施肥效果分析.doc

施肥效果分析 摘要本文研究了营养素对作物的产量的影响，分析了不同营养素对不同作物生长产量的差异，建立了施肥效果模型。并采用控制变量法和计算机数据拟合法建立了营养素对作物生长影响的模型。根据研究所所得的营养素与作物产量的数据，运用MATLAB得到营养素与作物产量关系的散点图。进一步运用拟合工具进行拟合数据，得到多项式的二次，三次函数和正弦函数一项，两项和三项函数。利用方差比较，得到N在三次多项式时拟合度最好，而P和K在二次多项式时拟合度最好。 本文最后总结了模型的优点和不足之处，并对施肥效果改进意见。关键词：散点图，方差比较，拟合方程，控制变量 一问题重述作物生长所需的营养素主要是氮（N）、钾（K）、磷（P）。为研究三种营养素对作物生长的影响，某作物研究所在该地区选取土豆与生菜做了一定数量的实验，实验过程中当一个营养素的施肥量变化时，总将另二个营养素的施肥量保持在最适宜植物生长状态。分析数据得出施肥量与产量之间关系，并对所得结果从应用价值与如何改进等方面作出估价。 二问题分析氮元素可促进植株茎叶的生长，更好的进行光合作用。磷元素具有一部分促根发育的作用还具有促进开花的作用。钾元素主要是促进果实的干物质积累，用来膨大果实。增加产量。由施肥量与产量的关系表格可得营养素对土豆生菜的产量有明显的促进作用。根据农业期刊Biology and fertility of soils，一般来说，产量W可以用营养素施肥量的多项正弦函数表示，故做拟合曲线并代入试验数据求得关系表达式；同时联想到Logistic函数的导函数曲线为二次多项式（也是随着自变量先增后减），因此作一次二次以及三次多项式拟合，并进行比较。 三基本假设每次试验独立且试验条件（如环境条件，种植密度，土壤条件）相同；由于数据由研究所提供，所以假设试验数据不会出现较大误差；三种元素的使施用量同作物产量有一定的函数关系，同一种元素对不同作物的作用表现为同一类的函数关系；忽略土壤中原有的N、P、K对作物生长的影响；三种元素对作物增长的作用是相互独立的； 四名词解释和符号说明名词解释:种植密度：单位面积作物种植量符号说明：（i=1,2,3.)多项式系数 ，正弦函数各项系数和常数项 五.模型建立和求解采用MATLAB2012b中配置的curve fitting tool（曲线拟合工具），直接输入数据，进行曲线拟合。二、N对作物生长的影响1.N对土豆生长的影响数据散点图 根据图像可进行大致估计，其函数关系应为二次（或更高次）多项式函数，此函数关系亦有可能为三角函数。根据实际，现假设其函数关系可能为二次多项式函数、三次多项式函数、一项正弦函数、两项正弦函数。根据已有数据建立回归模型，进行曲线拟合。（以下关于其他几组数据的讨论皆为这几种模型，不再赘述）（1） 二次多项式函数回归模型结果如下Linear model Poly2: f(x) = p1*x2 + p2*x + p3Coefficients (with 95% confidence bounds):（95%致信区间内的拟合常数） p1 = -0.0003395 (-0.0003886, -0.0002905) p2 = 0.1971 (0.1736, 0.2207) p3 = 14.74 (12.63, 16.85)Goodness of fit: SSE: 11.33 R-square: 0.9863 Adjusted R-square: 0.9824 RMSE: 1.272（2）三次多项式函数回归模型Linear model Poly3: f(x) = p1*x3 + p2*x2 + p3*x + p4Coefficients (with 95% confidence bounds): p1 = -2.666e-07 (-5.998e-07, 6.655e-08) p2 = -0.0001532 (-0.0003899, 8.347e-05) p3 = 0.1645 (0.1189, 0.2102) p4 = 15.71 (13.5, 17.91)Goodness of fit: SSE: 6.914 R-square: 0.9916 Adjusted R-square: 0.9875 RMSE: 1.073(3)单项正弦函数回归模型General model Sin1: f(x) = a1*sin(b1*x+c1)Coefficients (with 95% confidence bounds): a1 = 43.76 (42.27, 45.25) b1 = 0.004162 (0.003909, 0.004416) c1 = 0.3586 (0.3058, 0.4115)Goodness of fit: SSE: 10.32 R-square: 0.9875 Adjusted R-square: 0.984 RMSE: 1.214(4)两项正弦函数回归模型General model Sin2: f(x) = a1*sin(b1*x+c1) + a2*sin(b2*x+c2)Coefficients (with 95% confidence bounds): a1 = 92.31 (-1.453e+04, 1.472e+04) b1 = 0.006413 (-0.1468, 0.1596) c1 = -0.02898 (-27.93, 27.88) a2 = 50.29 (-1.459e+04, 1.469e+04) b2 = 0.008166 (-0.2279, 