Introduction

Dielectric spectroscopy is an important technique for analysing the physicochemical properties of media, which is used to obtain the ionic or electronic component properties of the object under test by studying the dielectric properties of the medium at different frequencies1. With the development of modern science and technology, dielectric spectroscopy has been widely used in materials science, chemical engineering, food engineering, biomedical and other fields2,3. In practical engineering, the object to be measured is often a mixed medium, and the macroscopic dielectric properties of the mixed medium are actually the feedback generated by the microscopic particles inside the medium to the external electric field action4. Due to the different microscopic particles within the mixed media, the media will produce a variety of internal polarisation type and degree5, at the same time, a variety of microscopic polarisation mechanism formed by the local electric field and the external electric field superimposed on each other, resulting in a new distribution of the electric field within the mixed media, so we can analyses the dielectric spectrum of the medium to analyse the medium to obtain the information of the internal microscopic level6.

Dielectric characterization is widely used in various fields because it can obtain physical information about the medium by analyzing its internal electrical properties7. González-Teruel et al.8 modelled a water-saturated clay mineral soil and analysed the dielectric properties of a mixture of water and clayey soil under different water content conditions; Pekonen et al.9 used FDTD simulation to fit the effective dielectric constant of the mixture effective dielectric constant was fitted; Schmidt and Webb10 proposed a new method for complex dielectric constant estimation based on a global integral equation; Shang et al.1 used complex dielectric constant with an artificial neural network for the identification of heavy metals in soils. However, the above research methods rely on model-based fitting and inverse extrapolation methods, and there are fewer studies using dielectric spectra for the analysis of mixed media components, and there is an urgent need for a method for unmixing research on mixed dielectric spectra.

In remote sensing spectral unmixing classification studies, deep learning has a good performance in abstract feature learning11. Based on the powerful feature extraction capability of deep learning, this study investigates a demixing algorithm for dielectric spectra, and realises the component determination of mixed media through feature extraction and abundance analysis of mixed dielectric spectra, which provides a new idea and a new method for the material analysis research of dielectric spectra.

Materials and methods

One-dimensional convolutional neural network

Convolutional neural network (CNN), a neural network that processes data with a grid-like structure12, extracts data features through locally associated convolutional kernels and reduces computational complexity through weight sharing13. Convolutional neural network mainly includes: input layer, convolution layer, activation layer, pooling layer, fully-connected layer, and output layer14, as shown in Fig. 1.

  1. 1.

    Convolutional layer

The mathematical expression of convolutional layer is given in Eq. (1):

(1)

where ω and b denote the weights and biases of the layer l, xl-1 is the feature mapping of the previous layer, xl is the feature mapping of the current layer, Mj is the set of feature mappings.

  1. 2.

    Activation layer

Activation functions are usually added to the convolutional and pooling layers, which is equivalent to normalising the output value of each neuron, and commonly used activation functions include Sigmoid, Tanh, Relu, etc15.

The sigmoid function is calculated using Eq. (2):

(2)

The Tanh function is calculated according to Eq. (3):

(3)

The Relu function is calculated according to Eq. (4):

(4)
  1. 3.

    Pooling layer

The pooling layer can not only prevent the phenomenon of overfitting, but also reduce the training parameters and computational cost of the network model. Average pooling and maximum pooling are two pooling operations commonly used in the construction of network models16, with mathematical expressions as in Eq. (5) and Eq. (6):

Average Pooling:

(5)

Maximum Pooling:

(6)
Fig. 1
figure 1

1D convolutional neural network.

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Dielectric theory

Mixed dielectric model

The commonly used mixed dielectric model is mainly divided into three types, the first is the empirical model obtained by fitting the measured data, the second is based on the microscopic composition of the mixed medium structure to derive the integrated apparent dielectric spectrum of the medium17, and the third is through the volume averaging of the dielectric constants of the components of the mixed medium, so as to derive the dielectric constant of the mixed medium.

The choice of dielectric mixing model plays a crucial role in the resolution of the mixed dielectric spectrum. Currently, the most representative and widely used dielectric mixing models are mainly volumetric mixing models. Volume mixing models are usually divided into three types, linear mixing models, root-mean-square mixing models and cube-root mixing models.

