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

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

Given a zero-mean $m$-dimensional time series, $\mathbf{Z}1, \mathbf{Z}_2, \ldots$, and $\mathbf{Z}_n$, its Fourier transform at the Fourier frequencies $\omega_p=2 \pi p / n,-[(n-1) / 2] \leq p \leq[n / 2]$, is $$\mathbf{Y}\left(\omega_p\right)=\frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n \mathbf{Z}_t \exp \left(-i \omega_p t\right) .$$
Then, the $m \times m$ sample spectrum matrix, which is also known as periodogram matrix, is simply the extension of Eqs. (9.9)-(9.11). Thus,

\begin{aligned} \widetilde{\mathbf{f}}\left(\omega_p\right) & =\mathbf{Y}\left(\omega_p\right) \mathbf{Y}^\left(\omega_p\right)=\left|\mathbf{Y}\left(\omega_p\right)\right|^2=\frac{1}{2 \pi n}\left|\sum_{t=1}^n \mathbf{Z}t \exp \left(-i \omega_p t\right)\right|^2 \ & =\frac{1}{2 \pi n}\left[\sum{t=1}^n \mathbf{Z}t \exp \left(-i \omega_p t\right)\right]\left[\sum{r=1}^n \mathbf{Z}r^{\prime} \exp \left(i \omega_p r\right)\right] \ & =\frac{1}{2 \pi n} \sum{t=1}^n \sum_{r=1}^n \mathbf{Z}t \mathbf{Z}_r^{\prime} e^{-i \omega_p(t-r)} \ & =\frac{1}{2 \pi} \sum{k=-(n-1)}^{(n-1)} \hat{\mathbf{\Gamma}}(k) e^{-i \omega_p k} \ & =\frac{1}{2 \pi}\left[\hat{\mathbf{\Gamma}}(0)+2 \sum_{k=1}^{(n-1)} \hat{\boldsymbol{\Gamma}}(k) e^{-i \omega_p k}\right]=\left[\widetilde{f}{i, j}\left(\omega_p\right)\right], \end{aligned} where $$\begin{gathered} \hat{\boldsymbol{\Gamma}}(k)=\left[\hat{\gamma}{i, j}(k)\right], \ \tilde{f}{i, j}\left(\omega_p\right)=\frac{1}{2 \pi} \sum{k=-(n-1)}^{(n-1)} \hat{\gamma}{i, j}(k) e^{-i \omega_p k}=y_i\left(\omega_p\right) y_j^\left(\omega_p\right), \end{gathered}$$
and
$$y_i\left(\omega_p\right)=\frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n Z_{i, t} e^{-i \omega_p t} .$$

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

Developed by Thompson (1982) for univariate processes and extended by Walden (2000) for multivariate processes, multitaper smoothing is another useful way to estimate power spectrum density that balances the bias and variance of nonparametric spectral estimation. The multitaper reduces estimation bias by averaging modified periodograms obtained using a family of mutually orthogonal tapers from the same sample data. Let $h_j(t)$ for $t=1, \ldots, n$ and $j=1, \ldots, n$, be $n$ orthonormal tapers such that
\begin{aligned} & \sum_{t=1}^n h_j^2(t)=1, \text { and } \ & \sum_{t=1}^n h_i(t) h_j(t)=0,(i \neq j) . \end{aligned}
From Eq. (9.45), we note that
\begin{aligned} \widetilde{\mathbf{f}}(\omega) & =\frac{1}{2 \pi} \sum_{k=-(n-1)}^{(n-1)} \hat{\boldsymbol{\Gamma}}(k) e^{-i \omega k} \ & =\frac{1}{\sqrt{2 \pi n}} \sum_{t=1}^n \mathbf{Z}t e^{-i \omega t} \frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n \mathbf{Z}t^{\prime} e^{i \omega t} \ & =\widetilde{\mathbf{Y}}(\omega) \widetilde{\mathbf{Y}}^(\omega), \end{aligned} where $\widetilde{\mathbf{Y}}(\omega)=\frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n \mathbf{Z}t e^{-i \omega t}$ is the discrete Fourier transform of $\mathbf{Z}_t$. The multitaper power spectral estimator at frequency $\omega$ is $$\hat{\mathbf{f}}_M(\omega)=\frac{1}{K} \sum{j=1}^K \hat{\mathbf{Y}}j(\omega) \hat{\mathbf{Y}}_j^(\omega),$$
where $K$ is chosen through the method shown below and $\hat{\mathbf{Y}}_j(\omega)$ is the tapered Fourier transform such that
$$\hat{\mathbf{Y}}_j(\omega)=\frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n h_j(t) \mathbf{Z}_t \exp (-i \omega t) .$$

