# An Interesting Model with Unknown Orthogonal Design Matrix

Suppose the multivariate one-way anova model for the raw data , i.e. $$\label{Example_model_1} \mathbf{y}_{ij}=\mathbf{\mu}+\mathbf{z}_i+\mathbf{e}_{ij}, ~~ i=1,\ldots,m,~~j=1,\ldots,n_i,~~~~~~~~~~~~~~~~(1)$$ where $$\mathbf{\mu}$$ is the unknown overall mean of $$\mathbf{y}_{ij}$$, $$\mathbf{z}_i$$ is the unknown random effect at the $$ith$$ level, $$\mathbf{e}_{ij}$$ are the errors, and $$\mathbf{e}_{ij}\mid \mathbf{\Lambda}_0 \overset{iid}{\sim} N_k(\mathbf{0},\mathbf{\Lambda}_0), ~~ \mathbf{z}_i \mid \mathbf{\Lambda}_1 \overset{iid}{\sim} N_k(\mathbf{0},\mathbf{\Lambda}_1).$$ Howerever, the raw data is not observed; instead $$\mathbf{y}_{ij}^* = \mathbf{\Gamma} \mathbf{y}$$ is observed, where $$\mathbf{\Gamma}$$ is an unknown orthogonal matrix (i.e., the vector observations have been randomly rotated, with a common unknown rotation). So the model (1) becomes $$\label{model_1_1} \mathbf{y}^{*}_{ij}=\mathbf{\mu}^*+\mathbf{z}_i^{*}+\mathbf{e}_{ij}^{*}, ~~ i=1,\ldots,m,~~j=1,\ldots,n_i,~~~~~~~~~~~~~~(2)$$ where $$\mathbf{\mu}^*=\mathbf{\Gamma}\mathbf{\mu},$$ $$\mathbf{e}_{ij}^*\mid (\mathbf{\Lambda}_0,\mathbf{\Gamma}) \sim N_k(\mathbf{0},\mathbf{\Gamma}\mathbf{\Lambda}_0\mathbf{\Gamma}'),$$ and $$\mathbf{z}_i^* \mid (\mathbf{\Lambda}_1,\mathbf{\Gamma}) \sim N_k(\mathbf{0},\mathbf{\Gamma}\mathbf{\Lambda}_1\mathbf{\Gamma}').$$

Our aim is to estimate $$(\mathbf{\mu},\mathbf{\Gamma},\mathbf{\Lambda}_0,\mathbf{\Lambda}_1).$$

Question: I can't find the relative background of the above model. Please give me some reference or suggest.

• well, that's an unusual thing to ask for! Usually, you say "I have this application, what model would work?", you're asking "I have this model, what application would fit?". Where did you find that model? – Marcus Müller Jun 1 at 9:11
• Sometimes, Markus, theoretical people find a wonderful model and algorithm to solve it, yet lack of an application. – Laurent Duval Jun 1 at 12:49
• I fully agree! Hence, I was curious where xiaopai happened to find that model :) Also, I might have been putting too much thought into it; isn't practically the multi-user channel with each user's data being one entry in $\mu$, and $\mathbf \Gamma$ simply containing for example code sequences in a CDMA system? – Marcus Müller Jun 1 at 14:55
• @Laurent Duval Yes. We have a lack of application. – xiaopai833 Jun 2 at 1:39
• Thanks for your correction – xiaopai833 Jun 3 at 12:56

We find such signal models in radar literature. For example, $$\Gamma$$ represents the transmitted signal information matrix, $$\mu$$ is the deterministic unknown vector representing the reflections from the target, $$z_i$$ is a random vector generated by the clutter surrounding the target, and $$e_i$$ is the receiver noise. In statistical literature, such models are called linear mixed models. You can make several assumptions, such as orthogonal signal matrix, unknown clutter covariance matrix etc., and solve a hypothesis testing problem where the goal of the detector is to detect if there is a target or not. In other words, $$\mu = 0$$ vs $$\mu \ne 0$$.
• Thank you. Could you provide some references for me? Moreover, $\mathbf{\Gamma}$ is unknown here. Did you meet such cases? – xiaopai833 Jun 9 at 6:25
• Can't think of any at the moment. But it should be possible to estimate the signal subspace using $N$ number of snapshots, and find the eigenvectors corresponding to the dominant eigenvalues of the covariance matrix. – Maxtron Jun 10 at 19:18