### cs代写|机器学习代写machine learning代考|Neural networks and Keras

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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

## cs代写|机器学习代写machine learning代考|Neurons and the threshold perceptron

The brain is composed of specialized cells. These cells include neurons, which are thought to be the main information-processing units, and glia, which have a variety of supporting roles. A schematic example of a neuron is shown in Fig. 4.1a. Neurons are specialized in electrical and chemical information processing. They have an extensions called an axon to send signals, and receiving extensions called dendrites. The contact zone between the neurons is called a synapse. A sending neuron is often referred to as the presynaptic neuron and the receiving cell is a postsynaptic neuron. When an neuron becomes active it sends a spike down the axon where it can release chemicals called neurotransmitters. The neurotransmitters can then bind to receiving receptors on the dendrite that trigger the opening of ion channels. Ion channels are specialized proteins that form gates in the cell membrane. In this way, electrically charged ions can enter or leave the neuron and accordingly change the voltage (membrane potential) of the neuron. The dendrite and cell body acts like a cable and a capacitor that integrates (sums) the potentials of all synapses. When the combined voltage at the axon reaches a certain threshold, a spike is generated. The spike can then travel down the axon and affect further neurons downstream.

This outline of the functionality of a neuron is, of course, a major simplification. For example, we ignored the description of the specific time course of opening and closing of ion channels and hence some of the more detailed dynamics of neural activity. Also, we ignored the description of the transmission of the electric signals within the neuron; this is why such a model is called a point-neuron. Despite these simplifications, this model captures some important aspects of a neuron functionality. Such a model suffices for us at this point to build simplified models that demonstrate some of the informationprocessing capabilities of such a simplified neuron or a network of simplified neurons. We will now describe this model in mathematical terms so that we can then simulate such model neurons with the help of a computer.

Warren McCulloch and Walter Pitts were among the first to propose such a simple model of a neuron in 1943 which they called the threshold logical unit. It is now often

referred to as the McCulloch-Pitts neuron. Such a unit is shown in Fig. 4.2A with three input channels, although neurons have typically a much larger number of input channels. Input values are labeled by $x$ with a subscript for each channel. Each channel has an associated weight parameter, $w_{i}$, representing the “strength” of a synapse.
The McCulloch-Pitts neuron operates in the following way. Each input value is multiplied with the corresponding weight value, and these weighted values are then summed together, mimicking the superposition of electric charges. Finally, if the weighted summed input is larger than a certain threshold value, $w_{0}$, then the output is set to 1 , and 0 otherwise. Mathematically this can be written as
$$y(\mathbf{x} ; \mathbf{w})=\left{\begin{array}{cc} 1 & \text { if } \sum_{i}^{n} w_{i} x_{i}=\mathbf{w}^{T} \mathbf{x}>w_{0} \ 0 & \text { otherwise } \end{array}\right.$$
This simple neuron model can be written in a more generic form that we will call the perceptron. In this more general model, we calculate the output of a neuron by applying an gain function $g$ to the weighted summed input,
$$y(\mathbf{x} ; \mathbf{w})=g\left(\mathbf{w}^{T} \mathbf{x}\right)$$
where $w$ are parameters that need to be set to specific values or, in other words, they are the parameters of our parameterized model for supervised learning. We will come back to this point later regarding how precisely to chose them. The original McCulloch-Pits neuron is in these terms a threshold perceptron with a threshold gain function,
$$g(x)=\left{\begin{array}{l} 1 \text { if } x>0 \ 0 \text { otherwise } \end{array}\right.$$
This threshold gain function is a first example of a non-linear function that transforms the sum of the weighted inputs. The gain function is sometimes called the activation function, the transfer function, or the output function in the neural network literature. Non-linear gain functions are an important part of artificial neural networks as further discussed in later chapters.

