计算机代写|机器学习代写machine learning代考|COMP4702

2022年12月23日

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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
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计算机代写|机器学习代写machine learning代考|Logistic Regression

When developing regular linear regressors in Chapter 2, we wanted a model $f_\theta$ whose estimates $f_\theta\left(x_i\right)$ were as close as possible to the (real-valued) labels $y_i$. When adapting a linear regression algorithm to classification, we might instead seek models that associate positive values of $x_i \cdot \theta$ with positive labels $\left(y_i=1\right)$, and negative values of $x_i \cdot \theta$ with negative labels $\left(y_i=0\right)$.

If we could do so, we could write down the accuracy associated with a particular model:
$$\frac{1}{|y|} \sum_{i=1}^{|y|} \underbrace{\delta\left(y_i=0\right) \delta\left(x_i \cdot \theta \leq 0\right)}_{\text {label is negative and prediction is negative }}+\overbrace{\delta\left(y_i=1\right) \delta\left(x_i \cdot \theta>0\right)}^{\text {label is positive and prediction is positive }}$$
(here $\delta$ is an indicator function that returns 1 if the argument is true, 0 otherwise). The equation here, in spite of slightly confusing notation, is merely counting the number of times we correctly predict a positive score for a positively labeled instance, and a negative (or zero) score for a negatively labeled instance.

We now simply desire from our classifier $\theta$ that it maximizes the accuracy measured by Equation (3.1). Unfortunately, directly optimizing Equation (3.1) for $\theta$ is NP-hard (see, e.g., Nguyen and Sanner (2013)). To get a sense for why it is difficult, consider that the function in Equation (3.1) is essentially a step function (fig. 3.1, left), that is, it is flat (derivative zero) almost everywhere; it is therefore not amenable to techniques like gradient ascent as we saw in Section $2.5$.

So, to optimize the accuracy approximately, we would like a function that is similar to Equation (3.1), but is more straightforward to optimize.

Logistic Regression achieves this goal by converting the outputs of a linear function $x_i \cdot \theta$ to probabilities via a smooth function. Our intuition is that large values of $x_i \cdot \theta$ should correspond to high probabilities, and small (i.e., large negative) values of $x_i \cdot \theta$ should correspond to low probabilities.

计算机代写|机器学习代写machine learning代考|Other Classification Techniques

In our introduction to classification, we have only discussed a single classification technique: Logistic Regression. Our choice to explore this particular technique was largely a practical one: the idea of associating a probability with a particular outcome (as in eq. (3.5)) and estimating that probability via a differentiable function (to facilitate gradient ascent) will appear repeatedly as we develop more and more complex models.

However, the technique we have explored is only one class of approach to build classifiers. The specific choice to map binary labels to continuous probabilities via a smooth function has hidden assumptions and limitations, meaning that logistic regression is not the ideal classifier for every situation. Below we present a few alternatives, largely as further reading and to highlight specific situations where logistic regression may not be the preferable choice.

Support Vector Machines: While logistic regressors optimize a probability associated with a set of ohserved lahels, they do not explicitly minimize the number of mistakes made by the classifier. Support Vector Machines (SVMs) (Cortes and Vapnik, 1995) replace the sigmoid function in Figure $3.1$ with an expression that assigns zero cost to correctly classified examples, ${ }^1$ and a positive $\operatorname{cost}^2$ to incorrectly classified examples (in proportion to the confidence of the prediction $x \cdot \theta$ ). This distinction is fairly subtle: while every sample will influence the optimal value of $\theta$ for a logistic regressor, the solution found by an SVM is entirely determined by a few samples closest to the classification boundary, or those that are mislabeled. Conceptually it is appealing for a classifier to focus on the most ‘difficult’ samples in this way, though note that in many cases (and notably when building recommender systems) our goal is to optimize ranking performance rather than classification accuracy (as we will discuss in sec. 3.3.3), such that giving special attention to the most ambiguous examples is not necessarily desirable.

机器学习代考

计算机代写|机器学习代写machine learning代考|Logistic Regression

$$\frac{1}{|y|} \sum_{i=1}^{|y|} \underbrace{\delta\left(y_i=0\right) \delta\left(x_i \cdot \theta \leq 0\right)}_{\text {label is negative and prediction is negative }}+\overbrace{\delta\left(y_i=1\right) \delta\left(x_i\right.}^{\text {label is positive and predi }}$$
(这里 $\delta$ 是一个指示函数，如果参数为真则返回 1，否则返回 0)。尽 管符号有点令人困惑，但这里的等式只是计算我们正确预测正面标记 实例的正分数和负面标记实例的负 (或䨐) 分数的次数。

有限元方法代写

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。