28 多元平稳序列介绍

28.1 多维平稳序列的概念

沿时间变化的量经常有多个, 互相之间有相关性。 如第1章的北京地区洪涝灾害受灾面积\(X_t\)和成灾面积\(Y_t\)有正相关, 可以看成向量值时间序列 \[\begin{aligned} (X_t, Y_t), \ t \in \mathbb N \end{aligned}\]

向量值的时间序列称为多维(多元)时间序列。 仅介绍多维平稳序列。

定义28.1 (多维平稳序列) \(m\)维随机序列 \(\boldsymbol X_t = (X_{1t}, X_{2t}, \dots, X_{mt})^T\), \(t \in \mathbb Z\)平稳序列, 如果对任何\(t, n \in \mathbb Z\),

  • (1) \(E \boldsymbol X_t = \boldsymbol \mu = (\mu_1, \mu_2, \dots, \mu_m)^T\)\(t\)无关;
  • (2) \(\Gamma(n) = E[(\boldsymbol X_{t+n} - \boldsymbol \mu)(\boldsymbol X_t - \boldsymbol \mu)^T]\)\(t\)无关。

这时称\(\{\Gamma(n), n \in \mathbb Z \}\)为平稳序列\(\{ \boldsymbol X_t \}\)的自协方差函数(矩阵)。

定义 \[\begin{align} \gamma_{jk}(n) =& E[(X_{j,t+n} - \mu_j) (X_{kt} - \mu_k)] \tag{28.1} \end{align}\]\[\begin{align} \Gamma(n) =& (\gamma_{jk}(n))_{j,k=1,2,\dots,m}. \tag{28.2} \end{align}\]

定义自相关系数 \[\begin{align} \rho_{jk}(n) = \frac{\gamma_{jk}(n)}{\sqrt{ \gamma_{jj}(0) \gamma_{kk}(0) }} \tag{28.3} \end{align}\]\[\begin{aligned} R(n) = (\rho_{jk}(n))_{j,k=1,2,\dots,m} \end{aligned}\]\(\{ \boldsymbol X_t \}\)的自相关系数(矩阵)。

\(\{ \boldsymbol X_t \}\)标准化得 \[\begin{aligned} \boldsymbol Y_t =& \left( \frac{X_{1t} - \mu_1}{\sqrt{\gamma_{11}(0)}}, \frac{X_{2t} - \mu_2}{\sqrt{\gamma_{22}(0)}}, \dots, \frac{X_{mt} - \mu_m}{\sqrt{\gamma_{mm}(0)}} \right)^T \end{aligned}\]\(\{ R(n) \}\)是平稳序列\(\{ \boldsymbol Y_t \}\)的自协方差函数。

定理28.1 对任何\(n \in \mathbb Z\),

  • (1) \(\Gamma(-n) = [\Gamma(n)]^T\);
  • (2) \(|\gamma_{jk}(n)| \leq [ \gamma_{jj}(0) \gamma_{kk}(0) ]^{1/2}\);
  • (3) 非负定性:对任何实向量 \(\boldsymbol\alpha_1, \boldsymbol\alpha_2, \dots, \boldsymbol\alpha_n \in \mathbb R^m\), \[\begin{aligned} \sum_{k=1}^n \sum_{j=1}^n \boldsymbol\alpha_k^T \Gamma(k-j) \boldsymbol\alpha_j \geq 0. \end{aligned}\]

这些性质\(\{R(n)\}\)也满足。

证明: (1)  \[\begin{aligned} \Gamma(-n) =& E\left[(\boldsymbol X_{t-n} - \boldsymbol \mu)(\boldsymbol X_t - \boldsymbol \mu)^T\right] \\ =& \left\{ E \left[ (\boldsymbol X_t - \boldsymbol \mu) (\boldsymbol X_{t-n} - \boldsymbol \mu)^T \right]\right\}^T \\ =& \left[ \Gamma(n) \right]^T \end{aligned}\]

(2) 这就是Schwarz不等式。

(3) 记\(\boldsymbol Y_t = \boldsymbol X_t - \boldsymbol \mu\), 则 \[\begin{aligned} & E \left[ \sum_{j=1}^n \boldsymbol\alpha_j^T \boldsymbol Y_j \right]^2 \\ =& E \sum_{k=1}^n \sum_{j=1}^n (\alpha_k^T \boldsymbol Y_k) (\alpha_j^T \boldsymbol Y_j) \\ =& \sum_{k=1}^n \sum_{j=1}^n \boldsymbol\alpha_k^T E[\boldsymbol Y_k \boldsymbol Y_j^T] \boldsymbol\alpha_j \\ = & \sum_{k=1}^n \sum_{j=1}^n \boldsymbol\alpha_k^T \Gamma(k-j) \boldsymbol\alpha_j \geq 0 \end{aligned}\]

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例28.1 (多维白噪声) \(m\)维平稳序列\(\{\boldsymbol X_t \}\)满足 \[\begin{aligned} E \boldsymbol X_1 = \boldsymbol \mu, \quad \Gamma(n) = Q \delta_n \end{aligned}\] 就称\(\{\boldsymbol X_t \}\)\(m\)维白噪声, 简记为WN\((\boldsymbol\mu, Q)\)

这时\(\forall k, j\), 只要\(n \neq 0\),就有 \[\begin{aligned} E[(X_{kt} - \mu_k)(X_{j,t+n} - \mu_j)] = 0 \end{aligned}\] 另外,每个分量是一维白噪声。 特别地,当\(Q\)是对角阵时, 各分量是\(m\)个互不相关的白噪声。

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例28.2 (多维线性平稳列) \(\{ \boldsymbol\varepsilon_t, t\in \mathbb Z \}\)\(m\)维WN(0,\(Q\)), \(Q = E[\boldsymbol\varepsilon_t \boldsymbol\varepsilon_t^T]\)。 如果实矩阵列\(\{ C_j \}\)满足 \[\begin{aligned} \sum_{j=-\infty}^\infty C_j Q C_j^T < \infty \end{aligned}\] 就称 \[\begin{align} \boldsymbol X_t =& \sum_{j=-\infty}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \tag{28.4} \end{align}\]\(m\)维平稳线性序列

这时可以证明(28.4)中每个分量都是均方收敛的,并且 \[\begin{align} E \boldsymbol X_t =& 0, \\ \Gamma(n) =& E[\boldsymbol X_{t+n} \boldsymbol X_t^T] = \sum_{j=-\infty}^\infty C_{j+n} Q C_j^T \tag{28.5} \end{align}\]

例28.3 (多维MA模型) \(\{ \boldsymbol\varepsilon_t \}\)\(m\)维WN(0,\(Q\))。 如果\(B_1, B_2, \dots, B_q\)\(m \times m\)实矩阵, 满足 \[\begin{aligned} \mbox{det}(I_m + B_1 z + B_2 z^2 + \dots + B_q z^q) \neq 0, \ |z| < 1 \end{aligned}\] 就称 \[\begin{aligned} \boldsymbol X_t = \boldsymbol\varepsilon_t + B_1 \boldsymbol\varepsilon_{t-1} + \dots + B_q \boldsymbol\varepsilon_{t-q}, \ t \in \mathbb Z \end{aligned}\] 为一个\(m\)维MA(\(q\))序列

若进一步要求 \[\begin{aligned} \mbox{det}(I_m + B_1 z + B_2 z^2 + \dots + B_q z^q) \neq 0, \ |z| \leq 1 \end{aligned}\] 则称\(\{ \boldsymbol X_t \}\)为一个可逆的MA(\(q\))序列。

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矩阵系数多项式:

