The studying area is represented by the city of Machado, located in the micro region of Alfenas in the south of the Minas Gerais state. It has a territorial area of approximately 586 km² and is located in the geographic coordinates of latitude 21˚40'30''S and longitude 45˚55'12''W. Its main economic activity is the agriculture exploitation and coffee is the main economic product and a massive generator of employment [12] .

In order to use the statistic method determined for this research, a meteorological mean of 55 years of daily climate data was applied. The data is from to the climatological station of the INMET―National Institute of Meteorology located in Machado-MG.

The data obtained in INMET are related to the monthly and annual means of a historical series from 1961 to 2015, referring to temperature and precipitation. The organization and tabulation of the data were executed using an electronic speadsheet.

According to Marin et al. [2] the CHB can be determined through the local or sectional hydric availability information, the calculation of water deficit (DEF), the water excess (EXC) and the removal and recharging of groundwater.

In order to elaborate and estimate the CHB by Thornhwaite and Mather method [13] the balance between the inputs and outputs of water in the soil-plant system is made taking into account the storage capacity of soil water (CAD). The storage capacity of soil water (CAD) represents the maximum water availability that a certain kind of soil can retain depending on their physical characteristics.

To elaborate the CHB of this research, we used a model of an electronic spreadsheet developed by Rolim, Sentelhas and Barbieri [14] . When the CHB has only climatic purposes, the use of CAD in the soil being equivalent to 100 or 125 mm is recommended.

Firstly, monthly climatic potential evapotranspiration, in mm, was estimated through the Thornthwaite and Mather [13] method, in which the uncorrected monthly evapotranspiration potential was calculated (considering 12 hour days and a 30 days month). Then, it was multiplied by the correction factor, which is a dependent on the latitude and on the months of the year.

ETP = 16 ( 10 t / I ) a (1)

ETP = potential evapotranspiration for a 30 days month with a 12 hour insolation (mm), t is the average temperature of the month (˚C) and a is the cubic function of I (that can be calculated by the formula)

a = 6.75 × 10 − 7 I 3 − 7.71 × 10 − 5 I 2 + 1.792 × 10 − 2 I + 0.49239 (2)

I is the annual calorific value.

The value of I can be calculated by summing the 12 values of the monthly calorific indices (i), which can be calculated by the following formula:

i = ( t ′ / 5 ) 1 , 514 (3)

t’ is the common monthly average temperature (˚C).

After estimating the evapotranspiration, the calculation of Thornthwaite and Mather [13] started. To do so, some calculation factors used in CHB are necessary:

P-ETP: Calculates the difference between precipitation P and the estimated potential evapotranspiration (ETP) in order to collect positives and negative balances. In most areas, the most common is the occurrence of a rainy season followed by a drought. In the humid months P-ETP are positives, indicating excessive rainfall, while in the dry months P-ETP is negative, representing potential water loss. When the situation is the water recharge in the soil, that is, whenever the (PETP) ≥ 0, it has to be added to ARM (storage) of the previous period and through this new ARM, it is possible to calculate the new NAc (accumulated negative) by the following expression:

NAc = CAD ( ln ARM CAD ) (4)

When there is withdrawal of the water in the soil, that is, when the (P-ETP) < 0, it has to be accumulated and through the (P-ETP) we calculate the ARM, using the following expression:

First, we calculate the Nac by the equation:

NAc = NAc + NAc previous (5)

Then, we calculate the ARM:

ARM = CAD ( e − | NAc CAD | ) (6)

ALT―Storage Alteration

ALT = ARM − ARM anterior (7)

ALT > 0 There was replacement

ALT < 0 There was withdrawal of water from the soil

ETR―Real Evapotranspiration

If,

ETR = P + | ALT | (8)

If ( P − ETP ) ≥ 0 ,

ETR = ETP (9)

DEF―Hydric deficiency: refers to the amount that the soil-plant system did not evapotranspirate

DEF = ETP − ETR (10)

EXC―Water surplus: it is related to the water that the soil cannot retain or evapotranspirate

If ARM < CAD ,

EXC = 0 (11)

If ARM = CAD ,

EXC = ( P − ETP ) − ALT (12)

In order to do the climate classification by the Thornthwaite and Mather method [13] , we use indices calculated based on the CHB. The hydric index (I_n), the arid index (I_a) and the humidity index (I_u), connected to the hydric availability, are defined from the annual values. The climate types (Table 1) were defined based on the humidity index (I_u), while the subtypes were defined by the arid index (Table 2).

