral data.