\(A_1, A_2, \dots, A_p, B_1, B_2, \dots, B_q\)\(p+q\)\(m\times m\)实矩阵,记 \[\begin{aligned} A(z) =& I_m - A_1 z - A_2 z^2 - \dots - A_p z^p \\ B(z) =& I_m + B_1 z + B_2 z^2 + \dots + B_q z^q \end{aligned}\] 形如\(A(z), B(z)\)的多项式被称为矩阵系数多项式, 实际是\(m^2\)维向量值函数, 每个分量是多项式。

如果对任意满足 \[\begin{aligned} A(z)= C(z) A_1(z), \ B(z) = C(z) B_1(z) \end{aligned}\]\(m \times m\)矩阵系数多项式\(C(z)\)必有\(\mbox{det}(C(Z))=\)常数, 就称\(A(z)\)\(B(z)\)左互素的。

例28.4 (多维ARMA的不可识别性) \(\{ \boldsymbol\varepsilon_t \}\)是WN(0,\(Q\))。 称\(m\)维平稳序列\(\{ \boldsymbol X_t \}\)满足\(m\)维平稳可逆的ARMA(\(p,q\))模型, 如果对任何\(t \in \mathbb Z\)\[\begin{align} \boldsymbol X_t = \sum_{j=1}^p A_j \boldsymbol X_{t-j} + \varepsilon_t + \sum_{j=1}^q B_j \varepsilon_{t-j} \tag{28.6} \end{align}\] 其中多项式\(A(z), B(z)\)满足 \[\begin{aligned} (1) & A(z), B(z) \text{左互素}; \\ (2) & \mbox{det}(A(z)B(z)) \neq 0, \ |z| \leq 1. \end{aligned}\]

用推移算子把模型方程写成 \[\begin{align} A(\mathscr B) \boldsymbol X_t = B(\mathscr B) \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.7} \end{align}\]

对于一维ARMA,自协方差函数可以唯一决定模型参数; 对于多维ARMA不能。

在模型(28.7)中, 如果\(\mbox{det}(A(z)) = c\)是常数, 则\(A^{-1}(z)\)仍然是一个矩阵多项式,并且 \(\mbox{det}(A^{-1}(z))=c^{-1}\)。于是 \[\begin{aligned} \boldsymbol X_t = A^{-1}(\mathscr B) B(\mathscr B) \varepsilon_t = B_1(\mathscr B) \varepsilon_t, \ t \in \mathbb Z \end{aligned}\] 其中\(B_1(z)=A^{-1}(z)B(z)\)是矩阵系数多项式,满足 \[\begin{aligned} \mbox{det}(B_1(z)) = c^{-1} \mbox{det}(B(z)) \neq 0, \ |z| \leq 1 \end{aligned}\] 就有了两个模型。 所以ARMA模型参数是不唯一的。

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28.2 多维平稳序列的均值和自协方差函数的估计

28.2.1 均值的估计

\(\{ \boldsymbol X_t \}\)\(m\)维平稳序列, \(\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N\)是观测值。 均值\(\boldsymbol\mu\)的估计为 \[\begin{aligned} \hat{\boldsymbol\mu}_N = (\hat\mu_1, \hat\mu_2, \dots, \hat\mu_m)^T = \frac{1}{N} \sum_{t=1}^N \boldsymbol X_t. \end{aligned}\]

由分量的相应结论可得:

定理28.2 如果\(\{ \boldsymbol X_t \}\)的每个分量序列 \(\{ X_{jt}: t\in\mathbb Z \}\)都是严平稳遍历序列,则当\(N\to\infty\)时, \[\begin{aligned} \hat{\boldsymbol\mu}_N \to \boldsymbol\mu, \ \text{a.s.} \end{aligned}\]

定理28.3 如果自协方差函数\(\Gamma(n) \to 0\), 当\(n \to\infty\),则 \[\begin{aligned} E|\hat{\boldsymbol\mu}_N - \hat{\boldsymbol\mu}|^2 \to 0, \ \text{当$N\to\infty$时} \end{aligned}\] 其中\(|\hat{\boldsymbol\mu}_N - \hat{\boldsymbol\mu}|^2 = \sum_{j=1}^m (\hat\mu_j - \mu_j)^2\)

这是均方收敛,推出依概率收敛, 即\(\hat{\boldsymbol\mu}_N\)\(\boldsymbol\mu\)的相合估计。

证明\[\begin{aligned} E |\hat{\boldsymbol\mu}_N - \hat{\boldsymbol\mu}|^2 =& \sum_{j=1}^m E (\hat\mu_j - \mu_j)^2 \\ =& \sum_{j=1}^m E \left( \frac{1}{N} \sum_{t=1}^N X_{jt} - \mu_j \right)^2 \\ =& \sum_{j=1}^m \frac{1}{N^2} E \left( \sum_{t=1}^N (X_{jt} - \mu_j) \right)^2 \\ =& \sum_{j=1}^m \frac{1}{N^2} \sum_{l=1}^N \sum_{k=1}^N \gamma_{jj}(l-k) \\ =& \sum_{j=1}^m \frac{1}{N^2} \sum_{k=1-N}^{N-1}(N - |k|) \gamma_{jj}(k) \\ \leq& \sum_{j=1}^m \frac{1}{N} \sum_{k=1-N}^{N-1} |\gamma_{jj}(k)| \to 0, \ (N\to\infty) \end{aligned}\]

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定理28.4 如果 \[\begin{aligned} \sum_{k=0}^\infty | \gamma_{jj}(k) | < \infty, \ j=1,2,\dots,m \end{aligned}\] 则当\(N\to\infty\)\[\begin{aligned} N E | \hat{\boldsymbol\mu}_N - \boldsymbol\mu |^2 \to \sum_{j=1}^m \sum_{k=-\infty}^\infty \gamma_{jj}(k). \end{aligned}\]

证明 按定理28.3的证明,有 \[\begin{aligned} & N E | \hat{\boldsymbol\mu}_N - \boldsymbol\mu |^2 \\ =& \sum_{j=1}^m \frac{1}{N} \sum_{k=1-N}^{N-1}(N - |k|) \gamma_{jj}(k) \\ =& \sum_{j=1}^m \sum_{k=1-N}^{N-1} \gamma_{jj}(k) - \sum_{j=1}^m \frac{1}{N} \sum_{k=1-N}^{N-1} |k| \gamma_{jj}(k) \\ \to& \sum_{j=1}^m \sum_{k=-\infty}^{\infty} \gamma_{jj}(k). \end{aligned}\]

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定理28.5 \(\{\boldsymbol\varepsilon_t \}\)\(m\)维独立同分布的 WN(\(\boldsymbol\mu, Q\)), \(m \times m\)矩阵\(C_n = (c_{j,k}(n))\)的每个元素\(\{c_{j,k}(n)\}\)\(n\in\mathbb Z\)绝对可和, \(m\)维平稳序列\(\{ \boldsymbol X_t \}\)为线性平稳列 \[\begin{aligned} \boldsymbol X_t =& \sum_{j=-\infty}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \end{aligned}\] 如果 \[\begin{aligned} \Sigma =& \left( \sum_{j=-\infty}^\infty C_j \right) Q \left( \sum_{j=-\infty}^\infty C_j^T \right) \neq 0 \end{aligned}\]\[\begin{aligned} \sqrt{N}(\hat{\boldsymbol\mu}_N - \boldsymbol\mu) \stackrel{\mbox{d}}{\longrightarrow} \text{N}(0,\Sigma) \end{aligned}\]

(Brockwell and Davis 1987)

