## 数学代写|优化算法代写optimization algorithms代考| Complexity of Real Computation Processes

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。

• 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

## 数学代写|优化算法代写optimization algorithms代考|On the Computer Constructing Technology

Scheme of constructing (choice) of $T$-effective computational algorithm depends on many factors (class problems, input data, dimension and characteristics of the problems, computational resources that are available to the user, constrains (2.1), (2.2), and (2.3)); therefore, in the class problem $F$, it is advisable to distinguish multitude (subclasses) of problems that have common features in the context of computing [14]:

• One-off problems with a small amount of computing and moderate constraints on process time
• Problems (or series of problems) that are needed to be solved in real time
• Problems with a very large amount of computations that are needed to be solved in a practically reasonable amount of time (that cannot be achieved on traditional computing machines)

The performance of the conditions (2.1), (2.2), and (2.3) depending upon the statement of the problem can be achieved by choosing one of the following combinations of computing resources: $X,\left(X, I_{n}\right),(X, Y),\left(X, Y, I_{n}\right)$. In the first two situations, the possibilities of the computer are fixed. In the first situation, the information $I_{n}$ is also fixed; conditions (2.1), (2.2), and (2.3) are satisfied by the choice of the algorithm and its parameters; in the second one, it is still possible to select the set $I_{n}$ for this type of information operator. In the third situation, the information is fixed, and the parameters of the computer can be chosen besides the algorithm. In the fourth situation, all computing resources are used.

The first group of problems can be solved by the choice $X$ or $\left(X, I_{n}\right)$ of a regular sequential computer. Herewith, it is possible to devote three levels of detalization of the computing model. At the first level, there are algorithms that are focused on class $F$ solving problems using the information $I_{n}$. Herewith, there is support: approximation, stability, convergence of the approximate solution, the possibility to achieve a certain accuracy for the given input information, and the volume of computations as a function of the problem size (volume of input data). At this level, there is a possibility to discover the impossibility of computation of the $\varepsilon$-solution using specific input data, and there might be a possibility to clarify the class of problems and the requirements for the input information to provide a certain accuracy of the approximate solution, and it is possible (in this regard) to choose a new algorithm.
The second level (detalization) is related to the use of elements of the multitude $Y$ (machine word length, rounding rule) to compute the error estimate of rounding. Herewith, a multitude $A(\varepsilon)$ can be defined as conclusions in the case of the advisability of certain algorithms using from the multitude $A(\varepsilon)$ to save process time.
At the third level, where computational algorithm is a program for computing the $\varepsilon$-solution on a certain computer, time $T(\varepsilon)$ and memory $M(\varepsilon)[114]$ are estimated.
The variants $(X, Y)$ and $\left(X, Y, I_{n}\right)$ are specific to the second group of problems, for example, for digital signals processing and digital images processing using specialized computers. To achieve high rapid rates, the computer architecture is coherent with the computational algorithm $[131,277]$.

It is possible to use the third or fourth variants of the organization of computing to solve the problems of the third group. Herewith, the one purpose high-end computers [220] and computers of all purposes can be used [199].

## 数学代写|优化算法代写optimization algorithms代考|Specificity of Using Characteristic Estimates

In constructing real computational processes of computations, $\varepsilon$-solution is often used by some estimates of global error, its component and process time. Herewith, they distinguish estimates in the following way: a priori and a posteriori, majorizing and asymptotic, and determinate and stochastic. The possibility and advisability of these estimates using and the methods of their construction depend on the type, structure, and accuracy of a priori data, the problem, and the CA from that why the estimate is computed, and it also depends on the computational resources [114,238].

Majorizing a priori estimate guarantees the upper bound of the estimated deriv atives, and they are performed through known derivatives. Their computation does not require some significant computational expenses, but the value of estimates are often overrated; therefore, the conclusions based on them as for the possibility of computing of the solution under the conditions $(2.1)$ and $(2.2)$ may be false.

Asymptotic estimates approximate the estimated derivative. The variability of the parameter can be achieved by the desirable estimate proximity to the estimated derivative, but the computation of such estimates is related to significant computational expenses, and these estimates are usually a posteriori.

In the algorithmic support of solving problems under the conditions (2.1) and (2.2), given the properties of the estimates, it must be expected the possibility of computing of the various types of estimates of characteristics $E\left(E_{\mathrm{u}}, E_{\mu}, E_{\tau}\right)$ [238]. By the relaxed constraints (2.1) and (2.2), less precise and less complex (computational) estimates may be sufficient. By the tighten constrains (2.1) and (2.2), asymptotic (a posteriori) estimates are used. For example, the condition (2.2) may apply strict requirements to the accuracy of estimates of computational process parameters that are computed on the basis of errors estimate of the solution.

## 数学代写|优化算法代写optimization algorithms代考|Classes of Computational Problems, Informational

In the given technology of constructing problems solution per time that does not exceed the given $T$, available information plays a great significance. The more a priori information of different principles is known on the problem and algorithm uses it, the more accuracy effective and time it can be solved.

Note that the effectiveness of the algorithms is determined by the estimate of their characteristics so that the estimates should be of high quality (constants that are included in majorizing estimates of errors, accurate, estimates, unimprovable, etc.). And yet even high-quality estimates are constructed on a class of problems. And the wider this class is, the less suitable this estimate may be for a particular problem. Therefore, it is important to have a classification of problems that considers the additional a priori information. This will make a possibility to “select” such a class for a solved problem that is most likely to be used to obtain the required solution of a certain quality.

Consequently, the improvement of the quality of solving problems depends on the “narrowing” of the class of problems to which the solved problem belongs and the building of algorithms of such solving problems and the most accurate estimates of their characteristics.

However, it is not always possible to obtain $\varepsilon$-solution of some problems (although the total input information may be enough for this) using the given technology, or it cannot be checked that the solution was achieved. In these cases, it is important to have algorithms that are accuracy optimal (all available information on the problem is used as much as possible to improve accuracy) and a posteriori error estimates (that are more accurate next to a priori ones).

On the back of the accuracy optimal algorithm of this solving problem and a posteriori estimate of the error, it is often possible to obtain a solution that satisfies the user or draw a conclusion that it was not possible to obtain such a solution. We consider key principles of the problems classification and algorithms through the examples of some specific classes of problems of computational and applied mathematics.

## 数学代写|优化算法代写optimization algorithms代考|On the Computer Constructing Technology

• 计算量小、处理时间适度限制的一次性问题
• 需要实时解决的问题（或一系列问题）
• 需要在实际合理的时间内解决的大量计算问题（传统计算机无法实现）

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|优化算法代写optimization algorithms代考|Analytic Computational Complexity

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。

• 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

## 数学代写|优化算法代写optimization algorithms代考|Input Information, Algorithms, and Complexity

Consider the idealized computation model: Information $I_{n}(f)$ is given accurately, and the model $c$ is fixed. Here are some characteristics that are related to the lower and upper estimates of the error on the example of the passive pure minimax strategy $[102,253]$ (see Chap. 1 for more details):

• $\rho_{\mu}(F, a)=\sup {f \in F} \rho\left(E{\mu}\left(I_{n}(f)\right), a\right)$ is an error of algorithm $a \in A$ on the class of the problems $F$ using information $I_{n}(f)$ (global error [270]).
• $\rho_{\mu}(F, A)=\inf {a \in A} \rho{\mu}(F, a)$ is a lower boundary of error of algorithms of class $A$ in the class of problems $F$ using information $I_{n}(f)$ (radius of information [270])
If there is an algorithm $a_{0} \in A$ for which $\rho_{\mu}\left(F, a_{0}\right)=\rho_{\mu}(F, A)$, then it is called accuracy optimal in class $F$ using information $I_{n}(f)$.

The narrowing of the $F$ class that is provided by the incompleteness of information (in relation to $f \in F) F_{n}(f)=\left{\varphi: I_{n}(\varphi)=I_{n}(f), \varphi, f \in F\right}$ allows to introduce the characteristics that are equivalent to the mentioned one above: $\rho_{\mu}\left(F_{n}(f), a\right)$ and $\rho_{\mu}\left(F_{n}(f), A\right)$, which are also called the local error and the local radius of information [270], respectively.

Let $U(f)$ be the multitude of solving problems from $F_{n}(f)$, and $\gamma(f)$ is the center of this multitude (the Chebyshev center). The algorithm $a^{\gamma} \in A$ is called a central one if $a^{y}\left(I_{n}(f)\right)=\gamma(f)$. These algorithms are accuracy optimal. Their important quality is that they minimize the local error of the algorithm:
$$\inf {a \in A} \sup {\varphi \in F_{n}(f)} \rho_{\mu}\left(I_{n}(f), a\right)=\rho_{\mu}\left(I_{n}(f), a^{\gamma}\right)=\operatorname{rad} U(f) .$$
Note that $A(\varepsilon) \neq \varnothing$ only when $\rho_{\mu}\left(F_{n}(f), A\right)<\varepsilon$.
Consider that class $A$ contains stable congruent algorithms and
$$\rho_{\mu}\left(F_{n}(f), a\right) \rightarrow 0 \text { при } n \rightarrow \infty$$
The algorithm $a$ is called asymptotically accuracy optimal (accuracy order optimal) if the condition is performed:
$$\rho_{\mu}\left(F_{n}(f), A\right) / \rho_{\mu}\left(F_{n}(f), a\right) \rightarrow 1(\leq \text { const }), \quad n \rightarrow \infty$$
Let $A(\varepsilon) \neq \varnothing$. The value $T(F, a, \varepsilon)=\sup {f \in F} T\left(I{n}(f), a, \varepsilon\right)$ is called $\varepsilon$-complexity of the algorithm $a \in A(\varepsilon)$ on the class of problems $F$ using information $I_{n}(f)$; $T(F, A, \varepsilon)=\inf {a \in A(\varepsilon)} T(F, a, \varepsilon)$ is $\varepsilon$-complexity of class $F$ problems using algorithms $A(\varepsilon)$ and information $I{n}(f)$. If there is an algorithm $a^{0} \in A(\varepsilon)$ for which

$T\left(F, a^{0}, \varepsilon\right)=T(F, A, \varepsilon)$, then it is called complexity optimal algorithm in this computational model. As in the case of error, it is possible to enter characteristics $T\left(F_{n}(f), a, \varepsilon\right)$ and $T\left(F_{n}(f), A, \varepsilon\right)$ on the class $F_{n}(f)$ and definitions of the optimal, asymptotically optimal, and complexity order optimal of the algorithm.

The mentioned characteristics are about the so-called “worst”-case model. The estimates are focused on the “worst” (the most complex) problem of the class. Of course, these estimates are guaranteed, and they are achieved on any problem of class $\Pi$, but this problem can be atypical for a given class. Therefore, there are some possible situations where the $\varepsilon$-solution can be obtained at lower expenses $(T)$. Consequently, in order to minimize the computational complexity of the $\varepsilon$-solution construction, the current question of the problems of classes narrowing, the ways where input data is presented, and the use of a priori information on a problem is relevant.

## 数学代写|优化算法代写optimization algorithms代考|Computer Architecture and the Complexity

There is an opinion (see, for example, [218]) that the optimization of the mathematical support of applied problems and the progress of computing techniques make equal contributions to the increasing possibilities of complex solving problems, in a point of fact in decreasing the computational complexity.

Consider the effect of rounding of the numbers on the computational complexity. Hypothesis [22] on that it is enough to compute the estimate of function $f$ for obtaining the solution with accuracy $O(\varepsilon)$ and perform intermediate computations in implemented CA with $O\left(\ln \varepsilon^{-1}\right)$ binary digit bits found confirmation in solving many problems $[42,106,114]$.

