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Unless I am missing something, this can be seen from using Cauchy-Schwarz & Spectral Decomposition (as shown in pages 78-80 in Applied Multivariate Statistical Analysis, by Richard A. Johnson and Dean W. Wichern, 6th edition). First Cauchy-Schwarz: For two $p \times 1$ vectors $\mathbf{b}$ and $\mathbf{d}$, $$(\mathbf{b}' \mathbf{d})^2 \leq ... 1 Just do multiple linear regression for each of the n dimensions independently and take each set of m coefficients as a row in Q. Since square root is monotone for positive real numbers it is not hard to convince yourself that if for all i, Q_{i,*}\in \mathbb{R}^m minimizes$$ \sum_{v \in A} \left\|Q_{i,*}v - f(v)_i\right\|^2  Then $Q \in ... 1 Just notice that your unnormalized posterior factors as$u(\mu_1,\sigma_1) v(\mu_2,\sigma_2)$, for some suitable functions$u$and$v$, and that's enough to prove the desired independence. 1 Being a simple physicist, not a statistics expert, I'd take a simple approach. The two dimensions are of different natures. It would make sense to smooth along time with one algorithm, and smooth along wavelength with another. The actual algorithms I'd use: for wavelength, Savitzky-Golay with a higher order, 6 maybe 8. Along time, if that example is ... 0 This multivariate-regression method is the second of two ad hoc methods I've contrived. Again, I'd appreciate constructive criticism, questions, or other feedback about this method to help improve it or avoid inappropriate use, and I'm certainly interested in better, more principled strategies -- preferably posted as other answers. Before using this ... 3 It is certainly possible and not necessarily bad. It's not even unusual. You haven't told us what your variables are (it's usually useful for us to know that) but here's an abstract case. Model 1$Y = a + b_1X_1$Model 2$Y = a + b_1X_1 + b_2X_2$These models ask different questions. M1 asks about the relationship between Y and X, uncontrolled. M2 adds ... 1 This truncate-transform method is the first of two ad hoc methods I've contrived. I'd appreciate constructive criticism, questions, or other feedback about it to help improve it or avoid inappropriate use, and I'm certainly interested in better, more principled strategies -- preferably posted as other answers. Before using this method, we specify$\eta$... 2 This is called the re-transformation problem. I'm going to make your model a little simpler to talk about it:$\ln{Y} = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_2^2 + \epsilon$Now, that model does not make predictions for$Y$, it makes predictions for$\ln{Y}$. It is tempting to make predictions for$Y$by just taking predictions for$\ln{Y}$and ... 3 To Expand: Econometrics includes Multivariate Analysis as a tool (a mathematical one). At the same time it may include many other things, such as economic "fundamental" models. Econometrics is also a certain spin on (applied) statistics, just as biostatistics (one could say biometrics) or statistics in medicine, information theory or whatever field you can ... 2 Econometrics is a specialized branch of applied statistics. Multivariate analysis is a branch of mathematics that has a lot of applications to statistics. For a great econometrics intro (at the beginning PhD level), I recommend Mostly Harmless Econometrics. It doesn't cover everything by any means, but if you're starting from scratch with no stats ... 1 Yes. The constraint is truncating the distribution in one direction (given by$\vec{w}$). After changing coordinates,$S$can be expressed as the sum of a constant, a truncated normal distribution, and a normal distribution (from the sum of the other directions). Without any loss of generality rescale$S\$ so that the truncated distribution is standard, ...