# 数学代写|组合学代写Combinatorics代考|CS-E4555

## 数学代写|组合学代写Combinatorics代考|Probability Distribution

The PDF of JPDAS is defined on the Cartesian product space $\mathcal{X}^N$. It is evaluated by substituting a weighted Dirac delta train for $h(x)$ and taking the cross-derivative. (See Sect. B.5 of Appendix B.) As noted above, exactly $N$ objects are present with probability one, so delta trains of length less than or greater than $N$ yield zero probabilities. Let $\left{x_1, \ldots, x_N\right}$ denote a set of object states. For $\alpha=\left(\alpha_1, \ldots, \alpha_N\right)$, let
$$h_\delta(\alpha)=\sum_{n=1}^N \alpha_n \delta_{x_n}(x) .$$
Substituting into (4.11) gives the secular function; it is a multivariate polynomial in the components of $\alpha$,

\begin{aligned} \Psi_k^{\text {JPDAs }}\left(\alpha \mid \mathbf{y}k\right)=& \frac{c\left(\mathbf{y}_k\right)}{C_k(M, N)} \sum{\kappa=0}^{\min [N, M]}(N)\kappa\left(\sum{n=1}^N \alpha_n A\left(x_n\right)\right)^{N-\kappa} \ & \times S_\kappa^{(M)}\left(\sum_{n=1}^N \alpha_n B\left(x_n ; y_1\right), \ldots, \sum_{n=1}^N \alpha_n B\left(x_n ; y_M\right)\right) . \end{aligned}
The coefficient of the monomial $\prod_{n=1}^N \alpha_n=\alpha_1 \cdots \alpha_N$ is the conditional probability:
$$p_k\left(\left{x_1, \ldots, x_N\right} \mid \mathbf{y}_k\right)=\left[\alpha_1 \cdots \alpha_N\right] \Psi_k^{\text {JPDAs }}\left(\alpha \mid \mathbf{y}_k\right) .$$
The general expression for the coefficient is complicated and of little interest here, so it is omitted.

Example $1 M=1$. In this case, $\mathbf{y}k=\left{y_1\right}$ and $c\left(\mathbf{y}_k\right)=\exp \left(-\lambda_k^c\right) \lambda_k^c p_k^c\left(y_1\right)$. Since $\min {N, M}=1$, the coefficient (4.16) is the sum of two terms whose ESPs are $S_0^{(1)}(x)=1$ and $S_1^{(1)}(x)=x$. The normalization constant is \begin{aligned} C_k(1, N) &=\left(\exp \left(-\lambda_k^c\right) \lambda_k^c p_k^c\left(y_1\right)\right)\left(\int_X A(x) \mathrm{d} x\right)^N \ &+N\left(\exp \left(-\lambda_k^c\right) \lambda_k^c p_k^c\left(y_1\right)\right)\left(\int_X A(x) \mathrm{d} x\right)^{N-1}\left(\int_X B\left(x ; y_1\right) \mathrm{d} x\right) . \end{aligned} The $\kappa=0$ term, without $C_k(1, N)$, is \begin{aligned} T_k(0) & \equiv\left[\alpha_1 \cdots \alpha_N\right] c\left(\mathbf{y}_k\right)(N)_0\left(\sum{n=1}^N \alpha_n A\left(x_n\right)\right)^N \ &=\left(\exp \left(-\lambda_k^c\right) \lambda_k^c p_k^c\left(y_1\right)\right) N ! \prod_{n=1}^N A\left(x_n\right) \end{aligned}

## 数学代写|组合学代写Combinatorics代考|Intensity Function and Closing the Bayesian Recursion

Summary statistics for the superposed process are the same as for any single point process. The most commonly used statistic is the “count density” function, often termed the intensity function. The intensity is the expected number of objects per unit state space at each point $\bar{x} \in \mathcal{X}$. It has the same units as a probability density on $X$, but it is a very different function. This important distinction is discussed in Sect. B.6 of Appendix B.