Thus, in the building of the $\varepsilon$-solution, the program uses numeric arrays with a total volume of $N$ numbers, and then memory $O\left(N \ln \varepsilon^{-1}\right)$ is required to store them.
Next, there is a possibility to consider the example of separate classes of problems and how the rounding error affects the possibility of $\varepsilon$-solution computation and the complexity of CP.
Let
$$\begin{gathered} \rho\left(E_{\mathrm{H}}\left(I_{n}(f), a, c\right) \leq \varepsilon_{1}<\varepsilon\right. \ \rho\left(E_{\mu \tau}\left(I_{n}(f), a, c\right)>\varepsilon_{2}, \quad \varepsilon_{2}=\varepsilon-\varepsilon_{1}\right. \end{gathered}$$
where $E_{\mu \tau}=E_{\mu}+E_{\tau}, a \in A$, and the relations are performed (in a point of fact in the numerical integration of $\mathrm{ODE}$, the computation of integrals and other classes of problems) $[106,114]$;

$$E_{\mu}=O\left(n^{-p}\right), \quad E_{\tau}=O\left(n 2^{-\tau}\right),$$
where $p$ is the order of the numerical method accuracy, and $\tau$ is the length of the mantissa in binary number notation in the floating-point mode. Herewith,
$$\varepsilon_{\mu \tau}^{0}(\tau)=\min {n} \rho\left(E{\mu z}\left(I_{n}(f), a, c\right)=O\left(n_{0}^{-p}(\tau)\right),\right.$$
where $n_{0}(\tau)=O\left(2^{\tau /(p+1)}\right)$, and $E_{\mu}\left(n_{0}(\tau)\right)=O\left(n_{0}^{-p}(\tau)\right), E_{\tau}\left(n_{0}(\tau)\right)=O\left(n_{0}^{-p}(\tau)\right)$.
Method error predominates when $n \ll n_{0}$. It can be decreased using optimal sets $I_{n}$, increasing $n$ (considering (2.10)), using accuracy optimal and close to them CA, moving to another class of input data $I_{n}$ (to increase the order of accuracy) and relevant CA.

The rounding error predominates when $n \gg n_{0}$. The decreasing $\rho\left(E_{\mu \mathrm{r}}\right)$ can be achieved by immediate increasing $n$ (considering $(2.10)$ ) or using the same capabilities as when $n \gg n_{0}$ (except increasing), as well as by increasing of $\tau$. From the relations (2.9) and (2.10), it follows that the performance of the constrain $\varepsilon_{\mu \tau}^{0} \leq \varepsilon_{2}$ is related to the conditions:
$$n=O\left(\varepsilon_{2}^{-1 / p}\right), \quad \tau=O\left(\log \varepsilon_{2}^{-1}\right), \quad \varepsilon_{2} \rightarrow 0$$
Consider the case (2.6). Let $\varepsilon$-solution be computed on the one processor using a linear computational algorithm per hour:
$$T(\varepsilon)=T_{l}(\varepsilon)+T_{a}(\varepsilon),$$
where $T_{I}(\varepsilon)$ is the process time of computation of the set of functionals $I_{n}(f)$ (information complexity), and $T_{a}(\varepsilon)$ is the process time of CA implementation for given information $I_{n}(f)$ (combinatorial complexity). Thus,
$$\begin{gathered} T_{l}(\varepsilon)=n(\varepsilon) \beta_{f}(\varepsilon) \alpha(\varepsilon), \ T_{a}(\varepsilon)=n(\varepsilon) \beta_{a} \alpha(\varepsilon), \end{gathered}$$
where $\alpha(\varepsilon)$ is the time performance of the “mean” operation in the computation of $\varepsilon$ solution; $\beta_{f}(\varepsilon)$ is an average number of computation operations of the functional $i_{j}$; and $\beta_{a}$ is an average number of operations that are related to the use of one function in implementation of CA.

Note that $\beta_{f}(\varepsilon)$ does not depend on $\varepsilon$ if the functionals in $I_{n}(f)$ can be computed closely to the accurate arithmetical operations. Then,
$$T(\varepsilon)=n(\varepsilon) \beta \alpha(\varepsilon), \quad \beta=\beta_{f}+\beta_{a} .$$

## 数学代写|优化算法代写optimization algorithms代考|Optical Models of Computations

Optical Models of Computations Dynamic holography [102] is a promising way of implementing a variety of optical converters. For instance, the effect of the energy transfer of a beam of light into another coherent light of beam that goes in the other direction (courtesy of their transverse in a dynamic environment) is an optical analogue of the transistor. It is possible to control the temporal variations in its intensity by changing the intensity of the amplifying beam. In another variant of the optical analogue of the electronic transistor, such control is achieved by changing not the intensity but the phase of the intensity beam.

Another example can be an optical switching device that is similar to a high-speed electronic commutator that is an inherent part of the most important devices of the computing techniques.

The advantage of holography is the possibility of a single transformation of the most complex images and not just the primary plane or spherical waves.

For today, the experiments have been performed on the creation of optical of bistable devices that switch $10^{-12}$ with elements of optical fiber communication lines, the information of which is moved with the help of optical solutions with the duration of reaching $10^{-13}$. With such a switching time, the productivity of a digital optical processor $10^{5}-10^{6}$ has simultaneous channels that would be equal to $10^{18}$ operations per second; in other words, it is on six order higher than the productive potential of electronic schemes. Examples of primitive actions for an optical computer include addition and subtraction of mappings, computation of Fourier transform, mapping identification, and so on.

## 数学代写|优化算法代写optimization algorithms代考|Input Information, Algorithms, and Complexity

• ρμ(F,一种)=支持F∈Fρ(和μ(一世n(F)),一种)是算法错误一种∈一种关于问题的类别F使用信息一世n(F)（全局错误 [270]）。
• ρμ(F,一种)=信息一种∈一种ρμ(F,一种)是类算法的误差下界一种在问题类别中F使用信息一世n(F)（信息半径[270]）
如果有算法一种0∈一种为此ρμ(F,一种0)=ρμ(F,一种)，则称其为类中精度最优F使用信息一世n(F).

приρμ(Fn(F),一种)→0 在 n→∞

ρμ(Fn(F),一种)/ρμ(Fn(F),一种)→1(≤ 常量 ),n→∞

## 数学代写|优化算法代写optimization algorithms代考|Computer Architecture and the Complexity

ρ(和H(一世n(F),一种,C)≤e1<e ρ(和μτ(一世n(F),一种,C)>e2,e2=e−e1

eμτ0(τ)=分钟nρ(和μ和(一世n(F),一种,C)=这(n0−p(τ)),

n=这(e2−1/p),τ=这(日志⁡e2−1),e2→0

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|优化算法代写optimization algorithms代考| Algebraic Computing Complexity

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。

• 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

## 数学代写|优化算法代写optimization algorithms代考|Formal Computational Models

Turing machines (TM) [3] is a class of the most well-known formal models for the analysis of the problem of limited complexity. The problem is considered to be algorithmically solved if its solution can be built using the corresponding TM. It

should be noted that the class of problems that can be solved with TM is left to be solved moving from TM to another formal model [3,241, 271]. All problems of algebraic complexity are divided into two classes (the class $\mathrm{P}$ is a problem that can be solved with polynomial complexity on deterministic Turing machines (DTM), and the NP class is the class where the problems can be solved with polynomial complexity on nondeterministic Turing machines (NTM) [3]). As the characteristics of the computational complexity, computing time (number of steps that are necessary to use the solving problem of the algorithm) and memory (the amount of operating domain that is used by the algorithm) are used.

Here are some known relations between the time complexity $(T(n), n$ is the amount of input data) and by the amount of memory $(S(n))$ [249] that are obtained for TM.

Let DTIME $(T(n))$ (DSPACE $(S(n)))$ be a class of problem that suppose DTM per hour $T(n)$ (with a memory $(S(n)$ ). The classes of problems are determined likewise NTIME $(\cdot)$, NSPACE $(\cdot)$ for HTM. Then
$$\begin{gathered} \operatorname{DTIME}(T(n)) \subseteq \operatorname{NTIME}(T(n)) \ \operatorname{NTIME}(T(n)) \subseteq \operatorname{DTIME}\left(2^{O(T(n))}\right) \ \operatorname{DSPACE}(S(n)) \subseteq \operatorname{NSPACE}(S(n)) \ \operatorname{NSPACE}(S(n)) \subseteq \operatorname{DSPACE}\left(S^{2}(n)\right) \ \operatorname{NTIME}(T(n)) \subseteq \operatorname{DSPACE}(T(n)) \ \operatorname{DTIME}(T(n)) \subseteq \operatorname{DSPACE}\left(T(n) / \log _{2}(T(n))\right) \ \operatorname{NSPACE}(T(n)) \subseteq \operatorname{DTIME}\left(2^{O(S(n))}\right) \end{gathered}$$
An important example of complex problems is NP-complete problems. The problem $f$ is considered to be NP-completed if it belongs to the NP class and each NP problem can be polynomial complexity that is reduced to $f$. The central point in the theory of NP-completeness is whether or not the classes $\mathrm{P}$ and NP are congruent, in other words if the problem (from the class NP) is provided by practice that is related to problems (of class P) that can be solved. There are reasons to assume that the solution of the most complex problems of the NP class (NP-complete problems) requires (as it can be seen from the estimates) the deterministic exponential time; in other words, the classes P and NP are different. The NP-completeness of many problems is proved $[3,48]$. The difficulty is to prove that each NP problem can be polynomially transformed to this problem.

It should be noted that the definition of the NP class and the proof of the polynomial complexity of many “reset” problems had great practical importance. Together with practical valuation, it destroyed some illusions regarding the practical constructing of solving a problem that has a solution; it has been found that the existence of only one algorithm for solving a certain mass issue is not enough for

practice. On the other hand, the algorithms for which acceptable polynomial upper estimated were proved and found some practical use.

The basic possibility of classification by complexity is provided by the so-called theorems on the hierarchy. The hierarchy theorem for a given complexity (by time or memory) determines which decrease in the upper complexity estimate leads to the narrowing of the class of functions that can be computed with this complexity.

## 数学代写|优化算法代写optimization algorithms代考|Asymptotic Qualities of “Fast” Algorithms

The purpose of a lower complexity estimate construction is to prove that none of the algorithms in this computational model has less complexity of computation than the given function $\varphi(t)$. Unfortunately, the well-known “high” (nontrivial) lower estimates are perhaps the exception, not the rule.

The scheme of upper estimates of complexity constructing is as following. Based on some methods of solving problem, CA is built in a particular computational model, and it is proved that the computational complexity does not exceed some function from input data in the class. This function is called the upper estimate of the computational complexity of solving problem constructing.

There are several types of CA (which these estimates are implemented on). They are optimal, order optimal, and asymptotically optimal. Optimal CA corresponds to the case when the upper and lower boundaries are congruent. Two other types of CA concem, respectively, the estimates with the “accuracy to the multiplicative constant” and “accuracy to additive constants.” The practical use of algorithms is based on estimates that have an explicit specificity.

Consider these questions briefly. Let $A(0, X) \neq \varnothing A$ consider the computer model of sequential computations. Then
$$T\left(I_{n}(f), X, Y\right)=T_{I}\left(I_{n}(f), Y\right)+T_{a}(X, Y),$$
where $T_{I}(\cdot)=\sum_{1}^{r} \alpha_{i} n_{i}(n), T_{a}(\cdot)=\sum_{1}^{r} \alpha_{i} m_{i}(n, a), \alpha_{i}$ is a price of the $i$-operation from the model $c ; n_{i}(n), m_{i}(n, a)$ is the number of operations of the $i$-type that are necessary for the computation of the set of functionals $I_{n}(f)$ and the solution of the problem $f$ by the algorithm $a \in A$, provided that the set $I_{n}(f)$ is known; and $n$ is a number of functionals in the set. The values $T_{l}, T_{a}$ are called, respectively, informational and combinatorial (computational) complexities (solving computation) [270].

Note that the value $T$ depends essentially on $n$ and the character of the dependence $\left{n_{i}, m_{i}\right}$ from $n$. For example, by solving a system of $n$ linear equations, $A x=b$ by Gaussian elimination (for given $A, b) n_{i}=0, m_{i}=O\left(n^{3}\right), i=1,2$ (there is about the operations of addition and multiplication of two numbers).

In the general case, there is a possibility to assume that $n_{i}=O(n)$ (the functional $I_{n}(f)$ has a limited complexity) and $m_{i}(n)$ can be functions of $n$, for example, polynomial or exponential (or higher) complexity. Then the question arises on the

possibility of a solution computation with less computational complexity (see, for example, the class of NP-complete problems).

Of course, the character of dependence $m_{i}$ from $n$ is not determinative in the practical acceptability of the algorithm for solving a specific problem. It must be also considered that the constants in the functional dependences $m_{i}(n)$ can be that sort of algorithms with a lower order of complexity increasing, and advantage will be only for infinite values $n$. For example, offered algorithms of solving systems of linear algebraic equations for which $m_{i}=O\left(n^{\beta}\right), \beta<3$, have advantages over the complexity of Gaussian elimination for infinite values $n$. In addition, it is needed to pay attention to the possible loss of numerical stability of the algorithm. The fast Fourier transform (FFT) algorithm is used to multiply two numbers, and it has the complexity $O(n \log n)$ where $n$ is the number of binary digit bits for the number notation. The practical advantage of a high speed next to the traditional way of multiplication $\left(O\left(n^{2}\right)\right)$ is achieved for $n>100$.

