### 数学代写|组合学代写Combinatorics代考|MATH482

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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|组合学代写Combinatorics代考|Statement A—Object Existence and Detection

The first statement describes an extremely spartan setting in which the logical relationship between object existence and sensor detection is modeled but without specifying either the object state or measurement.

Statement $\mathbf{A}$. At most one object exists and, if it does exist, the sensor may or may not generate one measurement.

Let $N$ and $M$ denote the number of objects that exist and the number of measurements, respectively. Philosophers wrestle with the nature of existence, but here existence is strictly about counting: the object is said to exist if $N=1$ and not to exist if $N=0$. The number of objects is assumed to be a random ${0,1}$ integer, an assumption that justifies calling $\chi=\operatorname{Pr}{N=1}$ the object existence probability. The statement that at most one object exists is equivalent to $\operatorname{Pr}{N=0}=1-\chi$ and $\operatorname{Pr}{N \geq 2}=0$. By definition (see Appendix A), the GF for $N$ is
$$G_{N}(z)=\sum_{n=0}^{\infty} \operatorname{Pr}{N=n} z^{n}=1-\chi+\chi z,$$
where $z$ is the indeterminate variable of the GF for $N$. The linear equation (1.1) perfectly characterizes the statement that there is at most one object, but it says nothing about the number of measurements. It is the GF of the marginal distribution of $N$ for the joint random variable $(N, M)$.

## 数学代写|组合学代写Combinatorics代考|Statement B—Gridded Measurements

Statement A is “data starved,” meaning that most problems are formulated in a larger context in which objects have states and sensors produce measurements of the state. Statement B adds a point measurement to Statement A, which is otherwise unchanged. Statement $\mathrm{C}$ of the next subsection adds both a measurement and an object state.

The added measurement information is mapped into a counting problem. The mapping is done with a gridded measurement space. The same mapping works intuitively for continuous spaces, but technical details (see Appendix B) get in the way of the flow of ideass, so only griddéd spācês aree treaatēd héré. Laterr chapteers in the book do not use gridded spaces.

Statement B. At most one object exists and, if it does exist, the sensor may or may not generate a random measurement $Y$.

The measurement space $\mathcal{Y}$ is partitioned into a finite number of non-overlapping grid cells labeled $1, \ldots, R$. The measurement $Y$ is conceptualized as the nonnegative integer-valued random vector $Y_{1: R}=\left(Y_{1}, \ldots, Y_{R}\right)$, where $Y_{r}$ is the number of measurements in cell $r$. The random number of measurements $M$ is
$$M=\sum_{r=1}^{R} Y_{r}$$

## 数学代写|组合学代写Combinatorics代考|Statement A—Object Existence and Detection

$$G_{N}(z)=\sum_{n=0}^{\infty} \operatorname{Pr} N=n z^{n}=1-\chi+\chi z$$

## 数学代写|组合学代写Combinatorics代考|Statement B—Gridded Measurements

$$M=\sum_{r=1}^{R} Y_{r}$$

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## MATLAB代写

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