Principal Component Analysis, or PCA, is a fundamental dimensionality reduction technique in machine learning. It identifies new coordinate axes that better capture the variance in your data. Instead of using the original x and y axes, PCA finds principal components that align with the directions of maximum data spread, making it easier to analyze and visualize high-dimensional datasets.
To understand how PCA works, we need to see how different directions capture different amounts of variance. As we rotate a line through our data, we can measure how much the data spreads along that direction. The direction that captures the maximum variance becomes our first principal component. Watch as we rotate through different angles and see how the variance changes - the highest bar corresponds to the optimal direction.
To find principal components mathematically, we compute the covariance matrix of our data and perform eigenvalue decomposition. The covariance matrix captures how variables vary together. Its eigenvectors point in the directions of maximum variance, becoming our principal components. The eigenvalues tell us how much variance each component captures. The largest eigenvalue corresponds to the first principal component, which captures the most variance in the data.
PCA is a fundamental technique in machine learning and data analysis. It identifies the directions of maximum variance in high-dimensional data, allowing us to reduce dimensionality while preserving the most important information. The principal components shown here represent these optimal directions.
The PCA algorithm follows five main steps. First, we center the data by subtracting the mean. Then we compute the covariance matrix to understand how variables relate to each other. Next, we find eigenvalues and eigenvectors of this matrix. We sort these by eigenvalue magnitude, as larger eigenvalues correspond to directions of greater variance. Finally, we project our data onto the selected principal components.
Each principal component explains a certain percentage of the total variance in the data. The first component captures the most variance, followed by the second, and so on. The cumulative variance line shows how much total information we retain as we include more components. Typically, we choose enough components to explain 80-95% of the variance.
Here we see dimensionality reduction in action. The original high-dimensional data on the left is projected onto a lower-dimensional space on the right. The arrows show how each data point maps to its reduced representation. While we lose some information, we preserve the most important patterns and make the data much easier to work with.
Let's see PCA in action with a real example. We have house price data with four features: size, bedrooms, location score, and age. When we apply PCA, the first principal component combines these features in a meaningful way. PC1 might represent 'overall quality' - combining large size and good location while penalizing older homes. This transformation reveals the underlying patterns that drive house prices most strongly.