### 统计代写|数据科学代写data science代考| Circular PCA

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|数据科学代写data science代考|Circular PCA

Kirby and Miranda [5] introduced a circular unit at the component layer in order to describe a potential circular data structure by a closed curve. As illustrated in Fig. 2.4, a circular unit is a pair of networks units $p$ and $q$ whose output values $z_{p}$ and $z_{q}$ are constrained to lie on a unit circle
$$z_{p}^{2}+z_{q}^{2}=1 .$$
Thus, the values of both units can be described by a single angular variable $\theta$.
$$z_{p}=\cos (\theta) \quad \text { and } \quad z_{q}=\sin (\theta)$$
The forward propagation through the network is as follows: First, equivalent to standard units, both units are weighted sums of their inputs $z_{m}$ given by the values of all units $m$ in the previous layer.
$$a_{p}=\sum_{m} w_{p m} z_{m} \quad \text { and } \quad a_{q}=\sum_{m} w_{q m} z_{m} .$$
The weights $w_{p m}$ and $w_{q m}$ are of matrix $W_{2}$. Biases are not explicitly considered, however, they can be included by introducing an extra input with activation set to one.
The sums $a_{p}$ and $a_{q}$ are then corrected by the radial value
to obtain circularly constraint unit outputs $z_{p}$ and $z_{q}$
$$z_{p}=\frac{a_{p}}{r} \quad \text { and } \quad z_{q}=\frac{a_{q}}{r} .$$

## 统计代写|数据科学代写data science代考|Inverse Model of Nonlinear PCA

In this section we define nonlinear PCA as an inverse problem. While the classical forward problem consists of predicting the output from a given input, the inverse problem involves estimating the input which matches best a given output. Since the model or data generating process is not known, this is referred to as a blind inverse problem.

The simple linear PCA can be considered equally well either as a forward or inverse problem depending on whether the desired components are predicted as outputs or estimated as inputs by the respective algorithm. The autoassociative network models both the forward and the inverse model simultaneously. The forward model is given by the first part, the extraction

function $\Phi_{\text {extr }}: \mathcal{X} \rightarrow \mathcal{Z}$. The inverse model is given by the second part, the generation function $\Phi_{g e n}: \mathcal{Z} \rightarrow \hat{\mathcal{X}}$. Even though a forward model is appropriate for linear PCA, it is less suitable for nonlinear PCA, as it sometimes can be functionally very complex or even intractable due to a one-to-many mapping problem. Two identical samples $\boldsymbol{x}$ may correspond to distinct component values $\boldsymbol{z}$, for example, the point of self-intersection in Fig. 2.6B.

By contrast, modelling the inverse mapping $\Phi_{g e n}: \mathcal{Z} \rightarrow \hat{\mathcal{X}}$ alone, provides a numher of advantages: we direstly model the assumed data generation process which is often much easier than modelling the extraction mapping. We also can extend the inverse NLPCA model to be applicable to incomplete data sets, since the data are only used to determine the error of the model output. And, it is more efficient than the entire autoassociative network, since we only have to estimate half of the network weights.

Since the desired components now are unknown inputs, the blind inverse problem is to estimate both the inputs and the parameters of the model by only given outputs. In the inverse NLPCA approach, we use one single error function for simultaneously optimising both the model weights $\boldsymbol{w}$ and the components as inputs $z$.

## 统计代写|数据科学代写data science代考|The Inverse Network Model

Inverse NLPCA is given by the mapping function $\Phi_{g e n}$, which is represented by a multi-layer perceptron (MLP) as illustrated in Fig. 2.5. The output $\hat{\boldsymbol{x}}$ depends on the input $z$ and the network weighte $w \in W_{3}, W_{4}$.
$$\hat{\boldsymbol{x}}=\Phi_{g e n}(\boldsymbol{w}, \boldsymbol{z})=W_{4} g\left(W_{3} z\right)$$
The nonlinear activation function $g$ (e.g., tunh) is applied element-wise. Biases are not explicitly considered. They can be included by introducing extra units with activation set to one.

