### 统计代写|时间序列分析作业代写time series analysis代考|Bivariate Time Series Analysis

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• Foundations of Data Science 数据科学基础

## 统计代写|时间序列分析作业代写time series analysis代考|Elements of Bivariate Time Series Analysis

A multivariate random process is a set of scalar random processes:
$$\mathbf{x}{t}=\left[x{1, t}, \ldots, x_{M, t}\right]^{\prime},$$
where $t$ is time, $M$ the dimension of the process (the number of scalar components in $\mathbf{x}{t}$ ), and the strike means matrix transposition. The components of $\mathbf{x}{t}$ are characterized with respective scalar and joint PDFs. If all scalar PDFs are Gaussian, the joint PDFs are also Gaussian and the process $\mathbf{x}_{i}$ is Gaussian as well (e.g., Yaglom 1987). The properties of stationarity and ergodicity have the same meaning as in the case of scalar processes. In this part of the book, the time series will be regarded as Gaussian, which often agrees with climate observations (see Privalsky and Yushkov 2018, and Chap. 5) and with many other geophysical processes, especially those that do not contain quasi-periodic components such as daily and seasonal trends. If the process is not Gaussian, its analysis including the second statistical moments (covariance and spectral matrices) remains the same as in the Gaussian case.

The task of analyzing a sample record of a multivariate random process means obtaining the same statistical characteristics as for a scalar process plus the moments of the joint PDF. The quantities that need to be obtained include the multivariate stochastic difference equation, that is, an autoregressive (in our case) model in the time domain, and the spectral matrix in the frequency domain. This and the following chapters up to Chap. 14 are dedicated to the bivariate case $(M=2)$.

## 统计代写|时间序列分析作业代写time series analysis代考|Bivariate Autoregressive Models in Time Domain

The AR model of a bivariate zero mean time series $\mathbf{x}{t}=\left[x{1, t}, x_{2, t}\right]^{\prime}$ is

\begin{aligned} &x_{1, t}=\varphi_{11}^{(1)} x_{1, t-1}+\varphi_{12}^{(1)} x_{2, t-1}+\cdots+\varphi_{11}^{(p)} x_{1, t-p}+\varphi_{12}^{(p)} x_{2, t-p}+a_{1, t} \ &x_{2, t}=\varphi_{21}^{(1)} x_{1, t-1}+\varphi_{22}^{(1)} x_{2, t-1}+\cdots+\varphi_{21}^{(p)} x_{1, t-p}+\varphi_{22}^{(p)} x_{2, t-p}+a_{2, t} \end{aligned}
or
$$\mathbf{x}{t}=\sum{j=1}^{p} \boldsymbol{\Phi}{j} \mathbf{x}{t-j}+\mathbf{a}{t}$$ Here, $p$ is the AR order, $$\boldsymbol{\Phi}{j}=\left[\begin{array}{ll} \varphi_{11}^{(j)} & \varphi_{12}^{(j)} \ \varphi_{21}^{(j)} & \varphi_{22}^{(j)} \end{array}\right]$$
are the matrix AR coefficients and $\mathbf{a}{t}=\left[a{1, t}, a_{2, t}\right]^{\prime}$ is a bivariate white nose innovation sequence with a covariance matrix
$$\mathbf{P}{\mathbf{a}}=\left[\begin{array}{l} \mathrm{P}{11} \mathrm{P}{12} \ \mathrm{P}{21} \mathrm{P}{22} \end{array}\right]$$ where $\mathrm{P}{11}, \mathrm{P}{22}$ are the variances of $a{1, t}, a_{2, t}$ and $\mathrm{P}{12}=\mathrm{P}{21}=\operatorname{cov}\left[a_{1, t}, a_{2, t}\right]$ is their covariance. The notation for multivariate autoregressive model of order $p$ is $\mathbf{A R}(p)$.
Equations (7.4) and (7.5) represent a linear stochastic system, which possesses physically reasonable features:

• a time delay between the current values and system’s past behavior.
• each process may depend upon its own past.
• each process may depend upon the other process’s past.
The time series $x_{1, t}$ and $x_{2, t}$ will be regarded in what follows as the output and input of the linear system described with these equations. The frequency domain analysis and physical considerations allow one to verify in most cases whether this decision is correct. If it turns out that $x_{1, t}$ and $x_{2, t}$ are the input and output, the components of the time series can be interchanged.

