CN114897144A - Complex value time sequence signal prediction method based on complex value neural network - Google Patents

Abstract

The invention relates to a complex value time sequence signal prediction method based on a complex value neural network, which comprises the following steps: constructing a data set of the complex-valued time sequence signal, and dividing the data set into a training set and a test set; constructing a full-complex neural network and initializing parameters; training the fully-complex neural network by using a training set and a self-adaptive complex value orthogonal gradient descent method until a preset iteration number is reached, and stopping training to obtain a trained fully-complex neural network; and inputting the test set into the trained full-complex neural network to obtain a predicted value of the complex-value time sequence signal. The invention adopts the full-complex neural network to directly predict the complex value time sequence signal and adopts the self-adaptive complex value orthogonal gradient descent method used in training the full-complex neural network, thereby avoiding the damage to the internal relation of the complex value signal, accelerating the convergence speed of the training and improving the precision of predicting the complex value time sequence signal.

Description

Complex-valued time series signal prediction method based on complex-valued neural network

technical field

The invention relates to the technical field of time series signal prediction, in particular to a complex-valued time series signal prediction method based on a complex-valued neural network.

Background technique

Time series signal forecasting is a technology that predicts future data through known historical data. This technology is widely used in wind speed forecasting, speech analysis, fault diagnosis and stock market forecasting. The complex-valued time-series signal essentially reflects the trend of one or more complex-valued random variables changing with time. The complex-valued time-series signal prediction is to mine the variation law of the complex-valued time-series signal to estimate future data.

The classic time series parameter models are: Moving Average (MA), Auto Regressive (AR) and Auto Regression Moving Average (ARMA). The autoregressive moving average model ARMA(p,q) can be expressed as:

Among them, α _p represents the p-th order autoregressive coefficient, β _q is the q-th order moving average coefficient, assuming p is the autoregressive order, q is the moving average order, and ε _t is the white noise at time t. In fact, both the AR model and the MA model are special ARMA models. When α _i = 0, i = 1, 2, ..., p in the above formula, it is called MA(q) model; when β _j =0, j = 1, 2, ..., q, it is called AR( p) model. However, most of the traditional time series forecasting methods are realized by establishing statistical models based on assumptions, such as moving average method, autoregressive method, etc. These methods need to meet too many constraints, so the effect is not ideal in practical application. In recent years, with the rise of artificial intelligence technology, prediction methods based on machine learning have developed rapidly. Compared with traditional methods, methods based on machine learning have the advantages of high accuracy, strong robustness, and can handle nonlinear data well. Support vector machines, Bayesian networks, matrix factorization, artificial neural networks and other methods in machine learning are all good tools for time series signal prediction.

Among these methods, artificial neural network is a commonly used method. Commonly used artificial neural networks include real-valued neural networks and complex-valued neural networks. The traditional real-valued neural network needs to preprocess the complex-valued signal to convert it into a real-valued signal, and the decision boundary of the complex-valued neural network consists of two orthogonal hyperplanes, which has a higher generalization than the real-valued neural network. Therefore, it is a better choice to use a complex-valued neural network to establish a prediction model for complex-valued time series signals.

According to the different activation functions used, complex-valued neural networks can be divided into split complex-valued neural networks and full-complex complex-valued neural networks. However, the commonly used split complex-valued neural network at this stage needs to split the complex-valued signal into real and imaginary parts or amplitude and phase, which will destroy the intrinsic relationship of the complex-valued signal.

At the same time, in the training process of complex-valued neural network, the commonly used methods are: Complex Gradient Descent (CGD), Complex Least Mean Square (CLMS) and CRTRL (Complex-valued real-time recurrent learning), etc. However, these methods all have problems such as slow convergence speed, easy to fall into local minimum, and easy to be affected by saddle points. In order to solve these problems, some adaptive complex step algorithms have appeared in recent years, such as CBBM (Complex Barzilai–Borwein method), CALRT (Complex Adaptable Learning rate Tree) and SDTA (Selectable Direction Tree Algorithm). However, although these adaptive complex-step algorithms can greatly improve the convergence speed, they have a large amount of computation. Just like the CALRT algorithm, its training time increases dramatically due to the use of a double tree structure. Therefore, even if the structure of the general complex-valued neural network is not too large, the time cost added by using these methods is still insignificant compared to the traditional methods.

