### 统计代写|时间序列分析代写Time-Series Analysis代考|DSC425

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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

## 统计代写|时间序列分析代写Time-Series Analysis代考|Global Embedding Dimension

The next question about the data vector
$$\mathbf{y}(n)=[s(n), s(n-T), \ldots, s(n-(D-1) T)]$$
we need to address is the value of the integer “embedding dimension” D. Here is where Whitney’s and Takens’ results come into play. The key idea is that as we enlarge the dimension $D$ of the vector $\mathbf{y}(n)$ we eliminate step by step the intersections of orbits on the system attractor arising from our projection during the measurement process. If this is the case, then there might well be a global dimension allowing us to unfold a particular data set with particular coordinates as entries in $\mathbf{y}(n)$ at a dimension less than the sufficient dimension of the Whitney/Takens geometric result.

To examine this we need the notion of crossing of trajectories, and this we realize in the close analogy of neighbors in the state space which are a result of the dynamics-true neighbors-and neighbors in the state space which are a result of the projection during measurement-false neighbors [23]. If we select an embedding dimension $\mathrm{D}$, then it is a matter of an $\operatorname{order} \mathrm{N} \log (\mathrm{N})$ search among all the points $\mathbf{y}(n)$ in that space to determine the nearest neighbor to a point $\mathbf{y}(\mathrm{k})$. If this nearest neighbor is not a close neighbor in dimension $\mathrm{D}+1$, then its “neighborliness” to $\mathbf{y}(\mathrm{k})$ is the result of a projection from a higher dimensional space. This is a false nearest neighbor, and we wish to eliminate all of them. We accomplish this elimination of the false nearest neighbors by systematically examining the nearest neighbors in dimension $\mathrm{D}$ and their “neighborliness” in dimension D $+1$ for $\mathrm{D}=1,2, \ldots$ until there are no false nearest neighbors remaining. We call this integer dimension $d_E$.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Local or Dynamical Dimension

The embedding dimension we just selected is a global and average indicator of the number of coordinates needed to unfold the actual data $s(t)$ [24].

The global integer embedding dimension estimate tells us a minimum dimension $d_E$ in which we can place (embed) the signal from our source. This dimension can be larger than the number of degrees of freedom in the dynamics underlying the signal $s(t)$. Suppose that locally the dynamics happened to be a two-dimensional map $\left(x_n, y_n\right) \rightarrow\left(x_{n+1}, y_{n+1}\right)$ but the global structure of the dynamics placed this on the surface of an ordinary torus. To embed the points of the whole data set now lying on a torus, we would have to select $d_E=3$; however, if we wish to determine equations of motion (or a map) to describe the dynamics, we really need only the local dimension of 2 . This local dimension $d_L \leqslant d_E$, and is important when we wish to evaluate the Lyapunov exponent, as we do below, to characterize the dynamical system producing $s(t)$.

To determine $d_L$ we need to move beyond the geometry in the embedding theorem and ask a local question about the data in dimension $\mathrm{d}{\mathrm{E}}$ where we know there are no unwanted intersections of the orbit associated with $s(t)$ and itself. The notion is that data vectors of dimension $\mathrm{d} \leqslant \mathrm{d}{\mathrm{E}}$,
$$y_d(n)=[s(n), s(n-T), \ldots, s(n-(d-1) T)],$$
will map without ambiguity locally into other vectors of dimension $d \leqslant d_E$. We can test for this by forming a d-dimensional local map
$$\mathbf{y}{\mathrm{d}}(\mathrm{n}+1)=\mathbf{M}\left(\mathbf{y}{\mathrm{d}}(\mathrm{n})\right),$$ and asking whether this map accounts for the behavior of the actual data in $\mathrm{d} \leqslant \mathrm{d}{\mathrm{E}}$. For $\mathrm{d}$ too small, it will not. For $\mathrm{d}=\mathrm{d}{\mathrm{E}}$, it will. If for some intermediate $\mathrm{d}$, the map is accurate, this is an indication of a lower dimensional dynamics than $\mathrm{d}_{\mathrm{E}}$ needed to globally unfold the data.

To answer this question select a data vector $y(k)$ in $d_E$. Select $N_B$ nearest neighbors in phase space to $y(k): y^{(r)}(k) ; r=0,1,2, \ldots, N_B, y^{(0)}(k)=y(k)$. In $\mathrm{d}{\mathrm{E}}$ these are all true neighbors, but their actual time labels may or may not be near the time k. Choose in various ways a d-dimensional subspace of vectors $\mathbf{y}{\mathrm{d}}^{(r)}(\mathrm{k})$. There are $\left(\begin{array}{c}\mathrm{d}_{\mathrm{E}} \ \mathrm{d}\end{array}\right)$ ways to do this and all are worth examining.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Global Embedding Dimension

$$\mathbf{y}(n)=[s(n), s(n-T), \ldots, s(n-(D-1) T)]$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Local or Dynamical Dimension

$$y_d(n)=[s(n), s(n-T), \ldots, s(n-(d-1) T)],$$

$$\mathbf{y d}(\mathrm{n}+1)=\mathbf{M}(\mathbf{y d}(\mathrm{n})),$$

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## MATLAB代写

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