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一、生物学中的神经网络

一只猫大约有10亿个神经元，一只老鼠大约有7500万个神经元，一只蟑螂大约有100万个神经元，而人类有140亿个神经元。相比之下，许多人工神经网络包含的神经元要少得多，通常只有几百个，所以我们在短期内创建的人工大脑是没有危险的。

二、由生物神经元到人工神经元

过去我们在宏观角度模拟生物发明了很多东西，比如模拟鸟发明的飞机，模拟蝙蝠发明的雷达。随着对生物微观的深入研究，生物的神经元也被复制到了科学领域，就是人工神经网络。

上图是生物神经元示意图，我们可以理解Signal In是一个外部信号，比如有位美女正在吻你，cell body接收到这个信号后将其转换为Signal Out 传递至下一个神经元，这里的Signal Out也就是下一个神经元的Signal In了。就这样一层层传递至大脑中，大脑最终根据输入产生爽还是不爽的输出。至于到底爽不爽就取决于每一个神经元对信号的处理，对同样的吻，不同的人就会有不同的感受。

那么我们是否可以模拟这种方式构建出我们的人工神经网络呢？答案是肯定的。下图就是一个人工神经元的模型。

上图中X1、X2、X3是输入；是神经元本身；y就是输出。其中 就是一个函数，根据输入会产生特定的输出。我们之前讲的多维线性方程就可以是人工神经网络中某个神经元的函数。

三、机器学习相关概念      

        – 激活函数：将神经元的净输入信号转换成单一的输出信号，以便进一步在网络中传播。

        – 网络拓扑：描述了模型中神经元的数量以及层数和它们连接的方式。

上图中的多层网络就是一个神经网络，其中隐藏层可能有很多。我们熟知的谷歌大脑中隐藏层有3层，但是隐藏节点可能有很多。

        – 训练算法：指定如何设置连接权重，以便抑制或增加神经元在输入信号中的比重。

四、实例

4.1 数据

我们有如下数据。其中cement,slag,ash,water,superplastic,coarseagg,fineagg,age,strength分别代表：水泥、矿渣、灰、水、超塑性、粗集料、细集料、龄期、强度。我们要根据这些数据训练出一个神经网络模型，之后将新的数据输入，通过神经网络模型得出其强度。

