*Bounty: 100*

Consider the following problem. The dataset that I am considering has $n=1800$ units (high-end copying machines). Label the units $i = 1,dots,n$. Unit $i$ has $n_i$ recordings. It is of interest to model the use-rate for these copying machines. All machines are in the same building.

The following linear mixed effects model is used:

begin{equation}

begin{aligned}

X_i(t_{ij}) &= m_i(t)+ varepsilon_{ij} \

&= eta + z_i(t_{ij})w_i + varepsilon_{ij},

end{aligned}

end{equation}

where $eta$ is the mean, $z_i(t_{ij}) = [1, log(t_{ij})]$, $w_i = (w_{0i}, w_{1i})^top sim N(0,Sigma_w)$, $varepsilon_{ij} sim N(0, sigma^2)$, and

begin{equation}

Sigma_w =

begin{pmatrix}

sigma^2_1& rhosigma_1sigma_2 \

rhosigma_1sigma_2 & sigma^2_2

end{pmatrix}.

end{equation}

I can write this model in matrix form. More specifically, I have the model (I write this out for a reason)

begin{equation}

X = 1eta + Zw + varepsilon,

end{equation}

where

begin{equation}

X =

begin{pmatrix}

X_1\

vdots \

X_n

end{pmatrix} in mathbb{R}^N,

varepsilon =

begin{pmatrix}

varepsilon_1\

vdots \

varepsilon_n

end{pmatrix} in mathbb{R}^N,

1 =

begin{pmatrix}

1_{n_1}\

vdots \

1_{n_n}

end{pmatrix} in mathbb{R}^{N times p},

w =

begin{pmatrix}

w_1\

vdots \

w_n

end{pmatrix} in mathbb{R}^{2n},

end{equation}

where $N = sum_{i=1}^n n_i$. In addition,

begin{equation}

Z =

begin{pmatrix}

Z_1 & 0_{n_1 times 2} & dots & 0_{n_1 times 2} \

0_{n_2 times 2} & Z_2 & dots & 0_{n_2 times 2} \

vdots & & ddots & vdots \

0_{n_n times 2} & dots & & Z_n

end{pmatrix} in mathbb{R}^{N times 2n},

0_{n_i times 2} =

begin{pmatrix}

0 & 0 \

vdots& vdots \

0 & 0

end{pmatrix} in mathbb{R}^{2n_i}.

end{equation}

Furthermore, we have that

begin{equation}

begin{bmatrix}

w\

varepsilon

end{bmatrix} sim

N

begin{bmatrix}

begin{pmatrix}

0\

0

end{pmatrix},&sigma^2

begin{pmatrix}

G(gamma) & 0 \

0 & R(rho)

end{pmatrix}

end{bmatrix},

end{equation}

where $gamma$ and $rho$ are $r times 1$ and $s times 1$ vectors of unknown variance parameters corresponding to $w$ and $varepsilon$, respectively. Mathematically,

begin{equation}

G = frac{1}{sigma^2}

begin{pmatrix}

Sigma_w & dots & 0 \

vdots & ddots & vdots \

0 & dots & Sigma_w

end{pmatrix} in mathbb{R}^{2n times 2n},

R =

begin{pmatrix}

I_{n_1} & dots & 0 \

vdots & ddots & vdots \

0 & dots & I_{n_n}

end{pmatrix} in mathbb{R}^{N times N},

end{equation}

where $w_i sim N(0, Sigma_w)$, and $varepsilon_i sim N(0, sigma^2I_{n_i})$. Here $gamma = (sigma_1, sigma_2, rho)^top$ and $rho = sigma^2$.

Imagine I now obtain a dataset for a new building with $n$ units. But now, unit $i$ is in the same room as unit $i+1$ for $i = 1,3,5,dots, n-1$. How would I model the additional dependence between units in the same room? At first I thought to use the exact same model as above but changing $G$ to

begin{equation}

G = frac{1}{sigma^2}

begin{pmatrix}

Sigma_w & Sigma_{1,2} & dots &0& 0 \

Sigma_{1,2}& Sigma_w & dots &0& 0 \

vdots & vdots& ddots & vdots& vdots \

0 & 0& dots& Sigma_w & Sigma_{1799,1800} \

0 & 0& dots & Sigma_{1799,1800}& Sigma_w

end{pmatrix} in mathbb{R}^{2n times 2n},

end{equation}

where $Sigma_{i, i+1}$ is the covariance matrix which models the dependence between units $i$ and $i+1$ for $i = 1,3, dots, 1799$.

Is this a possible way to model the problem? I guess it would not be possible to use nlm in R to do it but it would be possible using an analytic solution.

What else could be done? I think a three level hierarchical model (instead of two level model) could also work, but I am not sure how to formulate a three level model.

Any advice on past modelling experiences and how to write down the three level model would be appreciated.

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