## cs代写|机器学习代写machine learning代考|Multilayer perceptron (MLP) and Keras

To represent more complex functions with perceptron-like elements we are now building networks of artificial neurons. We will start with a multilayer perceptron (MLP) as

shown in Fig.4.3. This network is called a two-layer network as it basically has two processing layers. The input layer simply represents the feature vector of a sensory input, while the next two layers are composed of the perceptron-like elements that sum up the input from previous layers with their associate weighs of the connection channels and apply a non-linear gain function $\sigma(x)$ to this sum,
$$y_{i}=\sigma\left(\sum_{j} w_{i j} x_{j}\right)$$
We used here the common notation with variables $x$ representing input and $y$ representing the output. The synaptic weights are written as $w_{i j}$. The above equation corresponds to a single-layer perceptron in the case of a single output node. Of course, with more layers, we need to distinguish the different neurons and weights, for example with superscipts for the weights as in Fig.4.3. The output of this network is calculated as
$$y_{i}=\sigma\left(w_{i j}^{\mathrm{o}} \sigma\left(\sum_{k} w_{j k}^{\mathrm{h}} x_{k}\right)\right) .$$
where we used the superscript “o” for the output weights and the superscript ” $h$ ” for the hidden weights. These formulae represent a parameterized function that is the model in the machine learning context.

## cs代写|机器学习代写machine learning代考|Representational learning

Here, we are discussing feedforward neural networks which can be seen as implementing transformations or mapping functions from an input space to a latent space, and from there on to an output space. The latent space is spanned by the neurons in between the input nodes and the output nodes, which are sometime called the hidden neurons. We can of course always observe the activity of the nodes in our programs so that these are not really hidden. All the weights are learned from the data so that the transformations that are implemented by the neural network are learned from examples. However, we can guide these transformations with the architecture. The latent representations should be learned so that the final classification in the last layer is much easier than from the raw sensory space. Also, the network and hence the representation it represents should make generalizations to previously unseen examples easy and robust. It is useful to pause for a while here and discuss representations.

## cs代写|机器学习代写machine learning代考|Neurons and the threshold perceptron

Warren McCulloch 和 Walter Pitts 在 1943 年率先提出了这样一个简单的神经元模型，他们称之为阈值逻辑单元。现在经常

McCulloch-Pitts 神经元以下列方式运作。每个输入值乘以相应的权重值，然后将这些权重值相加，模拟电荷的叠加。最后，如果加权求和输入大于某个阈值，在0，则输出设置为 1 ，否则设置为 0。数学上这可以写成
$$y(\mathbf{x} ; \mathbf{w})=\left{ 1 如果 ∑一世n在一世X一世=在吨X>在0 0 除此以外 \正确的。 吨H一世ss一世米pl和n和在r○n米○d和lC一个nb和在r一世吨吨和n一世n一个米○r和G和n和r一世CF○r米吨H一个吨在和在一世llC一个ll吨H和p和rC和p吨r○n.我n吨H一世s米○r和G和n和r一个l米○d和l,在和C一个lC在l一个吨和吨H和○在吨p在吨○F一个n和在r○nb是一个ppl是一世nG一个nG一个一世nF在nC吨一世○nG吨○吨H和在和一世GH吨和ds在米米和d一世np在吨, y(\mathbf{x} ; \mathbf{w})=g\left(\mathbf{w}^{T} \mathbf{x}\right) 在H和r和在一个r和p一个r一个米和吨和rs吨H一个吨n和和d吨○b和s和吨吨○sp和C一世F一世C在一个l在和s○r,一世n○吨H和r在○rds,吨H和是一个r和吨H和p一个r一个米和吨和rs○F○在rp一个r一个米和吨和r一世和和d米○d和lF○rs在p和r在一世s和dl和一个rn一世nG.在和在一世llC○米和b一个Cķ吨○吨H一世sp○一世n吨l一个吨和rr和G一个rd一世nGH○在pr和C一世s和l是吨○CH○s和吨H和米.吨H和○r一世G一世n一个l米CC在ll○CH−磷一世吨sn和在r○n一世s一世n吨H和s和吨和r米s一个吨Hr和sH○ldp和rC和p吨r○n在一世吨H一个吨Hr和sH○ldG一个一世nF在nC吨一世○n, g(x)=\左{ 1 如果 X>0 0 除此以外 \正确的。$$

## 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。