28.2.2 自协方差函数的估计

\(\{ \boldsymbol X_t \}\)\(m\)维平稳序列, \(\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N\)是观测值。 自协方差函数\(\Gamma(n)\)的估计为 \[\begin{aligned} \begin{cases} \hat\Gamma(n) = \frac{1}{N} \sum_{t=1}^{N-n} ( \boldsymbol X_{t+n} - \hat{\boldsymbol\mu}_N) ( \boldsymbol X_{t} - \hat{\boldsymbol\mu}_N)^T, & 0 \leq n \leq N-1 \\ \hat\Gamma(-n) = \hat\Gamma^T(n), & 1 \leq n \leq N-1 \end{cases} \end{aligned}\] 是一维情况的推广,具有良好统计性质。

\(\hat\gamma_{jk}(n)\)\(\hat\Gamma(n)\)的第\((j,k)\)元素, 则相关系数估计为 \[\begin{align} \hat\rho_{jk}(n) = \frac{\hat\gamma_{jk}(n)} {\sqrt{\hat\gamma_{jj}(0) \hat\gamma_{kk}(0)}} \tag{28.8} \end{align}\] 自相关系数矩阵的估计为 \[\begin{align} \hat R(n) = (\hat\rho_{jk}(n)). \tag{28.9} \end{align}\]

利用严平稳序列的遍历定理可得:

定理28.6 在定理28.5的条件下, 对固定的\(n\),当\(N\to\infty\)\[\begin{aligned} \hat\Gamma(n) \to \Gamma(n),\ \text{a.s.}, \quad \hat R(n) \to R(n), \ \text{a.s.} \end{aligned}\]

关于\(\hat R(n)\)的渐近分布有(见(Brockwell and Davis 1987)):

定理28.7 \(\{ Z_{1t} \}\)\(\{ Z_{2t} \}\)都是一维独立同分布的零均值白噪声, 彼此相互独立, \(\{a_j \}\)\(\{ b_j \}\)是绝对可和的实数列, \[\begin{aligned} X_{1t} =& \sum_{j=-\infty}^\infty a_j Z_{1,t-j}, \ t \in \mathbb N_+, \\ X_{2t} =& \sum_{j=-\infty}^\infty b_j Z_{2,t-j}, \ t \in \mathbb N_+, \end{aligned}\]

  • (1) 对\(k \geq 0\), 当\(N \to \infty\)时, \[\begin{aligned} \sqrt{N}\hat\rho_{12}(k) \stackrel{\text{d}}{\longrightarrow} \text{N}(0,\sigma_{11}^2) \end{aligned}\]
  • (2) 对\(h, k \geq 0\)\(h \neq k\)\[\begin{aligned} \sqrt{N}( \hat\rho_{12}(h), \hat\rho_{12}(k)) \stackrel{\text{d}}{\longrightarrow} \text{N}(0, \Sigma) \end{aligned}\] 其中 \[\begin{aligned} \Sigma =& \left(\begin{array}{cc} \sigma_{11}^2 & \sigma_{12}^2 \\ \sigma_{12}^2 & \sigma_{22}^2 \end{array}\right) \\ \sigma_{11}^2 =& \sum_{j=-\infty}^\infty \rho_{11}(j) \rho_{22}(j) \\ \sigma_{12}^2 =& \sum_{j=-\infty}^\infty \rho_{11}(j) \rho_{22}(j+k-h). \\ \end{aligned}\]

28.3 VAR模型

定义28.2 \(\{ \boldsymbol\varepsilon_t \}\)\(m\)维WN(0,\(Q\)), \(A_1, A_2, \dots, A_p\)\(m \times m\)实矩阵,使得 \[\begin{align} \det\left( I_m - \sum_{j=1}^p A_j z^j \right) \neq 0, \ |z| \leq 1 \tag{28.10} \end{align}\] 称如下的模型 \[\begin{align} \boldsymbol X_t = \sum_{j=1}^p A_j \boldsymbol X_{t-j} + \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.11} \end{align}\] 是一个\(m\)维AR(\(p\))模型; 如果平稳序列\(\{ \boldsymbol X_t \}\)满足\(m\)维AR(\(p\))模型, 就称\(\{ \boldsymbol X_t \}\)是一个\(m\)维的AR(\(p\))序列。

\[\begin{aligned} A(z) = I_m - \sum_{j=1}^p A_j z^j \end{aligned}\] 模型为 \[\begin{align} A(\mathscr B) \boldsymbol X_t = \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.12} \end{align}\]\(A(z)\)的伴随矩阵表示\(A^{-1}(z)\)后, 可知\(A^{-1}(z)\)的每个元素在\(|z| \leq 1\)有泰勒展式, 于是 \[\begin{align} A^{-1}(z) = \sum_{j=0}^\infty C_j z^j, \ |z| \leq 1. \tag{28.13} \end{align}\] 其中\(C_n=(c_{jk}(n))\), \(\{c_{jk}(n), n \in \mathbb N_+ \}\)以负指数速度趋于零。 类似一维情况可得 \[\begin{align} \boldsymbol X_t =& A^{-1}(\mathscr B) \boldsymbol\varepsilon_t = \sum_{j=0}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \tag{28.14} \end{align}\] 是模型(28.11)的唯一平稳解。

例28.5 (一维AR的马氏化) 将一维AR(\(p\))序列表示成VAR。

一维AR(\(p\)) \[\begin{align} X_t =& \sum_{j=1}^p a_j X_{t-j} + \varepsilon_t, \ t \in \mathbb Z, \tag{28.15} \\ \{ \varepsilon_t \} & \text{为WN}(0,\sigma^2) \end{align}\]\[\begin{aligned} \boldsymbol X_t =& (X_t, X_{t-1}, \dots, X_{t-p+1})^T \\ \boldsymbol \varepsilon_t =& (\varepsilon_t, 0, \dots, 0)^T \\ A =& \left(\begin{array}{ccccc} a_1 & a_2 & \cdots & a_{p-1} & a_p \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \end{array}\right) \end{aligned}\] 则得到\(p\)维AR(1)模型: \[\begin{align} \boldsymbol X_t = A \boldsymbol X_{t-1} + \boldsymbol\varepsilon_t, \ t \in \mathbb Z. \tag{28.16} \\ \end{align}\] 其中 \[\begin{aligned} & \det(I_p - Az) \\ =& \det\left(\begin{array}{ccccc} 1-a_1z & -a_2z & \cdots & -a_{p-1}z & -a_pz \\ -z & 1 & \cdots & 0 & 0 \\ 0 & -z & \cdots & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & -z & 1 \end{array}\right) \\ =& 1 - a_1 z - a_2 z^2 - \dots - a_p z^p \neq 0, \ |z| \leq 1. \end{aligned}\] 计算上面行列式时,按\(j=2, 3, \dots, p\)的顺序把第\(j\)列乘以\(z^{j-1}\)后加到第一列, 则第一列只有\((1,1)\)元素非零。

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28.3.1 多维AR序列的自协方差函数

由平稳解(28.14)可知 \[\begin{aligned} E[ \boldsymbol X_t \boldsymbol\varepsilon_{t+n}^T] = 0, n \geq 1. \end{aligned}\] 这体现出模型有因果性。 于是对\(n\geq 0\) \[\begin{align} \Gamma(n) =& E(\boldsymbol X_{t+n} \boldsymbol X_t^T) \\ =& E \left[ \left( \sum_{j=1}^p A_j \boldsymbol X_{t+n-j} + \boldsymbol\varepsilon_{t+n} \right) \boldsymbol X_t^T \right] \\ =& \sum_{j=1}^p A_j \Gamma(n-j) + E(\boldsymbol\varepsilon_{t+n} \boldsymbol X_t^T) \tag{28.17} \end{align}\] 从而 \[\begin{align} A(\mathscr B) \Gamma(n) = 0, \ n \geq 1. \tag{28.18} \end{align}\]

另外 \[\begin{align} E(\boldsymbol\varepsilon_t \boldsymbol X_t^T) = E \left( \sum_{j=0}^\infty \boldsymbol\varepsilon_t \boldsymbol\varepsilon_{t-j}^T C_j^T \right) = Q C_0^T = Q \tag{28.19} \end{align}\]

