# Inline Measurement of Particle Concentrations in Multicomponent Suspensions using Ultrasonic Sensor and Least Squares Support Vector Machines

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measurement Apparatus

**Figure 1.**(

**a**) Schematic diagram of the ultrasonic system for the measurement of suspensions concentration; (

**b**) The full time domain signals of pure water; (

**c**) The windowed, averaged and zero-padded time domain signals of (

**b**); (

**d**) The frequency spectrum of (

**c**).

#### 2.2. Analysis Method

- Signal pre-processing. Ultrasonic signals are windowed, averaged, zero-padded and mapped to the frequency domain to obtain the time and frequency domain features. Due to the relatively large differences, the features have to be auto-scaled by zero-mean normalization method before further analysis.
- Sample outliers. To reduce the signal noises and enhance the robustness of model, the outlier detection is very important. In inline applications, the outlier detection can determine whether the measurement is within the applicable scope of the model or not.
- Feature extraction. The feature extraction is completed by analyzing the importance of features in the model. A commonly-used method to calculate the importance of features is a combination of the scaled regression coefficients (SRC) and variable importance in the projection (VIP) [11,23], in which SRC is calculated automatically whilst constructing the regression model and VIP is based on calculating the projection part of single variable in relation to the target values.
- Model building and testing. The LS-SVM parameters are optimized at first, and then the model is trained and tested using the training and prediction subsets, respectively. The most appropriate model that produces the smallest prediction error can be obtained.
- Inline measurement. The optimized model with the optimal feature subset is employed for inline measurement of particle concentrations of intentionally-designed samples.

_{i}is Lagrange multiplier, b is the bias value, K(x, x

_{i}) is the kernel function and x

_{i}is the input data.

^{2}, play an important role in building a LS-SVM model with the RBF kernel featuring high prediction accuracy and stability [24]. In this study, an iterative grid search technique with 10-fold cross validation is employed to find out the optimal combination of (λ, σ

^{2}). After the three key factors are determined, the LS-SVM model is developed and the prediction performance is evaluated with training and prediction subsets, respectively.

^{2}). RMSE reflects the residual errors and provides a global idea of the difference between the observed and predicted values, which is expressed in the same unit as the target values [25]. In the paper, the RMSE of cross-validation (RMSEV) is used to decide the model parameters. However, it may lead to the over-fitting of the final model structure, so the RMSE of prediction subset (RMSEP) is used to check the final model. R

^{2}reflects the goodness of fit of the model.$\text{}{R}_{p}^{2}$ stands for the R

^{2}of prediction subset. They are given by:

#### 2.3. Sample Preparation

_{2}, with an average particle size of 19 μm and a density of 4260 kg/m

^{3}) and Kaolin (with an average particle size of 12 μm and a density of 2930 kg/m

^{3}). Both TiO

_{2}and Kaolin are prepared in nine groups from 0 g to 240 g, with an interval of 30 g. Seven hundred grams of pure water is added into possible combinations of TiO

_{2}and Kaolin to obtain 81 samples (including one sample of pure water). The component concentration is calculated as the ratio of the mass of component to that of the total mixture, and the component concentrations of Kaolin and TiO

_{2}are denoted as c

_{k}and c

_{t}respectively. For calibration and prediction, the whole sample set is partitioned into two subsets. The training and prediction subsets comprise 72 (89%) and nine (11%) samples, respectively. The training subset is applied to build the LS-SVM model and the prediction subset is used to evaluate the model performance. In addition, a subset of additional samples, namely the inline test subset, is used to assess the performance of inline measurement. The distribution of samples is shown in Figure 3.

**Figure 3.**The experimental design of the samples: the black solid squares represent the training subset; the red hollow squares stand for the prediction subset; the blue triangles denote the inline test subset.

#### 2.4. Ultrasonic Features

_{1}–T

_{7}and T

_{8}–T

_{12}represent the amplitude and distribution of the time domain signals x(n), respectively. Similarly, F

_{1}–F

_{7}and F

_{8}–F

_{12}describe the amplitude and distribution of the frequency spectrum X(k) , respectively.

_{s}is defined as the acoustic path length (2l) divided by the time of flight of the ultrasonic signal t

_{s}. The attenuation coefficients of suspensions for time and frequency features are calibrated by measuring those of pure water and calculated by the following Equation (5):

_{w}is one of the amplitudes of T

_{1}–T

_{7}and F

_{1}–F

_{6}in water, A

_{s}is one of the amplitudes of T

_{1}–T

_{7}and F

_{1}–F

_{6}corresponding to A

_{w}in suspensions.