  1. 1.

    Linear mixed models, also known as brown mixed models18:

    (7)
  2. 2.

    Root-mean-square mixed models, also known as complex refractive index method mixed models (CRIM)19:

    (8)
  3. 3.

    Cube root mixed model, also known as Looyenga mixed model20:

    (9)

where εm is the dielectric constant of the mixing medium, vi is the volume ratio of the components, and εi is the dielectric constant of the components.

Sample preparation and measurement

In the current research undertaking, three liquid media, namely water, ethanol, and isopropoxyethanol, which are widely employed in dielectric measurement and calibration protocols, were selected as the experimental substances. The density of water was measured as 1.0 g/cm2, while that of ethanol was 0.803 g/mL and the density of isopropoxyethanol was 0.903 g/cm2, with each possessing a concentration of 99.9%. Subsequently, two types of mixed solutions were prepared: (1) A water-isopropoxyethanol mixed solution was formulated with water volume ratios of 20%, 40%, 60%, and 80%. (2) A composite mixture of water, ethanol, and isopropoxyethanol was fabricated in various proportions (see Table 1).

Table 1 Water, iso, ethanol mixed media concentration ratio.
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The experiments were carried out at room temperature (23 ± 2) ℃. A Keysight E5071C vector network analyser was used to set the measurement frequency from 1 MHz to 6 GHz and the number of sampling points as 1001points, and an N connector was used to connect the extension cable of the vector network analyser to the probe, which was calibrated with open-circuit, short-circuit and 50Ω load calibrators after 2 h of powering up the vector network analyser. A 3.5 mm open-circuit probe was used for the measurement fixture. Throughout the measurement process, the temperature and humidity conditions were maintained at a stable level. Each sample underwent measurement procedures three times, while external electromagnetic interference was effectively mitigated. The configuration of the measurement system is illustrated in Fig. 2.

Fig. 2
figure 2

Schematic diagram of measuring system.

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Selection of volumetric mixing models

The real and imaginary parts of the dielectric constant can be transformed into each other under certain conditions21, therefore, in this study, the real part of the dielectric constant was selected to be fitted by using the volumetric mixing model, and the three volumetric mixing models were applied to two sets of pure dielectric spectra measured (water and isopropoxyethanol), and four mixing solutions with different measured water content volumetric ratios (20%, 40%, 60%, and 80%) were selected for validation and comparison.

In this study, the results of the three models are compared by two evaluation indexes: SAM (spectral angle) and RMSE (root mean square error), which are used to calculate the angle between high-dimensional vectors to determine the similarity of the two sets of spectral data, and the smaller the SAM, the higher the similarity of the two sets of data, and RMSE, which characterises the deviation of the two sets of data, and the smaller the RMSE, the smaller the deviation between the two sets of data. The SAM and RMSE values of the three hybrid models are calculated as follows:

Table 2 SAM for different dielectric mixing models.
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Table 3 RMSE for different dielectric mixing models.
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As shown in Tables 2 and 3, the SAM and RMSE values of Brown’s volumetric mixing model are smaller than those of the other two models. Therefore, Brown’s volumetric mixing model is selected to fit the dielectric spectra of the mixed solutions with different water content volume ratios in this study, which can provide reliable theoretical and data support for the study of the unmixing of mixed media.

Denoising dielectric spectrum data

Singular spectrum analysis (SSA) denoising algorithm

Traditional signal denoising methods have certain limitations, for example, low-pass filtering can be realized denoising is due to the high frequency and small amplitude of the general noise signal, the removal of noise is actually to remove the high-frequency part of the signal22. Based on the wavelet denoising method of threshold selection is based on the first layer of wavelet decomposition of the noise level is estimated, with the increase in noise level, the estimated threshold increases accordingly, the signal threshold will be below the threshold of the signal is considered as noise and removed, the resulting denoised signal will lose a lot of useful information, is not conducive to the subsequent data analysis23. Compared to the above noise reduction algorithms, singular spectrum analysis has the advantage of adaptive noise reduction24, which is independent of the signal model and is not affected by the spectral distribution of the noise25.

