Method and system for online detection of water content of chemical liquid by FTNIR based on MEMS

Abstract

The invention relates to a near-infrared online detection technology, and provides a method and a system for online detection of water content of chemical liquid based on a Fourier transform near-infrared spectrometer (FTNIR) of a micro electro mechanical system (MEMS). The device is characterized by consisting of a halogen lamp light source with adjustable output power, an FTNIR (Fourier Transform Network Infrared Spectroscopy) based on an MEMS (Micro Electro Mechanical System) structure, a cuvette or an immersion transmission probe and a computer. According to the invention, a self-built optical system based on near-infrared absorption is adopted to measure the moisture content of common chemical liquid. Near infrared spectrums of liquid samples with different moisture contents are collected, a partial least square method is adopted to establish a moisture concentration prediction model and verify the moisture concentration prediction model, and the verified model is used to calculate the moisture content of an unknown sample. The detection technology can be used for on-site in-situ detection of trace water content in common chemical liquid, and a low-cost test scheme for rapidly detecting the water content of the chemical liquid is provided.

Description

Technical Field

[0001] The invention belongs to the technical field of chemical liquid component detection, and specifically, uses the near-infrared absorption spectrum of water and a chemometric method to measure the water content of the chemical liquid. Background Art

[0002] The determination of moisture content in chemical liquids is of great significance for the quality control of many chemical processes and products. Accurate determination of moisture content can help engineers and operators adjust reaction conditions to ensure the selectivity, rate, quality and yield of the reaction. At the same time, the control of moisture content is crucial to the stability, shelf life, appearance quality and performance of many chemical products. In addition, by determining the moisture content, it is possible to manage and evaluate the stability of products during transportation and storage, reduce problems such as corrosion and degradation, increase the life of equipment and reduce maintenance costs, and also prevent artificial dilution to make illegal profits. Therefore, regular determination of the moisture content of chemical liquids is a necessary step to ensure the safety of the production process, stable product quality and normal operation of equipment.

[0003] Near infrared spectroscopy (NIR) spectrum lies between the visible spectrum and the mid-infrared spectrum, with a wavelength range of 780-2500nm (12820-4000cm-1). Its main spectral peaks reflect the frequency doubling and sum frequency absorption of hydrogen-containing groups such as CH, NH, and OH in molecules. Therefore, near infrared technology is more suitable for analyzing components that are directly or indirectly related to these groups. Traditional FTNIR has large size, power consumption, and high cost, which is not convenient for secondary development and difficult to meet the application requirements of on-site, real-time, and rapid detection in special environments. The FTNIR based on MEMS structure is not only small in size and low in cost, which is convenient for on-site in-situ detection, but also can be secondary developed to expand the application field. Since water has obvious absorption characteristics in the near infrared band, the FTNIR in-situ detection of water content in chemical liquids based on MEMS structure is a rapid detection technology.

[0004] At present, the moisture detection of chemical liquids mostly adopts the Karl Fischer method, which requires sampling in advance and pretreatment of the samples. In addition, the preparation of reagents is cumbersome, and it is necessary to use malodorous chemicals such as sulfur dioxide and pyridine, which pollutes the environment and harms the health of operators. Near-infrared spectroscopy has the advantages of fast analysis speed, simple instrumentation, non-destructiveness, and no need for sample pretreatment, and has obvious advantages in in-situ moisture detection.

[0005] At present, the most commonly used spectrometers for detecting moisture using near-infrared absorption technology include desktop Fourier transform spectrometers and dispersive spectrometers. The former is large in size and high in cost, and is only suitable for laboratory use, while the latter has low signal sensitivity and high cost. The FTNIR developed based on the MEMS manufacturing process is as small as about 10 cm in length and has low cost. Therefore, the development of a method and system for in-situ detection of the moisture content of chemical liquids by FTNIR based on MEMS has broad application prospects in the online non-destructive detection of moisture in chemical liquids. Summary of the invention

[0006] The purpose of the present invention is to provide a method and system for online detection of water content in chemical liquids by FTNIR based on MEMS in response to the needs of industrial production. The method has fast detection speed and good effect, and has broad application prospects in in-situ non-destructive detection of water content in chemical liquids.