cement,slag,ash,water,superplastic,coarseagg,fineagg,age,strength 141.3,212,0,203.5,0,971.8,748.5,28,29.89 168.9,42.2,124.3,158.3,10.8,1080.8,796.2,14,23.51 250,0,95.7,187.4,5.5,956.9,861.2,28,29.22 266,114,0,228,0,932,670,28,45.85 154.8,183.4,0,193.3,9.1,1047.4,696.7,28,18.29 255,0,0,192,0,889.8,945,90,21.86 166.8,250.2,0,203.5,0,975.6,692.6,7,15.75 251.4,0,118.3,188.5,6.4,1028.4,757.7,56,36.64 296,0,0,192,0,1085,765,28,21.65 155,184,143,194,9,880,699,28,28.99 151.8,178.1,138.7,167.5,18.3,944,694.6,28,36.35 173,116,0,192,0,946.8,856.8,3,6.94 385,0,0,186,0,966,763,14,27.92 237.5,237.5,0,228,0,932,594,7,26.26 167,187,195,185,7,898,636,28,23.89 213.8,98.1,24.5,181.7,6.7,1066,785.5,100,49.97 237.5,237.5,0,228,0,932,594,28,30.08 336,0,0,182,3,986,817,28,44.86 190.7,0,125.4,162.1,7.8,1090,804,3,15.04 312.7,0,0,178.1,8,999.7,822.2,28,25.1 229.7,0,118.2,195.2,6.1,1028.1,757.6,3,13.36 228,342.1,0,185.7,0,955.8,674.3,7,21.92 236,157,0,192,0,972.6,749.1,7,20.42 132,207,161,179,5,867,736,28,33.3 331,0,0,192,0,1025,821,28,31.74 310,143,0,168,10,914,804,28,45.3 304,76,0,228,0,932,670,90,49.19 425,106.3,0,153.5,16.5,852.1,887.1,91,65.2 166.1,0,163.3,176.5,4.5,1058.6,780.1,28,21.54 255,99,77,189,6,919,749,28,33.8 339,0,0,197,0,968,781,28,32.04 475,0,0,228,0,932,594,28,39.29 145.7,172.6,0,181.9,3.4,985.8,816.8,28,23.74 313,145,0,127,8,1000,822,28,44.52 331,0,0,192,0,1025,821,90,37.91 178,129.8,118.6,179.9,3.6,1007.3,746.8,28,39.16 165,0,143.6,163.8,0,1005.6,900.9,14,16.88 277.2,97.8,24.5,160.7,11.2,1061.7,782.5,14,47.71 325,0,0,184,0,1063,783,7,17.54 167,75.4,167,164,7.9,1007.3,770.1,28,41.41 194.7,0,100.5,170.2,7.5,998,901.8,14,24.28 246.8,0,125.1,143.3,12,1086.8,800.9,56,60.32 382,0,0,185,0,1047,739,28,37.42 149,118,92,183,7,953,780,28,23.52 531.3,0,0,141.8,28.2,852.1,893.7,91,59.2 387,20,94,157,11.6,938,845,56,61.46 193.5,290.2,0,185.7,0,998.2,704.3,7,17.2 326,0,138,199,11,801,792,28,40.68 337.9,189,0,174.9,9.5,944.7,755.8,28,49.9 425,106.3,0,153.5,16.5,852.1,887.1,3,33.4 200,200,0,190,0,1145,660,28,49.25 331,0,0,192,0,978,825,180,39 218.9,0,124.1,158.5,11.3,1078.7,794.9,56,37.27 234,156,0,189,5.9,981,760,28,39.3 309.9,142.8,111.2,167.8,22.1,913.9,651.2,28,38.22 350,0,0,186,0,1050,770,7,20.28 182,45.2,122,170.2,8.2,1059.4,780.7,100,48.67 480,0,0,192,0,936.2,712.2,7,34.57 295.7,0,95.6,171.5,8.9,955.1,859.2,14,35.23 250,0,95.7,187.4,5.5,956.9,861.2,100,42.35 178,129.8,118.6,179.9,3.6,1007.3,746.8,3,20.73 233.8,0,94.6,197.9,4.6,947,852.2,3,10.38 194.7,0,100.5,165.6,7.5,1006.4,905.9,100,37.34 379.5,151.2,0,153.9,15.9,1134.3,605,56,54.9 332.5,142.5,0,228,0,932,594,365,41.05 350,0,0,203,0,974,775,56,29.98 237,92,71,247,6,853,695,28,28.63 238.1,0,94.1,186.7,7,949.9,847,100,44.3 295.7,0,95.6,171.5,8.9,955.1,859.2,100,52.04 323.7,282.8,0,183.8,10.3,942.7,659.9,7,49.8 200,133,0,192,0,965.4,806.2,3,11.41 342,38,0,228,0,932,670,90,50.46 388.6,97.1,0,157.9,12.1,852.1,925.7,3,28.1 147.8,175.1,0,171.2,2.2,1000,828.5,28,26.92 290.4,0,96.2,168.1,9.4,961.2,865,56,45.08 500,0,0,200,0,1125,613,14,36.94 284,15,141,179,5.5,842,801,7,24.13 165,128.5,132.1,175.1,8.1,1005.8,746.6,3,19.42 218.2,54.6,123.8,140.8,11.9,1075.7,792.7,3,27.42 168.9,42.2,124.3,158.3,10.8,1080.8,796.2,56,39.15 190.3,0,125.2,161.9,9.9,1088.1,802.6,28,28.47 116,173,0,192,0,909.8,891.9,90,31.02 277,0,0,191,0,968,856,90,31.25 376,0,0,214.6,0,1003.5,762.4,100,43.06 236,0,0,194,0,968,885,14,12.84 273,0,90,199,11,931,762,28,32.24 500,0,0,140,4,966,853,28,67.57 212.5,0,100.4,159.3,8.7,1007.8,903.6,14,26.31 362.6,189,0,164.9,11.6,944.7,755.8,3,35.3 275.1,0,121.4,159.5,9.9,1053.6,777.5,3,23.8 139.6,209.4,0,192,0,1047,806.9,7,14.59 362.6,189,0,164.9,11.6,944.7,755.8,7,55.9 182,45.2,122,170.2,8.2,1059.4,780.7,14,21.5 427.5,47.5,0,228,0,932,594,180,41.84 183.9,122.6,0,203.5,0,959.2,800,3,4.9 475,0,59,142,1.9,1098,641,28,57.23 362.6,189,0,164.9,11.6,944.7,755.8,28,71.3 318.8,212.5,0,155.7,14.3,852.1,880.4,7,33.4 252,97,76,194,8,835,821,28,33.4 237.5,237.5,0,228,0,932,594,180,36.25 149.5,236,0,175.8,12.6,846.8,892.7,28,32.96 540,0,0,162,2.5,1055,676,28,61.89 313,0,113,178,8,1002,689,28,36.8 380,0,0,228,0,932,670,180,53.1 251.4,0,118.3,192.9,5.8,1043.6,754.3,3,11.98 436,0,0,218,0,838.4,719.7,28,23.85 155,0,143,193,9,877,868,28,9.74 281,0,0,186,0,1104,774,7,14.5 151.6,0,111.9,184.4,7.9,992,815.9,28,12.18 326.5,0,137.9,199,10.8,801.1,792.5,28,38.63 397,17.2,158,167,20.8,967,633,28,55.65 238,0,0,186,0,1119,789,7,12.05 158.6,148.9,116,175.1,15,953.3,719.7,28,27.68 385,0,0,186,0,966,763,28,31.35 302,0,0,203,0,974,817,14,18.13 192,288,0,192,0,929.8,716.1,3,12.79 155.6,243.5,0,180.3,10.7,1022,697.7,28,37.36 160,128,122,182,6.4,824,879,28,39.4 222.4,0,96.7,189.3,4.5,967.1,870.3,56,29.45 251.8,0,99.9,146.1,12.4,1006,899.8,14,33.36 331,0,0,192,0,978,825,7,16.26 250,0,95.7,191.8,5.3,948.9,857.2,56,39.64 155,183,0,193,9,877,868,28,23.79 213.5,0,174.2,154.6,11.7,1052.3,775.5,28,45.94 446,24,79,162,11.6,967,712,56,61.07 133,210,0,196,3,949,795,28,31.03 122.6,183.9,0,203.5,0,958.2,800.1,90,33.19 290.2,193.5,0,185.7,0,998.2,704.3,7,21.86 323.7,282.8,0,183.8,10.3,942.7,659.9,28,74.7 375,93.8,0,126.6,23.4,852.1,992.6,91,62.5 181.4,0,167,169.6,7.6,1055.6,777.8,28,27.77 298.2,0,107,209.7,11.1,879.6,744.2,28,31.88 236,0,0,193,0,968,885,7,9.99 236,0,0,193,0,968,885,365,25.08 339,0,0,197,0,968,781,3,13.22 387,20,94,157,14.3,938,845,56,56.34 166.8,250.2,0,203.5,0,975.6,692.6,90,43.57 387,20,94,157,13.9,938,845,28,46.68 