Multidimensional autoregression

What should we optimize?

Least-squares applications often present themselves as fitting goals such as

$$\bold 0 \approx \bold F \bold m - \bold d$$ (60)

$$\bold 0 \approx \bold m$$ (61)

To balance our possibly contradictory goals we need weighting functions. The quadratic form that we should minimize is

$$\min_m \quad (\bold F \bold m - \bold d)\T \bold A\T_n \bold A_n (\bold F \bold m - \bold d) + \bold m\T \bold A\T_m \bold A_m \bold m$$ (62)

where $\bold A\T_n \bold A_n$ is the inverse multivariate spectrum of the noise (data-space residuals) and $\bold A\T_m \bold A_m$ is the inverse multivariate spectrum of the model. In other words, $\bold A_n$ is a leveler on the data fitting error and $\bold A_m$ is a leveler on the model. There is a curious unresolved issue: What is the most suitable constant scaling ratio of $\bold A_n$ to $\bold A_m$ ?

Multidimensional autoregression