[0007] A MEMS-based FTNIR online monitoring system for the moisture content of chemical liquids comprises a halogen light source with adjustable output power, an FTNIR, a sample holder, and a computer. The characteristics are as follows: the halogen light source with adjustable output power is connected to a power supply to provide a stable light source, and the halogen light source with adjustable output power is connected to the sample holder through an optical fiber jumper with an SMA interface to connect the light source output end to the sample holder, and the sample holder is Figure 1 Cuvette holder in or Figure 2 The immersion transmission probe in the MEMS is connected to the sample holder through an optical fiber jumper with an SMA interface; the FTINR is connected to a computer through a data line for controlling the measurement process, recording data and performing spectral analysis.

[0008] A MEMS-based FTINR system for online detection of water content in chemical liquids, characterized in that: the FTNIR engine is small and compact and can be carried in one hand. The Michelson optical interferometer and the control circuit are built into a palm-sized housing. The near-infrared absorption spectrum can be measured by connecting to a computer via USB. Because it is small and easy to carry, the MEMS-based FTINR spectrometer can perform on-site online detection without bringing the measurement sample into the analysis room, and can perform continuous monitoring.

[0009] A MEMS-based FTINR system for online detection of water content in chemical liquids, characterized in that the sample holder is a quartz cuvette with a 10 mm optical path length or an immersion transmission probe with a fixed optical path length.

[0009] A method for online detection of water content of chemical liquid by FTINR based on MEMS, the method comprising the following steps:

[0010] Step 1: Establish near-infrared quantitative calibration model:

[0011] 1) Take the tested samples and divide them into a calibration set and a validation set in a ratio of 7:3;

[0012] 2) After the spectrometer is turned on, air is used as the reference spectrum, and then the near-infrared transmission measurement is performed on the liquid sample with known water content to obtain the absorbance spectrum of the calibration set sample;

[0013] 3) Preprocess the near-infrared transmission spectrum data and establish the optimal near-infrared quantitative correction model for moisture.

[0014] Step 2, determine the moisture content in the sample:

[0015] 1) Take the sample to be tested

[0016] 2) After the spectrometer is turned on, the air is used as the reference spectrum to perform near-infrared transmission measurement on samples with unknown moisture content to obtain the absorbance spectrum of the validation set samples;

[0017] 3) The moisture content of the sample to be tested is measured according to the near-infrared quantitative calibration model established in step 1.

[0018] The wavelength range of the spectrometer is 1100-2500nm, the measurement time interval of the near infrared spectrum is 6 or 10 seconds, and the collection time interval of the measured samples is 4 or 6 minutes.

[0019] Preferably, in steps 1 and 2, near-infrared spectral data of each sample is collected to obtain the absorbance value of the measured solution sample within the response band of the spectrometer, and each sample is collected 8 times or more, and the average spectrum is taken, such as Figure 2 As shown;

[0020] Preferably, in step 1, the collected near-infrared spectrum is preprocessed and then a prediction model is established. The preprocessing method is maximum normalization, unit vector normalization, Savitzky-Golay smoothing, Savitzky-Golay smoothing + first-order derivative and standard normal transformation SNV, or no preprocessing is required. In the present invention, the index value of the cross-validation root mean square error is used as the judgment standard for selecting which preprocessing method to use. The smaller the index value, the better the corresponding preprocessing method.

[0021] Preferably, the model in step 1 uses principal component regression analysis (PCR), partial least squares regression (PLSR) and support vector machine regression (SVMR) methods to model and predict the chemical liquid water volume samples. The principles of each algorithm are:

[0022] 1) PCR first uses principal component analysis (PCA) to regroup the variables into a new set of independent comprehensive variables, namely PCs, from which a few fewer PCs are taken to reflect the original variables as much as possible. Then, by correlating the spectrum with the chemical value, a calibration model between the two is established and the spectrum is predicted to obtain the corresponding chemical value.

[0023] 2) PLSR is based on ordinary multiple regression, borrowing the ideas of PCA and canonical correlation analysis, and effectively solves the problem of multicollinearity between independent variables. It has become a general method for establishing linear quantitative correction models in spectral analysis, and is particularly effective when the predicted variables are highly correlated or the number of predicted variables exceeds the number of samples.