As noted in Sect. B.6 of Appendix B, the intensity function at an arbitrarily specified point $\bar{x} \in X$ is the derivative of the GFL at $\bar{x}$ evaluated at one, not zero, so the secular function for the intensity at $\bar{x}$ employs the Dirac delta $h_\delta(x)=1+$ $\bar{\alpha} \delta_{\bar{x}}(x), \bar{\alpha} \in \mathbb{R}$. Using it in (4.11) gives
\begin{aligned} \Psi_k^{\mathrm{JPD} A S} &\left(\bar{\alpha} \mid \mathbf{y}k\right)=\Psi_k^{\mathrm{JPD} A S}\left(1+\bar{\alpha} \delta{\bar{x}} \mid \mathbf{y}k\right) \ =& \frac{c\left(\mathbf{y}_k\right)}{C_k(M, N)} \sum{\kappa=0}^{\min [N, M]}(N)k\left(\bar{\alpha} A(\bar{x})+\int{\mathcal{X}} A(x) \mathrm{d} x\right)^{N-\kappa} \ & \times S_k^{(M)}\left(\bar{\alpha} B\left(\bar{x} ; y_1\right)+\int_{\mathcal{X}} B\left(x ; y_1\right) \mathrm{d} x, \ldots, \bar{\alpha} B\left(\bar{x} ; y_M\right)+\int_X B\left(x ; y_M\right) \mathrm{d} x\right) . \end{aligned}
By inspection, this secular function is a (univariate) polynomial in $\bar{\alpha}$. The intensity at the point $\bar{x} \in \mathcal{X}$ is the coefficient of the linear term,
$$I_k^{\mathrm{JPDAS}}\left(\bar{x} \mid \mathbf{y}k\right)=[\bar{\alpha}] \Psi_k^{\mathrm{JPDAS}}\left(\bar{\alpha} \mid \mathbf{y}_k\right)$$ The general analytic expression for this coefficient is cumbersome and is omitted. The intensity function simplifies to $$I_k^{\text {JPDAs }}\left(\bar{x} \mid \mathbf{y}_k=\varnothing\right)=\frac{N \mu_k^{-}(\bar{x})\left(1-P d_k(\bar{x})\right)}{\int\chi \mu_k^{-}(x)\left(1-P d_k(x)\right) \mathrm{d} x}$$
when the scan measurement set is empty.
Since $\int_X I_k^{\text {JPDAS }}\left(x \mid \mathbf{y}_k\right) \mathrm{d} x=N$, dividing by $N$ gives the exact Bayes updated PDF as
$$\mu_k^{\mathrm{JPDSS}}\left(x \mid \mathbf{y}_k\right)=\frac{1}{N} I_k^{\mathrm{PPD} A}\left(x \mid \mathbf{y}_k\right) .$$
This step closes the Bayesian recursion for JPDAS.

## 数学代写|组合学代写Combinatorics代考|概率分布

JPDAS的PDF定义在笛卡尔积空间$\mathcal{X}^N$上。通过将$h(x)$替换为一个加权狄拉克δ列并取交叉导数来评估它。(见附录b B.5节)如上所述，恰好$N$对象的出现概率为1，因此长度小于或大于$N$的增量序列产生的概率为0。让$\left{x_1, \ldots, x_N\right}$表示一组对象状态。对于$\alpha=\left(\alpha_1, \ldots, \alpha_N\right)$，令
$$h_\delta(\alpha)=\sum_{n=1}^N \alpha_n \delta_{x_n}(x) .$$

\begin{aligned} \Psi_k^{\text {JPDAs }}\left(\alpha \mid \mathbf{y}k\right)=& \frac{c\left(\mathbf{y}_k\right)}{C_k(M, N)} \sum{\kappa=0}^{\min [N, M]}(N)\kappa\left(\sum{n=1}^N \alpha_n A\left(x_n\right)\right)^{N-\kappa} \ & \times S_\kappa^{(M)}\left(\sum_{n=1}^N \alpha_n B\left(x_n ; y_1\right), \ldots, \sum_{n=1}^N \alpha_n B\left(x_n ; y_M\right)\right) . \end{aligned}

$$p_k\left(\left{x_1, \ldots, x_N\right} \mid \mathbf{y}_k\right)=\left[\alpha_1 \cdots \alpha_N\right] \Psi_k^{\text {JPDAs }}\left(\alpha \mid \mathbf{y}_k\right) .$$

## 数学代写|组合学代写Combinatorics代考|强度函数和关闭贝叶斯递归

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