The bivariate autoregressive model described with Eqs. (7.4) or (7.5) provides valuable information about properties of the process $\mathbf{x}_{t}$. In particular, it shows

• the memory of the process $\mathbf{x}{f}$, which is defined with the autoregressive order $p$-the number of past values of $\mathbf{x}{f}$ that should be taken into account,
• in what way the time series $x_{1, t}$ depends upon its past values $x_{1, t-1}, \ldots, x_{1, t-p}$ : the coefficients $\varphi_{11}^{(j)}, j=1, \ldots, p$, provide quantitative measures of dependence between the current and past values of time series through their absolute values and their signs; the same is true for the time series $x_{2, t}$ and coefficients $\varphi_{22}^{(j)}$,
• in what way the component $x_{1, t}$ depends upon the past values of the time series $x_{2, t-j}, j=1, \ldots, p$; the coefficients $\varphi_{12}^{(j)}, j=1, \ldots, p$, provide quantitative

measures of dependence between the current value of $x_{1, t}$ and past values $x_{2, t-j}$; similar information about the dependence of $x_{2, t}$ upon $x_{1, t-j}$ is available through the coefficients $\varphi_{21}^{(j)}$ for the second component of the time series,

• the role played by the innovation sequence $\mathbf{a}{f}=\left[a{1, t}, a_{2, t}\right]^{\prime}$; its components $a_{1, t}, a_{2, t}$ do not depend upon their past and constitute the unpredictable part of the bivariate process; therefore, the ratios $\mathrm{P}{11} / \sigma{1}^{2}$ and $\mathrm{P}{22} / \sigma{2}^{2}$ define the statistical predictability (or persistence) of the process’ components $x_{1, t}$ and $x_{2, t}$ at the unit lead time: if the ratio is close to one, the time series predictability and persistence are low,
• the dependence between the innovation sequence components $a_{1, t}$ and $a_{2, t}$ expressed as the cross-correlation coefficient $\rho_{12}=\mathrm{P}{12} / \sqrt{\mathrm{P}{11} \mathrm{P}_{22}}$.

## 统计代写|时间序列分析作业代写time series analysis代考|Bivariate Autoregressive Models in Frequency Domain

The dependence of time series upon time also means that its behavior, including relations between its scalar components, may vary as a function of frequency. This is the reason why studying the frequency domain behavior of any time series, scalar or multivariate, is vital. The frequency-dependent properties of multivariate time series are determined through the spectral matrix $s(f)$ of the time series $\mathbf{x}{f}$. All frequency-dependent quantities can be estimated directly from the time series using nonparametric methods of spectral analysis but when an autoregressive model is used for the time domain it is more appropriate to estimate all required spectral characteristic through this time domain model. Besides, the nonparametric approach would lead to the loss of time domain information. Equation (7.5) can be rewritten as $$\mathbf{x}{t}=\left(\mathbf{I}-\boldsymbol{\Phi}{1} B-\cdots-\boldsymbol{\Phi}{p} B^{p}\right)^{-1} \mathbf{a}{t}$$ where I is a $(p \times p)$ identity matrix. By doing a Fourier transform of the last equation (i.e., by changing the backshift operator $B$ to $\mathrm{e}^{-i 2 \pi f \Delta t}$ ) and finding the square of the modulus of both sides of the equation, one receives the spectral matrix of the time series $\mathbf{x}{t}$ as
$$\mathbf{s}(f)=\frac{2 \mathbf{P}{\mathbf{a}} \Delta t}{\left|\mathbf{I}-\sum{j=1}^{p} \boldsymbol{\Phi}_{j} \mathrm{e}^{-i 2 \pi j f \Delta t}\right|^{2}}, 0 \leq f \leq 1 / 2 \Delta t$$
The elements of the matrix

$$\mathbf{s}(f)=\left[\begin{array}{l} s_{11}(f) s_{12}(f) \ s_{21}(f) s_{22}(f) \end{array}\right]$$
are the spectral densities $s_{11}(f), s_{22}(f)$ while $s_{12}(f)$ and $s_{21}(f)$ are complexly conjugated cross-spectral densities.