Features | Time Domain Features | Frequency Domain Features | ||
---|---|---|---|---|

Amplitude | Attenuation | Amplitude | Attenuation | |

Mean | ${T}_{1}=\frac{1}{N}{\displaystyle \sum}_{n=1}^{N}\left|x\left(n\right)\right|$ | ${\text{\alpha}}_{T1}$ | ${F}_{1}=\frac{1}{M}{\displaystyle \sum}_{k=1}^{M}\left|X\left(k\right)\right|$ | ${\text{\alpha}}_{F1}$ |

Standard deviation | ${T}_{2}=\sqrt{\frac{1}{N}{\displaystyle \sum}_{n=1}^{N}{\left(x\left(n\right)-{T}_{1}\right)}^{2}}$ | ${\text{\alpha}}_{T2}$ | ${F}_{2}=\sqrt{\frac{1}{M}{\displaystyle \sum}_{k=1}^{M}{\left(X\left(k\right)-{F}_{1}\right)}^{2}}$ | ${\text{\alpha}}_{F2}$ |

Root mean square | ${T}_{3}=\sqrt{\frac{1}{N}{\displaystyle \sum}_{n=1}^{N}{\left(x\left(n\right)\right)}^{2}}$ | ${\text{\alpha}}_{T3}$ | ${F}_{3}=\sqrt{\frac{1}{M}{\displaystyle \sum}_{k=1}^{M}{\left(X\left(k\right)\right)}^{2}}$ | ${\text{\alpha}}_{F3}$ |

Square mean root | ${T}_{4}={\left(\frac{1}{N}{\displaystyle \sum}_{n=1}^{N}\sqrt{\left|x\left(n\right)\right|}\right)}^{2}$ | ${\text{\alpha}}_{T4}$ | ${F}_{4}={\left(\frac{1}{M}{\displaystyle \sum}_{k=1}^{M}\sqrt{\left|X\left(k\right)\right|}\right)}^{2}$ | ${\text{\alpha}}_{F4}$ |

Energy | ${T}_{5}={\displaystyle \sum}_{n=1}^{N}{\left(x\left(n\right)\right)}^{2}$ | ${\text{\alpha}}_{T5}$ | ${F}_{5}={\displaystyle \sum}_{k=1}^{M}{\left(X\left(k\right)\right)}^{2}$ | ${\text{\alpha}}_{F5}$ |

Extremum | ${T}_{6}=\text{max}\left(x\left(n\right)\right)$ | ${\text{\alpha}}_{T6}$ | ${F}_{6}=\text{max}\left(X\left(k\right)\right)$ | ${\text{\alpha}}_{F6}$ |

${T}_{7}=\text{min}\left(x\left(n\right)\right)$ | ${\text{\alpha}}_{T7}$ | ${F}_{7}={f}_{\text{peak}}$ | ||

Latitude factor | ${T}_{8}=\frac{\text{max}\left(\left|x\left(n\right)\right|\right)}{{T}_{4}}$ | ${F}_{8}=\frac{\text{max}\left(\left|X\left(k\right)\right|\right)}{{F}_{4}}$ | ||

Crest factor | ${T}_{9}=\frac{\text{max}\left(\left|x\left(n\right)\right|\right)}{{T}_{3}}$ | ${F}_{9}=\frac{\text{max}\left(\left|X\left(k\right)\right|\right)}{{F}_{3}}$ | ||

Kurtosis | ${T}_{10}=\frac{{{\displaystyle \sum}}_{n=1}^{N}{\left(x\left(n\right)-{T}_{1}\right)}^{4}}{N{T}_{2}^{4}}$ | ${F}_{10}=\frac{{{\displaystyle \sum}}_{k=1}^{M}{\left(X\left(k\right)-{F}_{1}\right)}^{4}}{N{F}_{2}^{4}}$ | ||

Shape factor | ${T}_{11}=\frac{{T}_{3}}{{T}_{1}}$ | ${F}_{11}=\frac{{F}_{3}}{{F}_{1}}$ | ||

Impulse factor | ${T}_{12}=\frac{\text{max}\left(\left|x\left(n\right)\right|\right)}{{T}_{1}}$ | ${F}_{12}=\frac{\text{max}\left(\left|X\left(k\right)\right|\right)}{{F}_{1}}$ | ||

Other | ${v}_{\text{s}}=\frac{2l}{{t}_{s}}$ | X(f) (f=1.1, 1.6, 1.9, 2.2, 2.5, 2.8, 3.1, 3.6 MHz) | ${\text{\alpha}}_{f}\left(\text{X}\left(f\right)\right)$ |

## 3. Results and Discussion

#### 3.1. Data Pre-Processing

**Figure 4.**(

**a**) The scatter plot of principal components (PC1/PC2); (

**b**) The scatter plot of the leverage and the studentized residual.

_{8}–T

_{12}) and frequency distribution features (F

_{9}–F

_{12}) are insensitive or irregular in correlation to the particle concentrations.

