A deep dive into the intuition behind PCA with the Math and Code fully covered. PCA is the simplest of the true eigenvector-based multivariate analyses.

_PCA is the simplest of the true eigenvector-based multivariate analyses. It is most commonly used as a dimensionality-reduction technique, reducing the dimensionality of large data-sets while still explaining most of the variance in the data. _Seems cute, doesn’t it?

With this article, I strive to make the idea of PCA intuitive. To understand this article, you should know- Elementary Linear Algebra and High-School Statistics. So, let’s get started.

The curse of dimensionality refers to the unfavorable consequences of dealing with multi-dimensional studies. Let’s take a simple example- consider having sampled N data points, where each data point is a d dimensional vector. Now, for the same N, **data becomes sparse as we add dimensions**. Think of having randomly distributed N points on a line vs N points on a plane, which of the two will have a higher density? The answer is quite intuitive, the line (refer to these figures).

Fine, as we add dimensions to our data, we make it sparse, but why is that a problem? If we lack enough density in our data, we can never be sure of our predictions. For us to train a model to predict results with decent accuracy, the data must be well represented or we run the risk of overfitting. Although we live in the realm of big data, a compromise in the density can only be tackled by an exponential increase in the data (N), which might not be available. Another problem with high-dimensional data is that we can’t easily visualize data beyond 3 dimensions. **Distances lose meaning in higher dimensions**.

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Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data.

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