The other frequency-dependent quantities that characterize a bivariate random function of time (a bivariate time series) in the frequency domain are the coherence function
$$\gamma_{12}(f)=\frac{\left|s_{12}(f)\right|}{\left[s_{11}(f) s_{22}(f)\right]^{1 / 2}},$$
the coherent spectrum
$$s_{11.2}(f)=\gamma_{12}^{2}(f) s_{11}(f),$$
and the complex-valued frequency response function
$$h_{12}(f)=s_{12}(f) / s_{22}(f)$$
with its gain factor
$$g_{12}(f)=\left|s_{12}(f)\right| / s_{22}(f)$$
and phase factor
$$\phi_{12}(f)=\tan ^{-1}\left{\operatorname{Im}\left[s_{12}(f)\right] / \operatorname{Re}\left[s_{12}(f)\right]\right} .$$
The coherence function is dimensionless. $\operatorname{Here}, \operatorname{Im}(A)$ and $\operatorname{Re}(A)$ are the imaginary and real parts of a complex-valued quantity $A$.

The spectral densities or spectra $s_{11}(f)$ and $s_{22}(f)$ describe the behavior of the output and input scalar processes in the frequency domain. Their estimates obtained for the same time series as a scalar quantity and as a scalar component of a multivariate system may differ. The degree of discrepancy between them depends upon the system’s complexity, time series length, and the orders of the scalar and multivariate autoregressive models.

The coherence function or coherence given with Eq. (7.17) satisfies the condition $0 \leq \gamma_{12}(f) \leq 1$. It is dimensionless, and it describes the degree of linear interdependence between the time series. It usually changes with frequency and can be regarded as a frequency-dependent cross-correlation coefficient between $x_{1, t}$ and $x_{2, t}$.

In the case of time-invariant random vectors, the amount of information contained in vector $x_{1, n}$ about another such vector $x_{2, n}$ (and vice versa) is
$$J=-\log \left(1-r_{12}^{2}\right),$$

## 统计代写|时间序列分析作业代写time series analysis代考|Elements of Bivariate Time Series Analysis

X吨=[X1,吨,…,X米,吨]′,

## 统计代写|时间序列分析作业代写time series analysis代考|Bivariate Autoregressive Models in Time Domain

X吨=∑j=1p披jX吨−j+一种吨这里，p是 AR 订单，披j=[披11(j)披12(j) 披21(j)披22(j)]

• 当前值与系统过去行为之间的时间延迟。
• 每个过程可能取决于它自己的过去。
• 每个进程可能依赖于另一个进程的过去。
时间序列X1,吨和X2,吨在下文中将被视为用这些方程描述的线性系统的输出和输入。频域分析和物理考虑允许在大多数情况下验证该决定是否正确。如果事实证明X1,吨和X2,吨是输入和输出，时间序列的分量可以互换。

• 过程的记忆XF，这是用自回归顺序定义的p-过去值的数量XF应该考虑到的，
• 时间序列以什么方式X1,吨取决于其过去的价值观X1,吨−1,…,X1,吨−p: 系数披11(j),j=1,…,p，通过绝对值和符号提供时间序列当前值和过去值之间依赖性的定量测量；时间序列也是如此X2,吨和系数披22(j),
• 以什么方式组件X1,吨取决于时间序列的过去值X2,吨−j,j=1,…,p; 系数披12(j),j=1,…,p, 提供定量

• 创新序列所起的作用一种F=[一种1,吨,一种2,吨]′; 它的组成部分一种1,吨,一种2,吨不依赖于他们的过去并构成双变量过程的不可预测部分；因此，比率磷11/σ12和磷22/σ22定义过程组件的统计可预测性（或持久性）X1,吨和X2,吨在单位提前期：如果比率接近 1，则时间序列的可预测性和持久性较低，
• 创新序列组件之间的依赖关系一种1,吨和一种2,吨表示为互相关系数ρ12=磷12/磷11磷22.

## 统计代写|时间序列分析作业代写time series analysis代考|Bivariate Autoregressive Models in Frequency Domain

s(F)=2磷一种Δ吨|一世−∑j=1p披j和−一世2圆周率jFΔ吨|2,0≤F≤1/2Δ吨

C12(F)=|s12(F)|[s11(F)s22(F)]1/2,

s11.2(F)=C122(F)s11(F),

H12(F)=s12(F)/s22(F)

G12(F)=|s12(F)|/s22(F)

\phi_{12}(f)=\tan ^{-1}\left{\operatorname{Im}\left[s_{12}(f)\right] / \operatorname{Re}\left[s_{12} (f)\right]\right} 。\phi_{12}(f)=\tan ^{-1}\left{\operatorname{Im}\left[s_{12}(f)\right] / \operatorname{Re}\left[s_{12} (f)\right]\right} 。

Ĵ=−日志⁡(1−r122),

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