**3.1. Primary analysis of the initial voltammograms of fruits and their PCA models.**

Figure 1 shows that the "Lemon" sample is characterized by the maximum absolute values of the anode and cathode currents: the "Semerenko" variety has similar values, which according to Tables 1 and 2 have the most acidic, tartaric taste ("a sample with a pronounced sourness"). The pulp of these fruits has the highest electrical conductivity, the content of hydrogen ions – the most mobile ions in solution. The antipodes in this context are the apple variety "Economy" - with the lowest absolute values of currents; the voltammogram of the "Golden" variety is located next to it. Both of the latter varieties are characterized by a less sour taste and a greater sugar content compared to Lemon and "Semerenko". The voltammograms of the rest of the studied apples occupy a "conditionally" intermediate position between "Semerenko" and "Economy".

Most often, according to voltammetric data arrays, solutions with different total salt content, as well as with different electrical conductivity (different content of organic substances soluble in water), form the direction of the first main component along the maximum change in electrical conductivity, and often, as a consequence, along the change in the total mineralization of these solutions [7]. The same distribution of clusters along the first main component is observed for the studied fruit samples (Fig. 2.)

The mutual arrangement of the resulting clusters of fruits (for simplification, the location of fruits is further described in the article) confirms the primary conclusions drawn from the values of currents in Fig. 1. In Fig. 2 it can be seen that the Lemon occupies the leftmost position, then - "Semerenko" along the first main component, and the rightmost – "Econom" (Fig. 2b). As noted above, the number of main components, the coordination of which clusters are still distinguishable, can, in the first approximation, speak of the number of factors, chemical and/or physical, that determine the difference in the voltammetric behavior of the entire electrochemical system "electrode/double electric layer/solution" under the action of electric current [8–10].

Such factors may be – the difference in the total concentration of ions or their electrical mobility, which in total determines the electrical conductivity of the liquid medium, other things being equal; the presence of traces of electroactive substances that affect the flow of Faraday currents; the presence of substances adsorbed on the sensory surface of the electrodes; the presence of ligands that change the kinetics of electrochemical reactions and many other factors. For multisensory analysis, each factor hidden in the data array plays the role of a characteristic signal, according to the values of which and the number, the ability of the system to train the recognition of complex mixtures similar in composition is evaluated. The more such factors affect the type of voltammogram, the more signals we can use to improve recognition accuracy [11–13].

At the same time, the accuracy of modeling/data processing for extracting a useful analytical signal also plays an important role. PC-modeling of the studied VAGs showed that the proportion of the explained dispersion in the sum of the first two main components has already reached 97%, and this dispersion can only be due to the difference in the total concentration of ions in fruits; it is not a selective characteristic of juices.

In this regard, we propose to consider not only the stationary state of the voltammetric system according to the classical PCA, but also how it evolves with continuous operation from the moment of the beginning of the registration of voltammograms to the last 400-th point. We consider data in the format of voltampere time series using the quantitative method based on the fitting with the Fractional Ratio-Functions (FRFs).

### 3.2 The chosen fitting function

As a it was mentioned above in Introduction the basic aim of this paper is to find the corresponding fitting function that enables to describe quantitatively the principal components (1–3) that were used in the traditional PCA. As it is known the modern PCA can be characterized as qualitative method and systematic description of these components is absent. The proposed fitting function should be "universal", relatively simple and contain a minimal number of the fitting parameters. We tested many functions and found that the ratio of polynomials like

$$y(x) \cong F(x)=\frac{{{B_0}+\sum\limits_{{p=0}}^{P} {{C_{K+p - 1}}{x^p}} }}{{1+\sum\limits_{{k=0}}^{{K - 1}} {{C_k}{x^{k+1}}} }}$$

1

,

satisfies to the aforementioned criteria. Multiplying *y*(*x*) to denominator and extracting the mean value one can obtain the following linear relationship

$$\Delta \left[ {y(x)} \right]= - \sum\limits_{{k=0}}^{{K - 1}} {{C_k}\Delta \left[ {y(x) \cdot {x^{k+1}}} \right]+\sum\limits_{{p=1}}^{P} {{C_{K+p - 1}}\left[ {{x^p}} \right],} }$$

