计算机代写|机器学习代写machine learning代考|CS7641

2022年10月8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|Interpreting the Parameters of Linear Models

When analyzing the linear models developed so far, we have already talked about interpreting their parameters in terms of general trends, correlation, differences between groups, and so on.

While is tempting to casually interpret the meaning of various features, we must be careful and precise when doing so.

First, we should be precise about the interpretation of our slope and intercept terms. For example, when we modeled ratings as a function of review length (eq. (2.12)), we stated that under our model, ratings increased fractionally $\left(1.193 \times 10^{-4}\right)$ for every character of a review.

This interpretation makes sense given a model containing only a single features, but as soon as we incorporate multiple features we must be more careful. Consider, for example, the model from Equation (2.15), in which we included both the length and number of comments as predictors. We could no longer state that under this model, the rating increases (by $7.243 \times 10^{-5}$ ) for every character in the review. Precisely, we must interpret the parameters as follows: Our prediction of the rating increases by $7.243 \times 10^{-5}$ for every character in the review, assuming the other features remain unchanged. This definition is stated precisely in Figure 2.14.

Critically, features like review length and number of comments may be highly correlated (e.g., we may rarely see longer reviews without also seeing more comments). For example, when incorporating features based on polynomial functions (as in eq. (2.35)), or when dealing with one-hot encodings (as in eq. (2.39)), a feature cannot change without the other features changing.

Second, we should be clear when interpreting parameters that we are talking about predictions under a particular model rather than actual changes in the label $y_i$. These predictions can change as we include additional features; a feature that had previously been predictive may become less so in the presence of another (as we saw in Equation (2.15)). Likewise, we should be careful not to conclude that (e.g.) length is not related to the output variable, simply because another correlated feature has a stronger relationship.

计算机代写|机器学习代写machine learning代考|Fitting Models with Gradient Descent

So far, when solving regression problems, we looked for closed form solutions. That is, we set up a system of equations (eq. (2.3)) in $X, y$, and $\theta$, and attempted to solve them for $\theta$ (albeit approximately via the pseudoinverse).

As we begin to fit more complex models (including in Chapter 3), a closedform solution may no longer be available.

Gradient descent is an approach to search for the minimum value of a function, by iteratively finding bêtter solutions based on an initial starting point. The process (depicted in Figure 2.15) operates as follows:
(i) Start with an initial guess for $\theta$.
(ii) Compute the derivative $\frac{\partial}{\partial \theta} f(\theta)$. Here $f(\theta)$ is the MSE (or whatever criterion we are optimizing) under our model $\theta$.
(iii) Update our estimate of $\theta:=\theta-\alpha \cdot f^{\prime}(\theta)$.
(iv) Repeat Steps (ii) and (iii) until convergence.
During each iteration, the process now follows the path of steepest descent, and will gradually arrive at a minimum of the function $f_\theta \cdot{ }^{13}$

The above is a simple description of the procedure that omits many details. In practice, we will largely rely on high-level libraries to implement gradientbased methods (sec. 3.4.4). Briefly, to implement such techniques ‘from scratch,’ some of the main issues include:

• Given the starting point in Figure $2.15$, the algorithm would only achieve a local rather than a global optimum. To address this we could investigate ways to come up with a better initial ‘guess’ of $\theta$, or investigate variants of gradient descent that are less susceptible to local minima.
• The step size $\alpha$ (step (iii)) must be chosen carefully. If $\alpha$ is too small, the procedure will converge very slowly; if $\alpha$ is too large, the procedure may ‘overshoot’ the minimum value and obtain a worse solution during the next iteration. Again, other than carefully tuning this parameter, we could investigate optimization methods not dependent on choosing this rate (see e.g., quasi-Newton methods such as L-BFGS (Liu and Nocedal, 1989)).
• ‘Convergence’ as defined in Step (iv) is not well-defined. We might define convergence in terms of the change in $\theta$ (or $f_\theta(X)$ ) during two successive iterations, or alternately we may terminate the algorithm once we stop making progress on held-out (validation) data (see sec. 3.4.2).

机器学习代考

计算机代写|机器学习代写machine learning代考|拟合带有梯度下降的模型

. . . . .

(i)从$\theta$的初始猜测开始。
(ii)计算导数$\frac{\partial}{\partial \theta} f(\theta)$。这里$f(\theta)$是我们的模型$\theta$下的MSE(或任何我们正在优化的准则)。
(iii)更新我们对$\theta:=\theta-\alpha \cdot f^{\prime}(\theta)$的估计。
(iv)重复步骤(ii)和(iii)直到收敛。在每次迭代中，该过程现在遵循最陡峭的下降路径，并将逐渐到达函数$f_\theta \cdot{ }^{13}$ 的最小值

• 给定图$2.15$的起始点，算法只能实现局部最优，而不能实现全局最优。为了解决这个问题，我们可以研究出$\theta$更好的初始“猜测”的方法，或者研究不太容易受到局部极小值影响的梯度下降变量。
• 必须仔细选择步长$\alpha$ (step (iii))。如果$\alpha$太小，这个过程会收敛得非常慢;如果$\alpha$太大，过程可能会“超过”最小值，并在下次迭代中获得更糟糕的解决方案。同样，除了仔细调整这个参数，我们还可以研究不依赖于选择这个速率的优化方法(例如L-BFGS (Liu and Nocedal, 1989)等准牛顿方法)。
• 步骤(iv)中定义的“收敛”没有很好的定义。我们可以根据$\theta$(或$f_\theta(X)$)在两个连续迭代期间的变化来定义收敛，或者也可以在搁置(验证)数据停止取得进展时终止算法(见第3.4.2节)。

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。