\begin{document} \[ F_{j} = \frac{QS_{R} (\beta_{j} | \beta_{1}, \beta_{0})}{QM_{E} (x_{j}, x_{1})} \] \end{document}
where:
QSR - quadratic sum of the regression; and,
QME (xj, x1) - quadratic mean of the error in the model containing the variables x1 and xj.
The variables selected via StepWise were used to determine the multiple linear regression equation, for the simulation of wheat biomass yield to generate an equation of the following type:
(2)
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ y = b_{0} \pm b_{1}x \pm b_{2}x_{2} \pm b_{3}x_{3} \pm \cdots \pm b_{n}x_{n} \] \end{document}\begin{document} \[ y = b_{0} \pm b_{1}x \pm b_{2}x_{2} \pm b_{3}x_{3} \pm \cdots \pm b_{n}x_{n} \] \end{document}
Described in the matrix form as:
(3.1)
y
=
Y
1
Y
2
M
Y
n
(3.2)
X
=
1
X
11
X
12
⋯
X
p
1
1
X
21
X
22
⋯
X
p
2
M
M
M
⋯
M
1
X
1
n
X
2
n
⋯
X
p
n
(3.3)
β
=
β
0
β
1
M
β
n
(3.4)
ε
=
ε
1
ε
2
M
ε
n
These matrices were used to obtain the regression coefficients,
(4)
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ \hat{\beta} = (X'X)^{-1} X'Y \] \end{document}\begin{document} \[ \hat{\beta} = (X'X)^{-1} X'Y \] \end{document}
and the variance of these coefficients was obtained by the covariance matrix of the vector of the regression coefficients:
(5)
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ \hat{Cov}(\hat{\beta}) = (X'X)^{-1} \hat{\sigma}^{2} \] \end{document}\begin{document} \[ \hat{Cov}(\hat{\beta}) = (X'X)^{-1} \hat{\sigma}^{2} \] \end{document}
(6)
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ \hat{\sigma}^{2} = \frac{(Y - X \hat{\beta})(Y - X \hat{\beta})}{n - p - 1} \] \end{document}\begin{document} \[ \hat{\sigma}^{2} = \frac{(Y - X \hat{\beta})(Y - X \hat{\beta})}{n - p - 1} \] \end{document}
where:
n - number of equations; and,
p - number of parameters.
The test of hypothesis verified H0: βi = 0 vs Ha: βi ≠ 0, expressed by:
(7)
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ t = \frac{\hat{\beta}_{i} - \beta_{i}}{\sqrt{\hat{V}}(\hat{\beta}_{i})} \] \end{document}\begin{document} \[ t = \frac{\hat{\beta}_{i} - \beta_{i}}{\sqrt{\hat{V}}(\hat{\beta}_{i})} \] \end{document}
These analyses were made using the computer program Genes (Cruz, 2013).
Results and Discussion
For N application in the V3 stage, in 2012, the mean maximum temperature showed the highest value compared with the other years (Figure 1A). High temperatures without occurrence of rainfall were observed before and after fertilization, favoring losses of the nutrient through volatilization. Soil moisture conditions for the fertilization in V3/V6 and V3/E were adequate, because of rainfalls during the previous days. Although the rainfall volume was the lowest one compared with the historical average (Table 1), meteorological data, along with the reasonable yield, characterize 2012 as an intermediate year (IY) of cultivation. In 2013, maximum temperature for the N-fertilizer application in V3 was approximately 15 ºC and with favorable conditions of soil moisture, because of rainfalls during the days prior to fertilization (Figure 1B). This condition was also observed in the fertilization in V3/V6 and V3/E. According to Table 1, the total volume of rainfall was similar to the historical average, indicating adequate rainfall distribution along the cycle (Figure 1B). These conditions were decisive for the highest mean yield, characterizing 2013 as a favorable year (FY) for cultivation. In 2014, maximum temperature around 23 °C (Figure 1C) occurred during N-fertilizer application in V3. N application was followed by a significant volume of rainfalls (± 30 mm), a condition also observed close to grain harvest. When N was applied in V3/E, soil moisture was not adequate for fertilization. These facts justify the lower yield (Table 1), either by N loss through leaching in V3, reduced soil moisture in V3/E or damages caused by rainfalls during maturation (Figure 1C), characterizing 2014 as unfavorable year (UY) for cultivation.