133,200,0,192,0,927.4,839.2,3,6.88 162,214,164,202,10,820,680,28,30.65 262,111,86,195,5,895,733,28,33.72 213.7,98.1,24.5,181.7,6.9,1065.8,785.4,14,30.39 332.5,142.5,0,228,0,932,594,90,37.72 313.3,262.2,0,175.5,8.6,1046.9,611.8,7,42.8 302,0,0,203,0,974,817,180,26.74 183.9,122.6,0,203.5,0,959.2,800,90,32.53 500,0,0,200,0,1125,613,7,33.21 218.2,54.6,123.8,140.8,11.9,1075.7,792.7,28,55.51 387,20,94,157,11.6,938,845,28,46.68 380,95,0,228,0,932,594,270,41.15 213.5,0,174.2,159.2,11.7,1043.6,771.9,14,29.59 160,188,146,203,11,829,710,28,32.84 322,0,0,203,0,974,800,180,29.59 173.5,50.1,173.5,164.8,6.5,1006.2,793.5,14,33.01 252,0,0,186,0,1111,784,7,11.47 299.8,0,119.8,211.5,9.9,878.2,727.6,28,23.84 531.3,0,0,141.8,28.2,852.1,893.7,28,56.4 198.6,132.4,0,192,0,978.4,825.5,360,44.3 116,173,0,192,0,909.8,891.9,28,22.35 304,76,0,228,0,932,670,365,55.26 331,0,0,192,0,978,825,28,31.45 286.3,200.9,0,144.7,11.2,1004.6,803.7,28,67.7 349,0,0,192,0,1047,806,7,18.13 520,0,0,170,5.2,855,855,28,60.28 252.1,97.1,75.6,193.8,8.3,835.5,821.4,28,33.4 255.5,170.3,0,185.7,0,1026.6,724.3,7,17.24 446,24,79,162,11.6,967,712,28,44.42 172.4,13.6,172.4,156.8,4.1,1006.3,856.4,100,37.68 190.7,0,125.4,162.1,7.8,1090,804,56,35.34 322,149,0,186,8,951,709,28,52.42 446,24,79,162,11.6,967,712,28,57.03 178,129.8,118.6,179.9,3.6,1007.3,746.8,56,48.59 212.1,0,121.6,180.3,5.7,1057.6,779.3,3,12.47 276,116,90,180,9,870,768,28,44.28 393,0,0,192,0,940.6,785.6,90,48.85 238.1,0,0,185.7,0,1118.8,789.3,28,17.58 230,0,118.3,195.5,4.6,1029.4,758.6,3,10.03 389.9,189,0,145.9,22,944.7,755.8,3,40.6 157,236,0,192,0,935.4,781.2,90,43.38 296,0,0,192,0,1085,765,7,14.2 230,0,118.3,195.5,4.6,1029.4,758.6,28,24.48 359,19,141,154,10.9,942,801,28,59.49 374,189.2,0,170.1,10.1,926.1,756.7,56,63.4 102,153,0,192,0,887,942,7,7.68 202,11,141,206,1.7,942,801,3,9.85 350,0,0,203,0,974,775,14,22.53 236,157,0,192,0,972.6,749.1,28,32.88 252,0,0,185,0,1111,784,7,13.71 252.3,0,98.8,146.3,14.2,987.8,889,100,60.95 336.5,0,0,181.9,3.4,985.8,816.8,28,44.87 362.6,189,0,164.9,11.6,944.7,755.8,56,77.3 218.2,54.6,123.8,140.8,11.9,1075.7,792.7,56,61.99 315,137,0,145,5.9,1130,745,28,81.75 159,209,161,201,7,848,669,28,30.88 154.8,183.4,0,193.3,9.1,877.2,867.7,28,23.79 281,0,0,186,0,1104,774,28,22.44 231.8,0,121.6,174,6.7,1056.4,778.5,14,26.77 318.8,212.5,0,155.7,14.3,852.1,880.4,91,68.1 266,114,0,228,0,932,670,365,52.91 277,0,0,191,0,968,856,180,32.33 166.8,250.2,0,203.5,0,975.6,692.6,28,33.95 159.8,250,0,168.4,12.2,1049.3,688.2,28,39.46 397,0,0,185.7,0,1040.6,734.3,7,25.45 231.8,0,121.6,174,6.7,1056.4,778.5,100,45.84 164.6,0,150.4,181.6,11.7,1023.3,728.9,28,18.03 122.6,183.9,0,203.5,0,958.2,800.1,28,24.29 136.4,161.6,125.8,171.6,10.4,922.6,764.4,28,29.07 190,190,0,228,0,932,670,180,46.93 337.9,189,0,174.9,9.5,944.7,755.8,56,59.89 349,0,0,192,0,1047,806,90,39.49 184,86,190,213,6,923,623,28,22.93 424,22,132,178,8.5,822,750,7,39 379.5,151.2,0,153.9,15.9,1134.3,605,28,52.2 212,0,124.8,159,7.8,1085.4,799.5,3,19.52 156,243,0,180,11,1022,698,28,37.36 136,196,98,199,6,847,783,28,26.97 246.8,0,125.1,143.3,12,1086.8,800.9,100,66.42 203.5,135.7,0,185.7,0,1076.2,759.3,7,11.96 385,0,0,186,0,966,763,3,14.7 181.4,0,167,169.6,7.6,1055.6,777.8,3,13.62 254,0,0,198,0,968,863,90,26.94 359,19,141,154,10.9,942,801,3,23.64 230,0,118.3,195.5,4.6,1029.4,758.6,100,35.34 382,0,0,186,0,1047,739,7,24 220.8,147.2,0,185.7,0,1055,744.3,28,25.75 313.3,0,113,178.5,8,1001.9,688.7,28,36.8 167.4,129.9,128.6,175.5,7.8,1006.3,746.6,56,51.72 144,170,133,192,8,814,805,28,29.87 310,0,0,192,0,1012,830,90,35.76 425,106.3,0,151.4,18.6,936,803.7,56,64.9 108.3,162.4,0,203.5,0,938.2,849,3,2.33 214.9,53.8,121.9,155.6,9.6,1014.3,780.6,3,18.02 469,117.2,0,137.8,32.2,852.1,840.5,56,69.3 310,0,0,192,0,1012,830,28,27.83 522,0,0,146,0,896,896,7,50.51 236,0,0,194,0,968,885,3,6.47 213.7,98.1,24.5,181.7,6.9,1065.8,785.4,3,18 250.2,166.8,0,203.5,0,977.6,694.1,3,9.73 149,153,194,192,8,935,623,28,24.58 439,177,0,186,11.1,884.9,707.9,56,71.7 313,145,0,178,8,1002,689,28,41.05 427.5,47.5,0,228,0,932,594,90,41.54 165,0,143.6,163.8,0,1005.6,900.9,28,26.2 322.5,148.6,0,185.8,8.5,951,709.5,28,52.43 393,0,0,192,0,940,758,90,48.79 362.6,189,0,164.9,11.6,944.7,755.8,91,79.3 122.6,183.9,0,203.5,0,958.2,800.1,3,3.32 213.7,0,174.7,154.8,10.2,1053.5,776.4,28,40.15 200,0,0,180,0,1125,845,28,12.25 108.3,162.4,0,203.5,0,938.2,849,90,29.23 213.8,98.1,24.5,181.7,6.7,1066,785.5,28,40.23 290.4,0,96.2,168.1,9.4,961.2,865,28,34.74 153,239,0,200,6,1002,684,28,26.86 229.7,0,118.2,195.2,6.1,1028.1,757.6,56,31.35 251.4,0,118.3,192.9,5.8,1043.6,754.3,28,33.27 322,0,0,203,0,974,800,14,20.77 525,0,0,189,0,1125,613,270,67.11 389.9,189,0,145.9,22,944.7,755.8,56,79.4 252.3,0,98.8,146.3,14.2,987.8,889,14,42.29 475,118.8,0,181.1,8.9,852.1,781.5,3,37.8 259.9,100.6,78.4,170.6,10.4,935.7,762.9,28,49.77 310,0,0,192,0,1012,830,7,17.24 266,114,0,228,0,932,670,270,51.73 236.9,91.7,71.5,246.9,6,852.9,695.4,28,28.63 366,187,0,191.3,6.6,824.3,756.9,28,65.91 333,0,0,192,0,931.2,842.6,3,15.62 326,166,0,174,9,882,790,28,61.23 145.9,230.5,0,202.5,3.4,827,871.8,28,32.72 133,200,0,192,0,927.4,839.2,7,13.66 251.4,0,118.3,188.5,5.8,1028.4,757.7,56,36.97 500,0,0,200,0,1125,613,270,55.16 154.8,0,142.8,193.3,9.1,1047.4,696.7,28,12.46 349,0,0,192,0,1056,809,120,40.86 389.9,189,0,145.9,22,944.7,755.8,28,74.5 162,190.1,148.1,178.8,18.8,838.1,741.4,28,33.76 277.1,0,97.4,160.6,11.8,973.9,875.6,100,55.64 