SUMMARY OF THE INVENTION

Therefore, the technical problem to be solved by the present invention is to overcome the deficiencies in the prior art, and to provide a complex-valued time series signal prediction method based on a complex-valued neural network, which can avoid destroying the internal relationship of the complex-valued signal and reduce the complex steps of solving Long calculation time, speed up the convergence speed of training, and improve the prediction accuracy of complex-valued time series signals.

In order to solve the above-mentioned technical problems, the present invention provides a complex-valued time series signal prediction method based on a complex-valued neural network, comprising the following steps:

S1: Construct a data set of complex-valued time series signals and divide it into a training set and a test set;

S2: Build a fully complex neural network and initialize parameters;

S3: Use the adaptive complex-valued orthogonal gradient descent method and the training set to train the fully complex neural network, stop training until a preset number of iterations is reached, and obtain a fully trained fully complex neural network;

S4: Input the test set into the trained fully complex neural network to obtain the predicted value of the complex-valued time series signal.

Preferably, the fully complex neural network in S2 is a complex-valued forward single hidden layer neural network.

Preferably, in the complex-valued forward single-hidden layer neural network, the expression of the activation function o _l of the lth layer is:

o ₁ = h _l (w _l O _l-1 +b _l );

Among them, l=2,3, h _l (·) is the activation function of the lth layer, w _l is the weight between the (l-1)th layer and the lth layer, o _l-1 is the (l-th layer) 1) The output of the layer, b _l is the bias vector corresponding to o _l-1 .

Preferably, the loss function E of the output layer of the complex-valued forward single hidden layer neural network is:

其中x_j是输入向量，y_j是期望输出，

表示样本的输入空间是P维的复数空间，

表示输出空间是Q维的复数空间。where (·) ^H represents the conjugate transpose, o _j represents the final output of the complex-valued forward single-hidden layer neural network, and J is the total number of training samples; the training samples are expressed as

where x _j is the input vector, y _j is the desired output,

The input space representing the sample is a P-dimensional complex space,

Indicates that the output space is a Q-dimensional complex space.

Preferably, in S3, the fully complex neural network is trained using the adaptive complex-valued orthogonal gradient descent method and the training set, and in the t-th iteration of the adaptive complex-valued orthogonal gradient descent method, the full The expression of the parameters to be adjusted in the complex neural network is:

为第t次迭代中的复梯度向量，η_t为第t次迭代中的自适应复步长。where w ^(t) includes the weights and biases of each layer of neurons in the t-th iteration,

is the complex gradient vector in the t-th iteration, and η _t is the adaptive complex step size in the t-th iteration.

Preferably, the complex gradient vector in the t-th iteration is obtained by a Wirtinger operator.

Preferably, the adaptive complex step size η _t in the t-th iteration is decoupled into two parts, the real part and the imaginary part, and the iteration direction of the t-th iteration is expressed as:

where α _t is the real step size of the adaptive complex step size η _t of the t-th iteration, and β _t is the imaginary step size of the adaptive complex step size η _t of the t-th iteration,

Preferably, the fully complex neural network is trained using the adaptive complex-valued orthogonal gradient descent method and the training set in S3, and the specific process is:

自适应复步长的虚步长β的尺度因子序列为

M表示α的尺度因子个数，N表示β的尺度因子个数；S31: Initialize the parameter w ^(t) that needs to be adjusted, the number of nodes saved in each iteration is K, and the scale factor sequence of the real step size α of the adaptive complex step size is

The scale factor sequence of the imaginary step size β of the adaptive complex step size is

M represents the number of scale factors of α, and N represents the number of scale factors of β;

中的N个因子相乘；Traverse K candidate nodes, and compare the real step size α of each node with the scale factor sequence

Multiply the N factors in ;

S32: Find the parameters k _t and n _t that minimize the value of the loss function E at this time, where k _t is the index of the first-level child node that minimizes the value of the loss function E, and n _t is the one that minimizes the value of the loss function E. The index of the corresponding scale factor of the level child node;