2

where Δ[*f*(*x*)] = *f*(*x*) – mean(*f*). From (2) using the linear least squares method (LLSM) one can find the unknown constants *C**k* and *C**K*+*p*. The unknown parameter *B*0 is found as the mean value from the relationship

$${B_0}=mean\left( {y \cdot \left[ {1+\sum\limits_{{k=0}}^{{K - 1}} {{C_k}{x^{k+1}}} } \right] - \sum\limits_{{p=1}}^{P} {{C_{K+p - 1}}{x^p}} } \right)\,$$

3

,

The upper values for *K* and *P* in the corresponding sums are found from the condition that the value of the relative error from (3) should not exceed 5.0%. In our case this condition leads to the values *K* = *P* = 4. One can notice also that this elegant and simple procedure can be generalized for the case *x*→*f*(*x*), where *f*(*x*) is supposed to be known. Therefore, for our purposes the fitting function

$$F(x)=\frac{{{B_0}+{C_4}x+{C_5}{x^2}+{C_6}{x^3}+{C_7}{x^4}}}{{1+{C_0}x+{C_1}{x^2}+{C_2}{x^3}+{C_3}{x^4}}}$$

4

,

containing 10 parameters (*C*0-*C*7, *B*0, *Rg*(*y*)), where *Rg*(*y* ) = *Range*(*y*) = max(*y*) – min(*y*) will be used for the fitting of the basic components with accuracy less than 1%. One can notice that this function is rather flexible and can be used for the limited functions located in the interval [*y*min≅*B*0, *y*max≅*C*7/*C*3]. These ten fitting parameters can be used for standard "perception" in addition to being utilized with the fitting function (4). The function *y*(*x*) that will be utilized for the fitting purposes is normalized to its range value, i.e., to reduce potential outliers.

$${y_j}=\frac{{yi{n_j} - mean(yin)}}{{Range(y)}},\,\,{x_j}=\frac{j}{N},\,\,j=1,2,...,N$$

5

,

This normalized function is located within in the interval [-1,1] and convenient for comparison of the fitting results.

### 3.3. Results of combination of the conventional PCA and FRF and their interpretation

The study of the VAGs time series was carried out in two stages

1) search – to calculate the first three main components and the accounts/coordinates of the voltage-voltage time series of four samples of apples and lemon (Table.1);

2) structuring of VAGs by three main components and their quantitative description by the FRF method.

The time duration of operation of the entire electrochemical system with fruit samples corresponds to four hundred cycles of recording voltammograms, which occupies approximately 27 minutes of continuous operation of the potentiostat under the conditions of cyclic voltammetry (one voltammogram is recorded during 4 sec).

Each projection of the sample VAG is transformed by the PC method onto the main component 1, 2, 3 PCs, etc. Each PC is determined by the fraction of the dispersion currents in the measurement space corresponding to four cycles of the processed VAG. The realized fit is shown in Figs. 3–7.

It is important to emphasize that such peaks cannot be recognized in the original voltammogram, since they are obtained as a result of a long accumulation of small signals of their isolation and structuring with a quantitative description by the hybrid method as PCA + FRF.

It is important to note that the voltage curves for the first PC for all samples have a similar form –correlation coefficients calculated sequentially for each pair of fruit samples for all nine parameters achieve 0.99. At the same time, the share of the explained variance for the first PC achieves 99%.

The main difference in the voltage curves is observed for the second and third PCs, but the proportion of the explained variance is less than 0.1%.

Such low values of the explained dispersion in classical voltammetry would correspond to mean noise, while in the temporal approach, such voltammetric data become useful analytical signals and allow improving the resolution of the voltammetry method to distinguish samples of similar composition by the presence of useful micro-components.

The second and third components for the juices under study

are distinct, as can be seen from a comparison of their behaviors. The first component in all juices has a nearly identical form and only differs in terms of quantity. Tables 3–5 demonstrate their quantitative variations.

