Keywords: Inference for the spectrum
Consider an arbitrary set of (can be complexed-valued) observations $x_1, \dots, x_n$ made at times $1, \dots, n$ respectively. The vector $\mathbf{x} = (x_1, \dots, x_n)$ belongs to a n-dimensional complex space $\mathbb{C}^n$. If $u$ and $v$ are two elements of $\mathbb{C}^n$, define the inner product as $< u, v > = \sum_{i=1}^n u_i\bar{v}_i$
1. Definitions
1.1 Orthonormal set of this complex space
$\mathbf{x} = \sum_{j \in F_n} a_j \mathbf{e}_j$, where $\mathbf{e_j} = n^{-1/2} (e^{i\omega_j}, e^{i2\omega_j}, \dots, e^{in\omega_j})$, $F_n = \{j \in \mathbb{Z}: -\pi < \omega_j \le \pi\} = \{-[(n-1)/2], \dots, [n/2]\}$
Note that $< \mathbf{e}_j, \mathbf{e}_k > = n^{-1} \sum_{r=1}^n e^{ir(\omega_j - \omega_k)}$
$< \mathbf{e}_j, \mathbf{e}_k > = 0$ if $j =k$, otherwise $n^{-1} e^{i(\omega_j - \omega_k)}\frac{1- e^{in(\omega_j - \omega_k)}}{1- e^{i(\omega_j - \omega_k)}} = 0$ (等比数列求和, note $n(w_j - w_i) = n \frac{2\pi}{n}(j -k) = (j-k) 2\pi$, moreover $e^{k \cdot 2\pi i} = 1$)
Just like the inversion formula in the spectral analysis of time series, we can rewrite $x = \sum_{j \in F_n} a_j \mathbf{e}_j$, where $a_j = < \mathbf{x}, \mathbf{e}_j > = n^{-1/2} \sum_{t = 1}^n x_t e^{-it\omega_j}$. Note that $< \mathbf{x}, \mathbf{e}_j > = < \sum_{g=1}^n a_j e_j, e_j > = a_j$. This is also called discrete Fourier transformation of $\mathbf{x} \in R^n$.
Comment: 其实还应该证明下closed span, 同Fourier sequence。
1.2 The periodogram of x
The value $I(\omega_j)$ of the periodogram of $\mathbf{x}$ at frequency $w_j = 2\pi j/n, j \in F_n$ is defined as:
$I(\omega_j) = |a_j|^2 = |< x, e_j >|^2 = n^{-1} |\sum_{t=1}^n x_t e^{-itw_j}|^2$
Note that $|x|^2 = \sum_{j \in F_n} I(w_j)$.
Comment: Note that the definition of periodogram depends on the definition of random variables.
2. The Periodogram in terms of the sample autocovariance function
2.1 The estimation of periodogram
Theorem: If $\omega_j$ is any non-zero Fourier frequency, then $I(w_j) = \sum_{|k|< n} \hat{\gamma}(k) e^{-ik\omega_j}$, where $\hat{\gamma}(k) = n^{-1} \sum_{t=1}^{n-k} (x_{t + k} - m)(\bar{x}_{t} - \bar{m})$, $m = n^{-1} \sum_{t=1}^n x_t$
Proof: 易证。
2.2 Testing for the presence of Hidden Periodicities
$H_0$: The data is generated by a Gaussian white noise sequence.
$H_1$: The data is generated by a Gaussian white noise sequence with a superimposed deterministic periodic component.
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