Modelling colored noise with varying correlation length and variance with a large dataset

[smartass101]

The Gaussian random walk model is (according to my understanding) essentially an AR(1) (or very long MA) model. Yeas, I was thinking about soemthing like that, but I wasn’t sure how to relate the covariance between points with the innovation variance. Your formulation

@junpenglao

covariance from the data point right next to it.

got me thinking and I looked into it

Assuming something like an AR(1) model with a_1 = 1

x_i = x_{i-1} + u_i

Where u_i \sim \mathcal{N}(0, \sigma_i^2)

\mathrm{cov}(x_i, x_{i-1}) = \mathbf{E}\left( (x_i - \mathbf{E}x_i) (x_{i-1} - \mathbf{E}x_{i-1}) \right) =\mathbf{E}\left( (x_i - x_{i-1}) (x_{i-1} - x_{i-2}) \right)\\ = \mathbf{E}(u_i u_{i-1}) = \mathrm{cov}(u_i, u_{i-1})

So for u_i = u_{i-1} (homoscedastic) the covariance between points is the variance of the innovation.

Are you aware of some bibliography relating the covariance between points to the (co)variance of the innovations in a more general sense? I noticed that some text on computational statistics (e.g. Computational statistics, Gentle, 2099) ) mention that a MvNormal can be generated as essentially a MA sequence with coefficients given by something like the Cholesky factor of the MVNormal covariance, but they didn’t go into further detail.