166,260,0,183,13,859,827,28,37.91 277,116.8,91,190.6,7,946.5,665.6,28,43.58 425,106.3,0,153.5,16.5,852.1,887.1,3,33.4 439,177,0,186,11.1,884.9,707.9,28,66 540,0,0,173,0,1125,613,3,41.64 480,0,0,192,0,936.2,712.2,28,43.94 382,0,0,186,0,1047,739,28,37.42 475,0,0,228,0,932,594,90,42.23 143,169,143,191,8,967,643,28,29.72 203.5,305.3,0,203.5,0,963.4,630,90,51.86 374,189.2,0,170.1,10.1,926.1,756.7,91,64.9 181.9,272.8,0,185.7,0,1012.4,714.3,7,12.37 153,145,113,178,8,867,824,28,26.23 450.1,50,0,200,3,1124.4,613.2,28,39.38 238,0,0,186,0,1119,789,28,17.54 380,0,0,228,0,932,670,90,52.91 469,117.2,0,137.8,32.2,852.1,840.5,91,70.7 528,0,0,185,6.9,920,720,28,56.83 238.2,158.8,0,185.7,0,1040.6,734.3,28,26.91 186.2,124.1,0,185.7,0,1083.4,764.3,28,17.6 212.6,0,100.4,159.4,10.4,1003.8,903.8,28,37.4 475,0,0,228,0,932,594,270,42.13 425,106.3,0,153.5,16.5,852.1,887.1,3,33.4 198.6,132.4,0,192,0,978.4,825.5,3,9.13 491,26,123,201,3.9,822,699,7,37.92 277.1,0,97.4,160.6,11.8,973.9,875.6,56,51.04 342,38,0,228,0,932,670,180,52.12 387,20,94,157,13.9,938,845,7,45.9 152.6,238.7,0,200,6.3,1001.8,683.9,28,26.86 281,0,0,185,0,1104,774,28,22.44 198.6,132.4,0,192,0,978.4,825.5,90,38.07 491,26,123,210,3.9,882,699,28,55.55 290.4,0,96.2,168.1,9.4,961.2,865,14,34.67 144,0,175,158,18,943,844,28,15.42 246.8,0,125.1,143.3,12,1086.8,800.9,14,42.22 252.5,0,0,185.7,0,1111.6,784.3,28,19.77 236,0,0,193,0,968,885,180,24.1 540,0,0,173,0,1125,613,270,74.17 295,0,0,185,0,1069,769,7,14.8 150.7,0,185.3,166.7,15.6,1074.5,678,28,13.46 218.9,0,124.1,158.5,11.3,1078.7,794.9,28,30.22 233.8,0,94.6,197.9,4.6,947,852.2,14,22.14 249.1,0,98.8,158.1,12.8,987.8,889,100,51.06 339,0,0,185,0,1069,754,28,31.84 505,0,60,195,0,1030,630,28,64.02 230,0,118.3,195.5,4.6,1029.4,758.6,56,31.54 374,189.2,0,170.1,10.1,926.1,756.7,3,34.4 139.6,209.4,0,192,0,1047,806.9,360,44.7 148.1,0,136.6,158.1,16.1,1001.8,830.1,28,17.96 190.3,0,125.2,161.9,9.9,1088.1,802.6,100,40.39 376,0,0,214.6,0,1003.5,762.4,14,25.62 251.4,0,118.3,188.5,6.4,1028.4,757.7,28,32.66 157,236,0,192,0,935.4,781.2,7,16.89 304,76,0,228,0,932,670,28,47.81 167.4,129.9,128.6,175.5,7.8,1006.3,746.6,14,31.81 304,140,0,214,6,895,722,28,33.42 182,45.2,122,170.2,8.2,1059.4,780.7,3,7.32 252,0,0,185,0,1111,784,28,19.69 323.7,282.8,0,183.8,10.3,942.7,659.9,3,28.3 350,0,0,203,0,974,775,90,31.35 213.5,0,174.2,159.2,11.7,1043.6,771.9,100,52.96 286.3,200.9,0,144.7,11.2,1004.6,803.7,91,76.8 143.7,170.2,132.6,191.6,8.5,814.1,805.3,28,29.87 236,157,0,192,0,972.6,749.1,90,47.78 213.7,0,174.7,154.8,10.2,1053.5,776.4,14,33.73 181.4,0,167,169.6,7.6,1055.6,777.8,100,45.37 249.1,0,98.8,158.1,12.8,987.8,889,56,42.03 193.5,290.2,0,185.7,0,998.2,704.3,28,32.63 194.7,0,100.5,170.2,7.5,998,901.8,28,37.27 289,134,0,195,6,924,760,28,46.24 298,137,107,201,6,878,655,28,53.52 192,288,0,192,0,929.8,716.1,90,50.73 213.5,0,174.2,159.2,11.7,1043.6,771.9,56,51.26 190,190,0,228,0,932,670,28,40.86 173,116,0,192,0,946.8,856.8,90,32.1 213.5,0,174.2,154.6,11.7,1052.3,775.5,3,17.37 173.8,93.4,159.9,172.3,9.7,1007.2,746.6,28,37.81 284,120,0,168,7,970,794,28,40.93 135.7,203.5,0,185.7,0,1076.2,759.3,7,7.51 246.8,0,125.1,143.3,12,1086.8,800.9,28,52.5 136,162,126,172,10,923,764,28,29.07 139.6,209.4,0,192,0,1047,806.9,3,8.06 250,0,95.7,187.4,5.5,956.9,861.2,14,24.92 475,0,0,228,0,932,594,180,42.62 149,139,109,193,6,892,780,28,23.69 277,0,0,191,0,968,856,360,33.7 225,0,0,181,0,1113,833,7,11.17 289,133.7,0,194.9,5.5,924.1,760.1,28,46.25 480,0,0,192,0,936.2,712.2,90,54.32 153,102,0,192,0,888,943.1,90,26.32 231.8,0,121.6,174,6.7,1056.4,778.5,56,42.7 225,0,0,181,0,1113,833,28,17.34 190.3,0,125.2,166.6,9.9,1079,798.9,14,19.42 172.4,13.6,172.4,156.8,4.1,1006.3,856.4,28,33.69 266,114,0,228,0,932,670,180,48.7 212.5,0,100.4,159.3,8.7,1007.8,903.6,28,31.64 172.4,13.6,172.4,156.8,4.1,1006.3,856.4,56,35.85 388.6,97.1,0,157.9,12.1,852.1,925.7,28,50.7 277,0,0,191,0,968,856,28,25.97 141.3,212,0,203.5,0,971.8,748.5,3,4.83 339,0,0,197,0,968,781,90,35.17 289,0,0,192,0,913.2,895.3,90,32.07 154.8,0,142.8,193.3,9.1,877.2,867.7,28,9.74 480,0,0,192,0,936,721,28,43.89 198.6,132.4,0,192,0,978.4,825.5,28,28.02 194.7,0,100.5,165.6,7.5,1006.4,905.9,28,25.72 349,0,0,192,0,1056,809,7,9.01 366,187,0,191,7,824,757,28,65.91 236,0,0,194,0,968,885,90,21.95 305.3,203.5,0,203.5,0,965.4,631,28,43.38 213.8,98.1,24.5,181.7,6.7,1066,785.5,56,47.13 218.2,54.6,123.8,140.8,11.9,1075.7,792.7,14,35.96 238.2,158.8,0,185.7,0,1040.6,734.3,7,15.69 159,187,0,176,11,990,789,28,32.76 102,153,0,192,0,887,942,90,25.46 170.3,155.5,0,185.7,0,1026.6,724.3,7,10.73 425,106.3,0,153.5,16.5,852.1,887.1,7,49.2 342,38,0,228,0,932,670,365,56.14 173.5,50.1,173.5,164.8,6.5,1006.2,793.5,3,23.08 330.5,169.6,0,194.9,8.1,811,802.3,28,56.62 318.8,212.5,0,155.7,14.3,852.1,880.4,28,55.5 304.8,0,99.6,196,9.8,959.4,705.2,28,30.12 266,112,87,178,10,910,745,28,39.42 165,128.5,132.1,175.1,8.1,1005.8,746.6,14,33.09 362.6,189,0,164.9,11.6,944.7,755.8,3,35.3 139.6,209.4,0,192,0,1047,806.9,90,39.36 150,237,0,174,12,1069,675,28,37.43 310,142.8,0,167.9,10,914.3,804,28,45.3 148,0,182,181,15,839,884,28,15.52 203.5,305.3,0,203.5,0,963.4,630,28,41.68 212,0,124.8,159,7.8,1085.4,799.5,14,31.35 102,153,0,192,0,887,942,3,4.57 305.3,203.5,0,203.5,0,965.4,631,90,54.6 183.9,122.6,0,203.5,0,959.2,800,28,24.05 296,0,107,221,11,819,778,28,31.42 275.1,0,121.4,159.5,9.9,1053.6,777.5,100,58.61 424,22,132,178,8.5,822,750,56,65.7 166.1,0,163.3,176.5,4.5,1058.6,780.1,3,10.76 