Example

# Applying SVD

Use case
You're given a large dataset containing daily returns of various stocks over several years. You suspect some noise in the data due to microstructural effects. How might you useΒ Singular Value Decomposition (SVD) and to isolate the main components driving these returns, separating them from noise?

## Answer

To isolate the main components driving stock returns using SVD, follow these steps:
1. Standardize the Data: Make sure each stock's returns are centered with mean 0. This can help in emphasizing the co-movements between stocks.
2. Perform SVD: Decompose the returns matrix using SVD to obtain , and
3. Examine Singular Values: The diagonal entries of (the singular values) represent the importance of each corresponding component. Plot these singular values in descending order. Typically, for real-world data, you'll observe a scree plot β an initial sharp drop followed by a flattening. The point where the slope levels off represents the "noise level."
4. Select Main Components: Based on the scree plot, select a number of the largest singular values and their corresponding vectors in and . This number represents the significant components believed to capture the main dynamics in the data, while excluding the noise.
5. Reconstruct Reduced Data: Using the selected singular values and vectors, reconstruct a reduced-rank approximation of the returns matrix. This matrix represents the returns data with much of the noise filtered out.
6. Interpretation: The columns of (associated with the selected singular values) can be interpreted as the primary patterns or factors driving the returns. These can be market-wide movements, sectoral influences, or other systematic factors.
This approach makes the assumption that the largest components (in terms of variance) represent signal and the smaller ones represent noise. In some contexts, small but consistent patterns might also be of interest, so it's essential to understand the data and the domain when interpreting results.