## 数学代写|优化算法代写optimization algorithms代考|Accuracy and Complexity of Computations

The theory of analytic complexity is closely related to the theory of errors in the approximate solving problem. The value of the processing time is often determined by the requirements to the accuracy of the approximate solution; the relation of the components of the global error; the dependence of the error on the type, structure, volume of input data and their accuracy, bit grid of computer, and rounding rules; the type of error estimates; and the method of estimates constructing from below and from above. Therefore, there is a good reason to consider advisably these two characteristics: the error of the approximate solution and the process time [297, 301$]$.
Considering that it is difficult to build high lower and lower upper estimates in the given model of computation (when $E$ is a global error), some idealized models are considered that to consider only individual components of the global error (more often the errors of the method) and the influence of the individual components of computational models on error and complexity. For such incomplete models, it is possible to conclude the impossibility of constructing $\varepsilon$-solution based on this information.

The dependence of the approximate solution accuracy and the complexity of the $\varepsilon$-solution computation from the various components of the computational model will be considered next.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。

• 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

## 数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

Despite the achievements in the application software of modern computers, today there are many problems for which it is impossible to obtain a solution with given accuracy at limited computing resources. This is all about the problems of mathematical modeling, crystallography, radio astronomy, control of fleeting processes, cryptanalysis, and problems of high dimension.

As a rule, the solution of the applied problems is reduced to the solving typical classes of problems of computational and applied mathematics. Thus, it is important to create methods for building high-speed efficient algorithms for calculating $\varepsilon$-solutions of problems that use minimal computer memory for software. This will improve applied mathematical software and provide an opportunity to solve problems with less computing resources and reduce losses from the uncertainty of conclusions based on approximate solutions.

The main attention in the chapter is given to the creation of the elements of the complexity theory. With the use of it, this would be possible to construct effective complexity algorithms for computation of $\varepsilon$-solutions problems of numerical mathematics with limited computing resources.

Important results in the theory of computing optimization on the computing machinery were obtained by M. S. Bakhvalov, P. S. Bondarenko, V. V. Voievodin, H. Vozhniakovsky, V. V. Ivanov, M. P. Komeichuk, I. M. Molchanov, S. M. Nikolski, A. Sard, I. V. Sergienko, S. L. Sobolev, J. Traub, and others. These results allow estimating $\varepsilon$.

Computational complexity is less investigated than other characteristics. The complexity of the problem in time essentially depends on the computing model (computer architecture). A question of problem classes narrowing, the ways of input data presentation, and the complete use of a priori information on the problem are relevant for computational complexity minimizing of algorithm complexity of $\varepsilon$ solution constructing.

Today, many works are devoted to the study of the possibility of increasing the high speed of computing algorithms by paralleling the computations using traditional (with the focus on sequential computation) numerical methods. The general disadvantage of most of these studies lies in their obtainment of ideal computational models that lead to incomplete use of a priori information about the problem.

This chapter is devoted to the presentation of the general provisions of the complexity theory, statement of problems, algebraic and analytic complexity, and complexity of real computational processes. Key attention is given to the asymptotic qualities of “fast” algorithms, computer architecture, and the complexity and specificity of the characteristic estimate use. There are examples of the elements use of the complexity theory to the $\varepsilon$-solution construction of some practical important problems of computational and applied mathematics [285].

## 数学代写|优化算法代写optimization algorithms代考|General Provisions. Statement of the Problem

Let $F\left(I_{0}\right), A(X)$, and $C(Y)$ be the classes of problems of computational (or applied) mathematics, algorithms, and models of computing tools (computers), and $I_{0}$, $X, Y$ are a multitude of parameters that are dependent on the essentially suitable classes.

It is assumed that for the $\varepsilon$-solution constructing of the problem $f \in F$ (approximate solution, any error that does not exceed $\varepsilon \geq 0$ ), we use the algorithm $a \in A$ that was implemented on the computer $c \in C$ that is oriented on the use of information $I_{0}$ on class $F$ and information $I_{n}(f)$ on the specific class problem. The information (information operator) $I_{n}(f)$ can be given, for example, as a set of functionals $I_{n}(f)=\left(i_{1}(f), i_{2}(f), \ldots, i_{n}(f)\right)^{T}$ from the elements of the problem $f$.

Therefore, computation model is used for $\varepsilon$-solution construction that is described using $I_{0}, I_{n}(f), X, Y$.

The quality of the computational process (CP) of input data reduction, the result of which is $\varepsilon$-solution that is characterized by the computational complexity-the amount of a random computational resource that is necessary to the $\varepsilon$-solution constructing that is also called cost or expenses. The most widely used computing complexity characteristics is a processing time $T=T\left(I_{n}(f), X, Y, \varepsilon\right)$ and computer memory $M=M\left(I_{n}(f), X, Y, \varepsilon\right)$ that are required for $\varepsilon$-solution computing. Dependence of characteristics $T, M$ from $I_{0}$ is not specified since this information does not change.

They say that the problem has a restricted (algebraic) complexity (in this computational model) if there is an algorithm $a \in A$, by which it can be accurately solved $(\varepsilon=0)$ with limited computational complexity.

The problem has unrestricted (analytic) complexity if it cannot be solved precisely $(\varepsilon=0)$ in this computational model with restricted computational complexity.
A specific problem can have an algebraic or analytic complexity depending on input data and set of the computing model operations.

## 数学代写|优化算法代写optimization algorithms代考|solving problem computation of a system of linear algebraic

For example, solving problem computation of a system of linear algebraic equations by Gaussian elimination has an algebraic complexity providing that input data is given accurately and arithmetic operations are performed accurately either. If this condition is not performed, then the problem has an analytic complexity.

In real sets of operations, the great majority of problems of computational and applied mathematics are the problems of unlimited computational complexity; in other words, they are solved approximately ( $\varepsilon>0$ ). The exception is combinatorial and some algebraic problems [3].

The theory of analytic computational complexity is engaged in the optimization of the processes of approximate solving problems. The problems of algebraic complexity are used as an auxiliary in the theory of analytic complexity. On the other hand, the problems of algebraic complexity can have very high complexity and can be solved approximately [10].

The general situation of an approximate $\varepsilon$-solution of a problem constructing with constrained computing resources can be described by the following conditions $[14,106,114,237]$ :
$$\begin{gathered} E(I, X, Y) \leq \varepsilon, \ T(I, X, Y, \varepsilon) \leq T_{0}(\varepsilon), \ M(I, X, Y, \varepsilon) \leq M_{0}(\varepsilon), \end{gathered}$$
where $\varepsilon, T_{0}, M_{0}$ are the given numbers.
The quality of the approximate solution is characterized in the general case by the global error $\left(E\left(I_{n}(f), X, Y\right)\right)$, i.e., the sum of the three components: $E_{H}\left(I_{0}, I_{n}(f), Y\right)$ are the errors that are caused by inaccurate input information; $E_{\mu}\left(I_{0}, I_{n}(f), X\right)$ are the errors of the method; and $E_{z}\left(I_{n}(f), X, Y\right)$ are the errors through rounding $[106,114]$. Computations are often considered in the absence of some or all components of global error. All these can be some real computing situations or the results of idealization of computing conditions to simplify the research [106].
Thus, in the general case, it is needed to compute an approximate solving problem $f \in F$ using the model $I_{0}, I_{n}(f), X, Y$ under constraints (2.1), (2.2), and (2.3).

Further on, we will assume (if nothing other is not expected) that memory $M$ can be increased to the necessary volume; in other words, the constrain (2.3) can be removed but, apparently, by increasing the characteristic of $T$ (process time). This can be done, for example, by increasing a share of “slow” (disk) memory in the general structure of computer memory. Considering that within $\varepsilon \rightarrow 0, M_{0}(\varepsilon) \rightarrow \infty$ (for example, when it comes to rounding errors or errors in the method in stepwise algorithms), we will assume that $\varepsilon \geq \varepsilon_{0}>0$, where $\varepsilon_{0}$ is a given number.
Consider the problem of -solution finding (2.1), (2.2), and (2.3) [285].
Let $A(\varepsilon, X)(A=A(\varepsilon, X) \subseteq A(X))$ be a multitude of CA for which the condition (2.1) is used; in other words the algorithms for $\varepsilon$-solution computation for the given conditions. CA $A\left(\varepsilon, T_{0}\right)$ for which the conditions (2.1), (2.2) are used will be called $T$-effective, and $\left(A\left(\varepsilon, T_{0}\right) \subseteq A(\varepsilon, X)\right)$ is a multitude of $T$-effective CA.

## 数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

MS Bakhvalov, PS Bondarenko, VV Voievodin, H. Vozhniakovsky, VV Ivanov, MP Komeichuk, IM Molchanov, SM Nikolski, A. Sard, IV Sergienko, SL Sobolev, J. 特劳布等人。这些结果允许估计e.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|优化算法代写optimization algorithms代考|Improvement of the Lower Estimate of the Accuracy

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。

• 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

## 数学代写|优化算法代写optimization algorithms代考|Approximate Solving Problem by the Choice

Let $\widetilde{I}$ be any class of informational operators [285]. Assume that the class creates the informational operators of one type with different sets of functionals. For example, if a set of values of function is used, then their number or set of nodes can change their number or even the value of function is computed within constant $N$ (or both). Informational operators of different types (the value of the function and its derivatives, the coefficient of the factorize by certain basis, etc.) create different classes. It is possible to introduce the characteristics:
\begin{aligned} &\rho(\Pi, A, \widetilde{I})=\inf {I{N}(f) \in I} \rho\left(\Pi, A, I_{N}(P)\right)\left(\rho\left(\Pi, A, I_{N}(P)\right) \equiv \rho(\Pi, A)\right) \ &\rho(\Pi, \Lambda, \widetilde{I})=\inf _{A \in \Lambda} \rho(\Pi, A, \widetilde{I}) \end{aligned}
where $\rho=(\Pi, A, \tilde{I})$ is a lower boundary of the error of the algorithm $A \in \Lambda$ in the problem class $\Pi$ using information from class $\tilde{I}$, and $\rho(\Pi, \Lambda, \tilde{I})$ is a lower bound of the error of algorithms in the computing model $(\Pi, \Lambda, \widehat{I})$.

Information $I_{N}^{0}(P) \in \widetilde{I}$, for which the condition $\rho\left(\Pi, A, I_{N}^{0}(P)\right)=\rho(\Pi, A, \widetilde{I})$ is performed, is called an optimal in classes $\Pi, \tilde{I}$ by using the algorithm $A \in \Lambda$. If $\rho\left(\Pi, A^{0}, I_{N}^{0}(P)\right)=\rho(\Pi, \Lambda, \widetilde{I})$, then the algorithm $A^{0} \in \Lambda$ and the information $I_{N}^{0}(P) \in \widetilde{I}$ are called optimal in this computational model $(\Pi, \Lambda, \widetilde{I})$.

Likewise, it is possible to introduce the definition of complexity for the problem $P$ and the problem of class $\Pi$ and their characteristics:

• $T(\Pi, A, \widetilde{I}, \varepsilon)=\inf {I{N}(P) \in \widetilde{I}} T\left(\Pi, A, I_{N}(P), \varepsilon\right)$ is $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ in the problem of class $\Pi$ within the use of information $\tilde{I}$.
• $T(\Pi, \Lambda(\varepsilon), \tilde{I})=\inf _{A \in \Lambda(\varepsilon)} T(\Pi, A, \tilde{I}, \varepsilon)$ is $\varepsilon$-complexity of the problem in this computation model $(\Pi, \Lambda(\varepsilon), \widetilde{I})$.
• $T(P, A, \widetilde{I}, \varepsilon)$ is the $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ when the problem $P \in \Pi$ is solved using information $\tilde{I}$.
• $T(P, \Lambda(\varepsilon), \widetilde{I})$ is the $\varepsilon$-complexity of the problem $P$ by using the algorithms $\Lambda(\varepsilon)$ and information $\tilde{I}$, as well as the definition of complexity optimal algorithm and optimal information.