总之有\(m\)维情况下自协方差函数矩阵的Yule-Walker方程 \[\begin{align} \begin{cases} \Gamma(0) = \sum_{j=1}^p A_j \Gamma(-j) + Q, &\\ \Gamma(n) = \sum_{j=1}^p A_j \Gamma(n-j), & n \geq 1 \\ \end{cases} \tag{28.20} \end{align}\]

(28.20)的第二式写成分块矩阵形式: \[\begin{align} \left(\begin{array}{c} \Gamma^T(1) \\ \Gamma^T(2) \\ \vdots \\ \Gamma^T(n) \end{array}\right) = \left(\begin{array}{cccc} \Gamma(0) & \Gamma(1) & \cdots & \Gamma(p-1) \\ \Gamma(-1) & \Gamma(0) & \cdots & \Gamma(p-2) \\ \vdots & \vdots & & \vdots \\ \Gamma(-p+1) & \Gamma(-p+2) & \cdots & \Gamma(0) \end{array}\right) \left(\begin{array}{c} A_1^T \\ A_2^T \\ \vdots \\ A_p^T \end{array}\right) \tag{28.21} \end{align}\] 系数矩阵是向量 \[\begin{aligned} (\boldsymbol X_p^T, \boldsymbol X_{p-1}^T, \dots, \boldsymbol X_1^T)^T \end{aligned}\] 的协方差矩阵,如果它正定则\(A_1, A_2, \dots, A_p\)\(Q\)可以由 \[\begin{aligned} \Gamma(0), \Gamma(1), \dots, \Gamma(p) \end{aligned}\] 唯一决定,且满足最小相位条件(28.10)

28.3.2 多维AR序列的参数估计

28.3.2.1 Y-W方法

在Y-W方程(28.20)中, 把\(\Gamma(n)\)用样本自协方差函数\(\hat\Gamma(n)\)代替, 得到样本Y-W方程,可解得模型的Y-W估计 \[\begin{aligned} (\hat A_1, \hat A_2, \dots, \hat A_p, \hat Q). \end{aligned}\] 解法有类似一维时Levinson递推那样的递推公式, 见(谢衷洁 1990)。 只要Y-W方程中的系数矩阵正定, 则Y-W估计满足最小相位条件(28.10)

28.3.2.2 最小二乘

把观测值\(\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N\)满足的模型写成 \[\begin{align} \boldsymbol X_t^T = \sum_{j=1}^p \boldsymbol X_{t-j}^T A_j^T + \boldsymbol\varepsilon_t^T, \ t=p+1, p+2, \dots, N \tag{28.22} \end{align}\] 引入 \[\begin{aligned} \boldsymbol X_n =& \left(\begin{array}{cccc} \boldsymbol X_p^T & \boldsymbol X_{p-1}^T & \cdots & \boldsymbol X_1^T \\ \boldsymbol X_{p+1}^T & \boldsymbol X_p^T & \cdots & \boldsymbol X_2^T \\ \vdots & \vdots & & \vdots \\ \boldsymbol X_{n-1}^T & \boldsymbol X_{n-2}^T & \cdots & \boldsymbol X_{n-p} \end{array}\right)_{(n-p)\times mp} \\ \boldsymbol Y_n =& \left(\begin{array}{c} \boldsymbol X_{p+1}^T \\ \boldsymbol X_{p+2}^T \\ \vdots \\ \boldsymbol X_n^T \end{array}\right)_{(n-p)\times m} \ \boldsymbol E_n = \left(\begin{array}{c} \boldsymbol \varepsilon_{p+1}^T \\ \boldsymbol \varepsilon_{p+1}^T \\ \vdots \\ \boldsymbol \varepsilon_n^T \end{array}\right)_{(n-p)\times m} \\ \boldsymbol A =& \left(\begin{array}{c} \boldsymbol A_1^T \\ \boldsymbol A_2^T \\ \vdots \\ \boldsymbol X_p^T \end{array}\right)_{mp\times m} \end{aligned}\] 可以把(28.22)写成 \[\begin{aligned} \boldsymbol Y_n = \boldsymbol X_n \boldsymbol A + \boldsymbol E_n \end{aligned}\]\(A_1, A_2, \dots, A_p\)的最小二乘估计\(\hat{\boldsymbol A}\), 就是求\(\hat{\boldsymbol A}\)使得 \[\begin{aligned} S(\boldsymbol A) = (\boldsymbol Y_n - \boldsymbol X_n \boldsymbol A)^T (\boldsymbol Y_n - \boldsymbol X_n \boldsymbol A) \end{aligned}\] 最小。 这里\(S(\boldsymbol A)\)\(m \times m\)矩阵。对两个对称矩阵\(A\)\(B\), 当且仅当\(A-B\)非负定且不等于零矩阵时称\(A>B\)\(\hat{\boldsymbol A}\)满足方程 \[\begin{aligned} (\boldsymbol X_n^T \boldsymbol X_n) \hat{\boldsymbol A} = \boldsymbol X_n^T \boldsymbol Y_n \end{aligned}\]\((\boldsymbol X_n^T \boldsymbol X_n)\)可逆时 \[\begin{aligned} \hat{\boldsymbol A} = (\boldsymbol X_n^T \boldsymbol X_n)^{-1} \boldsymbol X_n^T \boldsymbol Y_n \end{aligned}\]

事实上,若\(\hat{\boldsymbol A}\)满足下式 \[\begin{aligned} (\boldsymbol X_n^T \boldsymbol X_n) \hat{\boldsymbol A} = \boldsymbol X_n^T \boldsymbol Y_n \end{aligned}\] 则对任何与\(\hat{\boldsymbol A}\)同阶的\(\boldsymbol B\),有 \[\begin{aligned} & (\boldsymbol Y_n - \boldsymbol X_n \boldsymbol B)^T(\boldsymbol Y_n - \boldsymbol X_n \boldsymbol B) \\ =& (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A} + \boldsymbol X_n \hat{\boldsymbol A} - \boldsymbol X_n \boldsymbol B )^T \\ & \cdot (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A} + \boldsymbol X_n \hat{\boldsymbol A} - \boldsymbol X_n \boldsymbol B ) \\ =& (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A})^T(\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A}) + (\boldsymbol X_n (\hat{\boldsymbol A} - \boldsymbol B))^T (\boldsymbol X_n (\hat{\boldsymbol A} - \boldsymbol B)) \\ \geq& (\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A})^T(\boldsymbol Y_n - \boldsymbol X_n \hat{\boldsymbol A}) \end{aligned}\]

