Error Correction of Weak Current Measurement System Based on Wavelet Denoising and Generalized Regression Neural Network

2.1 Wavelet denoising algorithm

Wavelets are orthogonal bases of finite length that can be used to represent a time series into a time-scale domain at different resolutions [13]. In this paper, the denoising capabilities of wavelet transforms were used to preprocess the weak current data to improve the error correction capabilities of the neural network model. The method of wavelet denoising on the basis of threshold processing, initialized by Dohono, can estimate the threshold value through wavelet transform coefficient and reconstruct signals with the retained wavelet coefficient [14]. The basic model of wavelet denoising can be set as:

$s(t)=f(t)+\delta n(t)$     (1)

Among them: s(t) is the signal containing noise; f(t) is the original signal; n(t) is the noise signal, δ is the noise intensity. Wavelet threshold denoising can be performed in the following 3 steps [15]:

(1) Perform wavelet transform on the signal with noise: selecting a wavelet base, determining the decomposition scale N, and then performing N-level wavelet decomposition on the signal s(t) to obtain wavelet decomposition coefficients W;

(2) Perform threshold processing on wavelet coefficient: calculating the wavelet threshold λ, selecting the appropriate threshold function(soft threshold or hard threshold), and quantizing the wavelet coefficients to obtain new wavelet coefficients W_λ;

(3) Reconstruct the signal: performing wavelet inverse transformation on the obtained wavelet coefficients W_λ to obtain the denoised signal.

The soft threshold function was obtained by improving the hard threshold function, and the denoising effect was smoother than that of the hard threshold function. The definition of the soft threshold function used in this study is shown in Eq. (2) [16]:

${{\overline{\text{w}}}_{j,k}}=\left\{ \begin{matrix}   sgn ({{w}_{j,k}})({{w}_{j,k}}-\lambda ),\left| {{w}_{j,k}} \right|\text{ }\lambda   \\   0,\left| {{w}_{j,k}} \right|\text{ }\lambda   \\\end{matrix} \right.$     (2)

In the formula, sgn(⋅) is sign function, $w_{j, k}$ and $\bar{w}_{j, k}$ represent the wavelet coefficients before and after the denoising process, respectively, and λ represents the threshold. General threshold adopted in this paper is as follows:

$\lambda =\delta \sqrt{2\ln X}$      (3)

Here X is signal length and $\delta=\frac{\operatorname{median}\left(\left|w_{j, k}\right|\right)}{0.6745}$  [17].

So far, there has not been mature theory to guide how to choose wavelet basis functions. However, in many practices of wavelet analysis, some relatively effective experience guidance can be obtained. To illustrate, for the processing of digital signals, Haar or Daubechies wavelet basis functions can often get better processing results, and Daubechies wavelet is an orthogonal wavelet function with compact support created by the famous wavelet scholar, Ingrid Daubechies [18]. Daubechies series wavelets are called dbN for short. Therefore, only those wavelet basis functions are considered in this paper. The determination of the best wavelet basis function and decomposition levels is discussed in Section 4.1.

In this paper, the wavelet threshold denoising method was used to preprocess the weak current signal sampling data, which can improve the validity and completeness of the data to a certain extent, thereby improving the generalization ability and accuracy of model prediction.

2.2 GRNN model

Generalized Regression Neural Network (GRNN) was put forward by American scholar Donald F. Specht in 1991. It owns strong nonlinear mapping ability, flexible network structure, as well as high fault tolerance and robustness, which is suitable for solving nonlinear problems [19]. In contrast with other iterative networks, GRNN possesses less training time (See Figure 1) [20].

In this work, the Keithley 6221 current source output current range was set to 1-700 nA, and the current output step value was 0.1 nA. The weak current digital readout system and PC acquisition software in the experimental system were used to continuously collect each point of current output by Keithley 6221 current source. The current of each point was continuously collected 200 times, and 1,400,000 weak current measurement data samples were obtained in total. These samples were used as the weak current data set of this experiment to verify the performance of the method followed in Section 2 of this paper. In order to meet the performance requirements of the simulation system for the simulation experiment, a computing system (64 bit 32 GB RAM, 3.6 GHz) is used for current measurement error correction applications in MATLAB Version 2016.