Analysis of noise reduction results

The results of SSA denoising on the dielectric spectra of mixed solutions with different volume ratios of water content are shown in Fig. 3:

Fig. 3
figure 3

The original dielectric spectrum of mixed solution with different water volume ratio and the dielectric spectrum of SSA after noise removal. (a) Comparison of the noise removal effect of mixed dielectric spectra with 80% water content by volume. (b) Comparison of the noise removal effect of mixed dielectric spectra with 60% water content by volume. (c) Comparison of the noise removal effect of mixed dielectric spectra with 40% water content by volume. (d) Comparison of the noise removal effect of mixed dielectric spectra with 20% water content by volume.

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The dielectric spectra of the mixed solutions composed of water, ethanol, and isopropoxyethanol in diverse proportions were subjected to SSA for denoising purposes. The resultant data are presented in Fig. 4.

Fig. 4
figure 4

Effect of noise removal by dielectric spectroscopy for a mixture of three media. (Water: Iso: Ethanol) Water: Iso: Ethanol is mixed in the following ratios: (a) W: I:E = 1:1:3 (b) W: I:E = 1:2:2 (c) W: I:E = 1:3:1 (d) W: I:E = 2:1:2 (e) W: I:E = 2:2:1 (f) W: I:E = 3:1:1.

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The denoising outcomes of the comprehensive apparent dielectric spectra for multiple sets of mixed solutions are depicted in Figs. 3 and 4. Specifically, the dielectric spectra of various mixed media were processed with SSA for noise reduction, there is a relatively obvious noise removal effect in the low-frequency band, and the characteristics of the original dielectric spectral data are preserved, which can be further realised to achieve a more effective analysis and research on the characteristics of the mixed media.

Design of 1D convolutional neural network

The network architecture of this study extracts the local features of sequence data through convolutional operations, and performs pooling operations to complete the down sampling, and finally combines and maps these features using a fully connected layer to produce accurate prediction results.

The network architecture design and parameters of this study are shown in Fig. 5.

Fig. 5
figure 5

Network architecture and parameters.

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In this study, the dataset is divided into a training set and a test set, where the test set constitutes 20% of the total data. This is achieved by calling the train_test_split function, which ensures the random division of the training and test sets, as well as obtaining an accurate estimate of the generalisation ability during model evaluation. The architecture of the model consists of a one-dimensional convolutional layer, a pooling layer and a fully connected layer. The final layer of the model is a fully connected layer with a linear activation function that is used to output the predictions.

In the compilation phase of the model, the mean square error (MSE) is used as a loss function to measure the difference between the predicted results and the true labels, and the Adam optimiser is chosen to optimise the weight parameters of the model.

The model is trained by iterating on the training set and the performance is evaluated on the test set. The training process of the model gradually decreases the value of the loss function and the model gradually learns the features of the data.

Data pre-processing and model training

In this study, three sets of measured pure dielectric spectra of isopropoxyethanol and water were used to generate training data using linear mixing models. The first group is the original data, and the data are generated by different mixing ratios from 0 to 100% (increasing with a gradient of 1%), with a total of 3 * 3 * 101 = 909 groups, and the second group is generated by using the SSA noise removal algorithm to firstly de-noise the three sets of dielectric spectral data of isopropoxy ethanol and water, and then using the linear mixing model to generate the data, with a total of 3 * 3 * 101 = 909 groups. Among them, Gaussian noise was added to the second data set to improve the robustness of the one-dimensional convolutional neural network. In the context of the dielectric mixing model involving three media, the first group was designed to represent the original dielectric spectrum mixing. Herein, the sum of the abundances of the three media was set to 1, and the dielectric spectrum of the mixed media was calculated iteratively with a gradient of 1%. This process yielded a total of 3 * 3 * (101 + 100 +… + 1) = 46,359 sets of dielectric spectrum data. Another set of data was subjected to noise removal and mixed processing, followed by the addition of Gaussian noise, also generating 46,359 sets of dielectric spectrum data. However, due to the presence of outliers in the low frequency band of the mixed dielectric spectra of the three liquid-phase media, the information of the dielectric spectra of the media components was masked by these outliers. Consequently, in the study of the unmixing of the dielectric spectra of the three liquid-phase media, it is necessary to remove the outliers and train the mixed dielectric spectra, thereby enabling the effective unmixing of the component information of the mixed media.