297.8,137.2,106.9,201.3,6,878.4,655.3,28,53.52 321.3,164.2,0,190.5,4.6,870,774,28,57.22 333,0,0,192,0,931.2,842.6,90,41.68 102,153,0,192,0,887,942,28,17.28 388.6,97.1,0,157.9,12.1,852.1,925.7,91,57.6 480,0,0,192,0,936,721,90,54.28 362.6,189,0,164.9,11.6,944.7,755.8,3,35.3 134.7,0,165.7,180.2,10,961,804.9,28,13.29 250,0,95.7,187.4,5.5,956.9,861.2,3,13.82 284,15,141,179,5.5,842,801,56,44.52 393,0,0,192,0,940.6,785.6,7,27.74 475,118.8,0,181.1,8.9,852.1,781.5,91,74.19 212.5,0,100.4,159.3,8.7,1007.8,903.6,100,42.92 251.4,0,118.3,188.5,5.8,1028.4,757.7,3,17.22 275.1,0,121.4,159.5,9.9,1053.6,777.5,28,51.33 166.8,250.2,0,203.5,0,975.6,692.6,3,6.9 173.8,93.4,159.9,172.3,9.7,1007.2,746.6,3,15.82 374,189.2,0,170.1,10.1,926.1,756.7,28,61.09 168,42.1,163.8,121.8,5.7,1058.7,780.1,28,24.24 425,106.3,0,153.5,16.5,852.1,887.1,28,60.29 331,0,0,192,0,1025,821,120,39.38 525,0,0,189,0,1125,613,14,48.4 446,24,79,162,10.3,967,712,28,53.39 300,0,120,212,10,878,728,28,23.84 213.8,98.1,24.5,181.7,6.7,1066,785.5,14,17.84 255,0,0,192,0,889.8,945,3,8.2 275.1,0,121.4,159.5,9.9,1053.6,777.5,14,38.77 382.5,0,0,185.7,0,1047.8,739.3,28,37.44 359,19,141,154,10.9,942,801,7,35.75 190.3,0,125.2,166.6,9.9,1079,798.9,56,31.72 290.2,193.5,0,185.7,0,998.2,704.3,28,33.04 380,95,0,228,0,932,594,90,40.56 165,128.5,132.1,175.1,8.1,1005.8,746.6,100,55.02 375,93.8,0,126.6,23.4,852.1,992.6,7,45.7 331,0,0,192,0,978,825,360,41.24 277,117,91,191,7,946,666,28,43.57 321,0,128,182,11,870,780,28,37.26 313.3,262.2,0,175.5,8.6,1046.9,611.8,56,64.9 522,0,0,146,0,896,896,28,74.99 446,24,79,162,11.6,967,712,56,56.14 339.2,0,0,185.7,0,1069.2,754.3,7,21.18 286.3,200.9,0,144.7,11.2,1004.6,803.7,7,38 288,192,0,192,0,932,717.8,90,50.53 212.6,0,100.4,159.4,10.4,1003.8,903.8,14,25.37 400,0,0,187,0,1025,745,28,43.7 155.2,183.9,143.2,193.8,9.2,879.6,698.5,28,28.99 349,0,0,192,0,1047,806,28,32.72 334,17.6,158,189,15.3,967,633,28,44.33 168,42.1,163.8,121.8,5.7,1058.7,780.1,100,39.23 362.6,189,0,164.9,11.6,944.7,755.8,56,77.3 255.5,170.3,0,185.7,0,1026.6,724.3,28,32.05 427.5,47.5,0,228,0,932,594,365,43.7 439,177,0,186,11.1,884.9,707.9,3,39.3 261.9,110.5,86.1,195.4,5,895.2,732.6,28,33.72 252,0,0,185,0,1111,784,28,19.69 485,0,0,146,0,1120,800,28,71.99 194.7,0,100.5,165.6,7.5,1006.4,905.9,3,12.45 148,175,0,171,2,1000,828,28,26.92 288,192,0,192,0,932,717.8,28,38.8 359,19,141,154,10.9,942,801,56,66.78 356,0,142,193,11,801,778,28,40.87 382,0,0,186,0,1111,784,7,11.47 439,177,0,186,11.1,884.9,707.9,91,73.3 264.5,111,86.5,195.5,5.9,832.6,790.4,28,41.54 190.7,0,125.4,162.1,7.8,1090,804,14,21.06 173.8,93.4,159.9,172.3,9.7,1007.2,746.6,100,50.94 317.9,0,126.5,209.7,5.7,860.5,736.6,28,40.06 288.4,121,0,177.4,7,907.9,829.5,28,42.14 425,106.3,0,153.5,16.5,852.1,887.1,91,65.2 359,19,141,154,10.9,942,801,3,25.12 237.5,237.5,0,228,0,932,594,365,39 133,200,0,192,0,927.4,839.2,28,27.87 427.5,47.5,0,228,0,932,594,28,37.43 162,207,172,216,10,822,638,28,39.84 251.8,0,99.9,146.1,12.4,1006,899.8,56,44.14 425,106.3,0,153.5,16.5,852.1,887.1,56,64.3 491,26,123,210,3.9,882,699,7,33.49 275,180,120,162,10.4,830,765,28,76.24 145.4,0,178.9,201.7,7.8,824,868.7,28,10.54 362.6,189,0,164.9,11.6,944.7,755.8,91,79.3 277.2,97.8,24.5,160.7,11.2,1061.7,782.5,3,30.45 212.5,0,100.4,159.3,8.7,1007.8,903.6,3,13.54 218.9,0,124.1,158.5,11.3,1078.7,794.9,100,46.23 297.2,0,117.5,174.8,9.5,1022.8,753.5,14,36.99 333,0,0,192,0,931.2,842.6,28,31.97 280,129,100,172,9,825,805,28,52.82 178,129.8,118.6,179.9,3.6,1007.3,746.8,100,49.99 339,0,0,185,0,1060,754,28,31.65 500,0,0,200,0,1125,613,1,12.64 362.6,189,0,164.9,11.6,944.7,755.8,7,22.9 375,93.8,0,126.6,23.4,852.1,992.6,3,29 382,0,0,185,0,1047,739,7,24 451,0,0,165,11.3,1030,745,28,78.8 251.8,0,99.9,146.1,12.4,1006,899.8,28,33.94 212,141.3,0,203.5,0,973.4,750,90,39.7 153,102,0,192,0,888,943.1,7,8.37 318.8,212.5,0,155.7,14.3,852.1,880.4,3,25.2 296,0,0,186,0,1090,769,28,25.18 313.8,0,112.6,169.9,10.1,925.3,782.9,28,38.46 164,0,200,181,13,849,846,28,15.09 297.2,0,117.5,174.8,9.5,1022.8,753.5,56,51.96 231.8,0,121.6,174,6.7,1056.4,778.5,3,15.44 203.5,305.3,0,203.5,0,963.4,630,7,19.54 425,106.3,0,153.5,16.5,852.1,887.1,56,64.3 310,143,111,168,22,914,651,28,33.69 349,0,0,192,0,1056,809,90,40.66 480,0,0,192,0,936,721,3,24.39 298.1,0,107.5,163.6,12.8,953.2,784,28,35.87 446,24,79,162,11.6,967,712,28,51.02 381.4,0,0,185.7,0,1104.6,784.3,7,14.54 475,0,0,162,9.5,1044,662,28,58.52 362.6,189,0,164.9,11.6,944.7,755.8,28,71.3 500,0,0,200,0,1125,613,3,26.06 425,106.3,0,153.5,16.5,852.1,887.1,91,65.2 261,100,78,201,9,864,761,28,32.4 359,19,141,154,10.9,942,801,56,68.75 237.5,237.5,0,228,0,932,594,90,33.12 393,0,0,192,0,940.6,785.6,3,19.2 286.3,200.9,0,144.7,11.2,1004.6,803.7,56,72.99 213.7,98.1,24.5,181.7,6.9,1065.8,785.4,56,50.77 173.8,93.4,159.9,172.3,9.7,1007.2,746.6,56,48.99 238.1,0,94.1,186.7,7,949.9,847,3,19.93 251.4,0,118.3,188.5,6.4,1028.4,757.7,14,24.43 387,20,94,157,13.9,938,845,56,47.97 531.3,0,0,141.8,28.2,852.1,893.7,7,46.9 304,76,0,228,0,932,670,180,50.95 525,0,0,189,0,1125,613,90,58.78 145,116,119,184,5.7,833,880,28,29.16 212,141.3,0,203.5,0,973.4,750,3,6.81 374,189.2,0,170.1,10.1,926.1,756.7,7,46.2 165,0,150,182,12,1023,729,28,18.03 424,22,132,168,8.9,822,750,28,72.1 150,236.8,0,173.8,11.9,1069.3,674.8,28,37.43 212,141.3,0,203.5,0,973.4,750,7,15.03 272.8,181.9,0,185.7,0,1012.4,714.3,7,19.77 284,15,141,179,5.5,842,801,28,43.73 116,173,0,192,0,909.8,891.9,3,6.28 260,101,78,171,10,936,763,28,49.77 