The aim is to find a function $\Phi_{g e n}$ which generates data $\hat{x}$ that approximate the observed data $\boldsymbol{x}$ by a minimal squared error $|\hat{\boldsymbol{x}}-\boldsymbol{x}|^{2}$. Hence, we search for a minimal error depending on $\boldsymbol{w}$ and $z: \min {w, z}\left|\Phi{g e n}(\boldsymbol{w}, \boldsymbol{z})-\boldsymbol{x}\right|^{2}$. Both the lower dimensional component representation $z$ and the model parameters $w$ are unknown and can be estimated by minimising the reconstruction error:
$$E(\boldsymbol{w}, \boldsymbol{z})=\frac{1}{2} \sum_{n}^{N} \sum_{i}^{d}\left[\sum_{j}^{h} w_{i j} g\left(\sum_{i}^{m} w_{j k} z_{k}^{n}\right)-x_{i}^{n}\right]^{2}$$
where $N$ is the number of samples and $d$ the dimensionality.
The error can be minimised by using a gradient optimisation algorithm, e.g., conjugate gradient descent [31]. The gradients are obtained by propagating the partial errors $\sigma_{i}^{n}$ back to the input layer, meaning one layer more than

usual. The gradients of the weights $w_{i j} \in W_{4}$ and $w_{j k} \in W_{3}$ are given by the partial derivatives:
$$\begin{array}{ll} \frac{\partial E}{\partial w_{i j}}=\sum_{n} \sigma_{i}^{n} g\left(a_{j}^{n}\right) \quad ; \quad & \sigma_{i}^{n}=\hat{x}{i}^{n}-x{i}^{n} \ \frac{\partial E}{\partial w_{j k}}=\sum_{n} \sigma_{j}^{n} z_{k}^{n} \quad ; \quad \sigma_{j}^{n}=g^{\prime}\left(a_{j}^{n}\right) \sum_{i} w_{1 j} \sigma_{i}^{n} \end{array}$$
The partial derivatives of linear input units $\left(z_{k}=a_{k}\right)$ are:
$$\frac{\partial E}{\partial z_{k}^{n}}=\sigma_{k}^{n}=\sum_{j} w_{j k} \sigma_{j}^{n}$$
For circular input units given by equations (2.6) and (2.7), the partial derivatives of $a_{p}$ and $a_{q}$ are:
$$\frac{\partial E}{\partial a_{p}^{n}}=\left(\bar{\sigma}{p}^{n} z{q}^{n}-\tilde{\sigma}{q}^{n} z{p}^{n}\right) \frac{z_{q}^{n}}{r_{n}^{3}} \quad \text { and } \quad \frac{\partial E}{\partial a_{q}^{n}}=\left(\bar{\sigma}{q}^{n} z{p}^{n}-\bar{\sigma}{p}^{n} z{q}^{n}\right) \frac{z_{p}^{n}}{r_{n}^{3}}$$

## 统计代写|数据科学代写data science代考|Circular PCA

Kirby 和 Miranda [5] 在组件层引入了一个圆形单元，以便通过闭合曲线描述潜在的圆形数据结构。如图 2.4 所示，一个圆形单元是一对网络单元p和q其输出值和p和和q被限制在单位圆上

## 统计代写|数据科学代写data science代考|The Inverse Network Model

X^=披G和n(在,和)=在4G(在3和)

∂和∂在一世j=∑nσ一世nG(一种jn);σ一世n=X^一世n−X一世n ∂和∂在jķ=∑nσjn和ķn;σjn=G′(一种jn)∑一世在1jσ一世n

∂和∂和ķn=σķn=∑j在jķσjn

∂和∂一种pn=(σ¯pn和qn−σ~qn和pn)和qnrn3 和 ∂和∂一种qn=(σ¯qn和pn−σ¯pn和qn)和pnrn3

## 广义线性模型代考

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## MATLAB代写

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