S33: Update the temporary weights according to the current k _t and n _t

递增排序，并将第2到K对参数保存为

作为下次迭代的候选节点；S34: According to the size of the loss function E, the

Sort ascending and save the 2nd to K pairs of parameters as

as a candidate node for the next iteration;

对应的最优节点基础上，分别将虚步长β与尺度因子

的M个因子相乘；S35: Temporary weights obtained in S33

On the basis of the corresponding optimal node, the virtual step size β and the scale factor are respectively

Multiply the M factors of ;

S36: Find the parameter m _t that minimizes the value of the loss function E at this time, where m _t is the index of the secondary child node that minimizes the value of the loss function E;

S37: Update the weight w ^(t) according to m _t , k _t and n _t at this time;

与S34中K-1个的参数对

合并，一共K对参数用作下次迭代的候选节点；S38: save parameter pair

Pairs with K-1 parameters in S34

Merge, a total of K pairs of parameters are used as candidate nodes for the next iteration;

S39: Repeat S31 to S38.

中的N个因子相乘，表达式为：Preferably, in said S31, the real step size α of each node is respectively associated with the scale factor sequence

The N factors in are multiplied, and the expression is:

表示第k个根节点的实步长，a_n表示第n个尺度因子，

表示与a_n对应的一级子节点的实步长；

表示第k个根节点参数

在使用上一步得到的实步长

更新之后的参数；

表示在使用参数

时得到的损失值，E_k,n为

的简化表示，X表示训练集数据，

表示训练集标签。in,

represents the real step size of the kth root node, a _n represents the nth scale factor,

Represents the real step size of the first _- level child node corresponding to an;

Represents the kth root node parameter

Using the real step size obtained in the previous step

The updated parameters;

Indicates that a parameter is being used

The loss value obtained when E _k,n is

A simplified representation of , where X represents the training set data,

represents the training set label.

的M个因子相乘，表达式为：Preferably, in the S35, the virtual step size β and the scale factor are respectively used

Multiplying the M factors of , the expression is:

表示与b_m对应的二级子节点的虚步长；

表示与b_m对应的二级子节点的参数，

表示在使用参数

时得到的损失值，

是

的简化表示。Among them, β ^(t-1) represents the imaginary step size determined by the t-1th iteration, b _m is the mth scale factor,

represents the virtual step size of the secondary child node corresponding to b _m ;

represents the parameter of the secondary child node corresponding to b _m ,

Indicates that a parameter is being used

The loss value obtained when

Yes

a simplified representation of .

The above-mentioned technical scheme of the present invention has the following advantages compared with the prior art:

1. The fully complex neural network is used to directly predict the complex-valued time series signal, and the complex-valued signal is processed as a whole. Compared with the traditional signal prediction method, it has strong nonlinear fitting ability, can directly process the complex-valued signal and accurately. high degree of advantage. It avoids breaking the internal relationship of complex-valued signals by splitting the complex-valued signal into the form of real and imaginary parts or amplitude and phase.

2. The present invention decouples the complex step size into two orthogonal parts, the real step size and the virtual step size. The search direction of the real step size is the negative gradient direction, while the search direction of the virtual step size is orthogonal to the negative gradient direction. It adjusts its size adaptively in two orthogonal directions respectively, and can search for suitable step sizes along these two directions without affecting each other. Compared with the traditional self-adaptive complex step size method, the calculation amount of solving the complex step size is greatly reduced on the basis of ensuring the convergence speed, and the convergence speed of training is accelerated.

3. The adaptive complex-valued orthogonal gradient descent method used in the present invention can help to escape from the saddle point and the local minimum point in the complex-valued neural network training process, and improve the prediction accuracy of the complex-valued time series signal.

Description of drawings

In order to make the content of the present invention easier to understand clearly, the present invention will be described in further detail below according to specific embodiments of the present invention and in conjunction with the accompanying drawings, wherein

FIG. 1 is a system block diagram of the present invention.

Figure 2 is a flow chart of the present invention.

FIG. 3 is a schematic diagram of the process of training a fully complex neural network by the adaptive complex-valued orthogonal gradient descent method in the present invention.

FIG. 4 is a flow chart of training a fully complex neural network by the adaptive complex-valued orthogonal gradient descent method in the present invention.

