# Generate synthetic data
t <- 500 # Number of days
m <- 100 # Number of stocks
h <- array(rnorm(m*t), c(m, t)) # Random normal returns
e <- h %*% t(h) / t # Correlation matrix
lambda_e <- eigen(e, symmetric = TRUE, only.values = TRUE) # Eigenvalues and Eigenvector
ee<- lambda_e$values # extract the valuesDenoising in Finance
2024-02-21
Deep dive
Simplifying Matrix Concepts: The 2D Vector
- A foundational element in matrix theory, the two-dimensional vector, embodies simplicity.
- Comprising just two components, it directly maps to coordinates on a 2D plane.
- This fundamental structure serves as a practical example of how matrices relate to geometric spaces.
Deep dive
Linear Transformation Essentials
- Linear transformations adjust vectors through matrix multiplication, stretching or compressing them along defined axes.
- The matrix \(\begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}\) shifts the x-axis to align with \(\begin{bmatrix} 3&1 \end{bmatrix}\) and the y-axis with \(\begin{bmatrix} 1&2 \end{bmatrix}\).
- These shifts are visualized by color-coded lines, showing how the transformation alters the coordinate system’s orientation.
Deep dive
Eigenvectors: Stability in Transformation
- Consider a vector \(\begin{bmatrix} -1&-1 \end{bmatrix}\) transforming to \(\begin{bmatrix} -4&-3 \end{bmatrix}\) under a matrix.
- Normally, transformations alter a vector’s direction, moving it off its span.
- Eigenvectors are exceptional; they stay on their span, simply scaling by the eigenvalue during transformation.
Deep dive
Eigenvector and Eigenvalue Complexity
- Eigenvectors and eigenvalues often involve complex, non-integer values, making precise calculation demanding.
- Scaling an original eigenvector retains its directionality, producing another scaled eigenvector.
Deep dive
Eigenvalues and Eigenvectors in 3D Transformations
- In 3D, a matrix transforms the x, y, and z axes, each axis adjustment representing the transformation’s effect.
- This specific relationship means eigenvectors and eigenvalues are exclusive to square matrices.
- Finding an eigenvector involves identifying its corresponding eigenvalue, through the equation \(Ax=\lambda x\), where \(A\) is the matrix and \(\lambda\) the eigenvalue.
- This equation underscores that multiplying an eigenvector by its matrix equates to scaling it by the eigenvalue.
Deep dive
Solving for Eigenvalues
- To solve for eigenvalues, rearrange the equation to isolate terms on one side, introducing an identity matrix \(I\) to equate matrix \(A\) with scalar \(\lambda\).
- Infinite “trivial solutions” exist, achievable by scaling an eigenvector by any scalar.
- To eliminate these trivial solutions, we utilize the determinant.
Deep dive
Understanding the Determinant
- The determinant quantifies how a transformation matrix stretches or compresses area.
- Consider a unit square in coordinate space (top figure). After transformation by \(\begin{bmatrix} 2&1\\0 &2\end{bmatrix}\), its area expands to four square units (bottom figure.
- This increase indicates the determinant of the matrix is four, reflecting the area’s stretching factor.
Deep dive
Determinant and Space Collapse
- A determinant of zero means the transformation collapses the square’s area to zero, indicating the axes’ vectors align on the same line.
- With a zero determinant, space is condensed into a line, becoming one-dimensional.
- Thus, for the equation to be solvable without trivial solutions, the matrix’s determinant must equal zero.
Deep dive
Finding Eigenvalues in 2D
- In 2D, determining eigenvalues involves solving a quadratic equation.
- For the matrix \(\begin{bmatrix}1&4\\3&2\end{bmatrix}\), the eigenvalues are 5 and -2.
- Thus, multiplying the matrix’s eigenvectors stretches their lengths by factors of 5 and -2, respectively.
Three dimensional data eigenvector visualisation
- For three or more dimension matrix, a different form of the determinant formula must be used
Deep dive
Retrieve the eigenvector
To find a matrix’s eigenvectors, insert each eigenvalue into the equation \(Ax = \lambda x\). For \(\lambda=5\), the eigenvector is \(\begin{bmatrix}1\\1\end{bmatrix}\); for \(\lambda=-2\), it’s \(\begin{bmatrix}-4\\3\end{bmatrix}\).
These eigenvectors and eigenvalues can reconstruct the original matrix, highlighting their crucial role in financial data science for simplifying and interpreting data transformations.
Visualizing Denoising Efficacy
- Visualize how well RMT approximates real data distributions through empirical and theoretical density plots.
# Visualize Marcenko-Pastur vs. empirical distribution
library(ggplot2)
empirical_pdf <- density(ee, width = 0.1, kernel = "gaussian")
marcenko_pastur_pdf <- mp_pdf(1, t, m, m)
ggplot() +
geom_line(data = tibble(eval = empirical_pdf$x,
pdf = empirical_pdf$y,
type = "Empirical PDF"), aes(x = eval, y = pdf, colour = type)) +
geom_line(data = tibble(eval = names(marcenko_pastur_pdf),
pdf = marcenko_pastur_pdf, type = "Marcenko-Pastur PDF"),
aes(x = as.numeric(eval), y = pdf, colour = type)) +
labs(title = "Marcenko-Pastur Distribution vs. Empirical Data", x = "Eigenvalues", y = "Density") +
theme(legend.title = element_blank())
Benefits of denoising
Key Takeaways
- The Marcenko-Pastur law offers a theoretical foundation for analyzing and denoising high-dimensional financial datasets.
- Understanding the eigenvalue distribution of covariance matrices is crucial for extracting meaningful information in both AI applications and financial trading.
- Leveraging this law can lead to more informed decision-making, improved model performance, and enhanced trading results.
