### 统计代写|随机过程代写stochastic process代考|MTH7090

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机过程代写stochastic process代考|Rotation, Stretching, Translation and Standardization

In two dimensions, rotating a Poisson-binomial process is equivalent to rotating its underlying lattice attached to the index space. Rotating the points has the same effect as rotating the lattice locations, because $F$ (the distribution attached to the points) belongs to a family of location-scale distributions [Wiki]. For instance, a $\pi / 4$ rotation will turn the square lattice into a centered-square lattice [Wiki], but it won’t change the main properties of the point process. Both processes, the original one and the rotated one, may be indistinguishable for all practical purposes unless the scaling factor $s$ is small, creating model identifiability [Wiki] issues. For instance, the theoretical correlation between the point coordinates $\left(X_h, Y_k\right)$ or the underlying lattice point coordinates $(h / \lambda, k / \lambda)$, measured on all points, remains equal to zero after rotation, because the number of points is infinite (this may not be the case if you observe points through a small window, because of boundary effects). Thus, a Poisson-binomial process has a point distribution invariant under rotations, on a macro-scale. This property is called anisotropy [Wiki]. On a micro-scale, a few changes occur though: for instance the twodimensional version of Theorem $4.1$ no longer applies, and the distance between the projection of two neighbor points on the $\mathrm{X}$ or $\mathrm{Y}$ axis, shrinks after the rotation.

Applying a translation to the points of the process, or to the underlying lattice points, results in a shifted point process. It becomes interesting when multiple shifted processes, with different translation vectors, are combined together as in Section 1.5.3. Theorem $4.1$ may not apply to the shifted process, though it can easily be adapted to handle this situation. One of the problems is to retrieve the underlying lattice space of the shifted process. This is useful for model fitting purposes, as it is easier to compare two processes once they have been standardized (after removing translations and rescaling). Estimation techniques to identify the shift are discussed in Section 3.4.

By a standardized Poisson-binomial point process, I mean one in its canonical form, with intensity $\lambda=1$, scaling factor $s=1$, and free of shifts or rotations. Once two processes are standardized, it is easier to compare them, assess if they are Poisson-binomial, or perform various machine learning procedures on observed data, such as testing, computing confidence intervals, cross-validation, or model fitting. In some way, this is similar to transforming and detrending time series to make them more amenable to statistical inference. There is also some analogy between the period or quasi-period of a time series, and the inverse of the intensity $\lambda$ of a Poisson-binomial process: in fact, $1 / \lambda$ is the fixed increment between the underlying lattice points in the lattice space, and can be viewed as the period of the process.

Finally, a two dimensional process is said to be stretched if a different intensity is used for each coordinate for all the points of the process. It turns the underlying square lattice space into a rectangular lattice, and the homogeneous process into a non-homogeneous one, because the intensity varies locally. Observed data points can be standardized using the Mahalanobis transformation [Wiki], to remove stretching (so that variances are identical for both coordinates) and to decorrelate the two coordinates, when correlation is present.

## 统计代写|随机过程代写stochastic process代考|Superimposition and Mixing

Here we are working with two-dimensional processes. When the points of $m$ independent point processes with same distribution $F$ and same index space $\mathbb{Z}^2$ are bundled together, we say that the processes are superimposed. These processes are no longer Poisson-binomial, see Exercise 14. Indeed, if the scaling factor $s$ is small and $m>1$ is not too small, they exhibit clustering around each lattice location in the lattice space. Also, the intensities or scaling factors of each individual point process may be different, and the resulting combined process may not be homogeneous. Superimposed point processes also called interlaced processes.
A mixture of $m$ point processes, denoted as $M$, is defined as follows:

• We have $m$ independent point processes $M_1, \ldots, M_m$ with same distribution $F$ and same index space $\mathbb{Z}^2$,
• The intensity and scaling factor attached to $M_i$ are denoted respectively as $\lambda_i$ and $s_i(i=1, \ldots, m)$,
• The points of $M_i(i=1, \ldots, m)$ are denoted as $\left(X_{i h}, Y_{i k}\right)$; the index space consists of the $(h, k)$ ‘s,
• The point $\left(X_h, Y_k\right)$ of the mixture process $M$ is equal to $\left(X_{i h}, Y_{i k}\right)$ with probability $\pi_i>0, i=1, \ldots, m$.
While mixing or superimposing Poisson-binomial processes seem like the same operation, which is true for stationary Poisson processes, in the case of Poisson-binomial processes, these are distinct operations resulting in significant differences when the scaling factors are very small (see Exercise 18). The difference is most striking when $s=0$. In particular, superimposed processes are less random than mixtures. This is due to the discrete nature of the underlying lattice space. However, with larger scaling factors, the behavior of mixed and superimposed processes tend to be similar.