FIG. 5 is a comparison diagram of a training process for training a complex-valued neural network using different training methods and reference circular signals, respectively, in an embodiment of the present invention.

FIG. 6 is a comparison diagram of a training process for training a complex-valued neural network using different training methods and a reference non-circular signal, respectively, in an embodiment of the present invention.

Detailed ways

The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and implement it, but the embodiments are not intended to limit the present invention.

As shown in the system block diagram of Fig. 1, the present invention adopts complex-valued neural network to predict complex-valued time series signals, and adopts adaptive complex-valued orthogonal gradient descent method to realize efficient training of complex-valued neural network. First, the signal values [x(k), x(k-1),...,x(k-P+1)] ^T of the first P moments are collected from the signal generator as the input of the complex-valued neural network, and the next moment The signal value x(k+1) is used as the expected output of the complex-valued neural network, and the loss function E(x) is constructed through the mean square error, and the loss function is optimized using the adaptive complex-valued orthogonal gradient descent method, so as to obtain a suitable The network model is finally used to realize the signal prediction.

Referring to the flowchart shown in FIG. 2, a method for predicting a complex-valued time series signal based on a complex-valued neural network of the present invention includes the following steps:

S1: Construct a dataset of complex-valued time series signals, and divide it into a training set and a test set of a complex-valued neural network. The method for constructing the complex-valued time series data set is to normalize the complex-valued time-series signals collected in reality or generated by simulation. In this embodiment, the complex-valued time-series signals are generated by a signal generator. Use MATLAB simulation to generate reference complex-valued circle and non-circular time series signals. The reference circle signal is generated by a stable AR (Autoregressive) process. The AR process is expressed as:

r(k)=1.79r(k-1)-1.85r(k-2)+1.27r(k-3)-0.41r(k-4)+n(k);

where n(k) is a complex-valued Gaussian circular white noise with mean 0 and variance 1. Among them, r(k-t), t=0, 1, 2, 3, 4 represents the signal value at time k-t. The reference non-circular signal is generated by the ARMA (Autoregressive Moving Average) process, and the ARMA process is expressed as:

r(k)=1.79r(k-1)-1.85r(k-2)+1.27r(k-3)-0.41r(k-4)+0.1r(k-5)+2n(k)+ 0.5n ^* (k)+n(k-1)+0.9n ^* (k-1);

Among them, n(k) needs to satisfy:

E{n(ki)n ^* (kj)}=δ(ij),

E{n(k-i)n(k-j)}=Cδ(i-j);

δ为狄拉克函数。将通过上述两个过程生成的复值时序信号中的连续3000个样本作为训练集，连续2000个样本作为测试集，然后进行归一化，构成所需的数据集。where n(k) is complex-valued Gaussian non-circular white noise, C=0.95. Among them,

δ is the Dirac function. Take 3000 consecutive samples in the complex-valued time series signal generated by the above two processes as the training set, and 2000 consecutive samples as the test set, and then normalize to form the required data set.

S2: The complex-valued neural network used in this embodiment is a fully complex neural network. A fully complex neural network is constructed and parameters are initialized; the fully complex neural network in this embodiment is a complex-valued forward single hidden layer neural network. Using the powerful function approximation ability of neural network to establish a prediction model can overcome the shortcomings of existing models in terms of adaptability and data feature extraction.

In the complex-valued forward single-hidden layer neural network, the expression of the activation function o _l of the lth layer is:

o _l = h _l (w _l o _l-1 +b _l );

Among them, l=2,3, h _l (·) is the activation function of the lth layer, w _l is the weight between the (l-1)th layer and the lth layer, and the weight w _l is a complex value, which is The real and imaginary parts of are randomly initialized, and the weights are adjusted by backpropagation during training. o _l-1 is the output of the (l-1)th layer, and b _l is the bias vector corresponding to o _l-1 . In this embodiment, the number of hidden layer neurons of the complex-valued forward single hidden layer neural network is 20, the connection weights and biases are randomly initialized, the activation function is the tanh function, and the loss function is the mean square error function.