0.2442) c2 = 2.783 (-42.74, 48.31)Goodness of fit: SSE: 3.436 R-square: 0.9958 Adjusted R-square: 0.9906 RMSE: 0.92682N对生菜生长的影响（1）二次多项式函数回归模型Linear model Poly2: f(x) = p1*x2 + p2*x + p3Coefficients (with 95% confidence bounds): p1 = -0.0002381 (-0.0003034, -0.0001729) p2 = 0.1013 (0.07535, 0.1273) p3 = 10.23 (8.295, 12.16)Goodness of fit: SSE: 9.541 R-square: 0.9249 Adjusted R-square: 0.9035 RMSE: 1.167（2）三次多项式函数回归模型Linear model Poly3: f(x) = p1*x3 + p2*x2 + p3*x + p4Coefficients (with 95% confidence bounds): p1 = -4.955e-10 (-6.719e-07, 6.709e-07) p2 = -0.0002379 (-0.0006361, 0.0001604) p3 = 0.1013 (0.03699, 0.1656) p4 = 10.23 (7.641, 12.82)Goodness of fit: SSE: 9.541 R-square: 0.9249 Adjusted R-square: 0.8874 RMSE: 1.261（3）单项正弦函数回归模型General model Sin1: f(x) = a1*sin(b1*x+c1)Coefficients (with 95% confidence bounds): a1 = 21.17 (19.79, 22.54) b1 = 0.004981 (0.004422, 0.00554) c1 = 0.511 (0.3986, 0.6235)Goodness of fit: SSE: 8.367 R-square: 0.9341 Adjusted R-square: 0.9153 RMSE: 1.093（4）两项正弦函数回归模型General model Sin2: f(x) = a1*sin(b1*x+c1) + a2*sin(b2*x+c2)Coefficients (with 95% confidence bounds): a1 = 21.4 (20.28, 22.53) b1 = 0.005007 (0.004152, 0.005862) c1 = 0.5232 (0.3647, 0.6817) a2 = -1.294 (-2.077, -0.511) b2 = 0.02706 (0.01633, 0.03779) c2 = -0.3913 (-2.352, 1.569)Goodness of fit: SSE: 1.127 R-square: 0.9911 Adjusted R-square: 0.98 RMSE: 0.5307建立表格比较各参数的拟合量，其中系数只比较方差和修改后的拟合优度 分类对象回归模型方差（SSE）改进后的拟合优度（AR-s）N的施用量同土豆产量的函数关系二次多项式11.330.9824三次多项式6.9140.9875单项正弦10.320.984两项正弦3.4360.9906N的施用量同生菜产量的函数关系二次多项式9.5410.9035三次多项式9.5410.8874单项正弦8.3670.9153两项正弦1.1270.98经比较应选择两项正弦。三、P对作物生长的影响1.P对土豆生长的影响（1）二次多项式函数回归模型Linear model Poly2: f(x) = p1*x2 + p2*x + p3Coefficients (with 95% confidence bounds): p1 = -0.0001378 (-0.0002491, -2.654e-05) p2 = 0.07186 (0.03322, 0.1105) p3 = 32.92 (30.41, 35.42)Goodness of fit: SSE: 16.09 R-square: 0.8645 Adjusted R-square: 0.8258 RMSE: 1.516（2）三次多项式函数回归模型Linear model Poly3: f(x) = p1*x3 + p2*x2 + p3*x + p4Coefficients (with 95% confidence bounds): p1 = 6.498e-07 (-4.929e-07, 1.793e-06) p2 = -0.0004684 (-0.00106, 0.0001229) p3 = 0.1141 (0.03088, 0.1973) p4 = 32 (29.09, 34.92)Goodness of fit: SSE: 12.17 R-square: 0.8975 Adjusted R-square: 0.8463 RMSE: 1.424（3）单项正弦函数回归模型General model Sin1: f(x) = a1*sin(b1*x+c1)Coefficients (with 95% confidence bounds): a1 = 42.3 (40.59, 