It is possible to introduce an optimization of nodes in numerical integrating as an example of such optimization: by optimization with accuracy for a fixed $N$ by computing $\varepsilon$-solution with $N=O\left(\varepsilon^{-1 / q}\right), q$ is the index of the smoothness of the subintegral function.

This case is about the optimization of choosing the functionals within the constrain of the same type of informational operator that is a set of values of the subintegral function.

Examples of the value optimization of the characteristics $E(\rho(\cdot))$ and $T$ moving to another class of informational operators are contained in [298].

Approximate Information There is known information (approximated) $I_{N \sigma}(P)$ instead of information (exact) $I_{N}(P)$ where $\sigma \geq 0$ characterizes the deviation of the approximate information from the exact one. It is possible to consider the characteristics for the approximate information $I_{N \sigma}(P)$ that are similar to those that were given above for $I_{N}(P)$ assuming that information $I_{N \sigma}(P)$ can be adjusted considering $I_{0}$-information about the problem of the class $\Pi$. Thus, the central algorithm [270] in this case decreases the effect of error of the information $I_{N o}(P)$ on the approximate solution. Examples of constructing these algorithms are given in $[33,106]$.

## 数学代写|优化算法代写optimization algorithms代考|Basic Approaches to Constructing the Accuracy

Consider the problem of the computation of the integral that looks
\begin{aligned} &I_{1}(\omega)=\int_{a}^{b} f(x) e^{-i \omega x} d x \ &I_{2}(\omega)=\int_{a}^{b} f(x) \sin \omega x d x \ &I_{3}(\omega)=\int_{a}^{b} f(x) \cos \omega x d x \end{aligned}
assuming that $f(x) \in F(F)$ is a certain class of functions, and $\omega$ is a certain real number $(\omega \mid \geq 2 \pi(b-a))$.

Let the information about $f(x)$ be given by $N$ values at nodes $\left{x_{i}\right}_{0}^{N-1}$ from its definition domain: $\left{f_{i}\right}_{0}^{N-1}=\left{f\left(x_{i}\right)\right}_{0}^{N-1}$, $\varepsilon_{i}$ characterizes the accuracy of the problem $f\left(x_{i}\right)=f_{i}:\left|\tilde{f}{i}-f{i}\right| \leq \varepsilon_{i}, i=\overline{0, N-1}$.

We concretize the general definition of the accuracy optimal algorithm that is given in the par. $1.4$ for the problem of the approximate computation $I(\omega)$ (we will understand one of the integrals $(1.20,1.21$, and $1.22)$ under $I(\omega))$ ).

Mark $R=R\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)$ as the result of the approximate computation $I(\omega)$ with quadrature formula $A$.
Introduce the characteristics
\begin{aligned} &V\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\rho(I(\omega), R) \ &V\left(F, A,\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\sup {f \in F} V\left(f, A,\left{x{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right) \ &V=V\left(F,\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\inf {A} V\left(F, A,\left{\varepsilon{i}\right}_{0}^{N-1}, \omega\right) \ &V(F, \omega)=V(F, 0, \omega) \end{aligned}

## 数学代写|优化算法代写optimization algorithms代考|The function f(x)

Definition 1.1 The function $f^{\pm}(x)$ is called majorizing (minorant) class of functions $F_{N}$ that are defined in some domain $D$ if:

1. $f^{+}(x) \geq f(x)\left(f^{-}(x) \leq f(x)\right)$ for all $m$,
2. $f^{+}(x) \in F_{N}\left(f^{-}(x) \in F_{N}\right)$.
The Chebyshev center $\left(y_{1}, \ldots, y_{N}\right)$ and the Chebyshev radius $\rho^{}(\omega)$ of domain of uncertainty of solving the problem $(1.20,1.21$, and $1.22$ ) can be defined as follows [102]: $$\left(y_{1}, \ldots, y_{m}\right),\left(y_{1}, \ldots, y_{m}\right)=F\left(x_{1}, \ldots, x_{n}\right) \ldots$$ The quadrature formula that computes $I^{}(\omega)$ will be called accuracy optimal, and $\rho^{}(\omega)$ is the error of introduction of the value domain of the integral $I(\omega)$ using $I^{}(\omega)$ or the optimal estimate of the error of numerical integration $I(\omega)$ on the class $F_{N}\left(\delta=\rho^{}(\omega)\right)$. The quadrature formula $R(\omega)$ of the computation $I(\omega)$ for which $$\sup {f \in F{N}}|R(\omega)-I(\omega)| \leq \rho^{}(\omega)+\eta, \eta>0 \text { and } \eta=o\left(\rho^{}\right), O\left(\rho^{}\right)$$
$\left(y_{1}, \ldots, y_{N}\right)$ is called asymptotically optimal or accuracy order optimal.
Within given information about the problem, any quadrature formula can’t give an accuracy less than $\rho^{}(\omega)$. For interpolation classes $\left(y_{1}, \ldots, y_{m}\right)=F\left(x_{1}, \ldots, x_{n}\right)$, the Chebyshev radius $\rho^{}(\omega)$ ) coincides with an optimal estimate $V_{1}$.

The use of the limiting function method for the estimate $V$ is based on the following statement [293].

Theorem $1.3$ Let $f(x) \in F$ ( $F$ is a class of limiting functions) on $f(x)$ the information about its value in $N$ nodes of a random grid, and there is at least one quadrature formula $A \in M$ such as that $I^{+}(\omega) \leq I(\omega) \leq \Gamma(\omega)$. Then the next estimate is valid for $V_{1}$ :

$$V_{1} \geq \sup {F{N} \in F} \rho^{}(\omega)$$ It follows from the definition of the estimates $V$ and $V_{1}$ : $$V \geq V_{1}$$ In the case of $F \equiv F_{N}$, we have $V=\rho^{}(\omega)$.
Remark 1.1 Similar statements are colligated on n-dimensional case [293, 298], and they are used to construct optimal error estimates and prove some optimal cubature formulae of computation of multidimensional integrals from highoscillating functions of the form
\begin{aligned} I_{1}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}}{n} f\left(x{1}, \ldots, x_{n}\right) \sin \omega x_{1} \cdot \ldots \cdot \sin \omega x_{n} d x_{1} \ldots d x_{n}, \ I_{2}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}}{n} f\left(x{1}, \ldots, x_{n}\right) \cos \omega x_{1} \ldots \ldots \cos \omega x_{n} d x_{1} \ldots d x_{n} \end{aligned}
in the case when $n>1, f(X)$ is a known function, $f(X)=f\left(x_{1}, \ldots, x_{n}\right) \in F(F$ is a certain class of functions $X=\left{x_{1}, \ldots, x_{n}\right}, \omega$ is a certain real number $(|\omega| \geq 2 \pi)$, and information about $f(X)$ is given by $N$ values in node points $\left{X_{i}\right}_{0}^{N-1}$ from its domain of definition: $\left{f_{i}\right}_{0}^{N-1}=\left{f\left(X_{i}\right)\right}_{0}^{N-1}$.

## 数学代写|优化算法代写optimization algorithms代考|Approximate Solving Problem by the Choice

\begin{aligned} &\rho(\Pi, A, \widetilde{I})=\inf {I {N}(f) \in I} \rho\left( \Pi, A, I_{N}(P)\right)\left(\rho\left(\Pi, A, I_{N}(P)\right) \equiv \rho(\Pi, A)\right ) \ &\rho(\Pi, \Lambda, \widetilde{I})=\inf _{A \in \Lambda} \rho(\Pi, A, \widetilde{I}) \end{aligned}

• $T(\Pi, A, \widetilde{I}, \varepsilon)=\inf {I {N}(P) \in \widetilde{I}} T\left(\Pi, A, I_{N}( P), \varrepsilon\right)一世s\伐普西隆−C这米pl和X一世吨是这F吨H和一种lG这r一世吨H米一个 \in \Lambda(\varepsilon)一世n吨H和pr这bl和米这FCl一种ss\π在一世吨H一世n吨H和在s和这F一世nF这r米一种吨一世这n\波浪号{I}$。
• 吨(圆周率,Λ(e),一世~)=信息一种∈Λ(e)吨(圆周率,一种,一世~,e)是e-此计算模型中问题的复杂性(圆周率,Λ(e),一世~).
• 吨(磷,一种,一世~,e)是个e- 算法的复杂性一种∈Λ(e)当问题磷∈圆周率使用信息解决一世~.
• 吨(磷,Λ(e),一世~)是个e- 问题的复杂性磷通过使用算法Λ(e)和信息一世~，以及复杂度最优算法和最优信息的定义。

## 数学代写|优化算法代写optimization algorithms代考|Basic Approaches to Constructing the Accuracy

\begin{aligned} &V\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right} _{0}^{N-1}, \omega\right)=\rho(I(\omega), R) \ &V\left(F, A,\left{\varepsilon_{i}\right}_{ 0}^{N-1}, \omega\right)=\sup {f \in F} V\left(f, A,\left{x {i}\right}_{0}^{N-1 },\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right) \ &V=V\left(F,\left{\varepsilon_{i}\right}_ {0}^{N-1}, \omega\right)=\inf {A} V\left(F, A,\left{\varepsilon {i}\right}_{0}^{N-1} , \omega\right) \ &V(F, \omega)=V(F, 0, \omega) \end{aligned}

## 数学代写|优化算法代写optimization algorithms代考|The function f(x)

1. F+(X)≥F(X)(F−(X)≤F(X))对全部米,
2. F+(X)∈Fñ(F−(X)∈Fñ).
切比雪夫中心(是1,…,是ñ)和切比雪夫半径ρ(ω)解决问题的不确定性域(1.20,1.21， 和1.22) 可以定义如下[102]：(是1,…,是米),(是1,…,是米)=F(X1,…,Xn)…计算的求积公式一世(ω)将被称为精度最优，并且ρ(ω)是积分值域引入的误差一世(ω)使用一世(ω)或数值积分误差的最优估计一世(ω)在课堂上Fñ(d=ρ(ω)). 求积公式R(ω)计算的一世(ω)为此支持F∈Fñ|R(ω)−一世(ω)|≤ρ(ω)+这,这>0 和 这=这(ρ),这(ρ)
(是1,…,是ñ)称为渐近最优或精度阶最优。
在有关问题的给定信息内，任何求积公式的准确度都不能低于ρ(ω). 对于插值类(是1,…,是米)=F(X1,…,Xn), 切比雪夫半径ρ(ω)) 与最优估计一致在1.

\begin{aligned}形式的高振荡函数计算多维积分的一些最优容积公式 I_{1}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}} {n} f\left(x {1}, \ldots, x_{n}\right) \sin \omega x_{1} \cdot \ldots \cdot \sin \omega x_{n} d x_{1} \ldots d x_{n}, \ I_{2} ^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}} {n} f\left(x {1}, \ldots, x_ {n}\right) \cos \omega x_{1} \ldots \ldots \cos \omega x_{n} d x_{1} \ldots d x_{n} \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。

• 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

## 数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

Input data about the problem and its quality is very important in many aspects. Mention some of them:

• The more qualitative information on the problem is, the more qualitative approximate solution we can compute.
• Maximum use of all available information on the problem allows to limit the class of solved problems, and thus, it increases the “potential ability” of the NM; the more accurate input data is, the more accurate estimates of error and the less uncertainly range of the approximate solving problem.
• The computer technology of solving problem with the fixed values of quality with accuracy and fast speed is based on the analysis of error estimates.

We stop on some aspects of identification and clarification of a priori information on a problem.

An appropriate a priori information on the problem is required for obtaining a problem solution of a high quality, for example, the order of the derivative, constants that constrain it, the Hölder constant, and the corresponding mark-for the problems of function recovery and functionals. Useful information may also be about geometric properties-convexity, monotonies, number of extremums, etc. Such information is necessary to obtain an error estimate of the finding solution. If this information is given with sufficient low accuracy, then the conclusions on the quality of solving the problem will be inaccurate.

Consequently, obtaining qualitative a priori information is important in solving applied problems. Such information can be obtained from specialists who have a good knowledge of the physical phenomenon that we are studying. This information can also be obtained by using algorithms for identifying and clarifying a priori information.

For example, if the function is approximated from Lipschitz interpolational class, $F \equiv C_{L, N,} \varepsilon$ [287], and not only $L$ and $\varepsilon$ are known, but an only approximation to them is known. In such cases, it is advisable to use methods of residual and quasisolutions for approximating function [203].

