The pH measured data were obtained from a pH sensor array with eight ruthenium dioxide pH electrodes. For example, each sensor was measured n times during a measurement period. The pH measured data were used as a mathematical or statistical method to obtain the average value (mean) for the data measured n times from each pH sensor. Let the n times measured data of the i^th sensor and the mean (μ) of n times measured data of the i^th sensor be as follows [15]: (1) x i ( k ) , k = 1 , 2 , ⋯ , n (2) μ i = 1 n ∑ k = 1 n x i , kwhere i is the number of sensors, k is the number of data measurements for each pH sensor.

In this study, the average data fusion of sensor array with eight pH electrodes have the same weighted coefficients (w₁ = w₂ =⋯= w₈). The sum of weighted coefficients is equal to 1 and the final average data fusion of sensor array is shown as follows [15]: (3) ∑ i = 1 8 w i = 1 (4) μ ADF = 1 8 ∑ i = 1 8 μ i

The current work obtained pH measured values from eight ruthenium dioxide pH sensors. Each sensor was measured n times during a measurement period. The pH measured data can be pre-processed using a mathematical or statistical method to obtain the mean (ȳ_i) and variance ( σ i 2 ) for the n times measured data from each pH sensor. The mean and variance equations are expressed in general form as follows [3]: (5) y ¯ i = 1 n ∑ j = 1 n y i j (6) σ i 2 = 1 n − 1 ∑ j = 1 n ( y i j − y ¯ i ) 2where i is the number of sensors, j is the number of data measurements for each pH sensor, y_ij is the j^th data from the i^th sensor, ȳ_i and σ i 2 are the mean and variance from the i^th sensor, respectively.

Here, we utilize the sensor array based on the minimum mean variance to proceed with measurement data fusion. First, we assume that all data from each sensor have the same mean and exclusion independent each other. We evaluated the weighted coefficients w_i (w₁, w₂, … w_n) for each pH sensor and the sum of weighted factors for each pH sensor is equal to unity. The estimated data fusion value μ_y can then be described as follows [3]: (7) ∑ i = 1 n w i = 1 (8) μ y = ∑ i = 1 n w i y iwhere w_i is the weighted coefficient of the i^th sensor, y_i is the measured data of the i^th sensor, μ_y is the final value after data fusion. After data fusion, the equation for the total mean variance is as follows [3]: (9) σ 2 = E [ ( Y − μ y ) 2 ] = E [ ( Y − ∑ i = 1 n w i y i ) 2 ] = E [ ( Y ∑ i = 1 n w i − ∑ i = 1 n w i y i ) 2 ] = E [ ( ∑ i = 1 n w i Y − ∑ w i y i ) 2 ] = E [ ( ∑ i = 1 n w i ( Y − y i ) ) 2 ] = ∑ i = 1 n w i 2 σ i 2

From Equation (6), this study can obtain the total of mean variance σ² which is related to each weighted coefficient in the multi-dimension second order function. According to the multi-dimension function theory, we can obtain the f function that consists of λ and w_i variables in the equation as follows [3]: (10) f ( w 1 , w 2 , ⋯ , w n , λ ) = ∑ i = 1 n w i 2 σ i 2 + λ ( ∑ i = 1 n w i − 1 )

The Lagrange multiplier method is used to evaluate the solution of Equation (9). Let the f function proceeds partial deviation of λ and w_i, respectively. The equations are obtained as follows [3]: (11) ∂ f ∂ w i = 2 w i σ i 2 − λ = 0 ( i = 1 , 2 , ⋯ , n ) (12) ∂ f ∂ λ = 1 − ∑ i = 1 n w i = 0

The solutions of the Equations (11) and (12) are evaluated and the expressed equation for w_i is obtained as follows [3]: (13) w i = 1 σ i 2 ( ∑ k = 1 n σ k − 2 ) ( i = 1 , 2 , ⋯ , n )

There are n sensors in the measurement system and the sensors are used to determine the analyte, respectively. The measured values of the i^th sensor in the k time are shown as follows [8]: (14) x i ( k ) , i = 1 , 2 , ⋯ , n

The measured values of each sensor acted as a fuzzy set. According to the fuzzy mathematic theory, we can closely measure between two fuzzy sets.

Definition 1: the approach degree measured values of the i sensor and j sensor at k time is shown as follows [8]: (15) σ i j ( k ) = min { x i ( k ) , x j ( k ) } / max { x i ( k ) , x j ( k ) }

Definition 2: the approach degree matrix between each sensor at k time is shown as follows [8]: (16) ∑ ( k ) = [ 1 σ 12 ( k ) ⋯ σ 1 n ( k ) σ 21 ( k ) 1 ⋯ σ 2 n ( k ) ⋮ ⋮ ⋱ ⋮ σ n 1 ( k ) σ n 2 ( k ) ⋯ 1 ]

Definition 3: the consistence measurement of the measured value between the i^th sensor and other sensors in the time of k is shown as follows [8]: (17) r i ( k ) = ∑ j = 1 n σ i j ( k ) / n

Considering measurement region dependability and defining the mean and variance of the i^th sensor are shown as follows [8]: (18) r ¯ i ( k ) = ∑ i = 1 k r i ( k ) / k (19) σ i 2 ( k ) = ∑ i = 1 k [ r i ( t ) − r ¯ i ( t ) ] 2 / k