28.3.3 VAR的预测

28.5证明了一维AR(\(p\))模型可以化为\(p\)维AR(1)模型。 进一步地,任何一个\(m\)维AR(\(p\))模型可以写成一个\(mp\)维的AR(1)模型。 考虑\(m\)维AR(\(p\))模型 \[\begin{align} \boldsymbol X_t = \sum_{j=1}^p A_j \boldsymbol X_{t-j} + \boldsymbol\varepsilon_t, \ t \in \mathbb Z \tag{28.23} \end{align}\]\[\begin{aligned} \boldsymbol Y_t =& \left(\begin{array}{c} \boldsymbol X_t \\ \boldsymbol X_{t-1} \\ \vdots \\ \boldsymbol X_{t-p+1} \end{array}\right)_{mp \times 1}, \quad \boldsymbol \eta_t = \left(\begin{array}{c} \boldsymbol \varepsilon_t \\ \boldsymbol 0 \\ \vdots \\ \boldsymbol 0 \end{array}\right)_{mp \times 1} \\ \boldsymbol A =& \left(\begin{array}{ccccc} A_1 & A_2 & \cdots & A_{p-1} & A_p \\ I_m & \boldsymbol 0 & \cdots & \boldsymbol 0 & \boldsymbol 0 \\ \boldsymbol 0 & I_m & \cdots & \boldsymbol 0 & \boldsymbol 0 \\ \vdots & \vdots & & \vdots & \vdots \\ \boldsymbol 0 & \boldsymbol 0 & \cdots & I_m & \boldsymbol 0 \end{array}\right)_{mp \times mp} \end{aligned}\]\(m\)维AR(\(p\))模型(28.23)可以写成\(mp\)维AR(1): \[\begin{aligned} \boldsymbol Y_t = \boldsymbol A \boldsymbol Y_{t-1} + \boldsymbol \eta_t, \ t \in \mathbb Z \end{aligned}\] 其中,系数矩阵\(\boldsymbol A\)满足 \[\begin{align} & \det(I_p - \boldsymbol A z) \\ =& \det\left(\begin{array}{ccccc} I_p - A_1z & -A_2z & \cdots & -A_{p-1}z & -A_pz \\ -I_p z & I_p & \cdots & 0 & 0 \\ 0 & -I_p z & \cdots & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & -I_p z & I_p \end{array}\right) \\ =& \det(I_p - A_1 z - A_2 z^2 - \dots - A_p z^p) \neq 0, \ |z| \leq 1. \tag{28.24} \end{align}\]\(\{ \boldsymbol \eta_t \}\)\(mp\)维零均值白噪声。

于是,只需要研究多维AR(1)的预报问题。 对\(m\)维随机向量\(\boldsymbol Y=(Y_1, Y_2, \dots, Y_m)^T\)定义 \[\begin{aligned} L(\boldsymbol Y | \boldsymbol Z) = [ L(Y_1 | \boldsymbol Z), L(Y_2 | \boldsymbol Z), \dots, L(Y_m | \boldsymbol Z) ] \end{aligned}\] 其中\(L(Y_k | \boldsymbol Z)\)\(\boldsymbol Z\)\(Y_k\)的最佳线性预测。 设\(\{ \boldsymbol X_t \}\)\(m\)维AR(1)序列,满足 \[\begin{aligned} \boldsymbol X_t = A \boldsymbol X_{t-1} + \boldsymbol \varepsilon_t, \ t \in \mathbb Z. \end{aligned}\] 考虑用\(\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n\)\(\boldsymbol X_{n+k}\)进行最佳线性预测。

利用平稳解的因果性, 即\(E(\boldsymbol X_t \boldsymbol \varepsilon_{n+k})=0\)(\(k \geq 1\))可得 \[\begin{aligned} & L(\boldsymbol X_{n+1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_n + \boldsymbol\varepsilon_{n+1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_n | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& A \, L(\boldsymbol X_n | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& A \boldsymbol X_n \end{aligned}\]\(k \geq 1\)\[\begin{aligned} & L(\boldsymbol X_{n+k} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_{n+k-1} + \boldsymbol\varepsilon_{n+k} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(A \boldsymbol X_{n+k-1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& A \, L(\boldsymbol X_{n+k-1} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& \cdots = A^k \boldsymbol X_n \end{aligned}\]

总之有 \[\begin{aligned} & L(\boldsymbol X_{n+k} | \boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_n) \\ =& L(\boldsymbol X_{n+k} | \boldsymbol X_n) = A^k \boldsymbol X_n, \ n,k \in \mathbb N_+ \end{aligned}\]

28.4 多维平稳序列的谱分析

28.4.1 多维平稳序列的谱函数

\(\{ \boldsymbol X_t = (X_{1t}, X_{2t})^T: \ t \in \mathbb Z \}\)是一个2维零均值平稳序列。 对给定复数\(z\)定义 \[\begin{aligned} Y_t = X_{1t} + z X_{2t}, \ t \in \mathbb Z \end{aligned}\] 易见\(E Y_t = 0\), \[\begin{aligned} \gamma_z(k) \stackrel{\triangle}{=}& E(Y_{t+k} \bar Y_t) = E[ (X_{1,t+k} + z X_{2,t+k}) (X_{1t} + \bar z X_{2t}) ] \\ =& \gamma_{11}(k) + z \gamma_{21}(k) + \bar z \gamma_{12}(k) + |z|^2 \gamma_{22}(k) \end{aligned}\] 都不依赖于\(t\),所以\(\{ Y_t \}\)是一个复值平稳序列。 设\(\{ Y_t \}\)有谱函数\(F_z\), 则对\(n \in \mathbb Z\), \[\begin{aligned} \gamma_z(n) = \int_{-\pi}^\pi e^{in\lambda} d F_z(\lambda), \ F_z(-\pi) = 0. \end{aligned}\]

\(z=\pm 1, \pm i\)时得 \[\begin{aligned} \gamma_1(n) =& \gamma_{11}(n) + \gamma_{21}(n) + \gamma_{12}(n) + \gamma_{22}(n) \\ \gamma_{-1}(n) =& \gamma_{11}(n) - \gamma_{21}(n) - \gamma_{12}(n) + \gamma_{22}(n) \\ \gamma_{i}(n) =& \gamma_{11}(n) + i \gamma_{21}(n) - i \gamma_{12}(n) + \gamma_{22}(n) \\ \gamma_{-i}(n) =& \gamma_{11}(n) - i \gamma_{21}(n) + i \gamma_{12}(n) + \gamma_{22}(n) \end{aligned}\] 可以解出 \[\begin{aligned} \gamma_{12}(n) =& \frac14[ \gamma_1(n) - \gamma_{-1}(n) + i \gamma_i(n) - i \gamma_{-i}(n)] \\ \gamma_{21}(n) =& \frac14[ \gamma_1(n) - \gamma_{-1}(n) - i \gamma_i(n) + i \gamma_{-i}(n)] \end{aligned}\]

\(F_{11}(\lambda)\)\(F_{22}(\lambda)\)分别表示\(\{ X_{1t} \}\)\(\{ X_{2t} \}\)的谱函数,并引入 \[\begin{aligned} F_{12}(\lambda) =& \frac14[ F_1(\lambda) - F_{-1}(\lambda) + i F_i(\lambda) - i F_{-i}(\lambda)] \\ F_{21}(\lambda) =& \frac14[ F_1(\lambda) - F_{-1}(\lambda) - i F_i(\lambda) + i F_{-i}(\lambda)] \end{aligned}\] 引入矩阵函数 \[\begin{align} \boldsymbol F(\lambda) =& \left(\begin{array}{cc} F_{11}(\lambda) & F_{12}(\lambda) \\ F_{21}(\lambda) & F_{22}(\lambda) \end{array}\right), \ \lambda \in [-\pi, \pi] \tag{28.25} \end{align}\] 则有 \[\begin{aligned} \Gamma(n) =& E(\boldsymbol X_{t+n} \boldsymbol X_t^T) = \int_{-\pi}^\pi e^{in\lambda} d \boldsymbol F(\lambda) \\ =& \left(\begin{array}{cc} \int_{-\pi}^\pi e^{in\lambda} d F_{11}(\lambda) & \int_{-\pi}^\pi e^{in\lambda} d F_{12}(\lambda) \\ \int_{-\pi}^\pi e^{in\lambda} d F_{21}(\lambda) & \int_{-\pi}^\pi e^{in\lambda} d F_{22}(\lambda) \end{array}\right), \ n \in \mathbb Z \tag{28.26} \end{aligned}\]