For the class $F \equiv C_{L, N, e}$ the approximating function is the solution of the problem:
$$\min {f \in F} \max {i} \varepsilon_{i^{*}}$$
Otherwise, the method of quasisolutions involves finding a function that deviates less from the given set of points $\left(x_{i}, \tilde{f}{i}\right), \tilde{f}{i}=f_{i}+\varepsilon_{i}, i=\overline{0, N-1}$.

The solving problem $(1.15)$ is a line spline $S(x, L)$ in which the maximal deviation from the given points $\left(x_{i}, \tilde{f}{i}\right), i=\overline{0, N-1}$ is the minimum [203]: \begin{aligned} &S(x, L)=\widehat{f}{i}+\frac{x-x_{i}}{x_{i+1}-x_{i}}\left(\widetilde{f}{i+1}-\tilde{f}{i}\right), x \in\left[x_{i}, x_{i+1}\right], i=\overline{0, N-1}, \ &\widehat{f}{i}=\frac{\tilde{f}{i}-\tilde{f}{i}}{2}, \tilde{f}{i}^{\pm}=\max {1 \leq j \leq N}\left[\pm\left(\tilde{f}{j} \mp L\left|x_{j}-x_{i}\right|\right)\right], i=\overline{0, N-1} \end{aligned}
It often happens that the quantitative a priori information that is used to define a class $F$ is given in the form of constraints on some functional. A uniform norm of the derivative is used a functional $\Phi(f)$ itself for classes $C_{L, N}$ and $C_{L, N,}$. We will approximate the function $f(x)$ by a function that is the solution of the following problem:
$$\min _{f \in F} \Phi(f)$$

## 数学代写|优化算法代写optimization algorithms代考|Accuracy Optimal Computational Algorithms

In the due form, the concept of the optimality of the solution or the algorithm is determined by some criteria. Such criteria can be a requirement of the solution to have a given error of a method or the algorithm to have the highest possible rate of convergence. Specific content of criteria has an important value for its use.

One of the main criteria for the optimality of approximate solving problem can be the requirement of its maximum accuracy (or minimum error) by the given resources that can be used in the solution process. The concept of the resource includes the amount and accuracy of input data of the problem, free use of computer memory, limit the time of computing on this computing machinery, the available supply of mathematical software of computing machinery, etc.

In such a statement, it is natural to consider the question of the “potential ability” of NM at the beginning of the study, in other words, on that maximum accuracy of the solution that can be achieved for this given input information on the problem.
Every CA of solving a certain problem uses only a finite number of input data on the problem, and thus, it automatically is a CA of solving the class of all those problems that have the same input data. On this multitude of problems, there are always two problems in the solution of which the worst and best limits of the optimized characteristics are achieved. Therefore, every, including the optimal one, CA of solving a problem that concerns us will have some “potential ability.” If, for example, there are two problems with the same input data, accurate solutions of which $x_{1}$ and $x_{2}$ are the elements of the metric space, moreover, the distance between them are
$$\rho\left(x_{1}, x_{2}\right) \geq d>0$$
Then a solution $x$ obtains for each CA their solving that have a property
$$\max {i=1,2} \rho\left(x, x{i}\right) \geq \frac{d}{2}$$
This means that there is no CA that would give a solution to the considered problem with a guaranteed accuracy of less than $d / 2$. If there is a need to improve the accuracy of solving the problem, it should be included by some additional information about it. Then the problem will belong to a new more “narrow” class of problems, and the CA of which solution will have a new more powerful “potential ability.” Similar considerations are valid for any other index (characteristics) of CA and problems.

## 数学代写|优化算法代写optimization algorithms代考|Time Optimal Computational Algorithms

The computational complexity of the problem essentially depends on the computing model (computer architecture). Assume that it is possible to use a computing model for the building $E$-solution that is based on the use of input information of $I$, informational operator of $L_{M}(I)$ that can be introduced, for example, in the form of some multitude of functionals, as well as on a multitude of parameters $X, Y$.

Define the time-optimal algorithms and similar to them algorithms. Let the problem $P(I) L_{N}(I)=I=\left(i_{1}, i_{2}, \ldots, i_{N}\right), I \in$ 田 be input data, and $M$ is a multitude of algorithms $A$ of solving problem with a given accuracy $\varepsilon, \varepsilon>0$, on fixed computing machinery, and $Q(A, I, \varepsilon)$ is a number of arithmetic operations that are required for this. Consider the characteristics
\begin{aligned} &Q_{N}(A, \varepsilon)=\sup {I \in \mathcal{J}} Q(A, I, \varepsilon) \ &Q{N}(\varepsilon)=\inf {A \in M} Q{N}(A, \varepsilon) \end{aligned}
Call an algorithm a time-optimal one on which $Q_{N}(\varepsilon)$ is achieved. If $Q_{N}\left(A^{}, \varepsilon\right)=Q_{N}(\varepsilon)+\xi, \xi>0$, then $A^{}$ is called time-optimal with accuracy within $\xi$. If $\xi=o\left[Q_{N}(\varepsilon)\right]$ or $\xi=O\left[Q_{N}(\varepsilon)\right]$, then it is asymptotically optimal or time-optimal in order algorithm, correspondingly.

The purpose of constructing a lower complexity estimate $Q_{N}(\varepsilon)$ is to prove that none of the algorithms in this computational model has a less complexity of computation than the current function $Q_{N}(\varepsilon)$. Unfortunately, the well-known “high” (nontrivial) lower estimates are most likely an exception to the rule.

The scheme of upper estimates of complexity constructing is like this. CA $A^{}$ is built based on a certain method of solving the problem in the current computing model, and it is proved that the computational complexity does not exceed $Q_{N}\left(A^{}, \varepsilon\right)$ within the input data from the class. $Q_{N}\left(A^{}, \varepsilon\right)$ is called the upper estimate of the computational complexity of CA $A^{}$ of solving problem obtaining.

## 数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

• 关于问题的定性信息越多，我们可以计算的定性近似解就越多。
• 最大限度地利用有关问题的所有可用信息可以限制已解决问题的类别，从而增加 NM 的“潜在能力”；输入数据越准确，误差估计越准确，近似求解问题的不确定性范围越小。
• 准确、快速地解决质量固定值问题的计算机技术是基于对误差估计的分析。

$$\min {f \in F} \max {i} \varepsilon_{i^{*}}$$

ρ(X1,X2)≥d>0

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。

• 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

## 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

Since every year the complexity of scientific and applied problems is increasing, there is an uprise of the need of a large amount of process of input data with the use of computing machinery for the numerical computation, and optimization of computations by all rational methods and means is a topical and important question. As a rule, the solution of the applied problems reduces to solving the typical problems of computational mathematics. Computational mathematics is a science about the methods of numerical solving problems of applied mathematics using modern computing machinery. The subject of computational mathematics is numerical methods (NM) or, that is the same thing, the multitude of computational algorithms (CA) and the question of their substantiation: the convergence and speed of NM convergence, their stability and error, the optimality on different criterion, the implementation time on the computing machinery, the necessary memory of computing machinery, etc.

The purpose of this chapter is to explain the main concepts and some results of computational mathematics, which are repeatedly used in the monograph, and at the same time, they have an independent value. The presentation of this information, in our opinion, is necessary for a confident possession of the given material.
In the first chapter:

• Certain supporting data in the theory of computing is given for determining the accuracy of the approximate solving problem, the quality of the used algorithm, and the comparative study of various algorithms according to some criteria.
• The principal features of the CA are determined (accuracy, execution time, necessary memory of the computing machinery), and the methods of their estimate are provided.
• There is a definition of optimal (inaccuracy and high speed), asymptotically optimal, and optimal in the order algorithms.
• The questions of identifying and clarifying a priori information on the problem are considered in cases when a priori information is not incorrectly set in the order to improve the quality of the obtained approximate solutions of a problem.

## 数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations

One of the main directions of development of modern computational mathematics is the improvement of the theory of error computations, the research of comparative study of CA, and the optimal algorithm development in the solving problems of computational and applied mathematics.

The solution of the majority of the problems with modern computing tools is based on the computational experiment (CE), which seamlessly unites the mathematical model (MM), the computational algorithm (CA), the computations on the computing machinery, and the experiment. CE is necessary for obtaining an adequate quantitative description of the studied phenomenon or process itself with the necessary accuracy for the least possible machine time. The process of observations and comparing them with the matched model of MM is conducted on the computing machinery to check the validity of MM. The implementation of MM is performed with the use of numerical methods (NM), which approximate the input model and make it suitable for practical use. The errors of this approximation, as well as the rounding errors in the implementation of NM on the computing machinery and the errors of measurement or observation of implementation of the studied process, must be considered in determining errors of MM or the adequacy of MM and in the process itself. The errors of input MM must be considered in the process of definition of the requirements for the accuracy of the solution of various problems that are solved within this model.

In this monograph, the most important computational aspects of the determination of the error of MM and constructing its optimal implementation are analyzed. The multifaceted approach is based on the analysis of the three main characteristics of the corresponding NM-accuracy, implementation time, and necessary memory on computing machinery. According to these characteristics, the comparative study and the optimization of the corresponding NM are performed.

Main Characteristics of CA Let the problem $P(I)$ is solved by the algorithm $A(X)$ on the computing machinery $C(Y)$, where $I, X, Y$ are finite multitudes (vectors) of parameters from which essentially $P, A, C$ are, respectively, dependent on. The components of the vector $I$ may include data on a priori qualities of solving a problem, for example, constant that constrains the absolute values of the order of the derivatives from the given functions, data on the accuracy of the input values, etc. A number of iterations of the algorithm, the degree of approximation, the step of the grid, etc., can be the vector $X$ components. Vector $Y$ can contain the number of cell digits of the computing machinery memory, the total volume of its RAM, the run time of the computing machinery, the quantity of the computing machinery

processors, the characteristics of the used operational systems and compiler programs, etc.

In the practice of computational solving problems on the computing machinery, the following are the characteristics of problems, algorithms, and computing machinery: $E(I, X, Y)$ is a global error of solution $E$ of the problem $P$ on the computing machinery $C$ with algorithm $A, T(I, X, Y)$ is the time that is required to obtain a solution of the problem, $M(I, X, Y)$ is a required computing machinery memory, and fef is a coefficient of technical and economic efficiency. Explain the characteristics of $T, M$, and fef..

Total time $T$ – The period of time from setting the problem $P(I)$ to its solution of CA $A(X)$ on computing machinery $C(Y)$ can be estimated as follows:
$$T=T_{1}+T_{2}+T_{3}+T_{4}$$
where $T_{1}$ is the time to set the problem and develop or select CA $A$ and computing machinery $C ; T_{2}$ is the programming time, transmission motion, and debugging $\mathrm{CA}$ $A ; T_{3}$ is the implementation time of $\mathrm{CA}, A$ on the computing machinery $C$; and $T_{4}$ is the time of interpretation of the obtained solution and its comprehension. Practically when estimating $T_{3}$, only the essential operators are often considered to the number and time of the operation performing of the computing machinery. In further detail, the given characteristic Tis described below and in the works $[2,85,97,114]$.

## 数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation

Optimization of computations lies in the optimization of one of the introduced characteristics (in a general way on $I, X, Y$ ) in accordance with certain constraints on others $[118,111]$.

The first primary problem is to minimize time $T(I, X, Y)$ while upholding the real (Re) constraints on $M, E$, and $f e f$ :
$$\begin{gathered} T(I, X, Y)=\min {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, E(I, X, Y) \leq E_{\mathrm{Re}}, f e f(I, X, Y) \leq f e f_{\mathrm{Re}} . \end{gathered}$$

The second primal problem is to minimize the global error $E(I, X, Y)$ with constraints on $M, T$, and $f e f$.
$$\begin{gathered} E(I, X, Y)=\min {l, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re},} \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad f e f(I, X, Y) \leq f e f_{\mathrm{Re}} \end{gathered}$$
The third primal problem is to maximize a coefficient of the technical and economic effectiveness $f e f(I, X, Y)$ :
$$\begin{gathered} f e f(I, X, Y)=\max {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad E(I, X, Y) \leq E_{\mathrm{Re}} \end{gathered}$$
We make some remarks on the mentioned statements of the primal problems. Let the computing machinery $C(Y)$ be fixed. Then $T, M, E$, and fef depend only on $I, X$. It is convenient to consider $I$ to be a random value and consider the probabilistic characteristics of the values $T, M, E$, and $f e f$, which will also be the characteristics of CA $A$ and will depend only on $X$. We designate each of the characteristics $T, M, E$, and fef through $H(I, X)$ and designate the frequency distribution accordingly to $H, I$ through $P(H), P(I)$. The essential characteristics of $\mathrm{CA} A(X)$ is the mathematical expectation $M_{H}(X)$ and the dispersion $D_{H}(X)$ :
\begin{aligned} &M_{H}(X)=\int_{G} H P(I) d I=\int_{-\infty}^{\infty} H P(H) d H \ &D_{H}(X)=\int_{G}\left(H-M_{H}\right)^{2} P(I) d I=\int_{-\infty}^{\infty}\left(H-M_{H}\right)^{2} P(H) d H \end{aligned}
where $G$ is a domain of possible values of $I$.

