# 统计计算|STAT206/STA 518/ STA 511/STA 6106/STAT 8070/STAT151Statistical Computing代写

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Consider maximization of the function $L(\mathbf{W}, \mathbf{H})$ in, written here without the matrix notation
$$L(\mathbf{W}, \mathbf{H})=\sum_{i=1}^{N} \sum_{j=1}^{p}\left[x_{i j} \log \left(\sum_{k=1}^{r} w_{i k} h_{k j}\right)-\sum_{k=1}^{r} w_{i k} h_{k j}\right] .$$
Using the concavity of $\log (x)$, show that for any set of $r$ values $y_{k} \geq 0$ and $0 \leq c_{k} \leq 1$ with $\sum_{k=1}^{r} c_{k}=1$,
$$\log \left(\sum_{k=1}^{r} y_{k}\right) \geq \sum_{k=1}^{r} c_{k} \log \left(y_{k} / c_{k}\right)$$

## MSTAT 502/STAT 316/MTH 513A/MATH 321/STAT210/STA 106 COURSE NOTES ：

For $m=1$ to $M$ :
(a) Fit a classifier $G_{m}(x)$ to the training data using weights $w_{i}$ –
(b) Compute
$$\operatorname{err}{m}=\frac{\sum{i=1}^{N} w_{i} I\left(y_{i} \neq G_{m}\left(x_{i}\right)\right)}{\sum_{i=1}^{N} w_{i}}$$
(c) Compute $\alpha_{m}=\log \left(\left(1-\operatorname{err}{m}\right) /\right.$ err $\left.{m}\right)$.
(d) Set $w_{i} \leftarrow w_{i} \cdot \exp \left[\alpha_{m} \cdot I\left(y_{i} \neq G_{m}\left(x_{i}\right)\right)\right], i=1,2, \ldots, N$.
Output $G(x)=\operatorname{sign}\left[\sum_{m=1}^{M} \alpha_{m} G_{m}(x)\right]$.

# 统计计算|MTH3045 Statistical Computing代写

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A variation on least squares scaling is the so-called Sammon mapping which minimizes
$$\sum_{i \neq i^{\prime}} \frac{\left(d_{i i^{\prime}}-\left|z_{i}-z_{i^{\prime}}\right|\right)^{2}}{d_{i i^{\prime}}} .$$
Here more emphasis is put on preserving smaller pairwise distances.
In classical scaling, we instead start with similarities $s_{i i^{r}}:$ often we use the centered inner product $s_{i i^{\prime}}=\left\langle x_{i}-\bar{x}, x_{i^{\prime}}-\bar{x}\right\rangle$. The problem then is to minimize
$$\sum_{i, i^{\prime}}\left(s_{i i^{\prime}}-\left\langle z_{i}-\bar{z}, z_{i^{\prime}}-\bar{z}\right\rangle\right)^{2}$$

## MTH3045 COURSE NOTES ：

We approximate each point by an affine mixture of the points in its neighborhood:
$$\min {W{i k}}\left|x_{i}-\sum_{k \in \mathcal{N}(i)} w_{i k} x_{k}\right|^{2}$$
over weights $w_{i k}$ satisfying $w_{i k}=0, k \notin \mathcal{N}(i), \sum_{k=1}^{N} w_{i k}=1 . w_{i k}$ is the contribution of point $k$ to the reconstruction of point $i$. Note that for a hope of a unique solution, we must have $K<p$.

Finally, we find points $y_{i}$ in a space of dimension $d<p$ to minimize
$$\sum_{i=1}^{N}\left|y_{i}-\sum_{k=1}^{N} w_{i k} y_{k}\right|^{2}$$
with $w_{i k}$ fixed.

# 统计计算|Statistical Computing代写 STAT 535

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Population regression function, or simply, the regression function:

For the Normal linear model
$$\mathrm{E}\left(Y_{i}\right)=\mu_{i}=\mathbf{x}{i}^{T} \boldsymbol{\beta} ; \quad Y{i} \sim \mathrm{N}\left(\mu_{i}, \sigma^{2}\right)$$
for independent random variables $Y_{1}, \ldots, Y_{N}$, the deviance is
$$D=\frac{1}{\sigma^{2}} \sum_{i=1}^{N}\left(y_{i}-\widehat{\mu}_{i}\right)^{2}$$

$$D_{0}=\frac{1}{\sigma^{2}} \sum_{i=1}^{N}\left[y_{i}-\widehat{\mu}{i}(0)\right]^{2}$$ and $$D{1}=\frac{1}{\sigma^{2}} \sum_{i=1}^{N}\left[y_{i}-\widehat{\mu}_{i}(1)\right]^{2} .$$

## STAT535 COURSE NOTES ：

If $\mathrm{E}(\mathbf{y})=\mathbf{X} \boldsymbol{\beta}$ and $\mathrm{E}\left[(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})^{T}\right]=\mathbf{V}$, where $\mathbf{V}$ is known, we can obtain the least squares estimator $\tilde{\beta}$ of $\beta$ without making any further assumptions about the distribution of $\mathbf{y}$. We minimize
$$S_{w}=(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})^{T} \mathbf{V}^{-1}(\mathbf{y}-\mathbf{X}) \boldsymbol{\beta}$$
The solution of
$$\frac{\partial S_{w}}{\partial \beta}=-2 \mathbf{X}^{T} \mathbf{V}^{-1}(\mathbf{y}-\mathbf{X} \beta)=0$$
is
$$\tilde{\boldsymbol{\beta}}=\left(\mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{X}\right)^{-1} \mathbf{X}^{T} \mathbf{V}^{-1} \mathbf{y}$$

# 统计计算|Statistical Computing代写STAT0030代考

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playwith (expr,
new = playwith.getoption ("new"),
title = NULL,
labels = NULL ,
data. points $=$ NULL,
viewport $=$ NULL,
parameters $=$ list (),
tools = list (),
init.actions = list(),
preplot.actions = list (),
update actions $=$ list () ,
$\cdots$,
width = playwith.getoption ("width"),
playwith (expr,
new = playwith. getoption ("new"),
title = NULL,
labels = NULL,
data. points = NULL,
viewport = NULL,
parameters = list(),
tools = list(),
init.actions = list(),
preplot.actions = list(),
update.actions = list(),
. ,
width = playwith.getoption ("width"),
height = playwith.getoption ("height"),
pointsize = playwith.getoption ("pointsize"),
eval.args = playwith.getoption ("eval.args"),
on. close = playwith.getoption ("on. close"),
modal = FALSE,
playstate = if (!new) playDevCur (),
plot. call,
main. function)
height = playwith.getoption ("height"),
pointsize = playwith.getoption ("pointsize"),
eval .args = playwith. getoption ("eval.args"),
on.close = playwith. getoption ("on. close"),
modal = FALSE,
playstate = if (!new) playDevCur(),
plot.call,
main. function)

## STAT0030 COURSE NOTES ：

> library (maps)
> library (maptools)
> library (RColorBrewer)
> install. packages ("classInt")
> library (classint)
$>$ ibrary (maps)
$>$ ibrary (maptools)
$>$ ibrary (RColorBrewer)
$>$ install. packages ("classInt")
$>1$ ibrary (classInt)
$>$ install. packages ("gpelib")
$>$ library (gpelib)
$>$ ibrary (mapdata)
> install. packages ("gpclib")
> library (gpelib)
> library (mapdata)