## 机器学习代写|机器学习代写machine learning代考|Logistic Regression

The logistic regression is a useful and traditional tool used to explain or predict a binary response based on information of explanatory variables. It models the conditional distribution of the response variable as a Bernoulli distribution with the probability of success given by
$$P\left(Y_i=1 \mid \boldsymbol{x}i\right)=p\left(\boldsymbol{x}_i ; \boldsymbol{\beta}\right)=\frac{\exp \left(\beta_0+\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}_0\right)}{1+\exp \left(\beta_0+\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}_0\right)} .$$ To estimate parameters under logistic regression, suppose that we have a set of data $\left(\boldsymbol{x}_i^{\mathrm{T}}, y_i\right), i=1, \ldots, n$ (training data), where $\boldsymbol{x}_i=\left(x{i 1}, \ldots, x_{i p}\right)^{\mathrm{T}}$ is a vector of features measurement and $y_i$ is the response measurement corresponding to the $i$ th drawn individual. To obtain the MLE of $\boldsymbol{\beta}$, first we need to build the likelihood function of the parameters of $\beta$. This is given by
\begin{aligned} L(\boldsymbol{\beta} ; \boldsymbol{y}) &=\prod_1^n p\left(\boldsymbol{x}i ; \boldsymbol{\beta}\right)^{y_i}\left[1-p\left(\boldsymbol{x}_i ; \boldsymbol{\beta}\right)\right]^{1-y_i}=\prod_i^n\left(\frac{p\left(\boldsymbol{x}_i ; \boldsymbol{\beta}\right)}{1-p\left(\boldsymbol{x}_i ; \boldsymbol{\beta}\right)}\right)^{y_i}\left[1-p\left(\boldsymbol{x}_i ; \boldsymbol{\beta}\right)\right]^1 \ &=\exp \left(\sum{i=1}^n y_i\left(\beta_0+\boldsymbol{x}i^{\mathrm{T}} \boldsymbol{\beta}_0\right)\right) \prod{i=1}^n \frac{1}{1+\exp \left(\beta_0+\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}_0\right)} \end{aligned}
and from here the log-likelihood is $\ell(\boldsymbol{\beta} ; \boldsymbol{y})=\log [L(\boldsymbol{\beta} ; \boldsymbol{y})]=\sum_{i=1}^n y_i\left(\beta_0+\boldsymbol{x}i^{\mathrm{T}} \boldsymbol{\beta}_0\right)-\sum{i=1}^n \log \left[1+\exp \left(\beta_0+\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}_0\right)\right]$