4.1 Denoising

In order to denoise the collected weak current signal and obtain a better denoising effect, it is necessary to select an appropriate wavelet basis function and wavelet decomposition scale N. For the sake of evaluating the effect of wavelet denoising algorithm, the following indicators were used as evaluation criteria:

(1) Signal Noise Ratio (SNR), the formula is defined as follows:

$SNR=10\log [\frac{\sum\limits_{i=1}^{n}{f(i)}}{\sum\limits_{i=1}^{n}{{{(\hat{f}(i)-f(i))}^{2}}}}]$     (5)

(2) Root Mean Square Error (RMSE), its formula is defined as:

$RMSE=\sqrt{\frac{1}{\text{n}}{{\sum\limits_{i=1}^{n}{[\hat{f}(i)-f(i)]}}^{2}}}$     (6)

where, f(i) is the original signal, $\hat{f}(i)$ is the signal after wavelet denoising, and n is the number of sampling points. After the signal was denoised by wavelet, the value of SNR became larger and the value of RMSE got smaller, indicating that it is closer to the original signal, which can prove that the wavelet basis function and decomposition levels have better denoising effect. Haar and Daubechies (dbN) as wavelet bases were commonly used. This paper used Haar, db2, db3, db4, db5, db6 wavelet basis functions and different decomposition levels to denoise the collected weak current data. The corresponding denoising results are shown in Table 1.

According to the experimental results in Table 1, db5 wavelet was used to decompose the weak current sampling data into four levels to realize denoising. The thresholds corresponding to the four levels are $\lambda_{\text {level } 1}=0.12, \lambda_{\text {level } 2}=0.22 , \lambda_{\text {level3 }}=2.00 \quad and \quad\lambda_{\text {level } 4}=1.49$  respectively. The SNR after denoised can reach 51.2041 which is higher than that of Haar wavelet and other Daubechies wavelets denoising. The RMSE after denoised is 1.1064, which is smaller than that of Haar wavelet and other Daubechies wavelets denoising. Therefore, this paper used db5 wavelet basis function to decompose the original weak current sampling data in four levels. Figure 5 shows the denoised result of the actual measurement of weak current data in the experimental system when Keithley 6221 output a standard current of 100 nA.

Table 1. The denoised effect of different wavelet bases and decomposition levels (the DL is decomposition level)

5.png

Figure 5. The effect of Wavelet threshold denoising

The blue curve in Figure 5 is the actual sampling data curve of the weak current, and the red curve is the weak current signal curve through wavelet denoising. It can be seen from Figure 5 that the db5 wavelet base is used to decompose the weak current signal into four levels to achieve the denoising effect, which can achieve the purpose of data preprocessing.

4.2 Current measurement error correction based on GRNN model

In this paper, the actual sample data of weak current signals were denoised as the input of GRNN training and testing. Besides, the current value output by Keithley 6221 was taken as the output value of GRNN training and the expected value of test output. It can be seen from Figure 5 that 20 current value samples can be obtained by continuously measuring the current value output from Keithley 6221, and the complete and effective sampling of one current value can be finished. Therefore, 20 consecutive samples were randomly selected from every 200 continuous current data samples in the dataset to obtain 140,000 current data samples, which constituted the dataset of GRNN model training and testing in this experiment. In the new data set, 70% of the 20 samples measured for each current value of Keithley 6221 output were used for the construction of GRNN model, and the remaining 30% of the data samples were used to test the performance of the constructed GRNN model.

In order to evaluate the performance of the proposed algorithm model, the mean square error (MSE), root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE) were used as evaluation criteria, and the corresponding formula was Eq. (7)-(10).

(1) Mean Square Error(MSE):

$MSE=\frac{1}{\text{n}}{{\sum\limits_{i=1}^{n}{[{{x}_{i}}-{{{\hat{x}}}_{i}}]}}^{2}}$     (7)

(2) Root Mean Square Error(RMSE):

$RMSE=\sqrt{\frac{1}{\text{n}}{{\sum\limits_{i=1}^{n}{[{{x}_{i}}-{{{\hat{x}}}_{i}}]}}^{2}}}$     (8)

(3) Mean Absolute Error(MAE):

$MAE=\frac{1}{\text{n}}\sum\limits_{i=1}^{n}{\left| {{x}_{i}}-{{{\hat{x}}}_{i}} \right|}$     (9)

(4) Mean Absolute Percentage Error(MAPE):

$MAPE=\frac{1}{\text{n}}\sum\limits_{i=1}^{n}{\left| \frac{{{x}_{i}}-{{{\hat{x}}}_{i}}}{{{x}_{i}}} \right|}\times 100%$    (10)

In the above formulas: x_i is the actual value, $\hat{x}_i$ is the model predicted value, and n is the number of samples.

