causality_for_spaceweather/notebooks/chapter0_ts/.ipynb_checkpoints/huang1998-checkpoint.md

非线性非平稳时间序列分析的EMD方法和Hilbert谱方法

瞬时频率

Hilbert transform: For any artitrary time series, X(t), we can always have its Hibert Transform, Y(t), as

$$ Y(t) = \frac{1}{\pi}P.V.\int_{-\infty}^{\infty}\frac{X(t')}{t-t'}dt' $$

With this definition, $X(t)$ and $Y(t)$ form the complex conjugate pair, so we can have an analytic signal, $Z(t)$, as

$$ Z(t) = X(t) + i Y(t)=a(t)e^{i\theta(t)} $$

in which

$$ a(t)=[X^2(t)+Y^2(t)]^{1/2},\ \ \theta(t)=\arctan\left(\frac{Y(t)}{X(t)}\right) $$

Then the instantaneous frequency is defined as

$$ \omega=\frac{d\theta(t)}{dt} $$

Intrinsic mode function (IMF) for which the instantaneous frequency can be defined everywhere.

定义瞬时频率的条件：

本征模函数与瞬时频率的关系？

本征模函数（Intrinsic mode functions）

Two conditions:

经验模态分解方法

Knowing the well-behaved Hilbert transforms of the IMF components is only the starting point.

这就是一个两重的迭代过程

分解的完备性和正交性

完备性根据如下公式可以得到保证：

$$ X(t) = \sum\limits_{i=1}^{n}c_i + r_n $$

正交性：

Orthogonality is a requirement only for linear decomposition systems; it would not make phhysical sense for a nonlinear decomposition as in EMD.

Fortunately, in most cases encountered, the leakage is small.

The Hilbert spectrum

首先对原始数据进行经验模态分解，得到IMF的各个分量，然后对IMF的各个分量进行Hilbert变换，并计算每个分量的瞬时频率，则可得到原始数据的一个如下展开：

$$ X(t)=\sum\limits_{j=1}^{n}a_j(t)exp\left(i\int \omega_j(t)dt\right) $$

这里我们不考虑最后的那个残量，因为它或者是一个单调的趋势函数，或者是一个常数。

对比Fourier变化：

$$ X(t)=\sum\limits_{j=1}^{\infty}a_je^{i\omega_j t} $$

The IMF represents a generalized Fourier expansion. The variable amplitude and the instantaneous frequency have not only greatly improved the efficiency of the expansion, but also enabled the expansion to accomodate non-stationary data.

根据上面的展开，可以得到振幅的时频分布：$H(\omega,t)$。如果振幅与能量密度有关系，一般振幅的平方可以用来表示Hilbert能量谱。

The marginal spectrum:

$$ h(\omega) = \int_0^T H(\omega,t)dt $$

The instantaneous energy:

$$ IE(t)=\int_{\omega}H^2(\omega,t)d\omega $$

定性评估时间序列的平稳性

mean marginal spectrum:

$$ n(\omega) = \frac{1}{T}h(\omega) $$

the degree of stationarity:

$$ DS(\omega) = \frac{1}{T}\int_0^T\left(1-\frac{H(\omega,t)}{n(\omega)}\right)^2dt $$

把平稳性定义成频率的函数是合理的，因为对于某些频率来说可能是非平稳的，但是对于其他的频率分量来说却是平稳的。

A modified statistical stationary：

$$ DSS(\omega, \Delta T)=\frac{1}{T}\int_0^{T}\left(1-\frac{\overline{H(\omega,t)}}{n(\omega)}\right)^2dt $$