I h = ( EXC / ETP ) × 100 (13)

I a = ( DEF / ETP ) × 100 (14)

I u = I h − ( 0.6 × I a ) (15)

In order to classify the thermal factor (TE), the climate types are defined based on the annual potential evapotranspiration (annual ETP). The subtypes depend on the percentage relation between the potential evapotranspiration in the

summer and the annual potential evapotranspiration (Table 3). ETP was used because it depends directly on the temperature [13] .

TE = ETP anual (16)

TE = ( ETP nover a ˜ o / ETP anual ) × 100 (17)

The non-parametric tendency test of Mann-kendall―MK [15] [16] , consists of comparing each value of the time series with the rest of the values, always in a sequential order, counting the number of the times that the rest of the terms are higher than the analyzed value. The method describes the tendency of a time data series. It is appropriated when the case could be assumed as monotonic, therefore, they do not present any seasonal cycle or another tendency in the data

[17] . The MK method is the most appropriate to analyze climate changes in climatological series. It is suggest by the World Meteorological Organization (OMM) to identify tendencies in time series [18] .

In MK test, the S statistic is calculated by the summing of all counts, as follows:

S = ∑ i = 1 n − 1 ∑ j = i + 1 n sgn ( x j − x i ) (18)

In which,

sgn ( x j − x i ) = { + 1 ;       if   x j ≥ x i 0 ;           if   x j = x i − 1 ;       if   x j < x i (19)

The S statistic tends to normality for a large n, with mean and variance given by:

E [ S i ] = 0 V a r ( s ) = n ( n − 1 ) ( 2 n + 5 ) − ∑ j = 1 p t j ( t j − 1 ) ( 2 t j + 5 ) 18 (20)

In which n is the size of the time series. Therefore, the statistic test Z is given by:

Z = { S − 1 ( V a r ( S ) ) 1 2       s e       S > 0 0                                   s e       S = 0 S + 1 ( V a r ( S ) ) 1 2       s e       S < 0 (21)

The considerable statistic tendency in the temporal series is measured by the Z value. This statistic is used to test the null hypothesis that the tendency does not exist. In Mann-Kendall test, a tendency is considered positive or negative, indicating a decrease or increase in the elements of the analyzed series, the case of Kandall’s Tau is negative or positive. The statistical significance was analyzed by the p-value test. The null hypothesis is not reject if p value is more or equal a; if p is less than a, the null hypothesis is rejected [15] [16] .

In addition to the MK test, I did the Pettitt non parametric statistic test in order to evaluate the occurrence of abrupt changes in the means of the historical series. According to Pettitt [11] , this test indicates if two samples from the same temporal series can be considered belonging to the same population.

The Petit test verifies two samples, X 1 , X 2 , ⋯ , X t e X t + 1 , X t + 2 , ⋯ , X T belonging to the same population, providing also information about the data homogeneity from the historical series analyzed. This statistic finds the point where an abrupt change in a temporal series occurred [11] .

The U_t,T statistic counts the times that a member of the first sample is higher than a member of the second sample. It can be written as:

U t , T = U t − 1 , T + ∑ j = 1 T sgn ( X i − X j ) (22)

for t = 2 , ⋯ , T

where: sgn(x) = 1 para x > 0; sgn(x) = 0 for x = 0; sgn(x) = −1 for x < 0.

The U_t,T statistic is calculated for the 1 < t < T values and the K(t) statistic from Pettitt test is the maximum absolute value for U_t,T. This statistic locates the changing point of a temporal series and its meaning. It can be described as:

k ( t ) = M A X 1 ≤ t ≤ T | U t , T | (23)

p ≅ 2 exp { − 6 k ( t ) 2 / T 3 + T 2 } (24)

The abrupt changing point is the time (t) where there is the maximum k(t). We can calculate the critical K values by the equation:

K c r i t = ± − ln ( p / 2 ) ( T 3 + T 2 ) 6 (25)

The significance level used was 5%.