\(\boldsymbol F(\lambda)\)为2维平稳序列\(\{ \boldsymbol X_t \}\)谱函数矩阵。 由于矩阵\(\boldsymbol F(\lambda)\)的每个元素都是\([-\pi, \pi]\)上分布函数的线性组合, 所以都是有界变差右连续函数。 另外\(\boldsymbol F(\lambda)\)还是Hermite矩阵: \[\begin{aligned} \boldsymbol F^*(\lambda) = \boldsymbol F(\lambda) \end{aligned}\] 其中星号表示共轭转置。

还可证明\(\boldsymbol F(\lambda)\)关于\(\lambda\)单调不减, 即\(\forall \lambda_1 < \lambda_2, \lambda_1, \lambda_2 \in [-\pi, \pi]\), \(\boldsymbol F(\lambda_2) - \boldsymbol F(\lambda_1)\)非负定。

\(\boldsymbol F(\lambda)\)的每个元素的实部和虚部都是连续函数, 并且除去有限个点外导函数连续时,称 \[\begin{align} \boldsymbol f(\lambda) =& \left(\begin{array}{cc} f_{11}(\lambda) & f_{12}(\lambda) \\ f_{21}(\lambda) & f_{22}(\lambda) \end{array}\right) \\ =& \left(\begin{array}{cc} F_{11}'(\lambda) & F_{12}'(\lambda) \\ F_{21}'(\lambda) & F_{22}'(\lambda) \end{array}\right), \ \lambda \in [-\pi, \pi] \tag{28.27} \end{align}\]\(\{ \boldsymbol X_t \}\)谱密度矩阵, 称\(f_{12}(\lambda)\)\(\{ X_{1t} \}\)\(\{ X_{2t} \}\)互谱密度

因为\(\boldsymbol F(\lambda)\)是Hermite矩阵,并且单调不减, 所以\(\boldsymbol f(\lambda)\)是Hermite非负定的。 这时 \[\begin{align} \Gamma(n) =& \int_{-\pi}^\pi e^{in\lambda} \boldsymbol f(\lambda) d\lambda \\ \stackrel{\triangle}{=}& \left(\begin{array}{cc} \int_{-\pi}^\pi e^{in\lambda} f_{11}(\lambda) d\lambda & \int_{-\pi}^\pi e^{in\lambda} f_{12}(\lambda) d\lambda \\ \int_{-\pi}^\pi e^{in\lambda} f_{21}(\lambda) d\lambda & \int_{-\pi}^\pi e^{in\lambda} f_{22}(\lambda) d\lambda \end{array}\right), \ n \in \mathbb Z \tag{28.28} \end{align}\]

定理28.8 \(m\)维平稳序列\(\{ \boldsymbol X_t \}\)有自协方差函数\(\{\Gamma(n) \}\), 则存在唯一的\(m \times m\)函数矩阵 \[\begin{aligned} \boldsymbol F(\lambda) = (F_{jk}(\lambda)), \ \lambda \in [-\pi, \pi], \end{aligned}\] 使得

  • (1) \(\Gamma(n) = \int_{-\pi}^\pi e^{in\lambda} d \boldsymbol F(\lambda)\), \(n \in \mathbb Z\);
  • (2) \(\boldsymbol F^*(\lambda) = \boldsymbol F(\lambda)\), \(\boldsymbol F(-\pi) = \boldsymbol 0\), 当\(\lambda_1 < \lambda_2\)\(\boldsymbol F(\lambda_2) - \boldsymbol F(\lambda_1)\)非负定;
  • (3) \(\boldsymbol F(\lambda)\)中的每个元素\(F_{jk}(\lambda)\)是有界变差和右连续的。

\(\boldsymbol F(\lambda)\)叫做\(\{ \boldsymbol X_t \}\)谱函数矩阵

如果\(\boldsymbol F(\lambda)\)的每个元素\(F_{jk}(\lambda)\)的实部和虚部都是连续函数, 且除去有限个点外导函数连续,就称 \[\begin{aligned} \boldsymbol f(\lambda) = (f_{jk}(\lambda)) \stackrel{\triangle}{=} (F_{jk}'(\lambda)) \end{aligned}\]\(\{ \boldsymbol X_t \}\)谱密度矩阵。 这时 \[\begin{align} \Gamma(n) =& \int_{-\pi}^\pi e^{in\lambda} \boldsymbol f(\lambda) d \lambda \\ =& \left( \int_{-\pi}^\pi e^{in\lambda} \boldsymbol f_{jk}(\lambda) d \lambda \right)_{m \times m}, \ n \in \mathbb Z \tag{28.29} \end{align}\]

称实变复值函数是绝对连续函数, 如果其实部和虚部都是绝对连续函数。 当\(\boldsymbol F(\lambda)\)的每个元素都是绝对连续函数时, \(\boldsymbol f(\lambda) = (F_{jk}'(\lambda))\)就是\(\{ \boldsymbol X_t \}\)谱密度矩阵

定理28.9 \(m\)维平稳序列\(\{ \boldsymbol X_t \}\)有自协方差函数 \(\Gamma(n) = (\gamma_{jk}(n))\), 如果 \[\begin{aligned} \sum_{n=-\infty}^\infty | \gamma_{jk}(n) | < \infty, \ j,k=1,2,\dots, m \end{aligned}\]\[\begin{aligned} \boldsymbol f(\lambda) =& \frac{1}{2\pi} \sum_{n=-\infty}^\infty \Gamma(n) e^{-in\lambda}, \ \lambda \in [-\pi, \pi] \end{aligned}\]\(\{ \boldsymbol X_t \}\)谱密度矩阵

28.4.2 多维平稳序列的谱表示

\(\{ \boldsymbol X_t = (X_{1t}, X_{2t}, \dots, X_{mt})^T\)\(m\)维零均值平稳序列, 则其每个分量\(\{ X_j(t) \}\)是一位零均值平稳序列,有谱表示 \[\begin{align} X_{jt} =& \int_{-\pi}^\pi e^{it\lambda} d Z_j(\lambda), \ t \in \mathbb Z \tag{28.30} \end{align}\]\(\{ Z_j(\lambda) \}\)\([-\pi, \pi]\)上右连续的正交增量过程。

\[\begin{align} \boldsymbol Z(\lambda) =& (Z_1(\lambda), Z_2(\lambda), \dots, Z_m(\lambda))^T, \ \lambda \in [-\pi, \pi], \tag{28.31} \end{align}\]\[\begin{align} \boldsymbol X_t =& \int_{-\pi}^\pi e^{in\lambda} d \boldsymbol Z(\lambda) \ \lambda \in [-\pi, \pi]. \tag{28.32} \end{align}\] 这称为\(m\)维平稳序列\(\{ \boldsymbol X_t \}\)谱表示

(28.32)中的\(\boldsymbol Z(\lambda)\)有下列性质:

  • (1) 正交增量性: 对\(-\pi \leq \lambda_1 < \lambda_2 \leq \lambda_3 < \lambda_4 \leq \pi\), \[\begin{aligned} E\left[ (\boldsymbol Z(\lambda_2) - \boldsymbol Z(\lambda_1)) (\boldsymbol Z(\lambda_4) - \boldsymbol Z(\lambda_3))^* \right] = 0. \end{aligned}\]
  • (2) 右连续性:当\(\delta \downarrow 0\)\[\begin{aligned} E\left[ (\boldsymbol Z(\lambda + \delta) - \boldsymbol Z(\lambda)) (\boldsymbol Z(\lambda + \delta) - \boldsymbol Z(\lambda))^* \right] \to 0. \end{aligned}\]
  • (3) \(\boldsymbol F(\lambda) = E[ \boldsymbol Z(\lambda) \boldsymbol Z^*(\lambda)]\)\(\{ \boldsymbol X_t \}\)的谱函数矩阵,满足 \[\begin{aligned} \boldsymbol F(\lambda_2) - \boldsymbol F(\lambda_1) =& E\left[ (\boldsymbol Z(\lambda_2) - \boldsymbol Z(\lambda_1)) (\boldsymbol Z(\lambda_2) - \boldsymbol Z(\lambda_1))^* \right], \ \lambda_1 < \lambda_2 \end{aligned}\]