The loss function E of the output layer of the complex-valued forward single hidden layer neural network is:

其中x_j是输入向量，y_j是期望输出，

表示样本的输入空间是P维的复数空间，

表示输出空间是Q维的复数空间。where (·) ^H represents the conjugate transpose, o _j represents the final output of the complex-valued forward single-hidden layer neural network, and J is the total number of training samples; the training samples are expressed as

where x _j is the input vector, y _j is the desired output,

The input space representing the sample is a P-dimensional complex space,

Indicates that the output space is a Q-dimensional complex space.

S3: Use the training set as the input set of the fully complex neural network, and use the Adaptive Orthogonal Complex-valued Gradient Descent (AOCGD) method to train the fully complex neural network until a preset number of iterations is reached (or other iterative stop conditions) to stop training, and get the fully complex neural network that has been trained. The preset number of iterations in this embodiment is 200 times.

其中，w^(t)包括第t次迭代中每层神经元的权值和偏置，

为第t次迭代中复梯度向量所述第t次迭代中复梯度向量

通过Wirtinger算子得到；η_t为第t次迭代中自适应复步长，且η_t是一个复数。所述第t次迭代中自适应复步长η_t解耦成实部和虚部两个部分，第t次迭代的迭代方向表示为：

其中，α_t为第t次迭代的自适应复步长η_t的实步长，所述β_t为第t次迭代的自适应复步长η_t的虚步长，

When using the adaptive complex-valued orthogonal gradient descent method and the training set to train the fully complex neural network, in the t-th iteration of the adaptive complex-valued orthogonal gradient descent method, the parameters that need to be adjusted in the fully complex neural network are The expression is:

where w ^(t) includes the weights and biases of each layer of neurons in the t-th iteration,

is the complex gradient vector in the t-th iteration said complex gradient vector in the t-th iteration

Obtained by the Wirtinger operator; η _t is the adaptive complex step size in the t-th iteration, and η _t is a complex number. In the t-th iteration, the adaptive complex step size η _t is decoupled into two parts, the real part and the imaginary part, and the iteration direction of the t-th iteration is expressed as:

where α _t is the real step size of the adaptive complex step size η _t of the t-th iteration, and β _t is the imaginary step size of the adaptive complex step size η _t of the t-th iteration,

As shown in Figure 3, the parameters adaptively adjust the size of α and β during training. For simplicity, each node of the first layer in FIG. 3 is called a root node, each node of the second layer is called a first-level child node, and each node of the third layer is called a second-level child node. The solid line arrow in the figure indicates that calculation needs to be performed, and the dotted line arrow indicates that the first-level child node is directly saved as the second-level child node.

As shown in Figure 4, the fully complex neural network is trained using the training set and the adaptive complex-valued orthogonal gradient descent method, and the specific process is as follows:

自适应复步长的虚步长β的尺度因子序列为

M表示α的尺度因子个数，N表示β的尺度因子个数。本实施例中α的尺度因子序列为[0.5,1,2]，β的尺度因子序列为[-10,-0.1,0,0.1,10]，每次迭代保存的候选节点数为5。α的初始值为0.25，β的初始值为0.1。S31: Initialize the parameter to be adjusted, that is, the weight w ^(t) , the number of nodes saved in each iteration is K, and the scale factor sequence of the real step size α of the adaptive complex step size is

The scale factor sequence of the imaginary step size β of the adaptive complex step size is

M represents the number of scale factors of α, and N represents the number of scale factors of β. In this embodiment, the scale factor sequence of α is [0.5, 1, 2], the scale factor sequence of β is [-10, -0.1, 0, 0.1, 10], and the number of candidate nodes saved in each iteration is 5. The initial value of α is 0.25, and the initial value of β is 0.1.

中的N个因子相乘，表达式为：Traverse K candidate nodes, and compare the real step size α of each node with the scale factor sequence

The N factors in are multiplied, and the expression is:

表示图3中第k个根节点的实步长，a_n表示第n个尺度因子，

表示与a_n对应的一级子节点的实步长；

表示图3中第k个根节点参数

在使用上一步得到的实步长

更新之后的参数；

表示在使用参数

时得到的损失值，E_k,n为

的简化表示，X表示训练集数据，

表示训练集标签。in,

represents the real step size of the kth root node in Figure 3, a _n represents the nth scale factor,

Represents the real step size of the first _- level child node corresponding to an;

Represents the kth root node parameter in Figure 3

Using the real step size obtained in the previous step

updated parameters;

Indicates that a parameter is being used

The loss value obtained when E _k,n is

A simplified representation of , where X represents the training set data,

represents the training set label.