Figure 1
Rainfall, maximum temperature and days of nitrogen application in the wheat cycle (A) 2012, (B) 2013 and (C) 2014
Table 1
Temperature and rainfall in the months of wheat cultivation and mean biomass yield
Year
Month
Temperature (ºC)
Rainfall (mm)
BYx (kg ha-1)
Class
Minimum
Maximum
Mean
Mean 25 years*
Occurred
2012
May
11.1
24.5
17.8
149.7
20.3
6096 b
IY
June
9.3
19.7
14.5
162.5
59.4
July
7.4
17.5
12.4
135.1
176.6
August
12.9
23.4
18.1
138.2
61.4
September
12
23
17.5
167.4
194.6
October
15
25.5
20.2
156.5
286.6
Total
-
-
-
909.4
798.9
2013
May
10.5
22.7
16.6
149.7
100.5
7058 a
FY
June
7.9
18.4
13.15
162.5
191
July
8.3
19.2
13.75
135.1
200.8
August
9.3
20.4
14.85
138.2
223.8
September
9.5
23.7
16.6
167.4
46.5
October
12.2
25.1
18.65
156.5
211.3
Total
-
-
-
909.4
973.9
2014
May
10.8
23.6
17.2
149.7
412
5245 c
UY
June
8.6
19
13.8
162.5
412
July
9.7
21.82
15.76
135.1
144
August
8.8
23.66
16.23
138.2
77.8
September
13.33
23.58
18.46
167.4
274.8
October
16.02
27.49
21.76
156.5
230.8
Total
-
-
-
909.4
1551.4
*Mean rainfall obtained from May to October from 1989 to 2014; Means followed by the same letter in the column do not differ at 0.05 probability level by Scott-Knott test; IY - Intermediate year; FY - Favorable year; UY - Unfavorable year; BYx - Biomass yield
Agriculture is one of the most important segments and the most dependent on natural conditions (Silva et al., 2008). Temperature and rainfall are the meteorological elements that most influence crop yield (Cordeiro et al., 2015). In wheat, a favorable climate is that with mild temperatures, good radiation index to favor tillering and grain filling, with occurrence of rains in small amounts and adequate soil moisture (Pereira et al., 2015).
The proposal of wheat biomass yield simulation per agricultural year does not contemplate efficient models, due to the strong variation between the cultivation years, interfering with N use for yield (Table 1 and Figure 1). Therefore, the cumulative effect of variability between unfavorable, intermediate and favorable years was considered to obtain the values of thermal time, rainfall and biomass yield in the crop cycle and the multiple linear regression coefficients for yield simulation. Thus, Table 2 shows the sum of meteorological values in each moment of cut, along with the mean biomass yields in each N supply condition in the systems of succession. In the soybean/wheat system, biomass yield was altered by the single and split N application at 60, 90 and 120 days after emergence at doses of 30 and 60 kg ha-1. In the maize/wheat system, biomass yield also showed a similar behavior to the previously mentioned. Regardless of cultivation system, split N application in V3/E was the least efficient, but the supply in a single dose (V3) and split in V3/V6 did not cause alteration. Highest biomass yields were obtained in the soybean/wheat system, strengthening the benefits of the vegetal cover with higher residual-N release on crop yield.
