PCA (Principal Components Analisys)

https://doi.org/10.1098/rsta.2015.0202 (opens in a new tab)

PCA is a powerful technique used to summarize and visualize spectra by reducing its dimensionality (number of points) while preserving the most important information. PCA identifies the directions (principal components) along which the data varies the most, allowing for a more compact representation of the data without losing much information.

• Number of components: Determines un how many dimensions data will be summarized. Dimensionality of the output transformed feature space. The lower, the less percentage of explained variance will be represented

Visual Example

A spectrum is a line drawn over hundreds of points. PCA allows you to summarize it in only 2 or 3 coordinates!

When applied to multiple spectra, you can effortesly identify groups of interest in your data.

Key Concepts

• Principal Components: The new set of variables calculated to summarize the data.

  Each point in the curve that defines our spectra is a dimension. That means that each spectrum is composed by hundreds of dimensions! We can decide to compress all the possible information in only 3 Principal Components (3 new calculated dimensions).

  
  

• Scores: The coordinates in the new principal component space for the original data points. They represent the values for the newly calculated dimensions.

  If we chose to calculate 3 Principal Components, each spectrum will be represented by 3 scores (values for PC1, PC2 and PC3). For each column, similar values indicate similar attributes. Data with similar scores are more alike.

  
  

• Loadings: The contributions of the original variables to each Principal Component. They
  represent the correlation between the original variables and the principal components, indicating how much each variable contributes to a particular principal component.

  For example. water spectra in NIR has two main peaks (1450 and 950 nm). That means that they will have the highest loadings (positive or negative) for PC1. This explains that these wavelengths are the most important in explaining the spectral variability.

  
  

• Explained variance: Each principal component captures a certain percentage of the total variance in the data. The first principal component (PC1) explains the largest amount of variance, followed by the second principal component (PC2), and so on. The number of components chosen determines the cumulative explained variance.

  For example, PC1 explains the 60% of the total variance, PC2 the 25% and PC3 the 15%. We can choose to work only with PC1 and PC2 knowing that we are keeping the 85% of the variability in only 2 dimensions.

Data Example

Let's break down the key concepts into a simple example dataset. Instead of spectra, we will use a classification of balls with different colors and sizes.

If we summarize this data using 2 PCs, we obtain the following Scores (or representation coordinates):

Loadings show the contribution of the original variables to each PC. For example, PC1 is a combination of both Size and Color, with Size having a larger contribution (loading of 0.894) compared to Color (loading of 0.447).

Finally, the Explained Variance shows that PC1 alone summarizes 80% of the total variance in the data.

Mathematical Explanation of PCA

1. Standardize the data:

This step creates a common scale by transforming the original variables to have a mean of zero and a standard deviation of one. Standardization ensures that all variables contribute equally to the analysis, regardless of their original scales or units. This is important because variables with larger values or variances might otherwise dominate the principal components.

$x_{ij}^{std} = \frac{x_{ij} - mu_j}{sigma_j}$

Where:
$x_{ij}$ is the value of the $j$-th variable for the $i$-th observation.
$\mu_j$ is the mean of the $j$-th variable.
$\sigma_j$ is the standard deviation of the $j$-th variable.

2. Calculate the covariance matrix

The covariance matrix is a square matrix that captures the pairwise covariances between all variables in the dataset. It provides a measure of how much two variables change together. A positive covariance indicates that the variables tend to increase or decrease together, while a negative covariance suggests that they move in opposite directions.

$\Sigma = \frac{1} {n - 1} \sum_{(i = 1)}^{n} (x_i - \mu)(x_i - \mu){T}$

Where:
$x_i$ is the $i$-th observation.
$\mu$ is the mean vector of the data.
$n$ is the number of observations.

3. Compute the eigenvectors and eigenvalues of the covariance matrix

An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a scalar multiple of itself. The scalar multiplier is called the eigenvalue corresponding to that eigenvector. In the context of PCA, the eigenvectors of the covariance matrix represent the principal components, which are the directions of maximum variance in the data. The corresponding eigenvalues indicate the amount of variance explained by each principal component.

$\Sigma v_j = \lambda_j v_j$

Where:
$v_j$ is the $j$-th eigenvector.
$\lambda_j$ is the corresponding eigenvalue.

4. Select the top $k$ eigenvectors based on their eigenvalues

After computing the eigenvectors and eigenvalues of the covariance matrix, the next step is to select the top $k$ eigenvectors that capture the most significant amount of variance in the data. This step is crucial for dimensionality reduction. igenvectors are typically sorted in descending order based on their corresponding eigenvalues. The eigenvector with the largest eigenvalue represents the first principal component, which captures the maximum variance in the data.

$V_k = [v_1, v_2, \ldots, v_k]$

Where:
$V_k$ is the matrix of the top $k$ eigenvectors.

5. Project the data onto the new subspace

The final step is to project the original data onto the new subspace defined by these eigenvectors. This projection transforms the data from the original high-dimensional space to a lower-dimensional space while preserving the most important information.

$Y = X V_k$

Where:
$X$ is the matrix of the standardized data.
$Y$ is the matrix of the projected data (scores).

The loadings, denoted as $v_j$, are the eigenvectors that define the principal components. They represent the contribution of each original variable to the principal components.

The loadings provide insight into how the original variables are combined to form the new dimensions in the reduced space. The explained variance for each principal component quantifies the amount of variance in the data that is captured by that component. It is calculated using the eigenvalues associated with each eigenvector:

$\frac{\lambda_j}{\sum_{i=1}^{p} \lambda_i}$

Where:
$p$ is the total number of variables.

The explained variance represents the proportion of the total variance in the data that is accounted for by each principal component. A higher explained variance indicates that a principal component captures a larger amount of the data's variability.