## 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

• 给出了计算理论中的一定支持数据，用于确定近似求解问题的准确性、所用算法的质量，以及按一定标准对各种算法进行比较研究。
• 确定了 CA 的主要特征（准确性、执行时间、计算机的必要内存），并提供了它们的估计方法。
• 在顺序算法中有最优（不准确和高速）、渐近最优和最优的定义。
• 在没有错误设置先验信息的情况下，考虑识别和澄清有关问题的先验信息的问题，以提高获得的问题近似解的质量。

## 数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations

CA的主要特点让问题磷(一世)由算法解决一种(X)在计算机器上C(是)， 在哪里一世,X,是是有限数量的参数（向量），它们本质上是磷,一种,C分别依赖于。向量的组成部分一世可能包括有关解决问题的先验质量的数据，例如，约束给定函数的导数阶的绝对值的常数、有关输入值准确性的数据等。 算法的多次迭代，逼近的程度，网格的步长等，可以是向量X组件。向量是可以包含计算机内存的单元位数，其RAM的总容量，计算机的运行时间，计算机的数量

## 数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation

F和F(一世,X,是)=最大限度一世,X,是 米(一世,X,是)≤米R和,吨(一世,X,是)≤吨R和,和(一世,X,是)≤和R和

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|机器学习代写machine learning代考|Mathematical Models of Learning

statistics-lab™ 为您的留学生涯保驾护航 在代写机器学习machine learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写机器学习machine learning代写方面经验极为丰富，各种代写机器学习machine learning相关的作业也就用不着说。

• 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

## 统计代写|机器学习代写machine learning代考|Mathematical Models of Learning

This chapter introduces different mathematical models of learning. A mathematical model of learning has the advantage that it provides bounds on the generalization ability of a learning algorithm. It also indicates which quantities are responsible for generalization. As such, the theory motivates new learning algorithms. After a short introduction into the classical parametric statistics approach to learning, the chapter introduces the PAC and VC models. These models directly study the convergence of expected risks rather than taking a detour over the convergence of the underlying probability measure. The fundamental quantity in this framework is the growth function which can be upper bounded by a one integer summary called the VC dimension. With classical structural risk minimization, where the VC dimension must be known before the training data arrives, we obtain $a$-priori bounds, that is, bounds whose values are the same for a fixed training error.

In order to explain the generalization behavior of algorithms minimizing a regularized risk we will introduce the luckiness framework. This framework is based on the assumption that the growth function will be estimated on the basis of a sample. Thus, it provides a-posteriori bounds; bounds which can only be evaluated after the training data has been seen. Finally, the chapter presents a PAC analysis for real-valued functions. Here, we take advantage of the fact that, in the case of linear classifiers, the classification is carried out by thresholding a realvalued function. The real-valued output, also referred to as the margin, allows us to define a scale sensitive version of the VC dimension which leads to tighter bounds on the expected risk. An appealing feature of the margin bound is that we can obtain nontrivial bounds even if the number of training samples is significantly less than the number of dimensions of feature space. Using a technique, which is known as the robustness trick, it will be demonstrated that the margin bound is also applicable if one allows for training error via a quadratic penalization of the diagonal of the Gram matrix.

## 统计代写|机器学习代写machine learning代考|Generative vs. Discriminative Models

In Chapter 2 it was shown that a learning problem is given by a training sample $z=(\boldsymbol{x}, \boldsymbol{y})=\left(\left(x_{1}, y_{1}\right), \ldots,\left(x_{m}, y_{m}\right)\right) \in(\mathcal{X} \times \mathcal{Y})^{m}=\mathcal{Z}^{m}$, drawn iid according to some (unknown) probability measure $\mathbf{P}{\mathrm{Z}}=\mathbf{P}{\mathrm{XY}}$, and a loss $l: \mathcal{Y} \times \mathcal{Y} \rightarrow \mathbb{R}$, which defines how costly the prediction $h(x)$ is if the true output is $y$. Then, the goal is to find a deterministic function $h \in \mathcal{Y}^{\mathcal{X}}$ which expresses the dependency implicitly expressed by $\mathbf{P}{\mathbf{Z}}$ with minimal expected loss (risk) $R[h]=\mathbf{E}{X Y}[l(h(\mathrm{X}), \mathrm{Y})]$ while only using the given training sample $z$. We have already seen in the first part of this book that there exist two different algorithmical approaches to tackling this problem. We shall now try to study the two approaches more generally to see in what respect they are similar and in which aspects they differ.

1. In the generative (or parametric) statistics approach we restrict ourselves to a parameterized space $\mathcal{P}$ of measures for the space $\mathcal{Z}$, i.e., we model the data generation process. Hence, our model is given by ${ }^{1} \mathcal{P}=\left{\mathbf{P}{\mathrm{Z} \mid \mathbf{Q}=\theta} \mid \theta \in \mathcal{Q}\right}$, where $\theta$ should be understood as the parametric description of the measure $\mathbf{P}{\mathrm{Z} \mid \mathbf{Q}=\theta}$. With a fixed loss $l$ each measure $\mathbf{P}{\mathbf{Z} \mid \mathbf{Q}=\theta}$ implicitly defines a decision function $h{\theta}$,
$$h_{\theta}(x)=\underset{y \in \mathcal{Y}}{\operatorname{argmin}} \mathbf{E}{Y \mid X=x, \mathbf{Q}=\theta}[l(y, Y)] \text {. }$$ In order to see that this function has minimal expected risk we note that $$R{\theta}[h] \stackrel{\text { def }}{=} \mathbf{E}{\mathbf{X Y |} \mathbf{Q}=\theta}[l(h(\mathrm{X}), \mathrm{Y})]=\mathbf{E}{\mathbf{X} \mid \mathbf{Q}=\theta}\left[\mathbf{E}{Y \mid X=x, \mathbf{Q}=\theta}[l(h(x), \mathrm{Y})]\right] \text {, }$$ where $h{\theta}$ minimizes the expression in the innermost brackets. For the case of zeroone loss $l_{0-1}(h(x), y)=\mathbf{I}{h(x) \neq y}$ also defined in equation $(2.10)$, the function $h{\theta}$ reduces to
$$h_{\theta}(x)=\underset{y \in \mathcal{Y}}{\operatorname{argmin}}\left(1-\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, \mathbf{Q}=\theta}(y)\right)=\underset{y \in \mathcal{Y}}{\operatorname{argmax}} \mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, \mathbf{Q}=\theta}(y),$$
which is known as the Bayes optimal decision based on $\mathbf{P}_{\mathrm{Z} \mid \mathbf{Q}=\theta}$.
2. In the discriminative, or machine learning, approach we restrict ourselves to a parameterized space $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$ of deterministic mappings $h$ from $\mathcal{X}$ to $\mathcal{Y}$. As a consequence, the model is given by $\mathcal{H}=\left{h_{\mathrm{w}}: \mathcal{X} \rightarrow \mathcal{Y} \mid \mathbf{w} \in \mathcal{W}\right}$, where $\mathbf{w}$ is the parameterization of single hypotheses $h_{\mathrm{w}}$. Note that this can also be interpreted as

a model of the conditional distribution of classes $y \in \mathcal{Y}$ given objects $x \in \mathcal{X}$ by assuming that $\mathbf{P}{Y \mid X=x, \mathcal{H}=h}=\mathbf{I}{y=h(x)}$. Viewed this way, the model $\mathcal{H}$ is a subset of the more general model $\mathcal{P}$ used in classical statistics.

The term generative refers to the fact that the model $\mathcal{P}$ contains different descriptions of the generation of the training sample $z$ (in terms of a probability measure). Similarly, the term discriminative refers to the fact that the model $\mathcal{H}$ consists of different descriptions of the discrimination of the sample $z$. We already know that a machine learning method selects one hypothesis $\mathcal{A}(z) \in \mathcal{H}$ given a training sample $z \in \mathcal{Z}^{m}$. The corresponding selection mechanism of a probability measure $\mathbf{P}_{\mathbf{z} \mid \mathbf{Q}=\theta}$ given the training sample $z$ is called an estimator.

## 统计代写|机器学习代写machine learning代考|Classical PAC and VC Analysis

In the following three subsections we will only be concerned with the zero-one loss $l_{0-1}$ given by equation $(2.10)$. It should be noted that the results we will obtain can readily be generalized to loss function taking only a finite number values; the generalization to the case of real-valued loss functions conceptually similar but will not be discussed in this book (see Section $4.5$ for further references).

The general idea is to bound the probability of “bad training samples”, i.e., training samples $z \in \mathcal{Z}^{m}$ for which there exists a hypothesis $h \in \mathcal{H}$ where the deviation between the empirical risk $R_{\text {emp }}[h, z]$ and the expected risk $R[h]$ is larger than some prespecified $\varepsilon \in[0,1]$. Setting the probability of this to $\delta$ and solving for $\varepsilon$ gives the required generalization error bound. If we are only given a finite number $|\mathcal{H}|$ of hypotheses $h$ then such a bound is very easily obtained by a combination of Hoeffding’s inequality and the union bound.

Theorem 4.6 (VC bound for finite hypothesis spaces) Suppose we are given a hypothesis space $\mathcal{H}$ having a finite number of hypotheses, i.e., $|\mathcal{H}|<\infty$. Then, for any measure $\mathbf{P}{\mathrm{Z}}$, for all $\delta \in(0,1]$ and all training sample sizes $m \in \mathbb{N}$, with probability at least $1-\delta$ over the random draw of the training sample $z \in \mathcal{Z}^{m}$ we have $$\mathbf{P}{Z^{w}}\left(\exists h \in \mathcal{H}:\left|R[h]-R_{\operatorname{emp}}[h, \mathbf{Z}]\right|>\varepsilon\right)<2 \cdot|\mathcal{H}| \cdot \exp \left(-2 m \varepsilon^{2}\right)$$ Proof Let $\mathcal{H}=\left\{h_{1}, \ldots, h_{|\mathcal{H}|}\right\}$. By an application of the union bound given in Theorem A.107 we know that $\mathbf{P}_{Z^{m}}\left(\exists h \in \mathcal{H}:\left|R[h]-R_{\text {emp }}[h, \mathbf{Z}]\right|>\varepsilon\right)$ is given by
$$\mathbf{P}{Z^{m}}\left(\bigvee{i=1}^{|\mathcal{H}|}\left(\left|R\left[h_{i}\right]-R_{\mathrm{emp}}\left[h_{i}, \mathbf{Z}\right]\right|>\varepsilon\right)\right) \leq \sum_{i=1}^{|\mathcal{H}|} \mathbf{P}{Z^{m}}\left(\left|R\left[h{i}\right]-R_{\mathrm{emp}}\left[h_{i}, \mathbf{Z}\right]\right|>\varepsilon\right) .$$