376,0,0,214.6,0,1003.5,762.4,28,31.97 379.5,151.2,0,153.9,15.9,1134.3,605,7,47.1 233.8,0,94.6,197.9,4.6,947,852.2,56,27.66 181.9,272.8,0,185.7,0,1012.4,714.3,28,27.94 500,0,0,200,0,1125,613,90,47.22 167,75.4,167,164,7.9,1007.3,770.1,14,32.9 212.6,0,100.4,159.4,10.4,1003.8,903.8,3,13.33 297.2,0,117.5,174.8,9.5,1022.8,753.5,28,47.4 153.1,145,113,178.5,8,1001.9,688.7,28,25.56 173,116,0,192,0,946.8,856.8,28,24.28 525,0,0,189,0,1125,613,7,42.42 289,0,0,192,0,913.2,895.3,28,25.57 228,342.1,0,185.7,0,955.8,674.3,28,39.7 182,45.2,122,170.2,8.2,1059.4,780.7,56,43.5 144,136,106,178,7,941,774,28,26.14 155,183,0,193,9,1047,697,28,18.28 251.4,0,118.3,188.5,6.4,1028.4,757.7,3,13.12 475,118.8,0,181.1,8.9,852.1,781.5,56,72.3 190,190,0,228,0,932,670,270,50.66 157,236,0,192,0,935.4,781.2,28,33.66 355,19,97,145,13.1,967,871,28,44.03 202,11,141,206,1.7,942,801,56,23.25 446,24,79,162,11.6,967,712,56,55.25 425,106.3,0,151.4,18.6,936,803.7,91,66.7 362.6,189,0,164.9,11.6,944.7,755.8,7,55.9 425,106.3,0,153.5,16.5,852.1,887.1,28,60.29 190.7,0,125.4,162.1,7.8,1090,804,100,40.57 325,0,0,184,0,1063,783,28,30.57 251.4,0,118.3,188.5,6.4,1028.4,757.7,100,44.21 525,0,0,189,0,1125,613,180,61.92 212.5,0,100.4,159.3,8.7,1007.8,903.6,56,42.55 149,236,0,176,13,847,893,28,32.96 375,93.8,0,126.6,23.4,852.1,992.6,56,60.2 173.5,50.1,173.5,164.8,6.5,1006.2,793.5,100,56.06 165,0,143.6,163.8,0,1005.6,900.9,56,36.56 168,42.1,163.8,121.8,5.7,1058.7,780.1,3,7.75 157,214,152,200,9,819,704,28,33.05 272.6,0,89.6,198.7,10.6,931.3,762.2,28,32.25 251.8,0,99.9,146.1,12.4,1006,899.8,100,45.37 322,0,0,203,0,974,800,28,25.18 238.1,0,0,185.7,0,1118.8,789.3,7,10.34 424,22,132,168,8.9,822,750,56,74.36 339,0,0,197,0,968,781,180,36.45 331,0,0,192,0,978,825,3,13.52 133.1,210.2,0,195.7,3.1,949.4,795.3,28,28.94 212.1,0,121.6,180.3,5.7,1057.6,779.3,28,24.9 165,128.5,132.1,175.1,8.1,1005.8,746.6,56,53.72 145,0,134,181,11,979,812,28,13.2 277,0,0,191,0,968,856,3,11.36 339,0,0,197,0,968,781,365,38.89 143.6,0,174.9,158.4,17.9,942.7,844.5,28,15.42 210.7,316.1,0,185.7,0,977,689.3,7,21.82 362.6,189,0,164.9,11.6,944.7,755.8,56,77.3 260.9,100.5,78.3,200.6,8.6,864.5,761.5,28,32.4 305.3,203.5,0,203.5,0,965.4,631,3,14.14 166.1,0,163.3,176.5,4.5,1058.6,780.1,100,33.54 500,0,0,151,9,1033,655,28,69.84 475,0,0,228,0,932,594,7,38.6 229.7,0,118.2,195.2,6.1,1028.1,757.6,100,40.86 439,177,0,186,11.1,884.9,707.9,7,56.1 393,0,0,192,0,940,758,3,19.11 238.1,0,94.1,186.7,7,949.9,847,28,30.23 491,26,123,201,3.9,822,699,56,61.86 491,26,123,210,3.9,882,699,56,59.59 388.6,97.1,0,157.9,12.1,852.1,925.7,56,55.2 295.8,0,0,185.7,0,1091.4,769.3,7,14.84 250,0,95.7,187.4,5.5,956.9,861.2,56,38.33 339,0,0,197,0,968,781,14,27.04 332.5,142.5,0,228,0,932,594,180,39.78 167,75.4,167,164,7.9,1007.3,770.1,3,15.52 380,95,0,228,0,932,594,180,40.76 380,0,0,228,0,932,670,270,53.3 288,121,0,177,7,908,829,28,42.13 212.6,0,100.4,159.4,10.4,1003.8,903.8,100,47.74 200,133,0,192,0,965.4,806.2,90,43.25 387,20,94,157,13.9,938,845,3,25.51 296,0,0,186,0,1090,769,7,18.91 322.2,0,115.6,196,10.4,817.9,813.4,28,31.18 251.4,0,118.3,188.5,5.8,1028.4,757.7,14,29.93 446,24,79,162,11.6,967,712,3,35.36 304,76,0,228,0,932,670,270,54.38 310,0,0,192,0,1012,830,3,11.85 405,0,0,175,0,1120,695,28,52.3 321,164,0,190,5,870,774,28,57.21 275,0,0,183,0,1088,808,7,14.2 149,117.6,91.7,182.9,7.1,953.4,780.3,28,23.52 186.2,124.1,0,185.7,0,1083.4,764.3,7,8 446,24,79,162,11.6,967,712,7,39.3 401.8,94.7,0,147.4,11.4,946.8,852.1,7,54.1 331,0,0,192,0,1025,821,3,14.31 198.6,132.4,0,192,0,978.4,825.5,180,41.72 222.4,0,96.7,189.3,4.5,967.1,870.3,100,40.71 173.8,93.4,159.9,172.3,9.7,1007.2,746.6,14,29.55 255,0,0,192,0,889.8,945,28,18.75 255.3,98.8,77,188.6,6.5,919,749.3,28,33.8 322,0,116,196,10,818,813,28,31.18 266.2,112.3,87.5,177.9,10.4,909.7,744.5,28,39.42 480,0,0,192,0,936.2,712.2,3,24.4 250,0,95.7,191.8,5.3,948.9,857.2,3,8.49 190.7,0,125.4,162.1,7.8,1090,804,28,26.4 214.9,53.8,121.9,155.6,9.6,1014.3,780.6,14,38.6 277.2,97.8,24.5,160.7,11.2,1061.7,782.5,100,66.95 385,0,136,158,20,903,768,28,55.55 307,0,0,193,0,968,812,3,12.54 295.7,0,95.6,171.5,8.9,955.1,859.2,28,39.94 246.8,0,125.1,143.3,12,1086.8,800.9,3,23.52 525,0,0,189,0,1125,613,28,55.94 250,0,0,182,0,1100,820,7,12.73 251.4,0,118.3,192.9,5.8,1043.6,754.3,56,39.27 540,0,0,173,0,1125,613,14,59.76 190.3,0,125.2,166.6,9.9,1079,798.9,28,24.85 387,20,94,157,14.3,938,845,7,36.84 173,116,0,192,0,946.8,856.8,7,11.39 166.1,0,163.3,176.5,4.5,1058.6,780.1,56,28.63 310,0,0,192,0,970,850,28,27.92 254,0,0,198,0,968,863,3,9.31 190.3,0,125.2,161.9,9.9,1088.1,802.6,14,22.72 145,0,179,202,8,824,869,28,10.54 277.1,0,97.4,160.6,11.8,973.9,875.6,3,23.14 289,0,0,192,0,913.2,895.3,3,11.65 339.2,0,0,185.7,0,1069.2,754.3,28,31.9 200,0,0,180,0,1125,845,7,7.84 295.8,0,0,185.7,0,1091.4,769.3,28,25.22 230,0,118.3,195.5,4.6,1029.4,758.6,14,20.08 425,106.3,0,151.4,18.6,936,803.7,7,46.8 238.1,0,94.1,186.7,7,949.9,847,56,39.59 284,15,141,179,5.5,842,801,3,13.4 349,0,0,192,0,1047,806.9,3,15.05 152,178,139,168,18,944,695,28,36.35 160.2,188,146.4,203.2,11.3,828.7,709.7,28,35.31 202,11,141,206,1.7,942,801,7,15.07 194.7,0,100.5,170.2,7.5,998,901.8,3,12.18 156,178,187,221,7,854,614,28,29.41 222.4,0,96.7,189.3,4.5,967.1,870.3,14,24.45 143.8,136.3,106.2,178.1,7.5,941.5,774.3,28,26.15 168.9,42.2,124.3,158.3,10.8,1080.8,796.2,100,48.15 266,114,0,228,0,932,670,90,47.03 194.7,0,100.5,165.6,7.5,1006.4,905.9,56,33.96 349,0,0,192,0,1056,809,28,33.61 