Table 2
Meteorological variables and biomass yields at different moments of cut under the nitrogen supply conditions
Selected variables
N dose (kg ha-1)
Phenological stage
Moment of cut (DAE)
30
60
90
120
(2012+2013+2014)
Thermal time (degree d-1)
-
-
322
692
1075
1593
Rainfall (mm m-2)
-
-
280
426
613
867
Soybean/wheat system (2012+2013+2014)
Biomass yield (kg ha-1)
0
V3
433 a
2152 a
5043 a
6874 a
V3/V6
411 a
1972 a
5467 a
7187 a
V3/E
411 a
1940 a
5144 a
6734 a
30
V3
455 a
2765 a
6812 a
7215 a
V3/V6
464 a
2349 a
6496 a
7496 a
V3/E
438 a
1735 b
5876 b
6717 b
60
V3
440 a
2833 a
6623 a
7393 a
V3/V6
437 a
2868 a
6639 a
7841 a
V3/E
395 a
2351 b
5869 b
7033 b
120
V3
446 a
3564 a
7080 a
7961 a
V3/V6
458 a
3362 a
7212 a
8185 a
V3/E
416 a
3271 a
6825 a
7998 a
Maize/wheat system (2012+2013+2014)
Biomass yield (kg ha-1)
0
V3
274 a
1281 a
3157 a
5521 a
V3/V6
201 a
1286 a
3430 a
5278 a
V3/E
214 a
1053 a
3596 a
5140 a
30
V3
253 a
1761 a
5053 a
5848 a
V3/V6
250 a
1602 a
4871 a
5703 a
V3/E
212 a
1187 b
4212 b
5283 b
60
V3
279 a
2170 a
5283 a
6586 a
V3/V6
270 a
1989 a
5735 a
6117 a
V3/E
249 a
1432 b
4788 b
5311 b
120
V3
261 a
2245 a
6061 a
6600 a
V3/V6
257 a
2184 a
6111 a
6584 a
V3/E
237 a
1975 a
5957 a
6324 a
DAE - Days after emergence; N - Nitrogen (kg ha-1); V3 - Full (100%) N dose on third expanded leaf; V3/V6 - Split (70%/30%) N dose on third and sixth expanded leaves; V3/E - Split (70%/30%) N dose on third expanded leaf and beginning of grain filling; Means followed by the same letter in the column do not differ statistically at 0.05 probability level by Scott-Knott test
The chemical composition of the residues affects the dose and period of N supply (Siqueira Neto et al., 2010). Split application with the adjusted dose of N-fertilizer can increment wheat yield, provided that the conditions in the first application are not favorable (Espindula et al., 2010). Silva et al. (2008) observed no differences in wheat grain yield between the conditions of single and split N application, suggesting a single application under more-favorable conditions of soil moisture, to reduce the operating costs of the application.
In the indication of potential variables for inclusion in the multiple linear regression models, the mean square significance of the variables analysed by the StepWise technique is presented (Table 3). In each N supply condition, the inclusion of days of cycle, N doses and rainfall were applied in the simulation of wheat biomass yield through the multiple model.
Table 3
Identification of potential variables via StepWise technique to compose the multiple linear regression model for wheat biomass yield simulation
Source of variation
Mean square/StepWise Model
V3
V3/V6
V3/E
Soybean/wheat system (2012 + 2013 + 2014)
Regression
645508750*
723117769*
448795250*
Days of cycle (days)
1899534826*
2129886363*
1742899874*
N doses (Ndose)
25524628*
26858344*
21340402*
Thermal time (TT)
185376ns
275689ns
17932755ns
Rainfall (P)
11466795*
12608601*
13007953*
Error
2053370
3056952
2172296
Maize/wheat system (2012 + 2013 + 2014)
Regression
381279637*
493176333*
397702003*
Days of cycle (days)
1443593169*
1372638813*
1067642552*
N doses (Ndose)
33204366*
55227835*
48673854*
Thermal time (TT)
134971ns
1587386ns
1093195ns
Rainfall (P)
40916448*
51662350*
76799602*
Error
1570385
1797387
1273198
Days of cycle (Days) - Days of biomass cut (30, 60, 90 and 120 days); Thermal time (TT) degree d-1; Rainfall (P) mm m-2; Ndose - Nitrogen doses - 0, 30, 60, 120 kg N ha-1; V3 - Full (100%) N dose on third expanded leaf; V3/V6 - Split (70%/30%) N dose on third and sixth expanded leaves; V3/E - Split (70%/30%) N dose on third expanded leaf and beginning of grain filling; *Significant by F test at 0.05 probability level; ns Not significant by F test at 0.05 probability level
The identification of components that influence crop yield is decisive in the elaboration of efficient simulation models (Leal et al., 2015). The StepWise technique allows to select potential components for simulation through multiple linear regression (Balbinot Júnior et al., 2005; Mantai et al., 2016). Dalchiavon et al. (2012) selected through StepWise technique the number of panicles, panicle weight, number of spikelets per panicle and thousand-grain weight of rice to compose the multiple linear regression model in the simulation of grain yield. Mantai et al. (2016) simulated oat yield using the multiple model with the variables panicle harvest index, number of grains and spikelets per panicle and N.
Table 4 shows the multiple linear regression equations for the simulation of wheat biomass yield under the N supply conditions in the cultivation systems. This simulation used the values presented in Table 2 and the potential variables validated by the StepWise technique (Table 3). The observed values of biomass yield increased with the increment of N-fertilizer, regardless of system of succession. This trend of biomass growth was also obtained by the model of simulation through multiple regression. On the other hand, the results of biomass yield simulated by the proposed models in each N-fertilizer condition are very close to those observed in the actual cultivation conditions, thus validating an innovative proposal of simulation of wheat biomass yield for silage production.