Definition 4: the measurement of consistence dependability of the i^th sensor in the time of k is shown as follows [8]: (20) w i ( k ) = r ¯ i ( k ) / σ i 2 ( k )

Regularity equal to one then we have the following form [8]: (21) W i ( k ) = w i ( k ) / ∑ j = 1 n w j ( k )

Utilizing the measurement of consistence dependability for data fusion, to obtain the measured value of data fusion of all sensors in the k time is presented as follows [8]: (22) x f ( k ) = ∑ i = 1 n W i ( k ) x i ( k )

The coefficient of variance (CV), also named discrete coefficient, is used for different measurement data. The CV is the ratio of the standard deviation and mean value. The CV_i is presented as the coefficient of variance of measured data X_i, and the calculation of the CV_i is described as follows [9]: (23) C V i = σ i / μ i

To utilize the coefficient of variance for the data fusion of sensor array, the data process and equation are shown as follows [9]:

From Equation (23), calculate the coefficient of variance with measured data of sensor array (CV₁, CV₂, ⋯, CV_n).

The current study has obtained average data fusion (ADF) from each sensor with the same weighted coefficients (w₁ = w₂ =⋯= w₈). We used Equation (4) to derive the average data fusion with the pH measured data. The weighted coefficient of eight sensor array is 0.125. The data fusion results of grape wine, generic cola drink and bottled base water with average data fusion are 4.04, 5.11 and 7.62, respectively, and are shown in Table 2.

The measured pH data of the RuO₂ sensor array were used with self-adaptive data fusion (SADF). We used the Equation (13) to obtain the weighted coefficients (w_i) of sensor array. The weighted coefficients and the fusion results with weighted coefficients of SADF are shown in Table 3. The fusion results of grape wine, generic cola drink and bottled base water by using self-adaptive data fusion are 3.58, 4.67 and 7.44, respectively.

We used Equation (21) to obtain the weighted coefficients (w_i) of the sensor array with fuzzy set data fusion (FSDF). The fusion results and weighted coefficients (w_i) of every sensor are shown in Table 4. The fusion results of grape wine, generic cola drink and bottled base water with fuzzy set data fusion are 3.56, 4.68 and 7.30, respectively.

We used Equation (24) to obtain the weighted coefficients (w_i) of the sensor array with coefficient of variance data fusion (CVDF). Table 5 shows the fusion results of the coefficient of variance data fusion with weighted coefficients (w_i) of every sensor. The measured pH data of grape wine after coefficient of variance data fusion is 3.62. The measured pH data of generic cola drink after coefficient of variance data fusion is 4.79 and the measured pH data of bottled base water after coefficient of variance data fusion is 7.46.

The weighted coefficients of various data fusion methods are obtained from the measured pH data and used these measured pH data to compute the mean (μ), standard deviation (σ) and variance (σ²) with mathematic statistic functions. In this study, we investigated the various data fusion methods and applied for the measured pH values of an electrochemical pH sensor array. The fusion technology is only to use the measured data and added mathematic statistic formula to derive the solution. Table 6 shows the summary of pre-calculation, weighted coefficient and computational complexity with different data fusion methods. The mean (μ) value of measured data was obtained for the H sensor array in the average data fusion in advance. The average data fusion has the same weighted coefficient and the complexity of calculation is easy. The self-adaptive data fusion need to obtain the mean (μ) and variance (σ²) beforehand, the weighted coefficients of sensor array were obtained from the variance of each pH sensor. The approach degree (σ_ij) and consistent (r_i) were used to calculate the weighted coefficients of the fuzzy set data fusion. The coefficient of variance data fusion used the mean (μ) and standard deviation (σ) to derive the weighted coefficients for pH sensor array. According to the computational process of these data fusion methods, in which the fuzzy set data fusion uses the variance and matrix operations and is more difficult than the others. The computational complexity of self-adaptive and coefficient of variance data fusions are moderate with mean, standard deviation and variance statistic operation. The average data fusion has easy computation with arithmetic average to get the weighted coefficients.

In this study, we have performed a series of trials for commercial drinks using different data fusion methods with a RuO₂ pH sensor array. This section summarizes the fusion results in this experiment. We used the measured pH data of Table 1 for different data fusion methods to perform the data fusion. Tables 2–5 present the experiment results. Table 2 shows the final results of the pH measurement of grape wine, generic cola drink and bottled base water with a single sensor. The no. 6 sensor failed and its measured value is very different from other sensors. The fusion results with different data fusion methods are shown in Table 7 and compared with the measurement results of a commercial pH meter. From Table 7, we can conclude that the fusion result of average data fusion is more different from the commercial pH meter than the other data fusion methods. This phenomenon is due to the fact the 6th sensor of the pH sensor array failed; its measured data was incorrect and had the same weight coefficient as the others. As to the other data fusion methods, the 6th sensor has a smaller weighted coefficient, therefore the fusion results were fairly close to the measured value of the commercial pH meter. According to the experimental results, the conclusion was that the fusion results with self-adaptive, fuzzy set and coefficient of variance methods were superior to a single failed pH sensor and the average data fusion.