380,95,0,228,0,932,594,7,32.82 151,0,184,167,12,991,772,28,15.57 376,0,0,214.6,0,1003.5,762.4,3,16.28 212,0,124.8,159,7.8,1085.4,799.5,28,38.5 310,0,0,192,0,971,850.6,3,9.87 307,0,0,193,0,968,812,180,34.49 166,259.7,0,183.2,12.7,858.8,826.8,28,37.92 252.5,0,0,185.7,0,1111.6,784.3,7,11.48 475,118.8,0,181.1,8.9,852.1,781.5,28,68.3 229.7,0,118.2,195.2,6.1,1028.1,757.6,28,24.54 158.8,238.2,0,185.7,0,1040.6,734.3,7,9.62 139.9,132.6,103.3,200.3,7.4,916,753.4,28,36.44 349,0,0,192,0,1047,806,3,14.99 157,236,0,192,0,935.4,781.2,3,9.69 122.6,183.9,0,203.5,0,958.2,800.1,7,10.35 218.9,0,124.1,158.5,11.3,1078.7,794.9,14,26.05 331,0,0,192,0,978,825,90,37.23 295,0,0,185,0,1069,769,28,25.18 173.5,50.1,173.5,164.8,6.5,1006.2,793.5,56,53.77 374.3,0,0,190.2,6.7,1013.2,730.4,28,39.06 287.3,120.5,93.9,187.6,9.2,904.4,695.9,28,43.8 298.1,0,107,186.4,6.1,879,815.2,28,42.64 190,190,0,228,0,932,670,365,53.69 350,0,0,186,0,1050,770,28,34.29 303.6,139.9,0,213.5,6.2,895.5,722.5,28,33.42 350,0,0,203,0,974,775,28,27.34 168.9,42.2,124.3,158.3,10.8,1080.8,796.2,3,7.4 339,0,0,185,0,1069,754,7,21.16 252.3,0,98.8,146.3,14.2,987.8,889,3,21.78 380,95,0,228,0,932,594,365,43.7 500,0,0,200,0,1125,613,180,51.04 249.1,0,98.8,158.1,12.8,987.8,889,28,30.85 167,75.4,167,164,7.9,1007.3,770.1,56,53.46 381.4,0,0,185.7,0,1104.6,784.3,28,22.49 500,0,0,200,0,1125,613,28,44.09 212,141.3,0,203.5,0,973.4,750,28,33 255,0,0,192,0,889.8,945,7,10.22 220.8,147.2,0,185.7,0,1055,744.3,7,13.09 192,288,0,192,0,929.8,716.1,28,39.32 213.7,0,174.7,154.8,10.2,1053.5,776.4,100,50.08 190.3,0,125.2,161.9,9.9,1088.1,802.6,3,9.45 540,0,0,162,2.5,1040,676,28,79.99 362.6,189,0,164.9,11.6,944.7,755.8,91,79.3 214.9,53.8,121.9,155.6,9.6,1014.3,780.6,28,52.2 289,0,0,192,0,913.2,895.3,7,14.6 108.3,162.4,0,203.5,0,938.2,849,28,20.59 210.7,316.1,0,185.7,0,977,689.3,28,37.81 362.6,189,0,164.9,11.6,944.7,755.8,28,71.3 170.3,155.5,0,185.7,0,1026.6,724.3,28,25.73 140,164,128,237,6,869,656,28,35.23 540,0,0,173,0,1125,613,90,69.66 135.7,203.5,0,185.7,0,1076.2,759.3,28,18.2 150.9,0,183.9,166.6,11.6,991.2,772.2,28,15.57 469,117.2,0,137.8,32.2,852.1,840.5,7,54.9 290.4,0,96.2,168.1,9.4,961.2,865,100,48.97 167,75.4,167,164,7.9,1007.3,770.1,100,56.81 250.2,166.8,0,203.5,0,977.6,694.1,7,19.35 236,0,0,194,0,968,885,28,18.42 192,288,0,192,0,929.8,716.1,7,21.48 333,17.5,163,167,17.9,996,652,28,47.28 158.8,238.2,0,185.7,0,1040.6,734.3,28,21.07 252.3,0,98.8,146.3,14.2,987.8,889,28,50.6 446,24,79,162,11.6,967,712,7,38.02 254,0,0,198,0,968,863,365,29.79 349,0,0,192,0,1047,806,360,42.13 213.5,0,174.2,159.2,11.7,1043.6,771.9,3,15.61 251.4,0,118.3,188.5,5.8,1028.4,757.7,28,29.65 194.7,0,100.5,170.2,7.5,998,901.8,56,43.39 135,105,193,196,6,965,643,28,21.91 277.1,0,97.4,160.6,11.8,973.9,875.6,28,48.28 251.4,0,118.3,188.5,5.8,1028.4,757.7,100,43.58 387,20,94,157,14.3,938,845,3,22.75 525,0,0,189,0,1125,613,3,33.8 151,0,185,167,16,1074,678,28,13.46 146.5,114.6,89.3,201.9,8.8,860,829.5,28,19.99 362.6,189,0,164.9,11.6,944.7,755.8,91,79.3 313.3,145,0,178.5,8,1001.9,688.7,28,41.05 250.2,166.8,0,203.5,0,977.6,694.1,28,36.96 374,0,0,190,7,1013,730,28,39.05 146,230,0,202,3,827,872,28,33.06 168.9,42.2,124.3,158.3,10.8,1080.8,796.2,28,31.12 212.1,0,121.6,180.3,5.7,1057.6,779.3,14,20.92 400,0,0,187,0,1025,745,7,30.14 212.6,0,100.4,159.4,10.4,1003.8,903.8,56,44.4 275,0,0,183,0,1088,808,28,24.5 540,0,0,173,0,1125,613,180,71.62 376,0,0,214.6,0,1003.5,762.4,56,36.3 314,0,113,170,10,925,783,28,38.46 298,0,107,164,13,953,784,28,35.86 251.4,0,118.3,192.9,5.8,1043.6,754.3,100,40.15 296,0,106.7,221.4,10.5,819.2,778.4,28,31.42 251.4,0,118.3,192.9,5.8,1043.6,754.3,14,20.73 153,102,0,192,0,888,943.1,3,4.78 314,145.3,113.2,178.9,8,869.1,690.2,28,46.23 190.3,0,125.2,166.6,9.9,1079,798.9,100,33.56 446,24,79,162,11.6,967,712,3,23.35 307,0,0,193,0,968,812,90,32.92 516,0,0,162,8.3,801,802,28,41.37 168,42.1,163.8,121.8,5.7,1058.7,780.1,56,32.85 427.5,47.5,0,228,0,932,594,7,35.08 173.5,50.1,173.5,164.8,6.5,1006.2,793.5,28,38.2 333,0,0,192,0,931.2,842.6,7,23.4 339,0,0,197,0,968,781,7,20.97 152,0,112,184,8,992,816,28,12.18 213.8,98.1,24.5,181.7,6.7,1066,785.5,3,13.18 152.7,144.7,0,178.1,8,999.7,822.2,28,19.01 198.6,132.4,0,192,0,978.4,825.5,7,14.64 222.4,0,96.7,189.3,4.5,967.1,870.3,3,11.58 305,0,100,196,10,959,705,28,30.12 375,0,0,186,0,1038,758,7,26.06 297.2,0,117.5,174.8,9.5,1022.8,753.5,100,56.74 148.5,139.4,108.6,192.7,6.1,892.4,780,28,23.7 272.8,105.1,81.8,209.7,9,904,679.7,28,37.17 424,22,132,168,8.9,822,750,3,32.11 359,19,141,154,10.9,942,801,7,38.61 332.5,142.5,0,228,0,932,594,28,33.02 380,95,0,228,0,932,594,28,36.45 446,24,79,162,11.6,967,712,7,52.01 108.3,162.4,0,203.5,0,938.2,849,7,7.72 249.1,0,98.8,158.1,12.8,987.8,889,14,28.68 389.9,189,0,145.9,22,944.7,755.8,7,59.09 250,0,95.7,191.8,5.3,948.9,857.2,14,24.66 213.7,0,174.7,154.8,10.2,1053.5,776.4,3,17.57 165,128.5,132.1,175.1,8.1,1005.8,746.6,28,46.39 182,45.2,122,170.2,8.2,1059.4,780.7,28,31.27 469,117.2,0,137.8,32.2,852.1,840.5,28,66.9 154,174,185,228,7,845,612,28,24.34 250,0,0,182,0,1100,820,28,20.87 200,133,0,192,0,965.4,806.2,28,30.44 172.4,13.6,172.4,156.8,4.1,1006.3,856.4,14,29.75 164.2,0,200.1,181.2,12.6,849.3,846,28,15.09 382.5,0,0,185.7,0,1047.8,739.3,7,24.07 313,161,0,178,10,917,759,28,52.44 250,180,95,159,9.5,860,800,28,67.87 397,0,0,185,0,1040,734,28,39.09 139.7,163.9,127.7,236.7,5.8,868.6,655.6,28,35.23 190.3,0,125.2,161.9,9.9,1088.1,802.6,56,38.56 331,170,0,195,8,811,802,28,56.61 