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# 数学代写|金融数学代写Financial Mathematics代考|ACTS201 Analytics from Machine Learning Literature

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## 数学代写|金融数学代写Financial Mathematics代考|Analytics from Machine Learning Literature

The technical trading rules that are to be discussed in Chapter 6 involve a relatively small information set to carry out predictions. It treats each of the ‘ $n$ ‘ time series of asset returns as autonomous. As pointed out by Malkiel (2012) [258], these rules can be easily implemented by most market participants if such opportunities should emerge, and market efficiency would rule them out as winning strategies. More powerful prediction methods using a very large set of potential predictors and yet capable of avoiding overfitting are needed to take advantage of transient opportunities. We gave an overview of recent advances in high-dimensional regression in Chapter $4 .$ The classification techniques can also be formulated with a high-dimensional feature vector. In machine learning and in computer science the focus is to handle rapidly different types of data (including text). The literature in this field is vast and so richly detailed that a single chapter wouldn’t do justice to the topic. Consequently, here we will only briefly mention tools that are deemed to be relevant to trading and invite interested readers to further their knowledge with dedicated references such as Goodfellow, Bengio, and Courville (2016) [168] and Lopez de Prado (2018) [252].

Machine learning is a still-growing area of computer science that encompasses other well-established areas such as statistics, computational algorithms, control theory, etc. The focus in machine learning is on developing efficient algorithms for prediction or for classification using large data sets. The efficiency is gauged by predictive validation accuracy. The inferential aspects of statistical theory, such as standard error of the estimates, confidence intervals, etc., are generally not of much concern. Some areas where machine learning methods have led to significant contribution are classification, clustering and multi-dimensional regression. The attractiveness of these methods lies in the fact that they do not need any a priori theory to suggest which relevant variables to consider. Therefore with no prescription of variables, the variable or feature selection becomes an important process in machine learning. The statistical foundations of machine learning methods are also alternatively referred to as statistical learning methods. We provide a brief description of a select few methods in this section.

## 数学代写|金融数学代写Financial Mathematics代考|Neural Networks

This is one of the main methods of machine learning where the performance of a task is learned by analyzing training examples that have been hand-labeled in advance. The neural network in its simplest form can be represented as follows: Here, the square boxes contain ‘ $n$ ‘ dimensional input or feature vector, $X$ and ‘ $m$ ‘ dimensional output vector, $Y$ and the circular box indicates hidden or unknown layer in-between that connects the input to the output. Structurally, neural networks are a two-stage regression or classification model with intermediary layers, and conceptually the model is similar to the reduced-rank regression model 3.16. But, $Z$ is generally a non-linear function of $X$. In the ordinary least squares regression model, the goal is to get ‘ $m$ ‘ linear combinations of ‘ $X$ ‘ that best predicts ‘ $Y$ ‘. Here ‘ $r$ ‘ dimensional intermediaries can vary based on the assumption of ‘hidden’ layers, but the ‘key’ is that the output vector, ‘ $Y$ ‘, can be a non-linear function of ‘ $X$ ‘ via the hidden layers and ‘ $r$ ‘ can be larger than ‘ $n$ ‘. In its simplest form the neural network model can be written as follows:
\begin{aligned} &Z_{i}=\sigma\left(\beta_{i}^{\prime} X\right), \quad i=1,2, \ldots, r \ &Y_{j}=g_{j}(Z), \quad j=1,2, \ldots, m, \end{aligned}
where the function $\sigma(u)=\frac{1}{1+e^{-u}}$ is the sigmoid function. Note that this is the function used in logistic regression. In the regression set-up, $g_{j}(Z)=\alpha_{j}^{\prime} Z$, but in the $m$-class classification,
$$g_{j}(Z)=\frac{e^{\alpha_{j}^{\prime} Z}}{\sum_{l=1}^{m} e^{\alpha_{j}^{\prime} Z}}$$
called the softmax function, is used. In the set-up given in (4.56), we assume only one hidden layer, but in practical applications many layers are assumed which leads to non-uniqueness problems. As shown in Hastie, Tibshirani and Friedman (2009) [184] the neural network problem is closely related to a non-parametric method called projection pursuit regression. In its simplest form where the information flows in only one direction, it is called a feed-forward neural net, which is commonly used in many applications.

## 数学代写|金融数学代写Financial Mathematics代考|Neural Networks

$$Z_{i}=\sigma\left(\beta_{i}^{\prime} X\right), \quad i=1,2, \ldots, r \quad Y_{j}=g_{j}(Z), \quad j=1,2, \ldots, m,$$ 类分类，
$$g_{j}(Z)=\frac{e^{\alpha_{j}^{\prime} Z}}{\sum_{l=1}^{m} e^{\alpha_{j}^{\prime} Z}}$$

## 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。