## 机器学习代写|机器学习代写machine learning代考|Logistic Ridge Regression

Like the MLR, when there is strong collinearity, the variance of the MLE is severely affected and the true effects of the explanatory variables could be falsely identified (Lee and Silvapulle 1988). In a similar fashion as for the MLR, this could be judged directly from the asymptotic covariance matrix of $\widehat{\boldsymbol{\beta}}$. Moreover, in a common prediction context, when the number of features is larger than the number of observations $(p \gg n)$, the matrix design is not of full column rank and can cause overfitting, affecting the expected classification error (generalization error) when using the “MLE.” One way to avoid overfitting is by replacing the MLE with a regularized MLE as the Ridge MLE estimator of MLR. This is defined as
$$\widetilde{\boldsymbol{\beta}}s^R(\lambda)=\underset{\boldsymbol{\beta}_s}{\operatorname{argmax}}\left[\ell\left(\boldsymbol{\beta}_s ; \boldsymbol{y}\right)-\lambda \sum{j=1}^p \beta_{j s}^2\right],$$
where $\lambda$ is a hyperparameter that has a similar interpretation as in the MLR.
In the literature, there are some algorithms that approximate the Ridge estimation. For example, Genkin et al. (2007) used a cyclic coordinate descent optimization algorithm to approximate this. The one-dimensional optimization problem involved is solved by a modified Newton-Raphson method. Another method was proposed by Friedman et al. (2008) in a more general context. Given the current values of $\widetilde{\boldsymbol{\beta}}s(\lambda)$, the next update of coordinate $\beta_k$ is given by $$\beta{k s}=\frac{\sum_{i=1}^n w_i y_{i j}^* x_{i j}}{\sum_{i=1}^n w_i x_{i j}^2+\lambda}$$
with $y_{i j}^=y_i^-\widetilde{\mu}(\lambda)-\sum_{\substack{j-1 \ j \neq k}}^p x_{i j} \widetilde{\beta}{j s}(\lambda)$ for $k=1, \ldots, p$, and of $\mu$ is given by $$\mu=\frac{\sum{i=1}^n w_i e_i^}{\sum_{i=1}^n w_i}$$ with $e_i^=y_i^-\sum_{j-1}^p x_{i j s} \widetilde{\beta}{j s}(\lambda)$, where $y_i^=\widetilde{\beta}_0(\lambda)+\boldsymbol{x}_i^{\mathrm{T}} \tilde{\boldsymbol{\beta}}{0 s}(\lambda)+\frac{y_i-p\left(\boldsymbol{x}_i ; \tilde{\beta}_s(\lambda)\right)}{w_i}$ and $w_i=p\left(\boldsymbol{x}_i ; \tilde{\boldsymbol{\beta}}_s(\lambda)\right)\left[1-p\left(\boldsymbol{x}_i ; \tilde{\boldsymbol{\beta}}_s(\lambda)\right)\right], i=1, \ldots, n$, are pseudo responses and weights that change across the updates.

# 机器学习代考

## 机器学习代写|机器学习代写machine learning代考|Logistic Regression

. logsi . logsi

$$P\left(Y_i=1 \mid \boldsymbol{x}i\right)=p\left(\boldsymbol{x}i ; \boldsymbol{\beta}\right)=\frac{\exp \left(\beta_0+\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}_0\right)}{1+\exp \left(\beta_0+\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}_0\right)} .$$ 为了估计逻辑回归下的参数，假设我们有一组数据 $\left(\boldsymbol{x}_i^{\mathrm{T}}, y_i\right), i=1, \ldots, n$ (训练数据)，其中 $\boldsymbol{x}_i=\left(x{i 1}, \ldots, x{i p}\right)^{\mathrm{T}}$ 向量的特征是测量和 $y_i$ 响应测量是否与 $i$ 画出来的人。的MLE $\boldsymbol{\beta}$，首先我们需要建立参数的似然函数 $\beta$。这是由

，从这里log-likelihood是 $\ell(\boldsymbol{\beta} ; \boldsymbol{y})=\log [L(\boldsymbol{\beta} ; \boldsymbol{y})]=\sum{i=1}^n y_i\left(\beta_0+\boldsymbol{x}i^{\mathrm{T}} \boldsymbol{\beta}_0\right)-\sum{i=1}^n \log \left[1+\exp \left(\beta_0+\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}_0\right)\right]$

## 机器学习代写|机器学习代写machine learning代考|Logistic Ridge Regression

.

$$\widetilde{\boldsymbol{\beta}}s^R(\lambda)=\underset{\boldsymbol{\beta}s}{\operatorname{argmax}}\left[\ell\left(\boldsymbol{\beta}_s ; \boldsymbol{y}\right)-\lambda \sum{j=1}^p \beta{j s}^2\right],$$
，其中$\lambda$是一个超参数，具有与MLR中类似的解释。在文献中，有一些近似Ridge估计的算法。例如，Genkin等人(2007)使用循环坐标下降优化算法来近似这个问题。用改进的Newton-Raphson方法求解一维优化问题。弗里德曼等人(2008)在更普遍的背景下提出了另一种方法。给定$\widetilde{\boldsymbol{\beta}}s(\lambda)$的当前值，坐标$\beta_k$的下一次更新由$$\beta{k s}=\frac{\sum_{i=1}^n w_i y_{i j}^* x_{i j}}{\sum_{i=1}^n w_i x_{i j}^2+\lambda}$$

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