The software extension XLSTAT 2014.5.03, for Microsoft Office Excel, was used in order to analyse and organize the data.

From of the graphs of surplus and water deficit we can precisely establish the driest periods, the rainy seasons, the traffic conditions for supplies and machines, the best seasons for the development of vegetation and for the beginning of a recovery process from degraded areas through the hydric surplus and deficiency graphics. The CHB analysis shows that, in the region of Machado, the dry season, when the highest hydric deficits are observed, maintains close values for the different periods analyzed: a) Period (1961-1979), b) Period (1979-1998); c) Period (1998-2015); d) Period (1961-2015), varying between 9 to 16 mm in August, the most critical period regarding hydric deficiency (Table 4).

According to Matielo et al. [19] , these climate conditions are favorable to the development of the coffee-growing in the region of Machado. According to Monteiro et al. [20] , the climate characteristics of the region also favor the cultivation of other species, such as corn, beans and vegetables in general

The CHB shows a reduction in the hydric surplus for the specific a) Period (1961-1979), b) Period (1979-1998); c) Period (1998-2015); d) Period (1961-2015), mainly in February, October and November In August, the hydric deficit was more accentuated (Figure 1). The most critical reduction in the water surplus occurs in the month of November, which reduces the volume from 107.6 mm to 11.4 mm.

According to Cunha and Martins [21] , the climatic classification of Botucatu and São Manuel municipalities using the Köppen and Thornthwaite methodologies in air temperature and rainfall data over a period of 36 years (1971 to 2006) reinforces that both classifications showed similarities in the characterization of the climate, however, Köppen’s climatic classification loses in detail, since it does not differentiate climatic types, whereas Thornthwaite, besides taking into account temperature, precipitation and evapotranspiration, presents in detail the period of deficit and annual water surplus of a locality.

I calculated the index to do the climate classification through the information obtained in the CHB. The hydric index was (I_h = 66.65), the aridity index was (I_a = 1.72), and the humidity index was (I_u= 65.62). Regarding the thermal factor (TE), the climate types were defined by the potential annual evapotranspiration (ETP_anual = 930.77 mm) and by the percentual relation between the potential evapotranspiration in the summer and the potential annual evapotranspiration ((ETP in the summer/ETP annual) × 100 = 33.60%).

Thus, the climate classification of the region of Machado is a humid mesothermic clime, with little hydric deficit (B₃ r B ′ 3 a ′ ).

Craparo et al. [22] and Assad et al. [23] approaches climate changes and global warming as the beginning of a new geological configuration in the seeding and coffee-growing (Coffea arabica) in Brazil, bringing possible economic losses. In a pessimistic scenario, about 33% of the current coffee-growing areas can became unable or of climate high risks. The MAPA [24] shows that events such as hydric stress caused by the droughts, rain excess, low or high temperatures can bring serious damages to the agricultural and livestock activities.

In Figure 2, the normal monthly CHB and the CHB extract, for the region of Machado-MG, respectively, can be visualized according to the climatological normal of the region from the data series used in this study. The dry season, when the highest values of water deficit are observed, in July, August and September, registered a total annual hydric 16 mm deficit and in the rainy season a 620.4 mm surplus.

In Figure 3, the normal monthly CHB and the CHB extract, respectively, can be visualized simulating the climate changes presented through the tendency tests. The dry season in July, August and September and April presented a deficit and had an accentuated increase in October. Thus, the annual hydric deficit was 71.5 mm and the excess in the rainy season was 488.4 mm. The relationship between the tendencies in climate changes and the reduction of the volume of water in the hydric balance is evident (Table 8).

The climate classification in the region of Machado would continuous to be a humid mesothermic clime, with little hydric deficiency. However, its category would change from B3 r B’3 a’, to B2 r B’4.