In GRNN model, σ is the only parameter that needs to be adjusted; too large σ will lead to the smoother output of GRNN model, and the consequent error will increase; too small σ will lead to overfitting for the output of GRNN model. Therefore, the choice of σ value will have a decisive impact on the performance of GRNN model. In this paper, the grid search method was used to optimize the smoothing parameter, and the K-fold cross validation was carried out for the GRNN model to avoid overfitting and under fitting. MSE was used as the criteria for the performance evaluation of cross validation. The $\left[\sigma_{\text {Low }}, \sigma_{\text {High }}\right]$ was divided into N equal parts, and the σ value was iteratively selected in $\left[\sigma_{\text {Low }}, \sigma_{\text {High }}\right]$. The definition of σ value is shown in Eq. (11).

$\sigma ={{\sigma }_{Low}}+(k-1)\frac{{{\sigma }_{High}}-{{\sigma }_{Low}}}{N}$     (11)

6.png

Figure 6. The cross-validation cure for GRNN model (Optimal σ=0. 00027)

The optimal σ value was selected according to the minimum average MSE. In this paper, the optimization range of σ value was set as [0.00001,0.001], and the iteration times ‘m’ was 100. Figure 6 shows the effect of the smoothing parameter σ on the GRNN model. The curve has an obvious minimum value, and that is, the best value of the smoothing parameter σ is 0.00027.

In order to verify the effectiveness of the error correction model proposed in current measurement, the GRNN model was constructed by using the optimal parameter and training set, the weak current measurement value in the test set was used as the input of the constructed GRNN model, and the current correction output value of the constructed GRNN model can be obtained. Take 50 samples from the test set at fixed intervals to test the GRNN model. The corresponding results of correction output are shown in Figure 7(a), and the measurement errors without and with correction are displayed in Figure 7(b). It can be seen from Figure 7(b) that the measurement error of the weak current in the experimental system increases with the increase of the measured current, but the corrected output error of the GRNN model is not affected (the error is close to 0). Forasmuch as the above analyzed results, the GRNN model proposed in this work is effective in correcting the measurement error of weak current.

In this paper, the GRNN model was also compared with the least squares model and the BPNN model, and the performance of each model were evaluated via RMSE, MAE, and MAPE values. After least square fitting, BPNN model training and GRNN modelling were conducted through the same training dataset, 50 consecutive samples in the test set were randomly taken as the input of every correction model, and the corresponding correction output effect diagram and error comparison diagram are shown in Figures 8-10. Table 2 shows the corresponding correction output results of these several correction algorithms at different current measurement values. As can be seen from Figures 8-10 and Table 2, the error between the weak current value output by the least squares model and BPNN model correction and the standard value output by Keithley 6221 for weak current is smaller than the measurement error of the experimental system. However, the GRNN model obviously has a better correction effect on the measurement error of the experimental system.

The results of performance comparison for each correction method are clearly exhibited in Table 3 after taking the same training dataset to perform least squares fitting, BPNN model training, and GRNN model construction, respectively, and then taking the overall test set sample for weak current correction output. The standlone approach in Table 3 indicates that different correction models are fitted, trained and constructed by using the measured weak current data with noise as the dataset, and then the performances of corresponding models are tested. The wavelet denoising method in Table 3 shows that the same data as standlone approach were denoised and then were used as a dataset to fit, train and construct different correction models, separately. After that, the performances of corresponding models were tested. It can be seen from Table 3 that compared with preprocessing without wavelet threshold denoising, the use of wavelet threshold denoising to preprocess the original data improves the correction performance of each error correction model. Meanwhile, the weak current data sample was denoised by wavelet, and then the weak current measurement error was corrected by GRNN. The result corrected is that RMSE=0.0911 nA, MAE=0.0354 nA, MAPE=0.0078%. The RMSE value before corrected is 12.14 times as much as the RMSE value after corrected, and the RMSE value through the correction is smaller than that with the least square method and BPNN correction respectively. The experimental results show that the method of combining wavelet denoising and GRNN correction model shows better correction performances for the measurement error of the weak current measurement system than the least square method and BPNN model. Because the GRNN model has strong nonlinear mapping capabilities, high fault tolerance and robustness, the GRNN model has achieved very good results in the current measurement error correction of the system.