calc_inv: Calculate the _reduced inverse_ of a symmetric _matrix_ of...

In algoquant/HighFreq: High Frequency Time Series Management

Calculate the reduced inverse of a symmetric matrix of data using eigen decomposition.

Description

Calculate the reduced inverse of a symmetric matrix of data using eigen decomposition.

Usage

calc_inv(matrixv, dimax = 0L, singmin = 0)

Arguments

matrixv	A symmetric matrix of data.
dimax	An integer equal to the number of eigen values used for calculating the reduced inverse of the matrix matrixv (the default is dimax = 0 - standard matrix inverse using all the eigen values).
singmin	A numeric threshold level for discarding small eigen values in order to regularize the inverse of the matrix matrixv (the default is 0.0).

Details

The function calc_inv() calculates the reduced inverse of the matrix matrixv using eigen decomposition.

The function calc_inv() first performs eigen decomposition of the matrix matrixv. The eigen decomposition of a matrix \strong{C} is defined as the factorization:

\strong{C} = \strong{O} \, \Sigma \, \strong{O}^T

Where \strong{O} is the matrix of eigen vectors and \Sigma is a diagonal matrix of eigen values.

The inverse \strong{C}^{-1} of the matrix \strong{C} can be calculated from the eigen decomposition as:

\strong{C}^{-1} = \strong{O} \, \Sigma^{-1} \, \strong{O}^T

The reduced inverse of the matrix \strong{C} is obtained by removing eigen vectors with very small eigen values:

\strong{C}^{-1} = \strong{O}_{dimax} \, \Sigma^{-1}_{dimax} \, \strong{O}^T_{dimax}

Where \strong{O}_{dimax} is the matrix of eigen vectors that correspond to the largest eigen values \Sigma_{dimax}.

The function calc_inv() applies regularization to the matrix inverse using the arguments dimax and singmin.

The function calc_inv() applies regularization by discarding the smallest eigen values \Sigma_i that are less than the threshold level singmin times the sum of all the eigen values:

\Sigma_i < eigen\_thresh \cdot (\sum{\Sigma_i})

It also discards additional eigen vectors so that only the highest order eigen vectors remain, up to order dimax. It calculates the reduced inverse from the eigen decomposition using only the largest eigen values up to dimax. For example, if dimax = 3 then it only uses the 3 highest order eigen vectors, with the largest eigen values. This has the effect of dimension reduction.

If the matrix matrixv has a large number of small eigen values, then the number of remaining eigen values may be less than dimax.

Value

A matrix equal to the reduced inverse of the matrix matrixv.

Examples

## Not run: 
# Calculate ETF returns
retp <- na.omit(rutils::etfenv$returns[, c("VTI", "TLT", "DBC")])
# Calculate covariance matrix
covmat <- cov(retp)
# Calculate matrix inverse using RcppArmadillo
invmat <- HighFreq::calc_inv(covmat)
# Calculate matrix inverse in R
invr <- solve(covmat)
all.equal(invmat, invr, check.attributes=FALSE)
# Calculate reduced inverse using RcppArmadillo
invmat <- HighFreq::calc_inv(covmat, dimax=3)
# Calculate reduced inverse using eigen decomposition in R
eigend <- eigen(covmat)
dimax <- 1:3
invr <- eigend$vectors[, dimax] %*% (t(eigend$vectors[, dimax])/eigend$values[dimax])
# Compare RcppArmadillo with R
all.equal(invmat, invr)

## End(Not run)

algoquant/HighFreq documentation built on Oct. 26, 2024, 9:20 p.m.