数学代写|组合学代写Combinatorics代考|MA1510

2022年9月27日

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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数学代写|组合学代写Combinatorics代考|Closing the Bayesian Recursion

The Bayes posterior distribution (3.47) is not in the same form as the list given in (3.31) that defines the prior. It is approximated in the same “mean field” spirit as done in JPDA by using the single-object marginal distributions. The marginal distribution of object $j$ is defined as the sum over the set of existence events $\epsilon$ for which $\epsilon_j=1$ of the the term-by-term integral of the posterior (3.47) over all $x^n \in \mathcal{X}^n, n \neq j$. Indeed, the posterior probability that object $j$ exists and is in state $x^j$ is given by
\begin{aligned} p_k\left(x^j, \epsilon_j=1 \mid \mathbf{y}k\right) &=\sum{\epsilon^{\prime}: \epsilon^{\prime}=1} p_k\left(x^j, \mathbf{N}k=\epsilon^{\prime} \mid \mathbf{y}_k\right) \ &=\sum{\epsilon^{\prime}: \epsilon_j^{\prime}=1}\left[\int_{X^{\prime} \backslash X^{\prime}} p_k\left(x^{\epsilon^{\prime}}, \mathbf{N}k=\epsilon^{\prime} \mid \mathbf{y}_k\right) \mathrm{d} x^{\epsilon^{\prime}} \backslash \mathrm{d} x^j\right] . \end{aligned} A different way to do the same thing is to use the GFL of the marginal process. It is derived from (3.35) by substituting $h_n\left(x^n\right)=1, x^n \in \mathcal{X}^n, n \neq j$. Thus, \begin{aligned} &\Psi_k^{\text {IIDAM }}\left(h^j, g\right)=\exp \left(-\lambda_k^c+\lambda_k^c \int_y g(y) p_k^c(y) \mathrm{d} y\right) \ &\times\left[1-\chi_k^{j-}+\chi_k^{j-} \int{X^j} h^j\left(x^j\right) \mu_k^{j-}\left(x^j\right)\left(1-P d_k^j\left(x^j\right)+P d_k^j\left(x^j\right) \int_y g(y) p_k^j\left(y \mid x^j\right) \mathrm{d} y\right) \mathrm{d} x^j\right] \ &\times \prod_{\substack{n=1 \ n \neq j}}^N\left[1-\chi_k^{n-}+\chi_k^{n-} \int_{X^n} \mu_k^{n-}\left(x^n\right)\left(1-P d_k^n\left(x^n\right)+P d_k^n\left(x^n\right) \int_y g(y) p_k^n\left(y \mid x^n\right) \mathrm{d} y\right) \mathrm{d} x^n\right] \end{aligned}
Substituting the Dirac delta train of (2.22) into (3.50) gives the secularized marginal GFL, $\Psi_k^{\text {JIPD } ~}\left(h^j, \beta\right)$. It is a product of $N$ linear functions and an exponential of a linear function, so evaluating the cross-derivative using (C.37) and normalizing it gives the GFL of the Bayes posterior, $\Psi_k^{\text {IPDD } \omega}\left(h^j \mid \mathbf{y}_k\right)$. Substituting $h^j(\cdot)=\alpha_j \delta_x(\cdot)$ gives the secular form, $\Psi_k^{\text {IPPS }}\left(\alpha_j \mid \mathbf{y}_k\right)$. By inspection it is linear in $\alpha_j$.

数学代写|组合学代写Combinatorics代考|Resolution/Merged Measurement Problem

The problem of unresolved or merged measurements has received relatively little attention in the literature. In practice, the problem is usually ignored; all measurements are assumed resolved. In many scenarios, however, the issue is crucial and can become more serious than incorrect object-measurement assignments [15]. In [16], a hard threshold resolution model is developed for a fixed grid of resolution cells for the JPDA filter for $N=2$ objects. The idea is extended to MHT in [4]. In [17], the resolution function is switched from a hard ${0,1}$ threshold function to a probabilistic Gaussian function. The general unresolved tracking problem for $N \geq 2$ objects is addressed for both JPDA and MHT in [18-20].

