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# 统计代写|时间序列分析代写Time-Series Analysis代考|Spectral representations of multivariate time series processes

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## 统计代写|时间序列分析代写Time-Series Analysis代考|Spectral representations of multivariate time series processes

These univariate time series results can be readily generalized to the $m$-dimensional vector process. Let $\mathbf{Z}t=\left[Z{1, t}, Z_{2, t}, \ldots, Z_{m, t}\right]^{\prime}$ be a zero-mean jointly stationary $m$-dimensional vector process with the covariance matrix function, $\mathbf{\Gamma}(k)=\left[\gamma_{i, j}(k)\right]$, the spectral representation of $\mathbf{Z}_t$ is given by
$$\mathbf{Z}t=\int{-\pi}^\pi e^{i \omega t} d \mathbf{U}(\omega)$$
where $d \mathbf{U}(\omega)=\left[d U_1(\omega), d U_2(\omega), \ldots, d U_m(\omega)\right]^{\prime}$ is a $m$-dimensional complex-valued process with $d U_i(\omega)$, for $i=1,2, \ldots, m$, being both orthogonal as well as cross-orthogonal such that
$$E[d \mathbf{U}(\omega)]=\mathbf{0},-\pi \leq \omega \leq \pi$$
and
$$E\left{d \mathbf{U}(\omega)\left[d \mathbf{U}^(\lambda)\right]^{\prime}\right}=\mathbf{0}, \text { for all } \omega \neq \lambda$$ The spectral representation of the covariance matrix function is given by $$\boldsymbol{\Gamma}(k)=\int_{-\pi}^\pi e^{i \omega k} d \mathbf{F}(\omega),$$ where \begin{aligned} d \mathbf{F}(\omega) & =E\left{d \mathbf{U}(\omega)\left[d \mathbf{U}^(\omega)\right]^{\prime}\right} \ & =\left[E\left{d U_i(\omega) d U_j^*(\omega)\right}\right] \ & =\left[d F_{i, j}(\omega)\right], \end{aligned}
and $\mathbf{F}(\omega)$ is the spectral distribution matrix function of $\mathbf{Z}t$. The diagonal elements $F{i, i}(\omega)$ are the spectral distribution functions of the $Z_{i, t}$ and the off-diagonal elements $F_{i, j}(\omega)$ are the crossspectral distribution functions between the $Z_{i, t}$ and the $Z_{j, t}$.

If the covariance matrix function is absolutely summable in the sense that each of the $m \times m$ sequence $\gamma_{i, j}(k)$ is absolutely summable, then the spectrum matrix or the spectral density matrix function exists and is given by
\begin{aligned} \mathbf{f}(\omega) d \omega & =d \mathbf{F}(\omega) \ & =\left[d F_{i, j}(\omega)\right] \ & =\left[f_{i, j}(\omega) d \omega\right] . \end{aligned}

## 统计代写|时间序列分析代写Time-Series Analysis代考|The smoothed spectrum matrix

When $\mathbf{Z}_t$ is a multivariate Gaussian process with mean vector $\mathbf{0}$ and variance-covariance matrix $\boldsymbol{\Sigma}, \widetilde{\mathbf{f}}\left(\omega_p\right)$ has a distribution related to the sample variance-covariance matrix that is known as Wishart distribution with $n$ degrees of freedom, which is the multivariate analog of the Chi-square distribution. We refer readers to Goodman (1963), Hannan (1970), and Brillinger (2002) for further discussion of the properties of the periodogram and Wishart distribution.

Similar to the univariate extension of the sample spectral density discussed in Section 9.1, the sample spectrum matrix or periodogram matrix is also a poor estimate. So, we replace it by the following smoothed spectrum matrix
$$\hat{\mathbf{f}}(\omega)=\left[\hat{f}{i, j}(\omega)\right]$$ where $$\hat{f}{i, i}\left(\omega_p\right)=\hat{f}i\left(\omega_p\right) \sum{k=-M_i}^{M_i} W_i\left(\omega_k\right) \widetilde{f}_{i, i}\left(\omega_p-\omega_k\right),$$

$W_i(\omega)$ is a smoothing function, also known as kernel or spectral window, and $M_i$ is the bandwidth of the spectral window, and
$$\hat{f}{i, j}\left(\omega_p\right)=\sum{k=-M_{i, j}}^{M_{i, j}} W_{i, j}\left(\omega_k\right) \widetilde{f}{i, j}\left(\omega_p-\omega_k\right),$$ where $W{i, j}(\omega)$ is a spectral window, and $M_{i, j}$ is the corresponding bandwidth. Similar to the extension of the univariate smoothed spectrum, the smoothed spectrum matrix can also be approximated by the Wishart distribution.
Once $f_{i, i}(\omega)$ and $f_{i, j}(\omega)$ are estimated, we can estimate the co-spectrum, $c_{i, j}(\omega)$, the quadrature spectrum, $q_{i, j}(\omega)$, the cross-amplitude spectrum, $\alpha_{i, j}(\omega)$, phase spectrum, $\phi_{i, j}(\omega)$, the gain function, $G_{i, j}(\omega)$, and the squared coherency function, $K_{i, j}^2(\omega)$.

Note that the spectrum matrix is the Fourier transform of the covariance function, $\mathbf{\Gamma}(k)=$ $\left[\gamma_{i, j}(k)\right]$, and the sample spectrum matrix is
$$\widetilde{\mathbf{f}}\left(\omega_p\right)=\frac{1}{2 \pi} \sum_{k=-(n-1)}^{(n-1)} \hat{\mathbf{\Gamma}}(k) e^{-i \omega_p k}=\left[\widetilde{f}_{i, j}\left(\omega_p\right)\right] .$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Spectral representations of multivariate time series processes

$\mathbf{F}(\omega)$为$\mathbf{Z}t$的谱分布矩阵函数。其中对角元素$F{i, i}(\omega)$为$Z_{i, t}$的光谱分布函数，非对角元素$F_{i, j}(\omega)$为$Z_{i, t}$与$Z_{j, t}$之间的交叉光谱分布函数。

\begin{aligned} \mathbf{f}(\omega) d \omega & =d \mathbf{F}(\omega) \ & =\left[d F_{i, j}(\omega)\right] \ & =\left[f_{i, j}(\omega) d \omega\right] . \end{aligned}

## 统计代写|时间序列分析代写Time-Series Analysis代考|The smoothed spectrum matrix

$$\hat{\mathbf{f}}(\omega)=\left[\hat{f}{i, j}(\omega)\right]$$ where $$\hat{f}{i, i}\left(\omega_p\right)=\hat{f}i\left(\omega_p\right) \sum{k=-M_i}^{M_i} W_i\left(\omega_k\right) \widetilde{f}_{i, i}\left(\omega_p-\omega_k\right),$$

$W_i(\omega)$ 是平滑函数，又称核函数或谱窗，$M_i$为谱窗的带宽，而
$$\hat{f}{i, j}\left(\omega_p\right)=\sum{k=-M_{i, j}}^{M_{i, j}} W_{i, j}\left(\omega_k\right) \widetilde{f}{i, j}\left(\omega_p-\omega_k\right),$$其中$W{i, j}(\omega)$为光谱窗，$M_{i, j}$为对应带宽。与单变量平滑谱的扩展类似，平滑谱矩阵也可以用Wishart分布来近似。

## MATLAB代写

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