# Principal component analysis

Principal component analysis |
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**Principal Component Analysis** (PCA) is a statistical technique used to reduce the dimensionality of data sets. It is used to transform a set of correlated variables into a set of uncorrelated variables, or principal components. The aim of PCA is to identify patterns in data, detect underlying structure, and identify groups of observations. In management applications, PCA can be used to identify clusters of customer behavior and to reduce the amount of data needed to effectively describe a phenomenon. PCA can also be used to improve the accuracy of forecasting models by providing more efficient data transformation.

## Example of principal component analysis

- Principal component analysis can be used in customer segmentation. For example, if a company wants to understand customer behavior, they can use PCA to identify clusters of customers who have similar interests and preferences. This can help the company target specific customer segments with tailored marketing strategies.
- PCA can also be used to reduce the dimensionality of data sets. For example, if a researcher wants to study a particular phenomenon but has too much data to analyze, they can use PCA to transform the data into a smaller set of uncorrelated variables. This can make the analysis more manageable and interpretable.
- PCA can also be used to improve the accuracy of forecasting models. For example, by transforming data sets into uncorrelated variables, PCA can enable models to more accurately predict future outcomes. This can be especially useful for financial and economic forecasting.

## Formula of principal component analysis

Principal Component Analysis (PCA) is a statistical technique used to reduce the dimensionality of data sets. It is used to transform a set of correlated variables into a set of uncorrelated variables, or principal components. The aim of PCA is to identify patterns in data, detect underlying structure, and identify groups of observations.

The PCA algorithm can be represented mathematically as follows:

Given a set of $$N$$ observations of $$n$$ variables, $$X=(x_1,x_2,\ldots,x_n)^T$$, the goal of PCA is to find the linear combinations of the variables that maximize the variance of the data. This is achieved by finding the eigenvectors of the covariance matrix of $$X$$, $$\mathbf{C}_X=\mathbf{X}^T\mathbf{X}$$:

$$\begin{equation} \mathbf{C}_X \mathbf{v}_i = \lambda_i \mathbf{v}_i \end{equation}$$

where $$\mathbf{v}_i$$ is the $$i$$th eigenvector of $$\mathbf{C}_X$$ and $$\lambda_i$$ is the corresponding eigenvalue.

The eigenvectors are sorted in descending order according to the absolute values of their corresponding eigenvalues. The first $$k$$ eigenvectors, $$\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k$$, are used to form a reduced data set $$Z=(z_1,z_2,\ldots,z_k)^T$$:

$$\begin{equation} Z = \mathbf{X}\left[\mathbf{v}_1 \; \mathbf{v}_2 \; \cdots \; \mathbf{v}_k\right] \end{equation}$$

The principal components of the data are then given by the columns of the matrix $$V = \left[\mathbf{v}_1 \; \mathbf{v}_2 \; \cdots \; \mathbf{v}_k\right]$$.

The original data set can be reconstructed from the reduced data set using the inverse transformation

$$\begin{equation} \mathbf{X} = Z\mathbf{V}^T \end{equation}$$

## When to use principal component analysis

Principal Component Analysis (PCA) is a powerful tool for data analysis and exploration. It can be used for a variety of applications, including:

**Dimensionality Reduction**: PCA is used to reduce the number of dimensions in a data set while preserving as much of the original information as possible. This is especially useful when dealing with high-dimensional data, as it can help reduce the amount of processing time and storage space needed.**Feature Extraction**: PCA can be used to identify important features in a data set and reduce noise. This can help improve the accuracy of predictive models.**Clustering**: PCA can be used to identify clusters in a data set, allowing for better segmentation of the data and better understanding of customer behavior.**Visualization**: PCA can be used to create visuals that represent the data in a more intuitive manner, making it easier to spot patterns and correlations.**Forecasting**: PCA can be used to improve the accuracy of forecasting models by providing more efficient data transformation.

