select and apply an appropriate technique for the analysis of multivariate data to look for structure in such data or to achieve dimensionality reduction

这是一份Bath巴斯大学MA40090作业代写的成功案

When the units within missingness pattern $s$ are crossclassified only by their observed variables, the result is a table with counts that we shall denote by
$$
z_{O_{x}(y)}^{(s)}=\sum_{M_{s}(y) \in M_{x}} x_{y}^{(s)} \text { for all } O_{s}(y) \in O_{s} .
$$
The marginal probability that an observation falls within cell $O_{s}(y)$ of this table will be called
$$
\beta_{O_{x}(y)}=\sum_{M_{x}(y) \in M_{x}} \theta_{y}
$$

MA40090 COURSE NOTES ：

that is,
$$
x_{O_{x}(y)}^{(s)} \mid z_{O_{x}(y)}^{(s)}, \theta \sim M\left(z_{O_{x}(y)}^{(s)}, \gamma_{O_{x}(y)}\right)
$$
Notice that (7.34) is simply the portion of $\theta$ corresponding to $x_{O_{s}(y)}^{(s)}$, rescaled so that its elements sum to one. It follows that the conditional expectation of an element of $x^{(s)}$ is
$$
E\left(x_{y}^{(s)} \mid z^{(s)}, \theta\right)=z_{O_{x}(y)}^{(s)} \theta_{y} / \beta_{O_{x}(y)}
$$
The E-step consists of calculating for every $s=1, \ldots, S$ and summing the results,