数学代写|matlab代写|ON ESTIMATION METHODS

matlab是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|matlab代写|Beginnings of Estimation Theory

The first method for forming an optimal estimate from noisy data is the method of least squares. Its discovery is generally attributed to Carl Friedrich Gauss (1777-1855) in 1795. The inevitability of measurement errors had been recognized since the time of Galileo Galilei (1564-1642), but this was the first formal method for dealing with them. Although it is more commonly used for linear estimation problems, Gauss first used it for a nonlinear estimation problem in mathematical astronomy, which was part of a dramatic moment in the history of astronomy. The following narrative was gleaned from many sources, with the majority of the material from the account by Baker and Makemson [97]:
On January 1, 1801, the first day of the nineteenth century, the Italian astronomer Giuseppe Piazzi was checking an entry in a star catalog. Unbeknown to Piazzi, the entry had been added erroneously by the printer. While searching for the “missing” star, Piazzi discovered, instead, a new planet. It was Ceres – the largest of the minor planets and the first to be discovered-but Piazzi did not know that yet. He was able to track and measure its apparent motion against the “fixed” star background during 41 nights of viewing from Palermo before his work was interrupted. When he returned to his work, however, he was unable to find Ceres again.

Actually, it was not Bode, but Johann Tietz who first proposed this formula, in 1772 . At that time there were only six known planets. In 1781, Friedrich Herschel discovered Uranus, which fit nicely into this formula for $n=6$. No planet had been discovered for $n=3$. Spurred on by Bode, an association of European astronomers had been searching for the “missing” eighth planet for nearly 30 years. Piazzi was not part of this association, but he did inform Bode of his unintended discovery.

Piazzi’s letter did not reach Bode until March 20. (Electronic mail was discovered much later.) Bode suspected that Piazzi’s discovery might be the missing planet, but there was insufficient data for determining its orbital elements by the methods then available. It is a problem in nonlinear equations that Newton, himself, had declared as being among the most difficult in mathematical astronomy. Nobody had solved it and, as a result, Ceres was lost in space again.

Piazzi’s discoveries were not published until the autumn of 1801 . The possible discovery – and subsequent loss of a new planet, coinciding with the beginning of a new century, was exciting news. It contradicted a philosophical justification for there being only seven planets – the number known before Ceres and a number defended by the respected philosopher Georg Hegel, among others. Hegel had recently published a book in which he chastised the astronomers for wasting their time in searching for an eighth planet when there was a sound philosophical justification for there being only seven. The new planet became a subject of conversation in intellectual circles nearly everywhere. Fortunately, the problem caught the attention of a 24 -year-old mathematician at Göttingen named Carl Friedrich Gauss.

数学代写|matlab代写|Method of Least Squares

The following example of a least-squares problem is the one most often seen, although the method of least squares may be applied to a much greater range of problems.’In the meantime, the method of least squares had been discovered independently and published by Andricn-Maric Legendre (1752-1833) in France and Robert Adrian (1775-1855) in the United States [176]. [It had also been discovered and used before Gauss was born by the German-Swiss physicist Johann Heinrich Lambert (1728-1777).] Such Jungian synchronicity (i.e., the phenomenon of multiple, nearsimultaneous discovery) was to be repeated for other breakthroughs in estimation theory, as well-for the Wicner filter and the Kalman filter.