[0024] 3) SVMR is a regression algorithm based on support vector machine, the goal is to minimize the sum of squared errors between the model prediction value and the true value. At the same time, the margin is maximized. By introducing Lagrange multipliers and kernel functions, the optimization problem of SVMR can be transformed into a dual problem, which can be solved using convex optimization methods. The model prediction value can be calculated using the kernel function, and the bias term is calculated using samples from the support vector set. SVMR has good performance when dealing with nonlinear regression problems.

[0025] Preferably, in step 2, according to the water content measurement results, the coefficient of determination (R2) and the root mean square error of prediction (RMSEP) of the validation set are used as model evaluation indicators to select appropriate modeling parameters to obtain the optimal prediction model; when R 2 The value is close to 1 and the RMSEP value is close to 0, indicating that the established model has good stability and prediction accuracy.

[0026] The system of the present invention only takes a dozen seconds to analyze each sample, quickly gives feedback on the quality of the sample, and can quickly check and handle abnormalities based on the quality, avoiding the production of problematic products and reducing customer complaints. However, the traditional method for detecting water content, Karl Fischer (KF) titration, as a traditional method for determining the water content in the extractant, has a relatively cumbersome operation process and takes a long time, but products of unknown quality continue to be produced during this period, so the longer the detection time, the higher the risk.

[0027] The beneficial effects brought by the technology of the present invention are:

[0028] 1) By using the method described in the present invention, it is possible to monitor the moisture content of each point in the chemical liquid production process in real time online; abnormalities can be discovered and checked and remedied in the first place, thereby avoiding losses to the greatest extent.

[0029] 2) In the solution of the present invention, the near-infrared spectrometer is small in size and can be used for real-time detection in the factory, during transportation or when the product is delivered by the manufacturer. It is fast and non-destructive, does not require sample pretreatment, and can be measured immediately and the results can be received immediately.

[0030] The method and system provided by the present invention are simple to operate and low in cost, and can quickly and reliably predict the water content in chemical liquids while ensuring the accuracy of identification. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The above objects, features and advantages of the present invention will become more apparent through a more detailed description of exemplary embodiments of the present invention with reference to the accompanying drawings.

[0032] Figure 1 This is a schematic diagram of a MEMS-based FTINR online detection system for water content in chemical liquids. In the figure, 1 is a computer; 2 is a data line; 3 is FTNIR; 4 is a 600um fiber jumper; 5 is a cuvette holder; and 6 is a halogen light source with adjustable output power.

[0033] Figure 2 This is a schematic diagram of a FTNIR online detection system for water content in chemical liquids based on MEMS. In the figure, 1 is a computer; 2 is a data line; 3 is FTNIR; 6 is a halogen light source with adjustable output power; and 7 is an immersion optical fiber probe.

[0034] Figure 3 yes Figure 2 The probe part of the immersion transmission probe. (a) is the appearance of the probe; (b) is the cross-sectional view of the probe; (c) is the optical path diagram of the probe, 701 is the probe, 702 is the sample chamber, 703 is the lens, 704 is the reflector, 705 is the signal fiber, and 706 is the light source fiber.

[0035] Figure 4 It is the near infrared original absorption spectrum of the ethylene glycol solution obtained in the example.

[0036] Figure 5 It is a near-infrared absorption spectrum diagram after Savitzky-Golay smoothing + first-order derivative processing of the original spectrum in the embodiment.

[0037] Figure 6 It is the correlation curve between the spectral calculated value and the true value of the moisture content of the calibration set.

[0038] Figure 7 It is the correlation curve between the spectral predicted value and the true value of the moisture content of the sample to be tested. Figure 8 for Figure 1 The product diagram of the MEMS-based FTINR online detection system for moisture content in chemical liquids is shown. DETAILED DESCRIPTION

[0039] In order to make the technical solution of the present invention clearer and more specific, the present invention is further described below in conjunction with the accompanying drawings and specific implementations.