44.02) b1 = 0.002603 (0.001565, 0.003641) c1 = 0.8942 (0.7748, 1.014)Goodness of fit: SSE: 16.39 R-square: 0.862 Adjusted R-square: 0.8226 RMSE: 1.53（4）两项正弦函数回归模型General model Sin2: f(x) = a1*sin(b1*x+c1) + a2*sin(b2*x+c2)Coefficients (with 95% confidence bounds): a1 = 80.47 (-1.258e+04, 1.275e+04) b1 = 0.001878 (-0.168, 0.1717) c1 = 0.1624 (-73.26, 73.58) a2 = 20.07 (-4435, 4475) b2 = 0.00619 (-0.4728, 0.4852) c2 = 1.906 (-62.77, 66.59)Goodness of fit: SSE: 12.15 R-square: 0.8977 Adjusted R-square: 0.7698 RMSE: 1.7432.P对生菜生长的影响（1）二次多项式函数回归模型Linear model Poly2: f(x) = p1*x2 + p2*x + p3Coefficients (with 95% confidence bounds): p1 = -5.453e-05 (-8.1e-05, -2.805e-05) p2 = 0.0606 (0.04219, 0.07901) p3 = 6.876 (4.477, 9.274)Goodness of fit: SSE: 14.65 R-square: 0.9586 Adjusted R-square: 0.9467 RMSE: 1.447（2）三次多项式函数回归模型Linear model Poly3: f(x) = p1*x3 + p2*x2 + p3*x + p4Coefficients (with 95% confidence bounds): p1 = 1.05e-07 (-1.052e-08, 2.204e-07) p2 = -0.0001615 (-0.0002812, -4.178e-05) p3 = 0.08798 (0.05422, 0.1217) p4 = 5.687 (3.31, 8.064)Goodness of fit: SSE: 8.033 R-square: 0.9773 Adjusted R-square: 0.9659 RMSE: 1.157（3）单项正弦函数回归模型General model Sin1: f(x) = a1*sin(b1*x+c1)Coefficients (with 95% confidence bounds): a1 = 23.84 (22.02, 25.66) b1 = 0.002306 (0.001736, 0.002875) c1 = 0.3102 (0.1938, 0.4266)Goodness of fit: SSE: 17.22 R-square: 0.9513 Adjusted R-square: 0.9374 RMSE: 1.568（4）两项正弦函数回归模型General model Sin2: f(x) = a1*sin(b1*x+c1) + a2*sin(b2*x+c2)Coefficients (with 95% confidence bounds): a1 = 518 (-4.907e+06, 4.908e+06) b1 = 7.849e-05 (-0.7439, 0.7441) c1 = 0.005425 (-51.58, 51.59) a2 = 6.773 (-131.3, 144.8) b2 = 0.005607 (-0.03906, 0.05027) c2 = 0.4925 (-7.737, 8.722)Goodness of fit: SSE: 5.322 R-square: 0.985 Adjusted R-square: 0.9662 RMSE: 1.153 分类对象回归模型方差（SSE）改进后的拟合优度（AR-s）P的施用量同土豆产量的函数关系二次多项式16.090.8258三次多项式12.170.8463单项正弦16.390.8226两项正弦12.150.7698P的施用量同生菜产量的函数关系二次多项式14.650.9467三次多项式8.0330.9659单项正弦17.220.9374两项正弦5.3320.9662经比较，因三次多项式回归模型和两项正弦函数回归模型同文前的假设不符，故舍去，只讨论二次多项式回归模型和单项正弦函数回归模型。