then he could consider the problem of solving for that value of an estimate $\hat{x}$ (pronounced ” $x$-hat”) that minimizes the “estimated measurement error” $H \hat{x}-z$. He could characterize that estimation error in terms of its Euclidean vector norm $|H \hat{x}-z|$, or, equivalently, its square:
\begin{aligned} z^{2}(\hat{x}) &=|H \hat{x}-z|^{2} \ &=\sum_{i=1}^{m}\left[\sum_{j=1}^{n} h_{i j} \hat{x}{j}-z{i}\right]^{2}, \end{aligned}
which is a continuously differentiable function of the $n$ unknowns $\hat{x}{1}, \hat{x}{2}, \hat{x}{3}, \ldots, \hat{x}{n}$. This function $\varepsilon^{2}(\hat{x}) \rightarrow \infty$ as any component $\hat{x}{k} \rightarrow \pm \infty$. Consequently, it will achieve its minimum value where all its derivatives with respect to the $\hat{x}{k}$ are zero. There are $n$ such equations of the form
\begin{aligned} 0=& \frac{\partial \varepsilon^{2}}{\partial \hat{x}{k}} \ &=2 \sum{i=1}^{m} h_{i k}\left[\sum_{j=1}^{n} h_{i j} \hat{x}{j}-z{i}\right] \end{aligned}
for $k=1,2,3, \ldots, n$. Note that in this last equation the expression
$$\sum_{j=1}^{n} h_{i j} \hat{x}{j}-z{i}={H \hat{x}-z}_{i},$$
the $i$ th row of $H \hat{x}-z$, and the outermost summation is equivalent to the dot product of the $k$ th column of $H$ with $H \hat{x}-z$. Therefore Equation $1.7$ can be written as
\begin{aligned} 0 &=2 H^{\mathrm{T}}[H \hat{x}-z] \ &=2 H^{\mathrm{T}} H \hat{x}-2 H^{\mathrm{T}} z \end{aligned}
or
$$H^{\mathrm{T}} H \hat{x}=H^{\mathrm{T}} z,$$
where the matrix transpose $H^{\mathrm{T}}$ is defined as
$$H^{\mathrm{T}}=\left[\begin{array}{ccccc} h_{11} & h_{21} & h_{31} & \cdots & h_{m 1} \ h_{12} & h_{22} & h_{32} & \cdots & h_{m 2} \ h_{13} & h_{23} & h_{33} & \cdots & h_{m 3} \ \vdots & \vdots & \vdots & \ddots & \vdots \ h_{1 n} & h_{2 n} & h_{3 n} & \cdots & h_{m n} \end{array}\right]$$

数学代写|matlab代写|Gramian Matrix and Observability

For the examples considered above, observability does not depend upon the measurable data (z). It depends only on the nonsingularity of the Gramian matrix $(\mathscr{G})$, which depends only on the linear constraint matrix $(H)$ between the unknowns and knowns.

Observability of a set of unknown variables is the issue of whether or not their values are uniquely determinable from a given set of constraints, expressed as equations involving functions of the unknown variables. The unknown variables are said to be observable if their values are uniquely determinable from the given constraints, and they are said to be unobservable if they are not uniquely determinable from the given constraints.

The condition of nonsingularity (or “full rank”) of the Gramian matrix is an algebraic characterization of observability when the constraining equations are linear in the unknown variables. It also applies to the case that the constraining equations are not exact, due to errors in the values of the allegedly known parameters of the equations.

The Gramian matrix will be used in Chapter 2 to define observability of the states of dynamic systems in continuous time and discrete time.

数学代写|matlab代写|Beginnings of Estimation Theory

1801 年 1 月 1 日，即 19 世纪的第一天，意大利天文学家朱塞佩·皮亚齐正在检查星表中的条目。Piazzi 不知道，该条目是由打印机错误地添加的。在寻找“失踪”的恒星时，皮亚齐发现了一颗新行星。它是谷神星——最大的小行星，也是第一个被发现的——但皮亚齐还不知道。在他的工作被打断之前，他能够在巴勒莫的 41 晚观看期间跟踪和测量其在“固定”恒星背景下的明显运动。然而，当他回到工作岗位时，他再也找不到谷神星了。

Piazzi 的信直到 3 月 20 日才到达 Bode。（电子邮件是在很久以后才发现的。） Bode 怀疑 Piazzi 的发现可能是这颗失踪的行星，但没有足够的数据来通过当时可用的方法确定其轨道元素。这是一个非线性方程中的问题，牛顿本人曾宣称它是数学天文学中最困难的问题之一。没有人解决它，结果，谷神星再次迷失在太空中。

数学代写|matlab代写|Method of Least Squares

0=∂e2∂X^ķ =2∑一世=1米H一世ķ[∑j=1nH一世jX^j−和一世]

∑j=1nH一世jX^j−和一世=HX^−和一世,

0=2H吨[HX^−和] =2H吨HX^−2H吨和

H吨HX^=H吨和,

H吨=[H11H21H31⋯H米1 H12H22H32⋯H米2 H13H23H33⋯H米3 ⋮⋮⋮⋱⋮ H1nH2nH3n⋯H米n]

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。