[0040] 1. Design of near-infrared online detection device

[0041] Figure 1 It is a schematic diagram of a MEMS-based FTNIR system for online monitoring of the moisture content of chemical liquids. Figure 8 for Figure 1 The product schematic diagram of the MEMS-based FTINR online detection system for water content in chemical liquids is shown. In the figure, 1 is a computer; 2 is a data line; 3 is a MEMS-based FTNIR; 4 is a fiber jumper with a core diameter of 600 microns; 5 is a cuvette holder and can be replaced with Figure 2 7 is an immersion transmission probe shown in the figure; 6 is a halogen lamp light source with adjustable output power. In the experiment, a fiber jumper with a core diameter of 600 microns is used to connect the halogen lamp to the cuvette holder, and the cuvette holder to the spectrometer, and the positions of the halogen lamp, the cuvette holder, and the spectrometer are adjusted. After the halogen lamp is turned on. After the light emitted by it passes through the sample, a part of the light is absorbed, and the remaining light enters the spectrometer through the fiber jumper for analysis and is transmitted to the computer to obtain the original near-infrared absorption spectrum of ethylene glycol.

[0042] 2. Sample Preparation

[0043] Add water to the purchased ethylene glycol to prepare a mother liquor sample with a moisture content of 1%, and then add ethylene glycol thereto to prepare samples with contents of 0.95%, 0.9%, 0.85%, 0.8%, …, 0.1%. Then add water to the sample with a moisture content of 0.1% to prepare samples with contents of 0.09%, 0.08%, 0.07%, …, 0.01%. Finally, add ethylene glycol to the 0.01% sample to prepare samples with contents of 0.008%, 0.006%, 0.005%, 0.004%, and 0.002%.

[0044] The ethylene glycol moisture content data was collected on the above-mentioned near-infrared experimental platform to establish a near-infrared spectral correction model. The sample offline sampling data is shown in Table 1. The samples collected for prediction and calibration sets were 28 and 8 respectively (Table 1). In order to ensure the reliability of the spectral model, the maximum and minimum values ​​of the offline detection data were included in the set. The ethylene glycol moisture data and spectral data were randomly divided into a correction set and a prediction set. In this example, 28 groups of data were selected as the correction set for establishing the correction model, and 8 groups of data were selected as the prediction set for model verification, as shown in Table 1.

[0045] Table 1 Statistical table of water content in ethylene glycol

[0046]

[0047] The near-infrared spectrum was collected by transmission method, with a spectral range of 1100nm to 2500nm, 1000 scans, and a spectral resolution of 25cm -1 , signal-to-noise ratio 10000. Before spectrum acquisition, the sample components were mixed, and a reference spectrum with air as the background was first scanned to reduce the impact of environmental factors on sample acquisition. Then the near-infrared spectrum data of each sample was collected to obtain the absorbance value of the measured solution sample within the response band of the spectrometer. Each sample was collected 8 times, and the average spectrum was taken as the final modeling prediction data.

[0048] 3. Spectral Acquisition and Data Processing

[0049] The near-infrared spectral data and the corresponding moisture content obtained by the collection were preprocessed before modeling. Savitzky-Golay smoothing, Savitzky-Golay smoothing + first-order derivative processing, maximum normalization, unit vector normalization and standard normal transformation SNV were used to preprocess the spectra. PLSR, PCR and SVMR models were established for the calibration set data, and RMSEP and correlation coefficient R were used to calculate the relative humidity. 2 To evaluate the model, a model with good predictive ability should also have a high R 2 , smaller RMSEP. The modeling results are shown in Table 2.

[0050] Table 2 Prediction results of three models for ethylene glycol moisture content

[0051] Model <![CDATA[R 2 ]]> RMSEP PCR 0.999 0.000157 PLSR 0.996 0.0000959 SVMR 0.997 0.000163

[0052] As can be seen from Table 2, the correlation coefficients R 2 They are very close, both greater than 0.99. However, according to the smaller the value of the root mean square error (RMSEP), the better the prediction effect of the established model, so the linear model PLSR is selected for modeling and prediction.

[0053] 4. Results and Discussion

[0054] Figure 4 and Figure 5 They are the original near-infrared spectrum of the sample in the moisture correction set and the dynamic spectrum after Savitzky-Golay smoothing + first-order derivative processing. It can be seen that the overall spectral absorption signal of the sample has a strong correlation with the change in moisture. The higher the moisture, the stronger the absorption signal. This is mainly because the moisture in the sample increases the forward scattering of light and improves the absorption rate of near-infrared light. In order to improve the interference of baseline drift and peak overlap on spectral analysis, the original spectrum is processed by Savitzky-Golay smoothing + first-order derivative to make the spectral features clearer. Figure 4As shown in the figure, as the water content of the sample decreases, the OH frequency-doubled characteristic absorption band (1900-2100nm) of water molecules has regular and significant changes.