An unresolved object tracking filter is developed here using AC techniques for JPDA with $N=2$ objects of interest. It is referred to as the JPDA/Res filter. It is closely related to, but different from, the first application of $\mathrm{AC}$ techniques to model resolution problems in [21].

The JPDA filter assumes that a given object can generate at most one measurement per scan. It also assumes that a measurement originates from at most one object. That is, a measurement is either clutter-originated or induced by a single object of interest. In reality, sensors have limited resolution capability. The term resolution refers to the ability of the sensor to determine that two closely spaced objects are indeed distinct. Resolution depends on physical characteristics of the sensor, the signal processing algorithms, and the physical characteristics of the signal, such as relative object signal strength. In terms of “peaks” in the sensor response surface (see Sect. $2.3$ of Chap. 2), two objects are unresolved if there is only one peak in the surface and resolved if there are two peaks. Loosely speaking, closely spaced objects may have resolution issues. (Distance is measured in the sensor space, not the state space, since sensors may have observability limitations, e.g., they may measure angles only and not range.)

Resolution and detection are different phenomena-two objects can be resolvable, while one or both are undetectable. For example, they may be far apart in the measurement space (i.e., theoretically resolvable) but have weak signal strength (i.e., undetectable). Similarly, they can be unresolvable and undetectable.

组合学代考

数学代写|组合学代写Combinatorics代考|关闭贝叶斯递归

\begin{aligned} p_k\left(x^j, \epsilon_j=1 \mid \mathbf{y}k\right) &=\sum{\epsilon^{\prime}: \epsilon^{\prime}=1} p_k\left(x^j, \mathbf{N}k=\epsilon^{\prime} \mid \mathbf{y}k\right) \ &=\sum{\epsilon^{\prime}: \epsilon_j^{\prime}=1}\left[\int{X^{\prime} \backslash X^{\prime}} p_k\left(x^{\epsilon^{\prime}}, \mathbf{N}k=\epsilon^{\prime} \mid \mathbf{y}k\right) \mathrm{d} x^{\epsilon^{\prime}} \backslash \mathrm{d} x^j\right] . \end{aligned}给出。做同样事情的另一种方法是使用边缘过程的GFL。它由(3.35)通过替换$h_n\left(x^n\right)=1, x^n \in \mathcal{X}^n, n \neq j$派生而来。因此，\begin{aligned} &\Psi_k^{\text {IIDAM }}\left(h^j, g\right)=\exp \left(-\lambda_k^c+\lambda_k^c \int_y g(y) p_k^c(y) \mathrm{d} y\right) \ &\times\left[1-\chi_k^{j-}+\chi_k^{j-} \int{X^j} h^j\left(x^j\right) \mu_k^{j-}\left(x^j\right)\left(1-P d_k^j\left(x^j\right)+P d_k^j\left(x^j\right) \int_y g(y) p_k^j\left(y \mid x^j\right) \mathrm{d} y\right) \mathrm{d} x^j\right] \ &\times \prod{\substack{n=1 \ n \neq j}}^N\left[1-\chi_k^{n-}+\chi_k^{n-} \int_{X^n} \mu_k^{n-}\left(x^n\right)\left(1-P d_k^n\left(x^n\right)+P d_k^n\left(x^n\right) \int_y g(y) p_k^n\left(y \mid x^n\right) \mathrm{d} y\right) \mathrm{d} x^n\right] \end{aligned}

数学代写|组合学代写Combinatorics代考|分辨率/合并测量问题

JPDA过滤器假设给定的对象每次扫描最多只能生成一个测量值。它还假设测量最多起源于一个对象。也就是说，测量要么是由杂波引起的，要么是由单个感兴趣的对象引起的。实际上，传感器的分辨率有限。术语分辨率指的是传感器确定两个距离很近的物体确实不同的能力。分辨率取决于传感器的物理特性、信号处理算法和信号的物理特性，如相对物体信号强度。根据传感器响应面的“峰”(参见第二章的$2.3$节)，如果表面上只有一个峰，则两个对象无法解析;如果表面上有两个峰，则两个对象无法解析。粗略地说，间隔很近的物体可能存在分辨率问题。(距离是在传感器空间中测量的，而不是状态空间，因为传感器可能有可观察性的限制，例如，它们可能只测量角度而不是距离。

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。