_{8}–T

_{12}, and M5 contains all frequency domain features excluding F

_{9}–F

_{12}. M6 subset is the combination of M4 and M5 subsets. All subsets are used to build the model and find out the optimal feature subset that shows the best performance.

Subset | Time Features | Frequency Features | Count |
---|---|---|---|

M1 | ${T}_{1}-{T}_{12},{\text{\alpha}}_{T1}-{\text{\alpha}}_{T7},{v}_{\text{s}}$ | ${F}_{1}-{F}_{12},{\text{\alpha}}_{F1}-{\text{\alpha}}_{F6},\text{}X\left(f\right),{\text{\alpha}}_{f}\left(\text{X}\left(f\right)\right)$ | 54 |

M2 | ${T}_{1}-{T}_{12},{\text{\alpha}}_{T1}-{\text{\alpha}}_{T7},{v}_{\text{s}}$ | / | 20 |

M3 | / | ${F}_{1}-{F}_{12},{\text{\alpha}}_{F1}-{\text{\alpha}}_{F6},\text{}X\left(f\right),{\text{\alpha}}_{f}\left(\text{X}\left(f\right)\right)$ | 34 |

M4 | ${T}_{1}-{T}_{7},{\text{\alpha}}_{T1}-{\text{\alpha}}_{T7},{v}_{\text{s}}$ | / | 15 |

M5 | / | ${F}_{1}-{F}_{8},{\text{\alpha}}_{F1}-{\text{\alpha}}_{F6},\text{}X\left(f\right),{\text{\alpha}}_{f}\left(\text{X}\left(f\right)\right)$ | 30 |

M6 | ${T}_{1}-{T}_{7},{\text{\alpha}}_{T1}-{\text{\alpha}}_{T7},{v}_{\text{s}}$ | ${F}_{1}-{F}_{8},{\text{\alpha}}_{F1}-{\text{\alpha}}_{F6},\text{}X\left(f\right),{\text{\alpha}}_{f}\left(\text{X}\left(f\right)\right)$ | 45 |

#### 3.2. Optimization of Model Parameters

^{2}is the critical step to obtain the optimal LS-SVM model. In this study, the iterative grid search is performed to find the parameter range. For each combination of (λ, σ

^{2}) (grid point), the 10-fold cross-validation is used to calculate the prediction errors using the training set. The optimum parameters are selected which produce smallest prediction errors.

^{2}) is achieved for the LS-SVM models. The contour plot for the optimization process of parameters γ and σ

^{2}for M5 feature subset is shown in Figure 6. The search process has been completed through only two steps (two iterations). For the M5 feature subset, the optimal combinations of (λ, σ

^{2}) are found to be (1.1545 × 10

^{5}, 114.9) and (1.4978 × 10

^{5}, 125.5587) for c

_{k}and c

_{t}, respectively. In the same way, the optimal combinations of (λ, σ

^{2}) for other feature subsets can be found.

**Figure 6.**The contour plot of RMSEV versus γ and σ

^{2}in the grid search for (

**a**) c

_{k}and (

**b**) c

_{t}. The grids “·” and “+” are 10 × 10 in the first step and the second step, respectively. The color stand for the value of RMSEV.

_{k}and c

_{t}, is used to determine the optimal number of PCs. For each feature subset, the PLS model is built and $\overline{\text{RMSEP}}$ is calculated (Figure 7). It can be seen that all models start with high $\overline{\text{RMSEP}}$ values but decrease rapidly at first and then decrease smoothly. The $\overline{\text{RMSEP}}$ values become asymptotic around 10–15 factors, but the minimums are different. The optimum number of PCs which produces the smallest $\overline{\text{RMSEP}}$ can be easily found in Figure 7 and then used to construct the PLS model.