Table 3

Fitting parameters of the function (4) for the juices 3 and 6

Number of compt | Comp1 | Comp2 | Comp3 | Comp1 | Comp2 | Comp3 |

Relative Error(%) | 0,03747 | 17,7529 | 63,7105 | 0,04068 | 0,12981 | 0,13919 |

B0 | 0,16913 | -9,46213 | -34,2116 | 0,11818 | 0,37341 | 0,16907 |

C0 | 8,11723 | -4,27226 | 1,95597 | 7,93189 | 8,00363 | 9,93883 |

C1 | -27,2806 | 14,9545 | -4,46818 | -27,4002 | -23,7346 | -35,2311 |

C2 | 36,9696 | -20,1655 | 4,57968 | 38,3465 | 30,2547 | 48,5244 |

C3 | -17,2066 | 9,31807 | -2,02721 | -18,5011 | -14,0142 | -22,7972 |

C4 | -0,3485 | 1,6 | 0,359 | 0,13649 | -3,54208 | -2,69738 |

C5 | -0,39998 | 17,7529 | 63,7105 | -1,39157 | 11,4188 | 11,6746 |

C6 | 0,82885 | -9,46213 | -34,2116 | 1,36239 | -15,2824 | -18,1363 |

C7 | -0,24093 | -4,27226 | 1,95597 | -0,23858 | 7,26854 | 9,19892 |

Range(y) | 66,127 | 14,9545 | -4,46818 | 65,342 | 0,938 | 0,1247 |

Table 4

Fitting parameters entering into the fitting function (4) for the juices 10 and 11

Number of compt | Comp1 | Comp2 | Comp3 | Comp1 | Comp2 | Comp3 |

Relative Error(%) | 0,03052 | 0,16744 | 0,42011 | 0,03045 | 0,24769 | 0,07699 |

B0 | 0,09155 | -0,14259 | 0,0189 | 0,10689 | -0,47048 | 0,19988 |

C0 | 7,59441 | 9,20019 | 10,4472 | 7,52147 | 10,1994 | 8,3749 |

C1 | -24,8143 | -28,5136 | -35,131 | -24,2727 | -41,5857 | -25,3808 |

C2 | 33,5032 | 33,7284 | 46,3136 | 32,3632 | 61,3108 | 31,1795 |

C3 | -15,766 | -13,319 | -20,8077 | -15,0284 | -29,4204 | -13,308 |

C4 | 0,39937 | 0,61221 | -0,14671 | 0,35425 | 5,67151 | -1,37515 |

C5 | -2,37317 | 0,0274 | 0,40498 | -2,49591 | -20,3309 | 3,05985 |

C6 | 2,92486 | -1,76687 | -0,47413 | 3,28785 | 27,8569 | -2,96732 |

C7 | -1,07495 | 1,31273 | 0,19835 | -1,27628 | -12,9015 | 1,12187 |

Range(y) | 64,79 | 0,56 | 0,368 | 64,262 | 0,1974 | 0,0953 |

Table 5

Fitting parameters from (4) for the juice 14

Number of compt | Comp1 | Comp2 | Comp3 |

Relative Error(%) | 0,03271 | 0,05483 | 0,25097 |

B0 | 0,14697 | 0,23142 | 0,055 |

C0 | 8,0224 | 8,05781 | 11,0257 |

C1 | -25,0677 | -24,3187 | -39,7882 |

C2 | 31,9256 | 30,9655 | 56,03 |

C3 | -14,0635 | -14,0366 | -26,7843 |

C4 | -0,23843 | -2,44731 | -0,32147 |

C5 | -1,01329 | 8,18572 | 0,5152 |

C6 | 2,16285 | -10,7312 | -0,16493 |

C7 | -1,05408 | 4,81559 | -0,09568 |

Range(y) | 69,03 | 3,469 | 0,366 |

Interpretation of the obtained data on the example of samples 3 and 10: Since a copper electrode was connected to the port of the working electrode, the presence of peaks in the range of steps 0–200 are caused simultaneously by primary oxidation reactions on the copper electrode and reduction reactions on the silver electrode. This behavior, for example, is observed for apples No. 3 and No. 10. However, No. 10 is characterized at the same time by greater fluctuations in currents, apparently associated with lower acidity and electrical conductivity of the so-called conditional background pulp solution. According to the third main component, for sample No. 10, the voltammogram is noise, and for apple No. 3, additional characteristic signals are observed in the form of peaks superimposed on each other in the region of 200–600 steps of current registration corresponding to the potential range of + 1500..-1500 mV reverse sweep of potentials. Reversible reduction reactions on the copper electrode and oxidative processes on the silver electrode take place here.