[0055] Table 3. Partial least squares modeling parameters for ethylene glycol moisture under different pretreatment methods

[0056]

[0057] As can be seen from the table above, the modeling results of the near-infrared spectra after various preprocessing methods are improved compared with those without preprocessing. When Savitzky-Golay smoothing + first-order derivative is selected to preprocess the spectrum, the model has the smallest RMSEC and RMSEP values. The correlation coefficient of the calibration set (R 2 ) is 0.999, and the root mean square error of the calibration set (RMSEC) is 1.21*10 -5 , and the cross-validation correlation coefficient (R 2 ) is 0.999, and the root mean square error of prediction (RMSEP) is 3.52*10 -5 It can be seen that this system has a good prediction effect on the moisture content of chemical liquid above 0.01%. Overall, it is better than other pretreatment methods.

[0058] Figure 6 and Figure 7 The above model is used to verify the calibration set and validation set data. After Savitzky-Golay smoothing + first-order derivative processing, the prediction function of the model has been significantly improved. Under different moisture content conditions, R 2 All reached above 0.99.

[0059] The embodiments of the present invention have been described above, and the above description is exemplary, not exhaustive, and is not limited to the disclosed embodiments. Many modifications and changes will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments.

Claims

1. A system for online detection of water content in chemical liquids by FTNIR based on MEMS, as shown in FIG1 , comprising a computer (1), a data line (2), an FTNIR (3), an optical fiber jumper (4), a sample holder (5) and a halogen lamp light source with adjustable output power (6). The system is characterized by: The halogen light source (6) with adjustable output power is connected to a power supply to provide stable optical power, and the halogen light source (6) with adjustable output power connects the light source output end to a sample holder (5) via an optical fiber jumper (4) with an SMA interface. The sample holder is the cuvette holder (5) in FIG1 or the immersion optical fiber probe (7) in FIG2 , and the immersion optical fiber probe (7) comprises a probe (701), a sample chamber (702), a lens (703), a reflector (704), a signal optical fiber (705), and a light source optical fiber (706). The FTNIR (3) is connected to the sample holder (5) via an optical fiber jumper with an SMA interface, and the FTNIR (3) is connected to a computer (1) via a data line (2) for controlling the measurement process, recording data, and performing spectral analysis.

2. According to claim 1, a system for online detection of water content of chemical liquids by FTNIR based on MEMS, characterized in that: The wavelength response range of the FTNIR (3) is 1100-2500 nm; the cuvette holder (5) in FIG. 1 can be a quartz cuvette with an optical path of 2 mm, 5 mm or 10 mm.

3. The system for online detection of water content of chemical liquids by FTNIR based on MEMS according to claim 1 is characterized in that: The immersion fiber optic probe (7) in FIG. 2 can be directly inserted into the liquid to be tested. The complete immersion fiber optic probe includes a Y-type fiber bundle and a probe (701). The Y-type fiber bundle is divided into two optical fibers. One optical fiber is connected to a light source for irradiating the sample to be tested, and is called a light source fiber (706). The other optical fiber is connected to a spectrometer for collecting optical signals, and is called a signal fiber (705). The light of the halogen lamp light source (6) with adjustable output power is coupled into the light source fiber (705) through a standard SMA905 connector, and is collimated by a lens (703), and then transmitted to a reflector (704) through a sample chamber (702). The reflected light beam passes through the sample and returns and is coupled into the signal fiber (704), and then guided into the FTNIR (3) through the signal fiber (705) for measurement and analysis.

4. A system for online detection of water content in chemical liquids by FTNIR based on MEMS according to claim 3, characterized in that: The light emitted by the halogen lamp light source (6) with adjustable output power is transmitted to the liquid sample through the light source optical fiber (706) and the lens (703). In this process, the role of the optical fiber and the lens is to ensure that the directionality of the light beam and the size of the light spot are appropriate so as to form a stable light path in the liquid sample. After the light enters the liquid sample, some photons interact with the liquid molecules and are absorbed by the molecules to be tested in the sample, resulting in a decrease in light energy. The absorption intensity of the molecules to be tested depends on the light absorption characteristics, concentration and optical path length of the molecules. The light that is not absorbed will continue to propagate, and after being reflected by the reflector (704), it will pass through the sample again and be absorbed by the molecules to be tested again. After passing through the sample again, the light is focused by the lens (703) into the signal optical fiber (705) and transmitted to the FTNIR (3). The FTNIR (3) receives the transmitted light and converts it into an electrical signal. By comparing the incident light intensity (the light intensity before entering the liquid) with the transmitted light intensity (the light intensity after passing through the liquid), the absorbance can be calculated. The change in absorbance reflects the concentration of the absorbing substance in the liquid sample, thereby realizing the measurement of the absorption characteristics of the liquid.