Table 4
Multiple linear regression for the estimate of total biomass yield based on nitrogen doses, rainfall and days in each cultivation system
Stage
Dose (N)
Equation Y = f1 (x1, x2, ...)
BYO
BYE
(kg ha-1)
Soybean/wheat system (2012+2013+3014)
V3
0
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ BY = – 682.46 + 79.28_{days} + 8.02_{Ndose} – 2.29_{P} \] \end{document}\begin{document} \[ BY = – 682.46 + 79.28_{days} + 8.02_{Ndose} – 2.29_{P} \] \end{document}
6874
6846
30
7215
7086
60
7693
7327
120
7961
7808
V3/V6
0
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ BY = – 758.39 + 83.88_{days} + 8.23_{Ndose} – 2.40_{P} \] \end{document}\begin{document} \[ BY = – 758.39 + 83.88_{days} + 8.23_{Ndose} – 2.40_{P} \] \end{document}
7187
7226
30
7496
7473
60
7841
7720
120
8185
8214
V3/E
0
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ BY = – 2146.75 + 83.49_{days} + 12.26_{Ndose} – 1.44_{P} \] \end{document}\begin{document} \[ BY = – 2146.75 + 83.49_{days} + 12.26_{Ndose} – 1.44_{P} \] \end{document}
6734
6624
30
6717
6991
60
7033
7359
120
7998
8095
Maize/wheat system (2012+2013+3014)
V3
0
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ BY = – 756.28 + 86.15_{days} + 8.79_{Ndose} – 4.56_{P} \] \end{document}\begin{document} \[ BY = – 756.28 + 86.15_{days} + 8.79_{Ndose} – 4.56_{P} \] \end{document}
5421
5628
30
5848
5892
60
6586
6156
120
6600
6683
V3/V6
0
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ BY = – 838.08 + 87.99_{days} + 11.34_{Ndose} – 5.17_{P} \] \end{document}\begin{document} \[ BY = – 838.08 + 87.99_{days} + 11.34_{Ndose} – 5.17_{P} \] \end{document}
5278
5238
30
5703
5579
60
6117
5919
120
6584
6599
V3/E
0
\documentclass {article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ BY = – 201.04 + 88.77_{days} + 10.65_{Ndose} – 6.21_{P} \] \end{document}\begin{document} \[ BY = – 201.04 + 88.77_{days} + 10.65_{Ndose} – 6.21_{P} \] \end{document}
5140
5073
30
5283
5387
60
5311
5706
120
6324
6345
V3 - Full (100%) N dose on third expanded leaf; V3/V6 - Split (70%/30%) N dose on third and sixth expanded leaves; V3/E - Split (70%/30%) N dose on third expanded leaf and beginning of grain filling; Ndose - Nitrogen dose (0, 30, 60, 120 kg ha-1); P - Rainfall (mm); BY - Biomass yield (kg ha-1); BYO - Observed biomass yield (kg ha-1); BYE - Estimated biomass yield (kg ha-1)
Simulation through multiple linear regression can allow an efficient estimate of yield (Tsukahara et al., 2016). Mantai et al. (2016) estimated oat yield through multiple linear regression based on panicle harvest index, number of grains and spikelets per panicle and N use. Leilah & Al-Khateeb (2005) simulated through multiple regression wheat yield under drought condition, using the variables grain weight per panicle, panicle harvest index and panicle length. Using this model, Godoy et al. (2015) analysed soil attributes to simulate rice grain yield in the use of copper, nitrogen, iron and acid phosphatase.
Conclusions
Rainfall and nitrogen are potential variables to compose the multiple linear regression model.
Multiple linear regression models are efficient to simulate wheat biomass yield for silage during the cycle in the systems of succession.
Acknowledgments
To the Coordination for the Improvement of Higher Education Personnel (CAPES), National Council for Scientific and Technological Development (CNPq), Rio Grande do Sul Research Support Foundation (FAPERGS) and the Regional University of the Northwest of Rio Grande do Sul (UNIJUÍ), for the financial support to the research and scholarships of Scientific Initiation, Technological Initiation and Research Productivity.
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