Compute Forgetting Factor Required for Streaming Input Data
The growth in the QR decomposition can be seen by looking at the magnitude of the first element of the upper-triangular factor , which is equal to the Euclidean norm of the first column of matrix ,
To see this, create matrix as a column of ones of length and compute of the economy-size QR decomposition.
n = 1e4; A = ones(n,1);
R = fixed.qlessQR(A)
R = 100.0000
ans = 100
ans = 100
The diagonal elements of the upper-triangular factor of the QR decomposition may be positive, negative, or zero, but
fixed.qrAB always return the diagonal elements of as non-negative.
In a real-time application, such as when data is streaming continuously from a radar array, you can update the QR decomposition with an exponential forgetting factor where . Use the
fixed.forgettingFactor function to compute a forgetting factor that acts as if the matrix were being integrated over rows to maintain a gain of about . The relationship between and is .
m = 16; alpha = fixed.forgettingFactor(m); R_alpha = fixed.qlessQR(A,alpha)
R_alpha = 3.9377
ans = 4
If you are working with a system and have been given a forgetting factor , and want to know the effective number of rows that you are integrating over, then you can use the
fixed.forgettingFactorInverse function. The relationship between and is .
ans = 16