## Types of principal component analysis

**Classical PCA**: Classical PCA is a linear transformation technique that identifies the principal components of a data set. It is used to reduce the dimensionality of the data set and to identify clusters of observations in the data.**Kernel PCA**: Kernel PCA is a non-linear transformation technique that is used to reduce the dimensionality of the data set and to identify clusters of observations. It is based on the concept of kernels, which can be used to map the data into a higher dimensional space.**Incremental PCA**: Incremental PCA is a variation of PCA that allows for the incremental addition of data points into the data set. It is used for data sets that are too large to fit into memory all at once.**SVD-PCA**: SVD-PCA is a variation of PCA that uses singular value decomposition to identify the principal components of the data set. It is used to reduce the dimensionality of the data set and to identify clusters of observations.**Robust PCA**: Robust PCA is a variation of PCA that is used to identify the principal components of a data set in the presence of outliers. It is used to reduce the dimensionality of the data set and to identify clusters of observations.**Sparse PCA**: Sparse PCA is a variation of PCA that is used to identify the principal components of a data set with a sparse representation. It is used to reduce the dimensionality of the data set and to identify clusters of observations.

## Steps of principal component analysis

**Step 1**: Identify the variables of interest. PCA is used to transform a set of correlated variables into a set of uncorrelated variables. The variables of interest should be identified and selected before applying the PCA analysis.**Step 2**: Calculate the correlation matrix. The correlation matrix is a matrix of correlation coefficients between the variables of interest. This matrix is used to identify correlations between the variables and to determine the strength of the relationships.**Step 3**: Calculate the principal components. Principal components are linear combinations of the variables of interest. The components are calculated using the correlation matrix and a principal component analysis algorithm.**Step 4**: Interpret the components. The principal components are interpreted to determine the underlying structure of the data set. This information can be used to identify clusters of behavior, to reduce the amount of data needed to accurately describe a phenomenon, and to improve the accuracy of forecasting models.**Step 5**: Visualize the components. The components can be visualized using a variety of methods, such as scatter plots, biplots, and line graphs. These visualizations can help to identify patterns in the data and to further interpret the components.

## Advantages of principal component analysis

The advantages of Principal Component Analysis (PCA) include:

- It is an effective way to reduce the dimensionality of large data sets, allowing for more efficient data analysis.
- By reducing the dimensionality of data sets, it can also improve the accuracy of forecasting models.
- It can identify patterns and structure in data, which may lead to useful insights into customer behavior and other phenomena.
- It can also be used to identify clusters of related observations.
- It provides a more efficient way to visualize data, making it easier to identify relationships.

## Limitations of principal component analysis

Principal Component Analysis (PCA) is a useful tool for reducing the dimensionality of a data set and identifying patterns within it. However, it has some limitations. These include:

- PCA assumes that the underlying data is normally distributed, which may not always be the case.
- PCA is sensitive to outliers and can be affected by extreme values.
- PCA ignores the relative importance of variables and can be influenced by irrelevant variables.
- PCA is difficult to interpret and can be hard to explain to non-technical stakeholders.
- PCA is not an appropriate technique for all types of data sets and may not always provide the best results.

Principal Component Analysis (PCA) is a statistical technique used to reduce the dimensionality of data sets. Other approaches related to PCA include:

**Factor Analysis**: Factor Analysis is a statistical technique used to identify underlying relationships between observed variables. It is used to explain the variance underlying a set of observed variables by examining the correlations among them.**Independent Component Analysis (ICA)**: ICA is an unsupervised learning technique used to identify the underlying structure of a data set by maximising its statistical independence. It is useful for extracting meaningful features from complex data sets.**Canonical Correlation Analysis (CCA)**: CCA is a linear model used to examine the relationship between two or more sets of variables. It is used to identify which variables are most strongly correlated and to identify potential relationships between variables.**Multidimensional Scaling (MDS)**: MDS is a non-linear technique used to reduce the dimensionality of a data set by mapping observations onto a lower-dimensional space. It is useful for visualising complex data sets and for detecting clusters in the data.

In summary, Principal Component Analysis is a statistical technique used to reduce the dimensionality of data sets. Other approaches related to PCA include Factor Analysis, Independent Component Analysis, Canonical Correlation Analysis, and Multidimensional Scaling. These techniques can be used to identify patterns in data, detect relationships between variables, and improve the accuracy of forecasting models.

## Suggested literature

- Abdi, H., & Williams, L. J. (2010).
*Principal component analysis*. Wiley interdisciplinary reviews: computational statistics, 2(4), 433-459. - Bro, R., & Smilde, A. K. (2014).
*Principal component analysis*. Analytical methods, 6(9), 2812-2831.