S32: Find the parameters k _t and n _t that minimize the value of the loss function E at this time, expressed as:

Among them, k _t is the index of the first-level child node that minimizes the loss function E value, and n _t is the index of the corresponding scale factor of the first-level child node that minimizes the loss function E value; at this time, the real step is determined by the loss function. long α.

表示为：

S33: Update the temporary weights according to the current k _t and n _t

Expressed as:

递增排序，并将第2到K对参数保存为

作为下次迭代的候选节点；S34: According to the size of the loss function E, the

Sort ascending and save the 2nd to K pairs of parameters as

as a candidate node for the next iteration;

对应的最优节点基础上，分别将虚步长β与尺度因子

的M个因子相乘，表达式为：S35: Temporary weights obtained in S33

On the basis of the corresponding optimal node, the virtual step size β and the scale factor are respectively

Multiplying the M factors of , the expression is:

表示与b_m对应的二级子节点的虚步长；

表示与b_m对应的二级子节点的参数，

表示在使用参数

时得到的损失值，

是

的简化表示。Among them, β ^(t-1) represents the imaginary step size determined by the t-1th iteration, b _m is the mth scale factor,

represents the virtual step size of the secondary child node corresponding to b _m ;

represents the parameter of the secondary child node corresponding to b _m ,

Indicates that a parameter is being used

The loss value obtained when

Yes

a simplified representation of .

S36: Find the parameter m _t that minimizes the value of the loss function E at this time, expressed as:

m_t是使得损失函数E值最小的二级子节点的索引；

m _t is the index of the secondary child node that minimizes the loss function E value;

此时通过损失函数确定了虚步长β。S37: Update the weight w ^(t) according to m _t , k _t and n _t at this time, which is expressed as:

At this time, the virtual step size β is determined by the loss function.

与S34中K-1个的参数对

合并，一共K对参数用作下次迭代的候选节点；S38: save parameter pair

Pairs with K-1 parameters in S34

Merge, a total of K pairs of parameters are used as candidate nodes for the next iteration;

S39: Repeat S31 to S38.

S4: Input the test set into the trained fully complex neural network to obtain the predicted value of the complex-valued time series signal.

The invention decouples the complex step size into two orthogonal parts, the real step size and the virtual step size, and introduces an adaptive learning tree (Adaptive Learning Tree, ALRT) algorithm to adjust the real step size and the virtual step size adaptively. Adapt the complex-valued orthogonal gradient descent method to achieve efficient training of complex-valued neural networks. The complex-valued neural network trained by the above algorithm is applied to the prediction of complex-valued time series signals, and an effective prediction model can be established. Compared with the prior art, the above-mentioned technical solution of the present invention has the following advantages:

1. The fully complex neural network is used to directly predict the complex-valued time series signal, and the complex-valued signal is processed as a whole. Compared with the traditional signal prediction method, it has strong nonlinear fitting ability, can directly process the complex-valued signal and accurately. high degree of advantage. It avoids breaking the internal relationship of complex-valued signals by splitting the complex-valued signal into the form of real and imaginary parts or amplitude and phase.

2. The present invention decouples the complex step size into two orthogonal parts, the real step size and the virtual step size. The search direction of the real step size is the negative gradient direction, while the search direction of the virtual step size is orthogonal to the negative gradient direction. It adjusts its size adaptively in two orthogonal directions respectively, and can search for suitable step sizes along these two directions without affecting each other. Compared with the traditional self-adaptive complex step size method, the calculation amount of solving the complex step size is greatly reduced on the basis of ensuring the convergence speed, and the convergence speed of training is accelerated.

3. The adaptive complex-valued orthogonal gradient descent method used in the present invention can help to escape from the saddle point and the local minimum point in the complex-valued neural network training process, and improve the prediction accuracy of the complex-valued time series signal.