满足上述(1), (2), (3)的\(m\)维随机过程\(\{ \boldsymbol Z(\lambda) \}\)叫做右连续的正交增量过程

定理28.10 \(m\)维零均值平稳序列\(\{ \boldsymbol X_t \}\), 有右连续的正交增量过程 \(\{ \boldsymbol Z(\lambda) \}\)使得 \(\boldsymbol Z(-\pi) = \boldsymbol 0\)(28.32)成立。 如果正交增量过程\(\{ \boldsymbol\xi(\lambda) \}\)也满足上述的条件, 则 \[\begin{aligned} P(\boldsymbol\xi(\lambda) = \boldsymbol Z(\lambda)) = 1, \ \lambda \in [-\pi, \pi]. \end{aligned}\]

例28.6 (多维线性平稳序列的谱密度) \(\{ \boldsymbol \varepsilon_t \}\)\(m\)维WN(0, \(Q\)), \(Q = E[\boldsymbol\varepsilon_t \boldsymbol\varepsilon_t^T]\), \(\{ A_j \}\)是一列\(m \times m\)实值矩阵,满足 \[\begin{aligned} \sum_{j=-\infty}^\infty A_j Q A_j^T < \infty, \end{aligned}\]\[\begin{aligned} \boldsymbol X_t = \sum_{j=-\infty}^\infty A_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \end{aligned}\]\(m\)维零均值平稳序列。

自协方差函数为 \[\begin{aligned} \Gamma(n) = E[ \boldsymbol X_{t+n} \boldsymbol X_n ] = \sum_{j=-\infty}^\infty A_{j+n} Q A_j^T, \ n \in \mathbb Z \end{aligned}\]

本例中,对两个实矩阵\(A, B\), 用\(A \leq B\)表示\(A\)\(B\)的对应元素比较全部成立小于等于关系, 用\([A]\)表示把\(A\)的所有元素都取绝对值组成的矩阵。这时必有 \[ [AB] \leq [A] [B] \] 如果要求\(\sum_{j=-\infty}^\infty [A_j] < \infty\), 则 \[\begin{aligned} \sum_{n=-\infty}^\infty [\Gamma(n)] =& \sum_{n=-\infty}^\infty \left[\sum_{j=-\infty}^\infty A_{j+n} Q A_j^T \right] \\ \leq & \sum_{n=-\infty}^\infty \sum_{j=-\infty}^\infty [A_{j+n}] [Q] [A_j^T] \\ \leq& \left(\sum_{n=-\infty}^\infty [A_n] \right) [Q] \left(\sum_{j=-\infty}^\infty [A_j^T] \right) < \infty \end{aligned}\] 从而\(\{ \boldsymbol X_t \}\)有谱密度 \[\begin{aligned} \boldsymbol f(\lambda) =& \frac{1}{2\pi} \sum_{n=-\infty}^\infty \Gamma(n) e^{-in\lambda} \\ =& \frac{1}{2\pi} \sum_{n=-\infty}^\infty \sum_{j=-\infty}^\infty A_{j+n} Q A_j^T e^{-in\lambda} \\ =& \frac{1}{2\pi} \left( \sum_{k=-\infty}^\infty A_k e^{-ik\lambda} \right) Q \left( \sum_{j=-\infty}^\infty A_j e^{-ij\lambda} \right)^* \end{aligned}\]

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例28.7 (VARMA序列谱密度) 考虑平稳可逆的\(m\)维ARMA(\(p,q\))模型的谱密度。模型为 \[\begin{aligned} A(\mathscr B) \boldsymbol X_t = B(\mathscr B) \boldsymbol\varepsilon_t, \ t \in \mathbb Z, \end{aligned}\]

\(\det(A(z)) A^{-1}(z)\)\(A(z)\)的伴随矩阵, 仍是矩阵系数多项式。 于是\(A^{-1}(z) B(z)\)的每个元素是有理多项式。 将\(A^{-1}(z) B(z)\)的每个元素进行泰勒展开后得\(A^{-1}(z) B(z)\)泰勒展开式 \[\begin{aligned} A^{-1}(z) B(z) = \sum_{j=0}^\infty C_j z^j, \ |z| \leq 1. \end{aligned}\] 其中的系数矩阵满足 \[\begin{aligned} \sum_{j=0}^\infty [C_j] < \infty. \end{aligned}\]

由例28.6,平稳解 \[\begin{aligned} \boldsymbol X_t = A^{-1}(\mathscr B) B(\mathscr B) \boldsymbol\varepsilon_t = \sum_{j=0}^\infty C_j \boldsymbol\varepsilon_{t-j}, \ t \in \mathbb Z \end{aligned}\] 有谱密度矩阵 \[\begin{aligned} \boldsymbol f(\lambda) =& \frac{1}{2\pi} \left( \sum_{k=-\infty}^\infty C_k e^{-ik\lambda} \right) Q \left( \sum_{j=-\infty}^\infty C_j e^{-ij\lambda} \right)^* \end{aligned}\]

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例28.8 (二次相干函数) 设二维零均值平稳序列\(\{ \boldsymbol X_t \}\)有谱表示 \[\begin{aligned} \boldsymbol X_t = \int_{-\pi}^\pi e^{it\lambda} d \left(\begin{array}{c} Z_1(\lambda) \\ Z_2(\lambda) \end{array}\right) \end{aligned}\]

\(\lambda \in [-\pi, \pi]\), 对充分小的正数\(\Delta\lambda\), 定义 \[\begin{aligned} \Delta Z_k(\lambda) = Z_k(\lambda + \Delta\lambda) - Z_k(\lambda), \ k=1,2 \end{aligned}\] 则有 \[\begin{aligned} & \boldsymbol F(\lambda + \Delta\lambda) - F(\lambda) \\ =& E\left[ \left(\begin{array}{c} \Delta Z_1(\lambda) \\ \Delta Z_2(\lambda) \end{array}\right) (\Delta \bar Z_1(\lambda) , \Delta \bar Z_2(\lambda)) \right] \\ =& \left(\begin{array}{cc} E|\Delta Z_1(\lambda)|^2 & E[\Delta Z_1(\lambda) \Delta \bar Z_2(\lambda)] \\ E[\Delta \bar Z_1(\lambda) \Delta Z_2(\lambda)] & E|\Delta Z_2(\lambda)|^2 \end{array}\right) \end{aligned}\]

如果\(\boldsymbol F'(\lambda)\)连续,则有 \[\begin{aligned} d \boldsymbol F(\lambda) =& \boldsymbol f(\lambda) d\lambda = \left(\begin{array}{cc} E|d Z_1(\lambda)|^2 & E[d Z_1(\lambda) d \bar Z_2(\lambda)] \\ E[d \bar Z_1(\lambda) d Z_2(\lambda)] & E|d Z_2(\lambda)|^2 \end{array}\right) \end{aligned}\] \(d Z_1(\lambda)\)\(d Z_2(\lambda)\)的相关系数为 \[\begin{aligned} \rho_{12}(\lambda) =& \frac{E[d Z_1(\lambda) d \bar Z_2(\lambda)]} {\sqrt{E|d Z_1(\lambda)|^2 E|d Z_2(\lambda)|^2}} \\ =& \frac{f_{12}(\lambda) d\lambda} {\sqrt{f_{11}(\lambda) d\lambda \cdot f_{22}(\lambda) d\lambda}} \\ =& \frac{f_{12}(\lambda)}{\sqrt{f_{11}(\lambda) f_{22}(\lambda)}} \end{aligned}\] 其中规定\(0/0=0\)