The generated data set is divided into training set and test set, where the test set is 20% of the total data set. The division of the training and test sets was randomised to ensure that an accurate estimate of the generalisation ability was obtained for model evaluation. A Sequential model was then created, which is a linear stack of a series of network layers. The architecture of the model consists of a one-dimensional convolutional layer, a pooling layer and a fully connected layer. The setup of these layers can be adapted to feature extraction for abundance resolution by adjusting the number, size and activation function of the convolutional kernels. The final layer of the model is a fully connected layer with a linear activation function for outputting the predictions.

In the compilation phase of the model, Mean Squared Error (MSE) and Coefficient of Determination (R-square) are used as loss functions to measure the difference between the predicted results and the true labels, and the Adam optimiser is chosen to optimise the weight parameters of the model. The model is trained by iterative training, cycling through multiple epochs. in each epoch, the model is trained once on the training set and the performance is evaluated on the test set. During the training of the model, the value of the loss function gradually decreases and the model gradually learns the features of the data.

Data discussion

Model training results

In this study for the training of one-dimensional convolutional neural networks, the number of iterations epochs set to 15, in the first 9 times has converged to better, after 11 times MSE and R2 are stabilised and have a relatively good performance score, the model training loss function MSE plots in Fig. 6a. For the dielectric spectrum iteration process of the mixture of water, ethanol, and isopropoxyethanol, the number of epochs was designated as 40. It was observed that the optimal performance was achieved within the initial 11 iterations, and the MSE exhibited a relatively stable trend after 15 iterations. The MSE of the model training loss function is presented in Fig. 6b.

Fig. 6
figure 6

Plot of the loss function. (a) Loss function curve of the mixed solution comprising water and isopropoxyethanol. (b) Loss function curve of the mixed solution comprising water, ethanol and isopropoxyethanol.

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Mixed scale prediction results

20% of the dielectric spectrum generation data based on the linear hybrid model is used as a test set to validate the training results of the one-dimensional convolutional neural network, and finally 50 sets of data from the test set are randomly selected for evaluating the prediction effect of the model, and the prediction results are shown in Fig. 7.

Fig. 7
figure 7

Prediction results of test set data. (a) Prediction results of the test set data for the mixed solution of water and isopropoxyethanol. (b) Prediction results of the test set data for the mixed solution of water, ethanol and isopropoxyethanol.

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Finally, the abundance resolution of the dielectric spectral data of the measured mixed media with different water content volumes was carried out using the trained neural network, and the results are shown in Fig. 8.

Fig. 8
figure 8

Fitting relation curve of one-dimensional convolutional neural network to measured data. (a) The correlation among the measured data of the mixed solution composed of water and isopropoxyethanol was modeled and fitted. (b) The correlation among the measured data of the mixed solution composed of water, ethanol, and isopropoxyethanol was modeled and fitted.

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In this paper, the abundance resolution model for mixed dielectric spectra is proposed by using one-dimensional convolutional neural network for feature extraction of dielectric spectra. The results show that the model proposed in this study has a good effect on the abundance resolution of the measured mixed dielectric spectrum.

Conclusion

In this study, a one-dimensional convolutional neural network was used to implement deep feature extraction and analysis of mixed dielectric spectra for component content resolution of mixed media. The coefficient of determination (R²) values of the model for the two-component and three-component mixed solution test sets were both found to be 0.999. In the validation sets, the corresponding R² values were determined to be 0.9887 and 0.9786, which indicates that the 1D convolutional neural network exhibits a high degree of accuracy in mixed dielectric spectrum parsing, and provides an important tool and methodology for mixed media research. In addition, we use the singular spectral decomposition (SSA) denoising algorithm to process the measured dielectric spectrum data. After SSA denoising, the anomalous fluctuations of the dielectric spectrum data are effectively eliminated, which makes the processed dielectric spectrum more conducive to the in-depth analysis of the mixed dielectric spectrum. The study shows that the one-dimensional convolutional neural network is highly accurate in the analysis of hybrid dielectric spectra, and the SSA noise removal algorithm helps to improve the quality of the data and the analysis effect of the hybrid dielectric spectra, which provides a strong support for the research and application in related fields.