In order to further illustrate the beneficial effects of the present invention, in this embodiment, the Complex Gradient Descent (CGD), the Adaptive Orthogonal Complex-valued Gradient Descent (AOCGD), the Adaptive Orthogonal Step size algorithm CRTRL (Complex-valued real-time recurrent learning) and CBBM (Complex Barzilai–Borwein method) four training algorithms, respectively use the reference circular signal and the reference non-circular signal to train the complex-valued neural network, and the reference circular signal and the reference non-circular signal for prediction. The training results using different training methods and reference circular signals are shown in Figure 5, the training results predicted using different training methods and reference non-circular signals are shown in Figure 6, and the prediction results of the reference circular signals are shown in Table 1. The prediction results of the circular signal are shown in Table 2.

Table 1 Prediction results of reference circular signals under different training methods

Table 2 Prediction results of benchmark non-circular signals under different training methods

In Figures 5 and 6, the abscissa represents the number of iterations, and the ordinate represents the loss value of the loss function E. It can be seen from Figures 5 and 6 that the loss value of the present invention using the adaptive complex-valued orthogonal gradient descent method decreases the fastest , the fastest convergence rate. Tables 1 and 2 respectively use the mean absolute error (Mean Absolute Error, MAE), the standard root mean square error (normalized Root Mean Squared Error, nRMSE) and the coefficient of determination R ² three outcome evaluation indicators to evaluate the prediction results. The smaller the value and the larger the R ² , the more accurate the prediction result. It can be seen from Table 1 and Table 2 that the prediction results obtained by using the present invention have the highest accuracy.

As will be appreciated by those skilled in the art, the embodiments of the present application may be provided as a method, a system, or a computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) having computer-usable program code embodied therein.

The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present application. It will be understood that each flow and/or block in the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to the processor of a general purpose computer, special purpose computer, embedded processor or other programmable data processing device to produce a machine such that the instructions executed by the processor of the computer or other programmable data processing device produce Means for implementing the functions specified in a flow or flow of a flowchart and/or a block or blocks of a block diagram.

These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory result in an article of manufacture comprising instruction means, the instructions The apparatus implements the functions specified in the flow or flow of the flowcharts and/or the block or blocks of the block diagrams.

These computer program instructions can also be loaded on a computer or other programmable data processing device to cause a series of operational steps to be performed on the computer or other programmable device to produce a computer-implemented process such that The instructions provide steps for implementing the functions specified in the flow or blocks of the flowcharts and/or the block or blocks of the block diagrams.

Obviously, the above-mentioned embodiments are only examples for clear description, and are not intended to limit the implementation manner. For those of ordinary skill in the art, other different forms of changes or modifications can also be made on the basis of the above description. There is no need and cannot be exhaustive of all implementations here. However, the obvious changes or changes derived from this are still within the protection scope of the present invention.

Claims (10)

1. a complex-valued time series signal prediction method based on a complex-valued neural network, is characterized in that, comprises the following steps: S1: Construct a data set of complex-valued time series signals and divide it into a training set and a test set; S2: Build a fully complex neural network and initialize parameters; S3: Use the adaptive complex-valued orthogonal gradient descent method and the training set to train the fully complex neural network, stop training until a preset number of iterations is reached, and obtain a fully trained fully complex neural network; S4: Input the test set into the trained fully complex neural network to obtain the predicted value of the complex-valued time series signal. 2 . A complex-valued time series signal prediction method based on a complex-valued neural network according to claim 1 , wherein the fully complex neural network in the S2 is a complex-valued forward single hidden layer neural network. 3 . 3. a kind of complex-valued time series signal prediction method based on complex-valued neural network according to claim 2, is characterized in that: in described complex-valued forward single hidden layer neural network, the activation function _o1 of the 1st layer is The expression is: o _l = h _l (w _l o _l-1 +b _l ); Among them, l=2,3, h _l (·) is the activation function of the lth layer, w _l is the weight between the (l-1)th layer and the lth layer, o _l-1 is the (l-th layer) 1) The output of the layer, b _l is the bias vector corresponding to o _l-1 . 4. a kind of complex-valued time series signal prediction method based on complex-valued neural network according to claim 3, is characterized in that: the loss function E of the output layer of described complex-valued forward single hidden layer neural network is:

