# Time Series Prediction Based on Adaptive Weight Online Sequential Extreme Learning Machine
## 3 The Proposed Adaptive Weight Online Sequential Extreme Learning Machine
### Formula Derivation
\begin{array}{lcl}\boldsymbol{\beta}_{k+1}^{\prime}&=&\left(\mathbf{P}_{k}^{ \prime}-\mathbf{K}_{k+1}\mathbf{h}_{k+1}\mathbf{P}_{k}^{\prime}\right)\left[ \begin{array}{c}\mathbf{H}_{k}^{\prime}\\ \lambda_{k+1}\mathbf{h}_{k+1}\end{array}\right]^{T}\left[\begin{array}{c} \mathbf{T}_{k}^{\prime}\\ \lambda_{k+1}\mathbf{t}_{k+1}^{T}\end{array}\right]\\ &=&\left(\mathbf{P}_{k}^{\prime}-\mathbf{K}_{k+1}\mathbf{h}_{k+1}\mathbf{P}_{k }^{\prime}\right)\left(\mathbf{H}_{k}^{\prime T}\mathbf{T}_{k}^{\prime}+ \lambda_{k+1}^{2}\mathbf{h}_{k+1}^{T}\mathbf{t}_{k+1}^{T}\right)\\ &=&\mathbf{P}_{k}^{\prime}\mathbf{H}_{k}^{\prime T}\mathbf{T}_{k}^{\prime}- \mathbf{K}_{k+1}\mathbf{h}_{k+1}\mathbf{P}_{k}^{\prime}\mathbf{H}_{k}^{T} \mathbf{T}^{\prime}\\ &+\lambda_{k+1}^{2}\Big{(}\mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T}-\mathbf{ K}_{k+1}\mathbf{h}_{k+1}\mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T}\Big{)}\mathbf{t}_{k+1}^{T} \end{array}  \tag{26}\]  
Then we can simplify the \(\mathbf{P}_{k}^{\prime}\mathbf{h}(k+1)^{T}-\mathbf{K}_{k+1}\mathbf{h}(k+1) \mathbf{P}_{k}^{\prime}\mathbf{h}(k+1)^{T}\) in Equation (26) as:  
\[ \begin{array}{l}\mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T}-\mathbf{K}_{k+1} \mathbf{h}_{k+1}\mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T}\\ =\mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T}-\mathbf{K}_{k+1}\mathbf{h}_{k+1} \mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T}-\frac{\mathbf{K}_{k+1}}{\lambda_{k+ 1}^{2}}+\frac{\mathbf{K}_{k+1}}{\lambda_{k+1}^{2}}\\ =\mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T}-\mathbf{K}_{k+1}\bigg{(}\frac{I}{ \lambda_{k+1}^{2}}+\mathbf{h}_{k+1}\mathbf{P}_{k}^{\prime}\mathbf{h}_{k+1}^{T} \bigg{)}+\frac{\mathbf{K}_{k+1}}{\lambda_{k+1}^{2}}\\ =\frac{\mathbf{K}_{k+1}}{\lambda_{k+1}^{2}}\end{array}  \tag{27}\]  
Substituting Equation (27) into Equation (26), \(\boldsymbol{\beta}_{k+1}^{\prime}\) is able to be determined in a compact way as:  
\[\boldsymbol{\beta}_{k+1}^{\prime}=\boldsymbol{\beta}_{k}^{\prime}+\mathbf{K}_{k+1 }\big{(}\mathbf{t}_{k+1}^{T}-\mathbf{h}_{k+1}\boldsymbol{\beta}_{k}^{\prime} \big{)} \tag{28}\]  
**Proposed AWOS-ELM Algorithm:** If \(L\) and \(\mathrm{g}\):\(\Re\rightarrow\Re\) are given, the AWOS-ELM algorithm is able to be summarized as follows:  
Step 1. Initialization phase: choose the initial data chunk \(\aleph_{0}=\{(\mathbf{x}_{i},\mathbf{t}_{i})\}_{i=1}^{N_{0}}\), where \(N_{0}\geq L\). +++ ==ERROR: No output for this page==  
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the closed interval [\(-1\),\(1\)], and if the RBF function is selected, \(\omega_{i}\) is also obtained from the uniform distribution in [\(-1\),\(1\)], while \(b_{i}\) is determined according to the uniform distribution in the open interval (0,0.5) [15]. For the AWOS-ELM algorithm, we select the threshold \(\tau_{r}\) from the domain \(\{x|x=0.02+0.01k,k=0,1,2,\cdots 18\}\), and the best threshold is selected manually according to the prediction performance. In addition, the regulation parameter is set as \(C=10^{-5}\)[24]. In order to verify the prediction performance of AWOS-ELM with the existence of disturbance, Gaussian noise with a standard deviation of 0.0015 is added into the training samples. For the purpose of having a fair comparison, the hidden nodes number of OS-ELM is optimized by the cross validation method, afterwards the optimized hidden nodes number is extended to the AWOS-ELM algorithm.