## 统计代写|机器学习代写machine learning代考|Generative vs. Discriminative Models

1. 在生成（或参数）统计方法中，我们将自己限制在参数化空间中磷空间措施从，即我们对数据生成过程进行建模。因此，我们的模型由下式给出{ }^{1} \mathcal{P}=\left{\mathbf{P}{\mathrm{Z}\mid\mathbf{Q}=\theta}\mid\theta\in\mathcal{Q}\right } }{ }^{1} \mathcal{P}=\left{\mathbf{P}{\mathrm{Z}\mid\mathbf{Q}=\theta}\mid\theta\in\mathcal{Q}\right } }， 在哪里θ应该理解为度量的参数化描述磷从∣问=θ. 有固定损失l每个措施磷从∣问=θ隐式定义决策函数Hθ,
Hθ(X)=精氨酸是∈是和是∣X=X,问=θ[l(是,是)]. 为了看到这个函数具有最小的预期风险，我们注意到Rθ[H]= 定义 和X是|问=θ[l(H(X),是)]=和X∣问=θ[和是∣X=X,问=θ[l(H(X),是)]], 在哪里Hθ最小化最里面的括号中的表达式。对于 zeroone loss 的情况l0−1(H(X),是)=一世H(X)≠是也在等式中定义(2.10)， 功能Hθ减少到
Hθ(X)=精氨酸是∈是(1−磷是∣X=X,问=θ(是))=最大参数是∈是磷是∣X=X,问=θ(是),
这被称为基于贝叶斯最优决策磷从∣问=θ.
2. 在判别式或机器学习方法中，我们将自己限制在参数化空间中H⊆是X确定性映射H从X到是. 因此，模型由下式给出\mathcal{H}=\left{h_{\mathrm{w}}: \mathcal{X} \rightarrow \mathcal{Y} \mid \mathbf{w} \in \mathcal{W}\right}\mathcal{H}=\left{h_{\mathrm{w}}: \mathcal{X} \rightarrow \mathcal{Y} \mid \mathbf{w} \in \mathcal{W}\right}， 在哪里在是单个假设的参数化H在. 请注意，这也可以解释为

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|机器学习代写machine learning代考|The Relevance Vector Machine

statistics-lab™ 为您的留学生涯保驾护航 在代写机器学习machine learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写机器学习machine learning代写方面经验极为丰富，各种代写机器学习machine learning相关的作业也就用不着说。

• 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

## 统计代写|机器学习代写machine learning代考|The Relevance Vector Machine

In the last section we saw that a direct application of Bayesian ideas to the problem of regression estimation yields efficient algorithms known as Gaussian processes. In this section we will carry out the same analysis with a slightly refined prior $\mathbf{P}{\mathrm{w}}$ on linear functions $f{\mathrm{w}}$ in terms of their weight vectors $\mathbf{w} \in \mathcal{K} \subseteq \ell_{2}^{n}$. As we will

see in Section $5.2$ an important quantity in the study of the generalization error is the sparsity $|\mathbf{w}|_{0}=\sum_{i=1}^{n} \mathbf{I}{w{i} \neq 0}$ or $|\boldsymbol{\alpha}|_{0}$ of the weight vector or the vector of expansion coefficients, respectively. In particular, it is shown that the expected risk of the classifier $f_{\mathrm{w}}$ learned from a training sample $z \in \mathcal{Z}^{m}$ is, with high probability over the random draw of $z$, as small as $\approx \frac{\boldsymbol{w}{0}}{n}$ or $\frac{|\alpha|{0}}{m}$, where $n$ is the dimensionality of the feature space $\mathcal{K}$ and $\mathbf{w}=\sum_{i=1}^{m} \alpha_{i} \mathbf{x}{i}=\mathbf{X}^{\prime} \alpha$. These results suggest favoring weight vectors with a small number of non-zero coefficients. One way to achieve this is to modify the prior in equation (3.8), giving $\mathbf{P}{\mathbf{W}}=\operatorname{Normal}(\mathbf{0}, \boldsymbol{\Theta})$,
where $\boldsymbol{\Theta}=\operatorname{diag}(\theta)$ and $\theta=\left(\theta_{1}, \ldots, \theta_{n}\right)^{\prime} \in\left(\mathbb{R}^{+}\right)^{n}$ is assumed known. The idea behind this prior is similar to the idea of automatic relevance determination given in Example 3.12. By considering $\theta_{i} \rightarrow 0$ we see that the only possible value for the $i$ th component of the weight vector $w$ is 0 and, therefore, even when considering the Bayesian prediction $B a y e s_{z}$ the $i$ th component is set to zero. In order to make inference we consider the likelihood model given in equation (3.9), that is, we assume that the target values $t=\left(t_{1}, \ldots, t_{m}\right) \in \mathbb{R}^{m}$ are normally distributed with mean $\left\langle\mathbf{x}{i}, \mathbf{w}\right\rangle$ and variance $\sigma{t}^{2}$. Using Theorem A.28 it follows that the posterior measure over weight vectors $\mathbf{w}$ is again Gaussian, i.e.,
$\mathbf{P}{W \mid} \mathrm{X}^{m}=x, \mathrm{~T}^{\mathrm{m}}=t=\operatorname{Normal}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ where the posterior covariance $\mathbf{\Sigma} \in \mathbb{R}^{n \times n}$ and mean $\mu \in \mathbb{R}^{n}$ are given by $$\boldsymbol{\Sigma}=\left(\sigma{t}^{-2} \mathbf{X}^{\prime} \mathbf{X}+\boldsymbol{\Theta}^{-1}\right)^{-1}, \quad \boldsymbol{\mu}=\sigma_{t}^{-2} \boldsymbol{\Sigma} \mathbf{X}^{\prime} \boldsymbol{t}=\left(\mathbf{X}^{\prime} \mathbf{X}+\sigma_{t}^{2} \mathbf{\Theta}^{-1}\right)^{-1} \mathbf{X}^{\prime} \boldsymbol{t}$$
As described in the last section, the Bayesian prediction at a new test object $x \in \mathcal{X}$ is given by $B a$ ayes $_{z}(x)=\langle\mathbf{x}, \boldsymbol{\mu}\rangle$. Since we assumed that many of the $\theta_{i}$ are zero, i.e., the effective number $n_{\text {eff }}=|\theta|_{0}$ of features $\phi_{i}: \mathcal{X} \rightarrow \mathbb{R}$ is small, it follows that $\boldsymbol{\Sigma}$ and $\boldsymbol{\mu}$ are easy to calculate ${ }^{7}$. The interesting question is: Given a training sample $z=(\boldsymbol{x}, \boldsymbol{t}) \in(\mathcal{X} \times \mathbb{R})^{m}$, how can we “learn” the sparse vector $\boldsymbol{\theta}=\left(\theta_{1}, \ldots, \theta_{n}\right)^{\prime} ?$

## 统计代写|机器学习代写machine learning代考|Bayes Point Machines

The algorithms introduced in the last two sections solve the classification learning problem by taking a “detour” via the regression estimation problem. For each training object it is assumed that we have prior knowledge $\mathbf{P}{\mathbf{w}}$ about the latent variables $\mathrm{T}{i}$ corresponding to the logit transformation of the probability of $x_{i}$ being from the observed class $y_{i}$. This is a quite cumbersome assumption as we are unable to directly express prior knowledge on observed quantities such as the classes $\boldsymbol{y} \in \mathcal{Y}^{m}={-1,+1}^{m}$. In this section we are going to consider an algorithm which results from a direct modeling of the classes.

Let us start by defining the prior $\mathbf{P}{\mathbf{W}}$. In the classification case we note that, for any $\lambda>0$, the weight vectors $w$ and $\lambda w$ perform the same classification because $\operatorname{sign}(\langle\mathbf{x}, \mathbf{w}\rangle)=\operatorname{sign}(\langle\mathbf{x}, \lambda \mathbf{w}\rangle)$. As a consequence we consider only weight vectors of unit length, i.e., w $\in \mathcal{W}, \mathcal{W}={\mathbf{w} \in \mathcal{K} \mid|\mathbf{w}|=1}$ (see also Section 2.1). In the absence of any prior knowledge we assume a uniform prior measure $\mathbf{P}{\mathbf{W}}$ over the unit hypersphere $\mathcal{W}$. An argument in favor of the uniform prior is that the belief in the weight vector $w$ should be equal to the belief in the weight vector $-\mathbf{w}$

under the assumption of equal class probabilities $\mathbf{P}{\curlyvee}(-1)$ and $\mathbf{P}{\curlyvee}(+1)$. Since the classification $\mathbf{y}{-\mathbf{w}}=\left(\operatorname{sign}\left(\left\langle\mathbf{x}{1},-\mathbf{w}\right\rangle\right), \ldots, \operatorname{sign}\left(\left\langle\mathbf{x}{m},-\mathbf{w}\right\rangle\right)\right)$ of the weight vector $-\mathbf{w}$ at the training sample $z \in \mathcal{Z}^{m}$ equals the negated classification $-\mathbf{y}{\mathbf{w}}=$ $-\left(\operatorname{sign}\left(\left\langle\mathbf{x}{1}, \mathbf{w}\right\rangle\right), \ldots, \operatorname{sign}\left(\left\langle\mathbf{x}{m}, \mathbf{w}\right\rangle\right)\right)$ of $\mathbf{w}$ it follows that the assumption of equal belief in $\mathbf{w}$ and $-\mathbf{w}$ corresponds to assuming that $\mathbf{P}{\mathbf{Y}}(-1)=\mathbf{P}{\mathbf{Y}}(+1)=\frac{1}{2}$.

In order to derive an appropriate likelihood model, let us assume that there is no noise on the classifications, that is, we shall use the PAC-likelihood $l_{\mathrm{PAC}}$ as given in Definition 3.3. Note that such a likelihood model corresponds to using the zeroone loss $I_{0-1}$ in the machine learning scenario (see equations $(2.10)$ and $(3.2)$ ). According to Bayes’ theorem it follows that the posterior belief in weight vectors (and therefore in classifiers) is given by
\begin{aligned} f_{W \mid Z^{m}=z}(w) &=\frac{P_{Y^{m} \mid X^{m}=x, W=w}(y) f_{W}(w)}{P_{Y^{m} \mid X^{m}=x}(y)} \ &= \begin{cases}\frac{1}{P_{W}(V(z))} & \text { if } w \in V(z) \ 0 & \text { otherwise }\end{cases} \end{aligned}
The set $V(z) \subseteq \mathcal{W}$ is called version space and is the set of all weight vectors that parameterize classifiers which classify all the training objects correctly (see also Definition 2.12). Due to the PAC-likelihood, any weight vector which does not have this property is “cut-off” resulting in a uniform posterior measure $\mathbf{P}{\mathbf{W} \mid \mathbf{Z}^{m}=z}$ over version space. Given a new test object $x \in \mathcal{X}$ we can compute the predictive distribution $\mathbf{P}{Y \mid X=x, Z^{m}=z}$ of the class $y$ at $x \in \mathcal{X}$ by
$$\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, Z^{w}=z}(y)=\mathbf{P}{\mathrm{W} \mid \mathbf{Z}^{m}=z}(\operatorname{sign}(\langle\mathbf{x}, \mathbf{W}\rangle)=y) .$$
The Bayes classification strategy based on $\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, \mathrm{Z}^{\mathrm{m}}=z}$ decides on the class with the larger probability. An appealing feature of the two class case $\mathcal{Y}={-1,+1}$ is that this decision can also be written as $\operatorname{Bayes}{z}(x)=\operatorname{sign}\left(\mathbf{E}_{\mathbf{W} \mid \mathbf{Z}^{m}=z}[\operatorname{sign}(\langle\mathbf{x}, \mathbf{W}\rangle)]\right)$,
that is, the Bayes classification strategy effectively performs majority voting involving all version space classifiers. The difficulty with the latter expression is that we cannot analytically compute the expectation as this requires efficient integration of a convex body on a hypersphere (see also Figure $2.1$ and $2.8$ ). Hence, we approximate the Bayes classification strategy by a single classifier.

