*3.2. Shaping the Efficiency of Potatoes*

The total potato yield varied depending on the course of weather conditions in the years of the study. The highest value of this feature was recorded in 2015 (29.0 tha<sup>−</sup>1), and the lowest was in 2000 (16.9 tha<sup>−</sup>1). These fluctuations resulted mainly from the changing course of weather conditions during the growing season and differences in the level of soil factors. The regression analysis of the mean potato yield values for this region showed a curvilinear dependence of the fourth degree tuber yield on the years of the research (Figure 2). The coefficient of determination of this equation was 61.9%, which, according to Kranz [32], ensures its high credibility.

**Figure 2.** Variability of the total yield of potato tubers in Southeastern Poland in 2000–2019.

Potato yields in cultivar experiments were characterized by even greater yield variability in this part of Poland. In terms of the total and commercial yield, the yield of very early cultivars was the most diverse, ranging from 30 to 65 t·ha−<sup>1</sup> in the case of the total yield and from 37 to 62 t·ha−<sup>1</sup> in the case of the commercial yield. The most stable in yieldings were medium-late and late cultivars, and their total yield ranged from 40 to 58 t·ha<sup>−</sup>1, and the marketable yield ranged from 38 to 52 t·ha−<sup>1</sup> (Figure 3).

**Figure 3.** The total and trade yield of potato in Southeastern Poland in the Experimentals of Stations for Cultivar Assessment of Central Crop Research Centre, by groups of earliness of varieties.

Determining the trend of changes indicates periodic averages for the time series. Normal moving averages are used only for one-year data or for other time series with no seasonal variation. Two types of averages were used to smooth time series with moving averages: regular or centered. So, by using moving averages, you can smooth series containing only trend and random fluctuations. These fluctuations can be eliminated by replacing the original values of the series with a series of means calculated from several adjacent components of the time series. On the basis of the series y1, y2, ... , yn, two-, three- and five-period means can be calculated. These formulas were written as follows.

$$\mathbf{f}(\mathbf{z}) = \mathbf{q} + 1, \mathbf{q} + 2, \dots \\ \mathbf{n} - \mathbf{q} \tag{4}$$

The function (f (z)) belongs to holomorphic functions, where any function, f (z) with complex values, can be written as follows:

$$\mathbf{f}(\mathbf{z}) = \mathbf{P}(\mathbf{x}, \mathbf{y}) + \mathrm{i}\mathbf{Q}(\mathbf{x}, \mathbf{y}) \tag{5}$$

where x, y ∈ R, P (x, y) R and Q (x, y) ∈ R. It has been found that both real and unreal parts of holomorph functions are satisfied by CR equations (i.e., Cauchy Riemann) and are described above (derivation of CR formulas assuming a holomorphism of the function). In this way, random fluctuations have been eliminated to a greater extent from the time series. The new, smoothened series is four words shorter due to improved smoothing. The moving average values were recorded at the level of the middle period (for k = 3 at the level of the second period, and for k = 5 at the level of the third period, etc.). Hence, a secondary series of moving averages was obtained, which is shorter than the empirical series, a primary series of 2 for the 3-year mean, 4 for the 5-year mean, 6 values for the 7-year mean, etc., because only an odd number of words can be assigned a score to a specific period. This allows for further analysis to compare the original time series terms with the smoothened series terms. The longer the moving average used to smoothen the series, the better the smoothing is, but at the same time, the more the periods for which no trend values are obtained are lost. Hence, the selection of the length of the moving averages requires some moderation [5,20].