where (·) ^H represents the conjugate transpose, o _j represents the final output of the complex-valued forward single-hidden layer neural network, and J is the total number of training samples; the training samples are expressed as

where x _j is the input vector, y _j is the desired output,

The input space representing the sample is a P-dimensional complex space,

Indicates that the output space is a Q-dimensional complex space. 5. A complex-valued time series signal prediction method based on a complex-valued neural network according to claim 4, characterized in that: in said S3, an adaptive complex-valued orthogonal gradient descent method and the training set are used to train the said In the fully complex neural network, in the t-th iteration of the adaptive complex-valued orthogonal gradient descent method, the expression of the parameters to be adjusted in the fully complex neural network is:

where w ^(t) includes the weights and biases of each layer of neurons in the t-th iteration,

is the complex gradient vector in the t-th iteration, and η _t is the adaptive complex step size in the t-th iteration. 6 . The complex-valued time series signal prediction method based on a complex-valued neural network according to claim 5 , wherein the complex gradient vector in the t-th iteration is obtained by a Wirtinger operator. 7 . 7. a kind of complex-valued time series signal prediction method based on complex-valued neural network according to claim 5, is characterized in that: the self-adaptive complex step size η _t in the described t-th iteration is decoupled into real part and imaginary part There are two parts, and the iteration direction of the t-th iteration is expressed as:

where α _t is the real step size of the adaptive complex step size η _t of the t-th iteration, and β _t is the imaginary step size of the adaptive complex step size η _t of the t-th iteration,

8. A complex-valued time series signal prediction method based on a complex-valued neural network according to claim 7, characterized in that: in said S3, an adaptive complex-valued orthogonal gradient descent method and the training set are used to train the said Fully complex neural network, the specific process is: S31: Initialize the parameter w ^(t) that needs to be adjusted, the number of nodes saved in each iteration is K, and the scale factor sequence of the real step size α of the adaptive complex step size is

The scale factor sequence of the imaginary step size β of the adaptive complex step size is

M represents the number of scale factors of α, and N represents the number of scale factors of β; Traverse K candidate nodes, and compare the real step size α of each node with the scale factor sequence

Multiply the N factors in ; S32: Find the parameters k _t and n _t that minimize the value of the loss function E at this time, where k _t is the index of the first-level child node that minimizes the value of the loss function E, and n _t is the one that minimizes the value of the loss function E. The index of the corresponding scale factor of the level child node; S33: Update the temporary weights according to the current k _t and n _t

S34: According to the size of the loss function E, the

Sort ascending and save the 2nd to K pairs of parameters as

as a candidate node for the next iteration; S35: Temporary weights obtained in S33

On the basis of the corresponding optimal node, the virtual step size β and the scale factor are respectively

Multiply the M factors of ; S36: Find the parameter m _t that minimizes the value of the loss function E at this time, where m _t is the index of the secondary child node that minimizes the value of the loss function E; S37: Update the weight w ^(t) according to m _t , k _t and n _t at this time; S38: save parameter pair

Pairs with K-1 parameters in S34

Merge, a total of K pairs of parameters are used as candidate nodes for the next iteration; S39: Repeat S31 to S38. 9. A complex-valued time series signal prediction method based on a complex-valued neural network according to claim 8, wherein in said S31, the real step size α of each node is respectively associated with the scale factor sequence

The N factors in are multiplied, and the expression is:

in,

represents the real step size of the kth root node, a _n represents the nth scale factor,

Represents the real step size of the first _- level child node corresponding to an;

Represents the kth root node parameter

Using the real step size obtained in the previous step

updated parameters;

Indicates that a parameter is being used

The loss value obtained when E _{k, n} is E

A simplified representation of , where X represents the training set data,

represents the training set label. 10. A complex-valued time series signal prediction method based on a complex-valued neural network according to claim 8, characterized in that: in said S35, the imaginary step size β and the scale factor are respectively used

Multiplying the M factors of , the expression is:

Among them, β ^(t-1) represents the imaginary step size determined by the t-1th iteration, b _m is the mth scale factor,

represents the virtual step size of the secondary child node corresponding to b _m ;

represents the parameter of the secondary child node corresponding to b _m ,

Indicates that a parameter is being used

The loss value obtained when

Yes

a simplified representation of .

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