## 统计代写|机器学习代写machine learning代考|Estimating the Bayes Point

The main idea in computing the center of mass of version space is to replace the analytical integral by a sum over randomly drawn classifiers, i.e.,
$$\mathbf{w}{\mathrm{cm}}=\mathbf{E}{W \mid \mathbf{Z}^{\mathrm{w}}=z}[\mathbf{W}] \approx \frac{1}{K} \sum_{i=1}^{K} \mathbf{w}{i} \quad \mathbf{w}{i} \sim \mathbf{P}{\mathbf{W} \mid \mathbf{Z}^{m}=z}$$ Such methods are known as Monte-Carlo methods and have proven to be successful in practice. A difficulty we encounter with this approach is in obtaining samples $\mathbf{w}{i}$ drawn according to the distribution $\mathbf{P}{\mathbf{W} \mid \mathrm{Z}^{\mathrm{m}}=z \text {. Recalling }}$ that $\mathbf{P}{\mathrm{W} / \mathrm{Z}^{\mathrm{m}}=z}$ is uniform in a convex polyhedra on the surface of hypersphere in feature space we see that it is quite difficult to directly sample from it. A commonly used approach to this problem is to approximate the sampling distribution $\mathbf{P}{\mathrm{W} \mid \mathrm{Z}^{\mathrm{m}}=z}$ by a Markov chain. A Markov chain is fully specified by a probability distribution $\mathbf{P}{\mathrm{W}{1} \mathbf{W}{2}}$ where $f_{W_{1}} w_{2}\left(\left(w_{1}, w_{2}\right)\right)$ is the “transition” probability for progressing from a randomly drawn weight vector $\mathbf{w}{1}$ to another weight vector $\mathbf{w}{2}$. Sampling from the Markov chain involves iteratively drawing a new weight vector $w_{i+1}$ by sampling from $\mathbf{P}{\mathrm{W}{2} \mid \mathrm{W}{1}=\mathrm{w}{i} .}$ The Markov chain is called ergodic w.r.t. $\mathbf{P}{\mathbf{W} \mid \mathrm{Z}^{\mathrm{w}}=z}$ if the limiting distribution of this sampling process is $\mathbf{P}{\mathbf{w} \mid \mathbf{Z}^{m}=z}$ regardless of our choice of $\mathbf{w}{0}$. Then, it suffices to start with a random weight vector $w{0} \in \mathcal{W}$ and at each step, to obtain a new sample $\mathbf{w}{i} \in \mathcal{W}$ drawn according to $\mathbf{P}{\mathbf{w}{2} \mid \mathbf{w}{1}=\mathbf{w}{i-1} .}$. The combination of these two techniques has become known as the Markov-Chain-Monte-Carlo (MCMC) method for estimating the expectation $\mathbf{E}{\mathbf{W} \mid \mathbf{Z}^{w}=z}[\mathbf{W}]$.

We now outline an MCMC algorithm for approximating the Bayes point by the center of mass of version space $V(z)$ (the whole pseudo code is given on page 330). Since it is difficult to generate weight vectors that parameterize classifiers consistent with the whole training sample $z \in \mathcal{Z}^{m}$ we average over the trajectory of a ball which is placed inside version space and bounced like a billiard ball. As a consequence we call this MCMC method the kernel billiard. We express each position $\mathbf{b} \in \mathcal{W}$ of the ball and each estimate $\mathbf{w}{i} \in \mathcal{W}$ of the center of mass of $V(z)$ as a linear combination of the mapped training objects, i.e., $$\mathbf{w}=\sum{i=1}^{m} \alpha_{i} \mathbf{x}{i}, \quad \mathbf{b}=\sum{i=1}^{m} \gamma_{i} \mathbf{x}_{i}, \quad \alpha \in \mathbb{R}^{m}, \quad \boldsymbol{\gamma} \in \mathbb{R}^{m}$$

## 统计代写|机器学习代写machine learning代考|Bayes Point Machines

F在∣从米=和(在)=磷是米∣X米=X,在=在(是)F在(在)磷是米∣X米=X(是) ={1磷在(在(和)) 如果 在∈在(和) 0 除此以外

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|机器学习代写machine learning代考|Kernel Classifiers from a Bayesian Perspective

statistics-lab™ 为您的留学生涯保驾护航 在代写机器学习machine learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写机器学习machine learning代写方面经验极为丰富，各种代写机器学习machine learning相关的作业也就用不着说。

• 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

## 统计代写|机器学习代写machine learning代考|The Bayesian Framework

This chapter presents the probabilistic, or Bayesian approach to learning kernel classifiers. It starts by introducing the main principles underlying Bayesian inference both for the problem of learning within a fixed model and across models. The first two sections present two learning algorithms, Gaussian processes and relevance vector machines, which were originally developed for the problem of regression estimation. In regression estimation, one is given a sample of real-valued outputs rather than classes. In order to adapt these methods to the problem of classification we introduce the concept of latent variables which, in the current context, are used to model the probability of the classes. The chapter shows that the principle underlying relevance vector machines is an application of Bayesian model selection to classical Bayesian linear regression. In the third section we present a method which directly models the observed classes by imposing prior knowledge only on weight vectors of unit length. In general, it is impossible to analytically compute the solution to this algorithm. The section presents a Markov chain Monte Carlo algorithm to approximately solve this problem, which is also known as Bayes point learning. Finally, we discuss one of the earliest approaches to the problem the kernel trick to all these algorithms thus rendering them powerful tools in the application of kernel methods to the problem of classification learning.

In the last chapter we saw that a learning problem is given by the identification of an unknown relationship $h \in \mathcal{Y}^{\mathcal{X}}$ between objects $x \in \mathcal{X}$ and classes $y \in \mathcal{Y}$ solely on the basis of a given iid sample $z=(\boldsymbol{x}, \boldsymbol{y})=\left(\left(x_{1}, y_{1}\right), \ldots,\left(x_{m}, y_{m}\right)\right) \in$ $(\mathcal{X} \times \mathcal{Y})^{m}=\mathcal{Z}^{m}$ (see Definition 2.1). Any approach that deals with this problem starts by choosing a hypothesis space ${ }^{1} \mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$ and a loss function $l: \mathcal{Y} \times \mathcal{Y} \rightarrow \mathbb{R}$ appropriate for the task at hand. Then a learning algorithm $\mathcal{A}: \cup_{m=1}^{\infty} \mathcal{Z}^{m} \rightarrow \mathcal{H}$ aims to find the one particular hypothesis $h^{*} \in \mathcal{H}$ which minimizes a pre-defined risk determined on the basis of the loss function only, e.g., the expected risk $R[h]$ of the hypothesis $h$ or the empirical risk $R_{\text {emp }}[h, z]$ of $h \in \mathcal{H}$ on the given training sample $z \in \mathcal{Z}^{m}$ (see Definition $2.5$ and 2.11). Once we have learned a classifier $\mathcal{A}(z) \in \mathcal{H}$ it is used for further classification on new test objects. Thus, all the information contained in the given training sample is summarized in the single hypothesis learned.

## 统计代写|机器学习代写machine learning代考|The Power of Conditioning on Data

From a purely Bayesian point of view, for the task of learning we are finished as soon as we have updated our prior belief $\mathbf{P}{\mathrm{H}}$ into the posterior belief $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}$ using equation (3.1). Nonetheless, our ultimate goal is to find one (deterministic) function $h \in \mathcal{Y} \mathcal{X}^{\mathcal{X}}$ that best describes the relationship objects and classes, which is implicitly

expressed by the unknown measure $\mathbf{P}{Z}=\mathbf{P}{Y \mid X} \mathbf{P}{X}$. In order to achieve this goal, Bayesian analysis suggests strategies based on the posterior belief $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{m}}=z^{*}$ :

• If we are restricted to returning a function $h \in \mathcal{H}$ from a pre-specified hypothesis space $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$ and assume that $\mathbf{P}_{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}$ is highly peaked around one particular function then we determine the classifier with the maximum posterior belief.

Definition 3.6 (Maximum-a-posteriori estimator) For a given posterior belief $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}$ over a hypothesis space $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$, the maximum-a-posteriori estimator is defined by ${ }^{5}$ $\mathcal{A}{\mathrm{MAP}}(z) \stackrel{\text { def }}{=} \underset{h \in \mathcal{H}}{\operatorname{argmax}} \mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}(h)$ If we use the inverse loss likelihood and note that the posterior $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{m}=z}$ is given by the product of the likelihood and the prior we see that this scheme returns minimizer of the training error and our prior belief, which can be thought of as a regularizer (see also Subsection 2.2.2). The drawback of the MAP estimator is that it is very sensitive to the training sample if the posterior measure is multi modal. Even worse, the classifier $\mathcal{A}_{\text {MAP }}(z) \in \mathcal{H}$ is, in general, not unique, for example if the posterior measure is uniform.

• If we are not confined to returning a function from the original hypothesis space $\mathcal{H}$ then we can use the posterior measure $\mathbf{P}{\mathrm{H}{\mid \mathrm{Z}^{m}}=z}$ to induce a measure $\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, \mathrm{Z}^{m}=z}$ over classes $y \in \mathcal{Y}$ at a novel object $x \in \mathcal{X}$ by $$\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, \mathbf{Z}^{m}=z}(y)=\mathbf{P}_{\mathrm{H} \mid \mathrm{Z}^{m}=z}({h \in \mathcal{H} \mid h(x)=y})$$
This measure can then be used to determine the class $y$ which incurs the smallest loss at a given object $x$.

## 统计代写|机器学习代写machine learning代考|Bayesian Linear Regression

In the regression estimation problem we are given a sequence $\boldsymbol{x}=\left(x_{1}, \ldots, x_{m}\right) \in$ $\mathcal{X}^{m}$ of $m$ objects together with a sequence $t=\left(t_{1}, \ldots, t_{m}\right) \in \mathbb{R}^{m}$ of $m$ real-valued outcomes forming the training sample $z=(x, t)$. Our aim is to find a functional relationship $f \in \mathbb{R}^{\mathcal{X}}$ between objects $x$ and target values $t$. In accordance with Chapter 2 we will again consider a linear model $\mathcal{F}$
$\mathcal{F}={x \mapsto\langle\mathbf{x}, \mathbf{w}\rangle \mid \mathbf{w} \in \mathcal{K}}$,
where we assume that $\mathbf{x} \stackrel{\text { def }}{=} \phi(x)$ and $\phi: \mathcal{X} \rightarrow \mathcal{K} \subseteq \ell_{2}^{n}$ is a given feature mapping (see also Definition 2.2). Note that $\mathbf{x} \in \mathcal{K}$ should not be confused with the training sequence $\boldsymbol{x} \in \mathcal{X}^{m}$ which results in an $m \times n$ matrix $\mathbf{X}=\left(\mathbf{x}{1}^{\prime} ; \ldots ; \mathbf{x}{m}^{\prime}\right)$ when $\boldsymbol{\phi}$ is applied to it.

First, we need to specify a prior over the function space $\mathcal{F}$. Since each function $f_{\mathrm{w}}$ is uniquely parameterized by its weight vector $\mathbf{w} \in \mathcal{K}$ it suffices to consider a prior distribution on weight vectors. For algorithmic convenience let the prior distribution over weights be a Gaussian measure with mean $\mathbf{0}$ and covariance $\mathbf{I}{n}$, i.e., $\mathbf{P}{\mathrm{W}}=\operatorname{Normal}\left(\mathbf{0}, \mathbf{I}{n}\right)$. Apart from algorithmical reasons such a prior favors weight vectors $\mathbf{w} \in \mathcal{K}$ with small coefficients $w{i}$ because the log-density is proportional to $-|\mathbf{w}|^{2}=$ $-\sum_{i=1}^{n} w_{i}^{2}$ (see Definition A.26). In fact, the weight vector with the highest apriori density is $\mathbf{w}=\mathbf{0}$.

Second, we must specify the likelihood model $\mathbf{P}{T^{m} \mid X^{m}=x, W=w}$. Let us assume that, for a given function $f{\mathrm{w}}$ and a given training object $x \in \mathcal{X}$, the real-valued output $\mathrm{T}$ is normally distributed with mean $f_{\mathrm{w}}(x)$ and variance $\sigma_{t}^{2}$. Using the notion of an inverse loss likelihood such an assumption corresponds to using the squared loss, i.e., $l_{2}(f(x), t)=(f(x)-t)^{2}$ when considering the prediction task under a machine learning perspective. Further, it shall be assumed that the real-valued outputs $\mathrm{T}{1}$ and $\mathrm{T}{2}$ at $x_{1}$ and $x_{2} \neq x_{1}$ are independent. Combining these two requirements results in the following likelihood model:
$$\mathbf{P}{\mathrm{T}^{m} \mid \mathbf{X}^{\mathrm{m}}=x, \mathbf{W}=\mathbf{w}}(t)=\operatorname{Normal}\left(\mathbf{X w}, \sigma{t}^{2} \mathbf{I}_{m}\right) \text {. }$$

## 统计代写|机器学习代写machine learning代考|The Power of Conditioning on Data

• 如果我们仅限于返回一个函数H∈H从预先指定的假设空间H⊆是X并假设磷H∣从米=和在一个特定函数周围高度达到峰值，然后我们确定具有最大后验置信度的分类器。

• 如果我们不局限于从原始假设空间返回一个函数H那么我们可以使用后验测度磷H∣从米=和引发措施磷是∣X=X,从米=和过课是∈是在一个新奇的物体上X∈X经过磷是∣X=X,从米=和(是)=磷H∣从米=和(H∈H∣H(X)=是)
然后可以使用此度量来确定类别是在给定对象上产生最小的损失X.

F=X↦⟨X,在⟩∣在∈ķ,

## 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。