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# 统计代写|时间序列分析代写Time-Series Analysis代考|Calculations of feed-forward network configuration

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## 统计代写|时间序列分析代写Time-Series Analysis代考|Calculations of feed-forward network configuration

In a feed-forward multi-layered neural network, input signals are fed to the input layer, and they pass through hidden layers, then through the output layer, to yield a final output. When the signals are passing from one layer to the other, the respective connection strengths or weights are given to them for multiplication and to produce weighted vector inputs. Then, these weighted vectors are aggregated to generate a net input at an intermediate neuron. Subsequently, this generated net input passes through an activation function or the thresholding unit to transmit its net signal to the next layer and so forth to result in a final output.

Net input received by the hidden layer at the $j^{\text {th }}$ node is estimated as follows:
$$n e t h_j=\sum_{i=1}^{n i} w h_{j i} x_i \ldots$$
where $n i=$ no. of neurons in the input layer and $w h_{j i}=$ connecting strength between $i^{t h}$ node of the input layer and $j^{\text {th }}$ node of the hidden layer. The output of $j^{\text {th }}$ node of the hidden layer $h_j$ is given by
$$h_j=f\left(n e t h_j\right)$$
$f($.) is the usual sigmoidal transfer/activation function, and operated as follows
$$h_j=\frac{1}{1+e^{\left(-n e t h_j\right)}}$$
Further, the net input received by the output layer at $k^{\text {th }}$ node is estimated as follows:
$$n e t y_k=\sum_{j=1}^{n h} w o_{k j} h_j$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Error back propagation

The calculated error at the output side is back propagated to the input side layers through hidden layers to determine the updated or modified weights (Hario \& Jokinen, 1991). In a single input-output mapping, the sum square error (E) of the data set can be written as
$$\mathrm{E}=\frac{1}{2} \sum_{k=1}^{n o}\left(y_k-t_k\right)^2$$
where $y_k=$ estimated output value at the $k^{\text {th }}$ node and $t_k=$ target or observed output value at the same node or neuron. The incremental changes of the connecting strengths calculated in subsequent trials are subtracted from their previous strengths or weights and the deviations or errors are then propagated back to the input layers to minimize the error function in the next trial as follows.
\begin{aligned} & w o_{k j}^{\text {new }}=w o_{k j}^{\text {old }}-\Delta w o_{k j} \ & w h_{j i}^{\text {new }}=w h_{j i}^{o l d}-\Delta w h_{j i} \end{aligned}
$\Delta w o_{k j}$ and $\Delta w h_{j i}$ are increments in connecting strengths between the output layer and its corresponding hidden layer and hence,
$$\Delta w o_{k j}=\eta\left(\frac{\partial E}{\partial w o_{k j}}\right)$$
Applying the chain rule, $\frac{\partial E}{\partial w o_{k j}}$ is re-written as
$$\frac{\partial E}{\partial w o_{k j}}=\frac{\partial E}{\partial y_k} \frac{\partial y_k}{\partial n e t y_k} \frac{\partial n e t y_k}{\partial w o_{k j}}$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Calculations of feed-forward network configuration

$$n e t h_j=\sum_{i=1}^{n i} w h_{j i} x_i \ldots$$

$$h_j=f\left(n e t h_j\right)$$

$f($.)为常用的s型传递/激活函数，操作如下
$$h_j=\frac{1}{1+e^{\left(-n e t h_j\right)}}$$

$$n e t y_k=\sum_{j=1}^{n h} w o_{k j} h_j$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Error back propagation

$$\mathrm{E}=\frac{1}{2} \sum_{k=1}^{n o}\left(y_k-t_k\right)^2$$

\begin{aligned} & w o_{k j}^{\text {new }}=w o_{k j}^{\text {old }}-\Delta w o_{k j} \ & w h_{j i}^{\text {new }}=w h_{j i}^{o l d}-\Delta w h_{j i} \end{aligned}
$\Delta w o_{k j}$和$\Delta w h_{j i}$是输出层与其对应的隐藏层之间连接强度的增量，因此，
$$\Delta w o_{k j}=\eta\left(\frac{\partial E}{\partial w o_{k j}}\right)$$

$$\frac{\partial E}{\partial w o_{k j}}=\frac{\partial E}{\partial y_k} \frac{\partial y_k}{\partial n e t y_k} \frac{\partial n e t y_k}{\partial w o_{k j}}$$

## MATLAB代写

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