5. A method for online detection of water content in chemical liquids by FTNIR based on MEMS, characterized in that: The method comprises the following steps: Step 1, taking samples with known water content as a calibration set, collecting near-infrared absorption spectra of the calibration set samples, and obtaining absorbance values ​​of the measured samples within the response band of the spectrometer; Step 2, preprocessing the original near-infrared spectrum obtained in step 1 to determine the best preprocessing method; the preprocessing method is Savitzky-Golay smoothing, Savitzky-Golay smoothing + first-order derivative processing, maximum normalization, unit vector normalization and standard normal transformation SNV; Step 3, using the best spectral preprocessing method determined in step 2, applying a multivariate correction method, establishing a near-infrared quantitative correction model between the near-infrared absorption spectra of the calibration set samples and the measured values ​​of their corresponding moisture contents, using each model evaluation index to examine the model performance, and obtaining the optimal model; Step 4, collecting the near infrared spectrum of the unknown sample, and calculating the moisture content of the sample to be tested according to the chemometric model established in step 3.

6. The method for online detection of water content of chemical liquids by FTNIR based on MEMS according to claim 5, characterized in that: In step 1, the spectrum is collected as follows: carefully place the uniformly mixed ethylene glycol aqueous solution sample in a covered quartz cuvette with an optical path of 10 mm, scan the near-infrared spectrum, and obtain the original near-infrared absorption spectrum of the sample; the scanning conditions of the near-infrared spectrum are: the spectral range is 1100nm~2500nm, each sample is measured in parallel 8 times, and the average value is taken as the original spectrum of the sample; the near-infrared spectrum data of step 1 is measured by the transmission method for the ethylene glycol solution, and the air is used as the background when measuring the sample. The absorbance calculation formula of the ethylene glycol aqueous solution sample is as follows: Among them, I O Represents the initial intensity of the incident light; I t Indicates the intensity of transmitted light.

7. The method for online detection of water content of chemical liquids by FTNIR based on MEMS according to claim 5, characterized in that: In step 3, a multivariate calibration algorithm is used to establish a chemometric model, and the model performance is evaluated using model evaluation indicators. The multivariate calibration method is partial least squares method (PLSR), principal component regression analysis (PCR) or support vector machine regression (SVMR), where the SVMR algorithm selects radial basis as the kernel function, uses a grid search algorithm to optimize the penalty factor nu of the radial basis function and the kernel function parameter gamma, and selects the root mean square error as the objective function. Model evaluation indicators include: determination coefficient (R 2 ), the root mean square error of the calibration set and the validation set prediction (RMSEC, RMSEP), and select appropriate modeling parameters to obtain the optimal prediction model; when R 2 The values ​​of RMSEC and RMSEP were close to 1, and the RMSEP was close to 0, and the RMSEP was less than 2 times of the RMSEC, indicating that the established model had good stability and prediction accuracy.

8. The method for online detection of water content of chemical liquids by FTNIR based on MEMS according to claim 5, characterized in that: As described in step 4, the established model is used to analyze the moisture content of the sample to be tested online, the near-infrared absorption spectrum of the unknown sample is collected online, the spectrum data is input into the model, and the moisture content information of each sample can be obtained in real time through calculation.

9. The method for online detection of water content of chemical liquids by FTNIR based on MEMS according to claim 5, characterized in that: The established chemometric model was evaluated, including the root mean square error and the coefficient of determination; the calculation formula is as follows: Coefficient of determination: Root Mean Square Error: In the formula, y i represents the measured value of the moisture content of the measured sample, represents the predicted value of the moisture content of the measured sample, represents the mean value of the moisture content of the tested samples, and N represents the number of samples in the test set.

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