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

\begin{aligned} \widetilde{\mathbf{f}}\left(\omega_p\right) & =\mathbf{Y}\left(\omega_p\right) \mathbf{Y}^\left(\omega_p\right)=\left|\mathbf{Y}\left(\omega_p\right)\right|^2=\frac{1}{2 \pi n}\left|\sum_{t=1}^n \mathbf{Z}t \exp \left(-i \omega_p t\right)\right|^2 \ & =\frac{1}{2 \pi n}\left[\sum{t=1}^n \mathbf{Z}t \exp \left(-i \omega_p t\right)\right]\left[\sum{r=1}^n \mathbf{Z}r^{\prime} \exp \left(i \omega_p r\right)\right] \ & =\frac{1}{2 \pi n} \sum{t=1}^n \sum_{r=1}^n \mathbf{Z}t \mathbf{Z}r^{\prime} e^{-i \omega_p(t-r)} \ & =\frac{1}{2 \pi} \sum{k=-(n-1)}^{(n-1)} \hat{\mathbf{\Gamma}}(k) e^{-i \omega_p k} \ & =\frac{1}{2 \pi}\left[\hat{\mathbf{\Gamma}}(0)+2 \sum{k=1}^{(n-1)} \hat{\boldsymbol{\Gamma}}(k) e^{-i \omega_p k}\right]=\left[\widetilde{f}{i, j}\left(\omega_p\right)\right], \end{aligned} 在哪里$$\begin{gathered} \hat{\boldsymbol{\Gamma}}(k)=\left[\hat{\gamma}{i, j}(k)\right], \ \tilde{f}{i, j}\left(\omega_p\right)=\frac{1}{2 \pi} \sum{k=-(n-1)}^{(n-1)} \hat{\gamma}{i, j}(k) e^{-i \omega_p k}=y_i\left(\omega_p\right) y_j^\left(\omega_p\right), \end{gathered}$$

$$y_i\left(\omega_p\right)=\frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n Z_{i, t} e^{-i \omega_p t} .$$

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

\begin{aligned} & \sum_{t=1}^n h_j^2(t)=1, \text { and } \ & \sum_{t=1}^n h_i(t) h_j(t)=0,(i \neq j) . \end{aligned}

\begin{aligned} \widetilde{\mathbf{f}}(\omega) & =\frac{1}{2 \pi} \sum_{k=-(n-1)}^{(n-1)} \hat{\boldsymbol{\Gamma}}(k) e^{-i \omega k} \ & =\frac{1}{\sqrt{2 \pi n}} \sum_{t=1}^n \mathbf{Z}t e^{-i \omega t} \frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n \mathbf{Z}t^{\prime} e^{i \omega t} \ & =\widetilde{\mathbf{Y}}(\omega) \widetilde{\mathbf{Y}}^(\omega), \end{aligned}其中$\widetilde{\mathbf{Y}}(\omega)=\frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n \mathbf{Z}t e^{-i \omega t}$是$\mathbf{Z}_t$的离散傅里叶变换。频率$\omega$处的多锥度功率谱估计为$$\hat{\mathbf{f}}_M(\omega)=\frac{1}{K} \sum{j=1}^K \hat{\mathbf{Y}}j(\omega) \hat{\mathbf{Y}}_j^(\omega),$$

$$\hat{\mathbf{Y}}_j(\omega)=\frac{1}{\sqrt{2 \pi n}} \sum{t=1}^n h_j(t) \mathbf{Z}_t \exp (-i \omega t) .$$

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