通常称 \[\begin{aligned} K_{12}^2(\lambda) = |\rho_{12}(\lambda)|^2 = \frac{|f_{12}(\lambda)|^2}{f_{11}(\lambda) f_{22}(\lambda)} \end{aligned}\]\(\{ \boldsymbol X_t, t \in \mathbb Z \}\)二次相干函数\(K_{12}^2(\lambda)\)描述了\(\{ X_{1t} \}\)\(\{ X_{2t} \}\) 在角频率\(\lambda\)处的线性相关性强弱。 如果\(K_{12}^2(\lambda)=1\), 说明\(\{ X_{1t} \}\)\(\{ X_{2t} \}\)在角频率\(\lambda\)处线性相关。

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例28.9 (线性滤波的二次相干函数) 设平稳序列\(\{ X_{1t} \}\)有谱密度\(f(\lambda)\), \(\{ h_j \}\)是绝对可和线性滤波器, 输出过程为 \[\begin{aligned} X_{2t} = \sum_{j=-\infty}^\infty h_j X_{1, t-j}, \ t \in \mathbb Z \end{aligned}\]

\(\{X_{2t}\}\)有谱密度(见定理6.4\[\begin{aligned} f_{22}(\lambda) =& |h(\lambda)|^2 f(\lambda) \end{aligned}\] 其中 \[\begin{aligned} h(\lambda) = \sum_{j=-\infty}^\infty h_j e^{-ij\lambda}. \end{aligned}\]

计算得 \[\begin{aligned} \gamma_{12}(n) =& E(X_{1, t+n} X_{2t}) \\ =& \sum_{j=-\infty}^\infty h_j E(X_{1,t+n} X_{2,t-j}) \\ =& \sum_{j=-\infty}^\infty h_j \gamma_{11}(n+j) \\ =& \sum_{j=-\infty}^\infty h_j \int_{-\pi}^\pi e^{i(n+j)\lambda} f(\lambda) d\lambda \\ =& \int_{-\pi}^\pi e^{in\lambda} h(-\lambda) f(\lambda) d\lambda \end{aligned}\] 所以\(\{ X_{1t} \}\)\(\{ X_{2t} \}\)的互谱密度为 \[\begin{aligned} f_{12}(\lambda) = h(-\lambda) f(\lambda), \ \lambda \in [-\pi, \pi]. \end{aligned}\] 于是2维平稳序列\(\boldsymbol X_t = (X_{1t}, X_{2t})^T\)有退化的谱密度矩阵 \[\begin{aligned} \boldsymbol f(\lambda) =& \left(\begin{array}{cc} f(\lambda) & h(-\lambda)f(\lambda) \\ h(\lambda)f(\lambda) & |h(\lambda)|^2 f(\lambda) \end{array}\right) \end{aligned}\] 这时二次相干函数 \[\begin{aligned} K^2(\lambda) \equiv 1, \ \forall \lambda \in [-\pi, \pi]. \end{aligned}\]

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28.4.3 谱密度矩阵的估计

二维零均值平稳序列\(\{ \boldsymbol X_t \}\)的观测值\(\boldsymbol X_1, \boldsymbol X_2, \dots, \boldsymbol X_N\)的周期图定义为 \[\begin{aligned} I_N(\lambda) =& \frac{1}{2\pi N} \left( \sum_{j=1}^N \boldsymbol X_j e^{-ij\lambda} \right) \left( \sum_{j=1}^N \boldsymbol X_j e^{-ij\lambda} \right)^*. \end{aligned}\] 周期图\(I_n(\lambda)\)的值是\(2 \times 2\)方阵, 可以对周期图平滑得到谱密度矩阵估计。

\(\lambda_j = 2j\pi / N\), 对\(\lambda \in [0, \pi]\), 用\(g(N, \lambda)\)表示\(\{ \lambda_j: 0 \leq j \leq N/2 \}\)中距离\(\lambda\)最近的\(\lambda_j\)(若左右两个距离相等取左边一个)。 取\(M_N = O(\sqrt{N})\)\(\lim_{N \to\infty} M_N = \infty\)。 设\(W_N(k)\)为满足下列条件的实值权函数: \[\begin{aligned} (1) & W_N(k) = W_N(-k), \ W_N(k) \geq 0, \ |k| \leq M_N; \\ (2) & \sum_{|k| \leq M_N} W_N(k) = 1; \\ (3) & \lim_{N \to\infty} \sum_{|k| \leq M_N} W_N^2(k) = 0. \end{aligned}\] 谱密度矩阵的平滑周期图估计为 \[\begin{aligned} \hat{\boldsymbol f}(\lambda) =& \sum_{|k| \leq M_N} W_N(k) I_N(g(N,\lambda) + \lambda_k), \ \lambda \in [0,\pi], \\ \hat{\boldsymbol f}(\lambda) =& [\hat{\boldsymbol f}(-\lambda) ]^T, \ \lambda \in [-\pi, 0]. \end{aligned}\]

\[\begin{aligned} \hat{\boldsymbol f}(\lambda) =& \left(\begin{array}{cc} \hat f_{11} & \hat f_{12} \\ \hat f_{21} & \hat f_{22} \end{array}\right) \end{aligned}\] 二次相干函数\(K_{12}^2(\lambda)\)的估计定义为 \[\begin{aligned} \hat K_{12}^2(\lambda) =& \frac{| \hat f_{12}(\lambda) |^2} {\hat f_{11}(\lambda) \hat f_{22}(\lambda)}. \end{aligned}\]

28.5 附录:补充

28.5.1 多维ARMA不可辨识的例子

只要\(\mbox{det}(A(z))\)为不依赖于\(z\)的非零常数, 则多维ARMA模型参数不能从\(\{ \Gamma(n) \}\)唯一确定。

例如,考虑2维\(VAR(1)\)。 模型为 \[ \boldsymbol X_t = A \boldsymbol X_{t-1} + \boldsymbol\varepsilon_t \] 其中 \[ A = \left(\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right) \] 特征多项式的行列式为 \[\begin{aligned} \mbox{det}(I - A z) =& \left| \begin{array}{cc} 1 - a_{11} z & -a_{12}z \\ -a_{21}z & 1 - a_{22}z \end{array}\right | \\ =& 1 - (a_{11} + a_{22}) z + (a_{11} a_{22} - a_{12} a_{21}) z^2 \end{aligned}\]\(z\)\(z^2\)系数等于零, 则 \[ a_{22} = -a_{11}, \ a_{12} a_{21} = - a_{11}^2 \]\(a_{11} \neq 0\), \(a_{22} = -a_{11}\), 取\(a_{12} \neq 0\), 取\(a_{21} = - \frac{a_{11}^2}{a_{12}}\)则有\(\mbox{det}(I - A z)=1\)不依赖于\(z\)。 这时 \[\begin{aligned} (I - A z)^{-1} =& \left( \begin{array}{cc} 1 - a_{22} z & a_{21}z \\ a_{12}z & 1 - a_{11}z \end{array}\right) \\ =& I + \left( \begin{array}{cc} - a_{22} & a_{21} \\ a_{12} & - a_{11} \end{array}\right) z \end{aligned}\]\[ B = \left( \begin{array}{cc} - a_{22} & a_{21} \\ a_{12} & - a_{11} \end{array}\right) \] 则模型可以写成 \[ \boldsymbol X_t = (I - A \mathscr B)^{-1} \boldsymbol\varepsilon_t = (I + B \mathscr B) \boldsymbol\varepsilon_t \] 又可以写成一个2维MA(1)模型。

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References

Brockwell, P. J., and R. A. Davis. 1987. Time Series: Theory and Methods. Springer-Verlag.
谢衷洁. 1990. 时间序列分析. 北京大学出版社,.