#### 3.3. Model Training and Testing

^{2}, the LS-SVM models for c

_{k}and c

_{t}are trained for each feature subset using the training subset, respectively. To investigate the prediction ability of the model, the model is used to measure the particle concentrations in prediction subset, which doesn’t contribute to the model building. The performance of the LS-SVM model (${R}_{\text{P}}^{2}$ and RMSEP) is calculated and showed in Table 3. For the comparison among six feature subsets, the best performance is achieved with M5 subset. In the optimal model and selection of features, RMSEP of the LS-SVM model is 0.31 wt% for Kaolin and 0.34 wt% for TiO

_{2}. The maximum absolute residual errors $\text{max}\left(\left|y-\widehat{y}\right|\right)$) in prediction subset of Kaolin and TiO

_{2}are less than 0.45 wt% and 0.56 wt%, respectively. Based on the results above, it can be concluded that the combination of ultrasonic sensor and the LS-SVM model is a reliable and accurate method for the prediction of particle concentrations in multicomponent suspensions.

^{2}of the optimal PLS model, namely 0.52 wt% and 0.995 for c

_{k}, 0.67 wt% and 0.989 for c

_{t}, are larger and less than those of the LS-SVM model, respectively, which indicates that the prediction performance of the LS-SVM model is better than that of the PLS model. The PLS model is suited to the linear system, while the LS-SVM model is commonly used in the nonlinear system. Hence, the reason why the LS-SVM model produces higher prediction accuracy than the PLS model may be the presence of nonlinear relationship between ultrasonic features and component concentrations.

**Table 3.**The performance of the partial least square (PLS) model and the least squares support vector machines (LS-SVM) model.

Feature Subset | PLS | LS-SVM | ||||||
---|---|---|---|---|---|---|---|---|

RMSEP_c_{k} | ${\mathit{R}}_{P}^{\mathbf{2}}$_c_{k} | RMSEP_c_{t} | ${\mathit{R}}_{P}^{\mathbf{2}}$_c_{t} | RMSEP_c_{k} | ${\mathit{R}}_{P}^{\mathbf{2}}$_c_{k} | RMSEP_c_{t} | ${\mathit{R}}_{P}^{\mathbf{2}}$_c_{t} | |

M1 | 0.0068 | 0.993 | 0.0075 | 0.988 | 0.0057 | 0.994 | 0.0081 | 0.984 |

M2 | 0.0060 | 0.994 | 0.0098 | 0.977 | 0.0065 | 0.993 | 0.0091 | 0.980 |

M3 | 0.0058 | 0.996 | 0.0092 | 0.980 | 0.0056 | 0.994 | 0.0156 | 0.943 |

M4 | 0.0055 | 0.996 | 0.0089 | 0.982 | 0.0048 | 0.996 | 0.0088 | 0.982 |

M5 | 0.0069 | 0.993 | 0.0124 | 0.964 | 0.0031 | 0.999 | 0.0034 | 0.998 |

M6 | 0.0052 | 0.995 | 0.0067 | 0.989 | 0.0030 | 0.999 | 0.0041 | 0.996 |

**Figure 8.**The predicted values vs. the actual values using (

**a**) the optimal PLS model and (

**b**) the optimal LS-SVM model.

#### 3.4. Inline Measurement

_{t}shows more fluctuant changes than c

_{k}. Compared with the offline predicted results in Figure 8, the error of inline concentration measurement is larger, which is mainly caused by the dynamic measurement along with the fluctuation of signals. The noise disturbance cannot be eliminated through the mean method in consideration of real-time measurement, and accordingly the accuracy of concentration measurement is reduced. Nevertheless, the errors of concentration measurements of Kaolin and TiO

_{2}in inline test subset are less than ±0.65 wt% and ±0.80 wt% respectively, which meet the industrial requirements.

**Figure 9.**The result of inline measurement: (

**a**) The real-time measurement concentration; (

**b**) The XY error bar graph of the averaged c

_{k}and c

_{t}in 5 min.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Zhan, X.; Jiang, S.; Yang, Y.; Liang, J.; Shi, T.; Li, X.
Inline Measurement of Particle Concentrations in Multicomponent Suspensions using Ultrasonic Sensor and Least Squares Support Vector Machines. *Sensors* **2015**, *15*, 24109-24124.
https://doi.org/10.3390/s150924109

**AMA Style**

Zhan X, Jiang S, Yang Y, Liang J, Shi T, Li X.
Inline Measurement of Particle Concentrations in Multicomponent Suspensions using Ultrasonic Sensor and Least Squares Support Vector Machines. *Sensors*. 2015; 15(9):24109-24124.
https://doi.org/10.3390/s150924109

**Chicago/Turabian Style**

Zhan, Xiaobin, Shulan Jiang, Yili Yang, Jian Liang, Tielin Shi, and Xiwen Li.
2015. "Inline Measurement of Particle Concentrations in Multicomponent Suspensions using Ultrasonic Sensor and Least Squares Support Vector Machines" *Sensors* 15, no. 9: 24109-24124.
https://doi.org/10.3390/s150924109