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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

## 英国补考|组合学代写Combinatorics代考|Joint Integrated Probabilistic Data Association Filter

The JIPDA filter “integrates” a detection capability into JPDA by extending the multiple object continuous state space of JPDA to a multiple object discrete-continuous stâte spãcé. Thé discreeté componnént énảblês JIPDA to modèl objéct existéncee. It is, like JPDA, a classical Bayesian estimator and is conditioned on all the available measurements. It was first derived in [9-11]. A random set style derivation was given in [12, 13]. The AC derivation of JIPDA was first given in [14] and also later in [7]. I et $N$ denote the specified number of IPDA-style nhject GFI . models, so that it is also the maximum number of objects. The JPDA notation for objects and measurements is retained for JIPDA. The IPDA existence models and notation given in Chap. 2 are extended to multiple objects, with indices added to make everything object-specific.

The existence of object $n$ is modeled as a continuous-time two-state Markov chain, $N^n(t)$, with states in the set $\mathcal{B}={0,1}$. If $N^n(t)=1$, object $n$ is said to exist at time $t$; if $N^n(t)=0$, object $n$ is said not to exist. Objects are independent, by assumption, so the Markov chains are independent. The existence variable $N_k^n \equiv N^n\left(t_k\right)$ for object $n$ at time $t_k$ is a discrete-time Markov chain on $\mathcal{B}$. Existence variables are written with subscripts or superscripts or both, so they should not be confused with the specified number of object models $N$.

The transition probability matrix for object $n$ is row stochastic,
$$A_{k-1}^n=\left(\begin{array}{cc} \pi_{k-1}^{0 n} & 1-\pi_{k-1}^{0 n} \ 1-\pi_{k-1}^{1 n} & \pi_{k-1}^{1 n} \end{array}\right),$$
where $\pi_{k-1}^{0 n}$ and $\pi_{k-1}^{1 n}$ are the probabilities that the chain stays in state 0 and 1 , respectively, when transitioning from scan $k-1$ to scan $k$. The value $\chi_0^n=0.5$ is a common choice for the prior probability of existence. The predicted existence probability $\chi_k^{n-} \equiv \operatorname{Pr}\left{N_k^n=1 \mid \mathbf{y}{1: k-1}\right}$ is determined by the Markov chain via the vector-matrix product, as in (2.36). The posterior probability that object $n$ exists at time $t_k$ is $\chi_k^n \equiv \operatorname{Pr}\left{N_k^n=1 \mid \mathbf{y}{1: k}\right}, k \geq 1$.

It is important to note that existence probabilities can be state dependent. This case is treated in Sect. 4.5.2 of Chap. 4.

## 英国补考|组合学代写Combinatorics代考|Integrated State Space

The state space for JIPDA with at most $N$ objects is complicated not because it is a discrete-continuous space, but because objects that do not exist cannot have a continuous state space. The complication is seen even in its simplest form for $N=1$, as discussed in Sect. 2.6.1. The “integrated” JIPDA state space is the union of $2^N$ Cartesian products:
$$\mathcal{L}^N=\bigcup_{\kappa=0}^N \bigcup_{1 \leq n_1<\cdots<n_k \leq N} \mathcal{X}^{n_1} \times \cdots \times \mathcal{X}^{n_k},$$
where for $\kappa=0$ the union is taken to be the singleton set ${\varnothing}$ consisting only of the empty set $\varnothing$. For $N=1$, the IPDA space is $\mathcal{L}^1={\varnothing} \cup \mathcal{X}^1$. For $N=2$, it is $\mathcal{L}^2-{\varnothing} \cup \mathcal{X}^1 \cup \mathcal{X}^2 \cup\left(\mathcal{X}^1 \times \mathcal{X}^2\right)$, and for $N-3$ it is
$\mathcal{L}^3={\varnothing} \cup X^1 \cup X^2 \cup X^3 \cup\left(X^1 \times X^2\right) \cup\left(X^1 \times X^3\right) \cup\left(X^2 \times X^3\right) \cup\left(X^1 \times X^2 \times X^3\right)$
To see that (3.30) is the JIPDA space for $N \geq 1$, note that there are $\left(\begin{array}{l}N \ k\end{array}\right)$ ways for $\kappa \geq 1$ out of $N$ objects to exist. The continuous state space of each combination of $\kappa$ existing objects is the Cartesian product of their state spaces. Arranging the product in the same order as the object indices gives (3.30).