四、K对作物生长的影响1.K对土豆生长的影响（1）二次多项式函数回归模型Linear model Poly2: f(x) = p1*x2 + p2*x + p3Coefficients (with 95% confidence bounds): p1 = -6.995e-05 (-0.0001502, 1.028e-05) p2 = 0.07498 (0.02194, 0.128) p3 = 24.41 (17.85, 30.98)Goodness of fit: SSE: 109.6 R-square: 0.8155 Adjusted R-square: 0.7628 RMSE: 3.957（2）三次多项式函数回归模型Linear model Poly2: f(x) = p1*x2 + p2*x + p3Coefficients (with 95% confidence bounds): p1 = -6.995e-05 (-0.0001502, 1.028e-05) p2 = 0.07498 (0.02194, 0.128) p3 = 24.41 (17.85, 30.98)Goodness of fit: SSE: 109.6 R-square: 0.8155 Adjusted R-square: 0.7628 RMSE: 3.957（3）单项正弦函数回归模型General model Sin1: f(x) = a1*sin(b1*x+c1)Coefficients (with 95% confidence bounds): a1 = 44.55 (39.71, 49.4) b1 = 0.001817 (0.0007315, 0.002903) c1 = 0.5927 (0.3898, 0.7956)Goodness of fit: SSE: 116.5 R-square: 0.8039 Adjusted R-square: 0.7479 RMSE: 4.079（4）两项正弦函数回归模型General model Sin2: f(x) = a1*sin(b1*x+c1) + a2*sin(b2*x+c2)Coefficients (with 95% confidence bounds): a1 = 172.6 (-3.369e+04, 3.404e+04) b1 = 0.005038 (-0.08892, 0.099) c1 = -0.4575 (-35.13, 34.22) a2 = 133.3 (-3.373e+04, 3.4e+04) b2 = 0.005948 (-0.1048, 0.1167) c2 = 2.346 (-39.07, 43.76)Goodness of fit: SSE: 16.49 R-square: 0.9722 Adjusted R-square: 0.9375 RMSE: 2.0312.K对生菜生长的影响（1）二次多项式函数回归模型Linear model Poly2: f(x) = p1*x2 + p2*x + p3Coefficients (with 95% confidence bounds): p1 = -7.189e-07 (-2.723e-05, 2.579e-05) p2 = 0.005115 (-0.01241, 0.02264) p3 = 16.23 (14.06, 18.4)Goodness of fit: SSE: 11.96 R-square: 0.4514 Adjusted R-square: 0.2947 RMSE: 1.307（2）三次多项式函数回归模型Linear model Poly3: f(x) = p1*x3 + p2*x2 + p3*x + p4Coefficients (with 95% confidence bounds): p1 = 5.573e-08 (-9.867e-08, 2.101e-07) p2 = -5.47e-05 (-0.0002068, 9.742e-05) p3 = 0.01825 (-0.02254, 0.05905) p4 = 15.69 (12.96, 18.42)Goodness of fit: SSE: 10.59 R-square: 0.5146 Adjusted R-square: 0.2718 RMSE: 1.328（3）单项正弦函数回归模型General model Sin1: f(x) = a1*sin(b1*x+c1)Coefficients (with 95% confidence bounds): a1 = 25.7 (-289.3, 340.7) b1 = 0.0002512 (-0.005675, 0.006178) c1 = 0.6841 (-9.241, 10.61)Goodness of fit: SSE: 11.97 R-square: 0.4513 Adjusted R-square: 0.2946 RMSE: 1.307（4）两项正弦函数回归模型General model Sin2: f(x) = a1*sin(b1*x+c1) + a2*sin(b2*x+c2)Coefficients (with 95% confidence bounds): a1 = 18.5 (14.73, 22.26) b1 = 0.0006226 (-0.00216, 0.003405) c1 = 1.139 (0.6399, 1.637) a2 = 1.351 (-0.4026, 3.104) b2 = 0.01788 (0.01255, 0.0232) c2 = -3.12 (-5.036, -1.204)Goodness of fit: SSE: 5.559 R-square: 0.7451 Adjusted R-square: 0.4266 RMSE: 1.179 分类对象回归模型方差（SSE）改进后的拟合优度（AR-s）K的施用量同土豆产量的函数关系二次多项式176.20.7035三次多项式39.890.8993单项正弦116.50.7479两项正弦16.490.9375K的施用量同生菜产量的函数关系二次多项式11.960.2947