## 2 Blind Source Separation Techniques
### Complexity Pursuit Algorithm (CP)
#### 2.3.2 Measuring Complexity Using Signal Predictability  
Simple exponential smoothing (SES) is an effective forecasting method for data points that do not exhibit any trend or seasonal behavior. It is based on the idea of using a weighted average of the past data points to make predictions about the future. The weights are exponentially decreasing, giving more importance to the most recent data points. SES is most effective as a forecasting method for quasi-periodic signals. Formally, the exponential smoothing equation is  
\[\hat{\mathbf{y}}(t)=\alpha\mathbf{y}(t-1)+(1-\alpha)\hat{\mathbf{y}}(t-1),\cdots 0\leq \alpha\leq 1, \tag{8}\]  
where \(\hat{\mathbf{y}}(t)\) is the forecast for the next period, \(\alpha\) is the smoothing constant, \(\mathbf{y}(t-1)\) stands for the observed value of the series in period \(t-1\), and \(\hat{\mathbf{y}}(t-1)\) is the old forecast for period \(t-1\). This equation is based on the idea of weighting the most recent observation \(\mathbf{y}(t-1)\) with a weight \(\alpha\) and the most recent forecast \(\hat{\mathbf{y}}(t-1)\) with a weight of \(1-\alpha\). Equation (8) can be expanded as follows:  
\[\hat{\mathbf{y}}(t)=\alpha\mathbf{y}(t-1)+\alpha(1-\alpha)\mathbf{y}(t-2)+\alpha(1-\alpha )^{2}\mathbf{y}(t-3)+\cdots+\alpha(1-\alpha)^{t-1}\mathbf{y}(1). \tag{9}\]  
* Long-term predictor \(\overline{\mathbf{y}}(t)\): \(\alpha\) is in the range of 0 to 0.3, when \(\alpha\) close to \(\overline{\mathbf{y}}(t)\approx 0\);
* Short-term predictor \(\hat{\mathbf{y}}(t)\): \(\alpha\) is in the range of 0.4 to 1.  
In general, short-term forecast is more accurate than long-term forecast. Thus, the exponential smoothing method is more suitable for the recent prediction.  
According to the SES method, a predictability measure \(F\) is defined according to the extracted signal \(\mathbf{y}_{i}=\mathbf{w}_{i}^{\mathrm{T}}\mathbf{x}\). For extracting the source signal \(\mathbf{y}_{i}=\mathbf{q}_{i}\) from a set of \(M\) signal mixtures \(\mathbf{x}(t)=[x_{1}(t),...,x_{M}(t)]^{\mathrm{T}}\), it is necessary to find the weight vector \(\mathbf{w}_{i}\) that maximizes the predictability \(F\) of \(\mathbf{y}_{i}\), so as to minimize its complexity.  
[PERSON] [26] proposed a simple and robust measure of signal complexity, termed temporal predictability, which is defined as  
\[F(\mathbf{y}_{i})=\log\frac{V(\mathbf{y}_{i})}{U(\mathbf{y}_{i})}=\log\frac{\sum\limits_{ t=1}^{N}\left(\overline{y}_{i}(t)-y_{i}(t)\right)^{2}}{\sum\limits_{t=1}^{N} \left(\hat{y}_{i}(t)-y_{i}(t)\right)^{2}}, \tag{10}\]  
where \(\overline{y}_{i}(t)\) is the long-term predictor, and \(\hat{\mathbf{y}}_{i}(t)\) is the short-term predictor.  
\[\begin{array}{l}\overline{y}_{i}(t)=\alpha_{L}y_{i}(t-1)+(1-\alpha_{L}) \overline{y}_{i}(t-1),\cdots 0\leq\alpha_{L}\leq 1\\ \hat{y}_{i}(t)=\alpha_{S}y_{i}(t-1)+(1-\alpha_{S})\hat{y}_{i}(t-1),\cdots 0 \leq\alpha_{S}\leq 1\end{array} \tag{11}\]  
The parameter \(\alpha\) can be defined using the half-life parameter \(h\), where \(\alpha_{S}\) and \(\alpha_{L}\) correspond to the different ranges of \(\alpha\). The formula for \(\alpha\) is  
\[\alpha=2^{-1/h}, \tag{12}\]  
where \(h_{S}=1\), and \(h_{L}\) is an arbitrary value (e.g., 900,000) as long as \(h_{L}>>h_{S}\)[26].