A general probability distribution on $\mathcal{L}^N$ is a PDF/PMF hybrid comprising a collection of $\Sigma_{\kappa=1}^N\left(\begin{array}{c}N \ k\end{array}\right)=2^N-1$ continuous distributions, as well as a discrete distribution on the $2^N$ elements of $\mathcal{L}^N$. It is shown below that the exact Bayesian posterior distribution on $\mathcal{L}^N$ is a list of this kind. For scan $k$, the prior probability distribution for JIPDA is not specified in this manner, but as a list of $N$ ordered pairs that give the existence probability and continuous PDF of each object,

$$\left{\left(\chi_{k-1}^n, \mu_{k-1}^n\left(x_{k-1}^n\right)\right): n=1, \ldots, N\right} .$$
This list corresponds to the GFL
$$\Psi_{k-1}^{\text {IрФA }}\left(h^{1: N}\right)=\prod_{n=1}^N\left(1-\chi_{k-1}^n+\chi_{k-1}^n \int_{X^n} h^n\left(x_{k-1}^n\right) \mu_{k-1}^n\left(x_{k-1}^n\right) \mathrm{d} x_{k-1}^n\right) .$$
Objects are independent, so the parameters specify a probability distribution over the integrated JIPDA space $\mathcal{L}^N$. The parameters $\chi_k^{n-}$ and $\mu_k^{n-}\left(x_k^n\right)$ of the predicted processes are determined, independently, using object-specific Markov chains $A_{k-1}^n$ and Markovian motion models $p_k^n\left(x_k^n \mid x_{k-1}^n\right)$.

# 组合学代考

## 英国补考|组合学代写combinatorics代考|联合集成概率数据关联滤波器

JIPDA滤波器将JPDA的多对象连续状态空间扩展为一个多对象离散连续stâte spãcé，从而将检测能力“集成”到JPDA中。Thé discreeté componnént énảblês JIPDA到modèl objéct existéncee。它像JPDA一样，是一个经典的贝叶斯估计量，并以所有可用的测量为条件。它最早起源于[9-11]。[12,13]给出了随机集风格的推导。JIPDA的AC推导首先在[14]中给出，随后在[7]中也给出。I et $N$表示ipda风格的nhject GFI的指定数量。模型，使它也是对象的最大数目。对象和测量的JPDA符号为JIPDA保留。第二章中给出的IPDA存在模型和表示法被扩展到多个对象，并添加了索引，使所有内容都特定于对象

$$A_{k-1}^n=\left(\begin{array}{cc} \pi_{k-1}^{0 n} & 1-\pi_{k-1}^{0 n} \ 1-\pi_{k-1}^{1 n} & \pi_{k-1}^{1 n} \end{array}\right),$$

## 英国补考|组合学代写combinatorics代考|综合状态空间

JIPDA的状态空间(最多$N$个对象)是复杂的，并不是因为它是一个离散-连续空间，而是因为不存在的对象不能有连续的状态空间。对于$N=1$，即使以最简单的形式也可以看到这种复杂性，如第2.6.1节所述。“集成的”JIPDA状态空间是$2^N$笛卡尔积的并集:
$$\mathcal{L}^N=\bigcup_{\kappa=0}^N \bigcup_{1 \leq n_1<\cdots<n_k \leq N} \mathcal{X}^{n_1} \times \cdots \times \mathcal{X}^{n_k},$$
，其中对于$\kappa=0$，并集被认为是仅由空集$\varnothing$组成的单例集${\varnothing}$。对于$N=1$, IPDA空间是$\mathcal{L}^1={\varnothing} \cup \mathcal{X}^1$。对于$N=2$，它是$\mathcal{L}^2-{\varnothing} \cup \mathcal{X}^1 \cup \mathcal{X}^2 \cup\left(\mathcal{X}^1 \times \mathcal{X}^2\right)$，对于$N-3$，它是
$\mathcal{L}^3={\varnothing} \cup X^1 \cup X^2 \cup X^3 \cup\left(X^1 \times X^2\right) \cup\left(X^1 \times X^3\right) \cup\left(X^2 \times X^3\right) \cup\left(X^1 \times X^2 \times X^3\right)$

$\mathcal{L}^N$上的一般概率分布是一个PDF/PMF混合分布，包括$\Sigma_{\kappa=1}^N\left(\begin{array}{c}N \ k\end{array}\right)=2^N-1$连续分布的集合，以及$\mathcal{L}^N$上$2^N$元素上的离散分布。如下图所示，$\mathcal{L}^N$上确切的贝叶斯后验分布就是这类列表。对于扫描$k$, JIPDA的先验概率分布不是以这种方式指定的，而是$N$有序对的列表，给出每个对象的存在概率和连续PDF

$$\left{\left(\chi_{k-1}^n, \mu_{k-1}^n\left(x_{k-1}^n\right)\right): n=1, \ldots, N\right} .$$

$$\Psi_{k-1}^{\text {IрФA }}\left(h^{1: N}\right)=\prod_{n=1}^N\left(1-\chi_{k-1}^n+\chi_{k-1}^n \int_{X^n} h^n\left(x_{k-1}^n\right) \mu_{k-1}^n\left(x_{k-1}^n\right) \mathrm{d} x_{k-1}^n\right) .$$

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assignmentutor™您的专属作业导师
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