Several of the concepts discussed in Section $1.5$ are illustrated in Figure 2, representing a realization of $m$ superimposed shifted stretched Poisson-binomial processes, called $m$-interlacing. For each individual process $M_i, i=1, \ldots, m$, the distribution attached to the point $\left(X_{i h}, X_{i k}\right)$ (with $h, k \in \mathbb{Z}$ ) is
$$P\left(X_{i h}<x, Y_{i k}<y\right)=F\left(\frac{x-\mu_i-h / \lambda}{s}\right) F\left(\frac{y-\mu_i^{\prime}-k / \lambda^{\prime}}{s}\right), \quad i=1, \ldots, m$$
This generalizes Formula (2). The parameters used for the model pictured in Figure 2 are:

• Number of superimposed processes: $m=4$; each one displayed with a different color,
• Color: red for $M_1$, blue for $M_2$, orange for $M_3$, black for $M_4$,
• scaling factor: $s=0$ (left plot) and $s=5$ (right plot),
• Intensity: $\lambda=1 / 3$ ( $\mathrm{X}$-axis) and $\lambda^{\prime}=\sqrt{3} / 3$ ( $\mathrm{Y}$-axis),
• Shift vector, $\mathrm{X}$-coordinate: $\mu_1=0, \mu_2=1 / 2, \mu_3=2, \mu_4=3 / 2$,
• Shift vector, Y-coordinate: $\mu_1^{\prime}=0, \mu_2^{\prime}=\sqrt{3} / 2, \mu_3^{\prime}=0, \mu_4^{\prime}=\sqrt{3} / 2$,
• $F$ distribution: standard centered logistic with zero mean and variance $\pi^2 / 3$.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Superimposition and Mixing

• 我们有 $m$ 独立点过程 $M_1, \ldots, M_m$ 具有相同的分布 $F$ 和相同的索引空间 $\mathbb{Z}^2$ ，
• 附加的强度和比例因子 $M_i$ 分别记为 $\lambda_i$ 和 $s_i(i=1, \ldots, m)$,
• 的要点 $M_i(i=1, \ldots, m)$ 表示为 $\left(X_{i h}, Y_{i k}\right)$ ；索引空间由 $(h, k)$ 的，
• 重点 $\left(X_h, Y_k\right)$ 混合过程 $M$ 等于 $\left(X_{i h}, Y_{i k}\right)$ 有概率 $\pi_i>0, i=1, \ldots, m$. 虽然混合或諿加泊松二项式过程看起来像是相同的操作，对于平稳泊松过程也是如此，但在泊松二 项式过程的情况下，这些是不同的操作，当缩放因子非常小时会导致显着差异 (参见练习 18). 当 $s=0$. 特别是，咺加过程的随机性低于混合过程。这是由于底层晶格空间的离散性质。然而，对 于较大的比例因子，混合过程和䝁加过程的行为往往相似。
本节中讨论的几个概念 $1.5$ 如图 2 所示，代表了一种实现 $m$ 呾加的偏移拉伸泊松二项式过程，称为 $m$ – 交 错。对于每个单独的过程 $M_i, i=1, \ldots, m$, 分布附加到点 $\left(X_{i h}, X_{i k}\right.$ ) (和 $h, k \in \mathbb{Z}$ ) 是
$$P\left(X_{i h}<x, Y_{i k}<y\right)=F\left(\frac{x-\mu_i-h / \lambda}{s}\right) F\left(\frac{y-\mu_i^{\prime}-k / \lambda^{\prime}}{s}\right), \quad i=1, \ldots, m$$
这推广了公式 (2)。用于图 2 中所示模型的参数是:
• 㠬加进程数： $m=4$; 每一个都以不同的颜色显示，
• 颜色: 红色为 $M_1$ ，蓝色为 $M_2$ ，橙色为 $M_3$ ，黑色为 $M_4$ ，
• 比例因子: $s=0$ (左图) 和 $s=5$ (右图),
• 强度: $\lambda=1 / 3$ ( $\mathrm{X}$-轴) 和 $\lambda^{\prime}=\sqrt{3} / 3$ (Y-轴)，
• 移位向量， $\mathrm{X}$-协调: $\mu_1=0, \mu_2=1 / 2, \mu_3=2, \mu_4=3 / 2$,
• 移位向量， Y坐标: $\mu_1^{\prime}=0, \mu_2^{\prime}=\sqrt{3} / 2, \mu_3^{\prime}=0, \mu_4^{\prime}=\sqrt{3} / 2$,
• $F$ 分布: 均值和方差为零的标准中心逻辑 $\pi^2 / 3$.

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