229.7,0,118.2,195.2,6.1,1028.1,757.6,14,22.32 425,106.3,0,153.5,16.5,852.1,887.1,7,49.2 385,0,0,186,0,966,763,1,6.27 238.1,0,94.1,186.7,7,949.9,847,14,25.69 325.6,166.4,0,174,8.9,881.6,790,28,61.24 337.9,189,0,174.9,9.5,944.7,755.8,91,67.8 222.4,0,96.7,189.3,4.5,967.1,870.3,28,24.89 212,0,124.8,159,7.8,1085.4,799.5,100,47.82 181.4,0,167,169.6,7.6,1055.6,777.8,56,35.57 531.3,0,0,141.8,28.2,852.1,893.7,56,58.8 337.9,189,0,174.9,9.5,944.7,755.8,7,35.1 302,0,0,203,0,974,817,28,21.75 313.3,262.2,0,175.5,8.6,1046.9,611.8,3,28.8 332.5,142.5,0,228,0,932,594,7,30.28 300,0,0,184,0,1075,795,7,15.58 337.9,189,0,174.9,9.5,944.7,755.8,3,24.1 388.6,97.1,0,157.9,12.1,852.1,925.7,7,34.9 491,26,123,201,3.9,822,699,3,29.55 165,0,143.6,163.8,0,1005.6,900.9,3,14.4 252.3,0,98.8,146.3,14.2,987.8,889,56,55.83 349,0,0,192,0,1056,809,3,15.87 427.5,47.5,0,228,0,932,594,270,43.01 249.1,0,98.8,158.1,12.8,987.8,889,3,15.36 203.5,135.7,0,185.7,0,1076.2,759.3,28,22.63 132,206.5,160.9,178.9,5.5,866.9,735.6,28,33.31 387,20,94,157,11.6,938,845,3,34.77 425,106.3,0,151.4,18.6,936,803.7,3,36.3 312.7,144.7,0,127.3,8,999.7,822.2,28,44.52 375,93.8,0,126.6,23.4,852.1,992.6,28,56.7 218.2,54.6,123.8,140.8,11.9,1075.7,792.7,100,63.53 300,0,0,184,0,1075,795,28,26.85 162,190,148,179,19,838,741,28,33.76 310,0,0,192,0,970,850,180,37.33 310,0,0,192,0,1012,830,120,38.7 401.8,94.7,0,147.4,11.4,946.8,852.1,56,73.7 213.7,98.1,24.5,181.7,6.9,1065.8,785.4,100,53.9 153,145,113,178,8,1002,689,28,25.56 273,105,82,210,9,904,680,28,37.17 214.9,53.8,121.9,155.6,9.6,1014.3,780.6,56,53.96 172.4,13.6,172.4,156.8,4.1,1006.3,856.4,3,21.29 516,0,0,162,8.2,801,802,28,41.37 116,173,0,192,0,909.8,891.9,7,10.09 331,0,0,192,0,1025,821,7,17.44 212.1,0,121.6,180.3,5.7,1057.6,779.3,100,39.61 155,0,143,193,9,1047,697,28,12.46 401.8,94.7,0,147.4,11.4,946.8,852.1,3,41.1 279.8,128.9,100.4,172.4,9.5,825.1,804.9,28,52.83 158.4,0,194.9,219.7,11,897.7,712.9,28,8.54 265,111,86,195,6,833,790,28,41.54 237.5,237.5,0,228,0,932,594,270,38.41 290.4,0,96.2,168.1,9.4,961.2,865,3,22.5 425,106.3,0,153.5,16.5,852.1,887.1,56,64.3 277.1,0,97.4,160.6,11.8,973.9,875.6,14,41.89 213.5,0,174.2,154.6,11.7,1052.3,775.5,14,33.7 146,173,0,182,3,986,817,28,23.74 425,106.3,0,153.5,16.5,852.1,887.1,7,49.2 500.1,0,0,200,3,1124.4,613.2,28,44.13 295.7,0,95.6,171.5,8.9,955.1,859.2,56,48.72 213.5,0,174.2,154.6,11.7,1052.3,775.5,56,51.43 141.9,166.6,129.7,173.5,10.9,882.6,785.3,28,44.61 491,26,123,201,3.9,822,699,28,57.92 148.1,0,182.1,181.4,15,838.9,884.3,28,15.53 313.3,262.2,0,175.5,8.6,1046.9,611.8,91,66.6 362.6,189,0,164.9,11.6,944.7,755.8,56,77.3 250,0,95.7,191.8,5.3,948.9,857.2,100,41.16 212,0,124.8,159,7.8,1085.4,799.5,56,45.08 148,0,137,158,16,1002,830,28,17.95 393,0,0,192,0,940.6,785.6,28,39.6 286.3,200.9,0,144.7,11.2,1004.6,803.7,3,24.4 233.8,0,94.6,197.9,4.6,947,852.2,100,34.56 213.7,0,174.7,154.8,10.2,1053.5,776.4,56,46.64 254,0,0,198,0,968,863,180,27.63 475,0,0,228,0,932,594,365,41.93 379.5,151.2,0,153.9,15.9,1134.3,605,91,56.5 355.9,0,141.6,193.3,11,801.4,778.4,28,40.87 153,145,0,178,8,1000,822,28,19.01 318,0,126,210,6,861,737,28,40.06 159.1,186.7,0,175.6,11.3,989.6,788.9,28,32.77 285,190,0,163,7.6,1031,685,28,53.58 379.5,151.2,0,153.9,15.9,1134.3,605,3,28.6 314,145,113,179,8,869,690,28,46.23 331,0,0,192,0,879,825,3,13.52 424,22,132,178,8.5,822,750,28,62.05 159,149,116,175,15,953,720,28,27.68 318.8,212.5,0,155.7,14.3,852.1,880.4,56,66.1 168,42.1,163.8,121.8,5.7,1058.7,780.1,14,17.82 385,0,0,186,0,966,763,7,23.22 212.1,0,121.6,180.3,5.7,1057.6,779.3,56,34.2 194.7,0,100.5,165.6,7.5,1006.4,905.9,14,24.99 239.6,359.4,0,185.7,0,941.6,664.3,28,39.44 310,0,0,192,0,970,850,360,38.11 181.4,0,167,169.6,7.6,1055.6,777.8,14,21.6 310,0,0,192,0,970,850,90,34.68 375,0,0,186,0,1038,758,28,38.21 310,0,0,192,0,970,850,7,14.99 297,0,0,186,0,1040,734,7,30.96 295.7,0,95.6,171.5,8.9,955.1,859.2,3,22.95 475,118.8,0,181.1,8.9,852.1,781.5,7,55.6 144,15,195,176,6,1021,709,28,15.34 446,24,79,162,10.3,967,712,56,54.77 321.4,0,127.9,182.5,11.5,870.1,779.7,28,37.27 350,0,0,203,0,974,775,7,18.13 313.3,145,0,178.5,8,867.2,824,28,40.93 316.1,210.7,0,185.7,0,977,689.3,7,24.44 362.6,189,0,164.9,11.6,944.7,755.8,7,55.9 397,0,0,185.7,0,1040.6,734.3,28,33.08 540,0,0,173,0,1125,613,28,67.31 313.3,262.2,0,175.5,8.6,1046.9,611.8,28,59.8 355,19,97,145,12.3,967,871,28,55.45 307,0,0,193,0,968,812,28,27.53 312.9,160.5,0,177.6,9.6,916.6,759.5,28,52.45 153.6,144.2,112.3,220.1,10.1,923.2,657.9,28,16.5 214.9,53.8,121.9,155.6,9.6,1014.3,780.6,100,56.63 143,169.4,142.7,190.7,8.4,967.4,643.5,28,29.73 362.6,189,0,164.9,11.6,944.7,755.8,3,35.3 153,102,0,192,0,888,943.1,28,17.96 194.7,0,100.5,170.2,7.5,998,901.8,100,44.28 142,167,130,174,11,883,785,28,44.61 272.8,181.9,0,185.7,0,1012.4,714.3,28,31.38 133,200,0,192,0,927.4,839.2,90,36.59 387,20,94,157,11.6,938,845,7,41.67 389.9,189,0,145.9,22,944.7,755.8,91,82.6 332.5,142.5,0,228,0,932,594,270,40.27 190,190,0,228,0,932,670,90,42.33 275.1,0,121.4,159.5,9.9,1053.6,777.5,56,56.85 166.1,0,163.3,176.5,4.5,1058.6,780.1,14,25.48 469,117.2,0,137.8,32.2,852.1,840.5,3,40.2 356,119,0,160,9,1061,657,28,59 307,0,0,193,0,968,812,365,36.15 200,133,0,192,0,965.4,806.2,7,17.17 313,145,0,178,8,867,824,28,44.39 251.8,0,99.9,146.1,12.4,1006,899.8,3,21.02 218.9,0,124.1,158.5,11.3,1078.7,794.9,3,15.34 277,0,0,191,0,968,856,14,21.26 520,0,0,175,5.2,870,805,28,51.02 277.2,97.8,24.5,160.7,11.2,1061.7,782.5,28,63.14 401.8,94.7,0,147.4,11.4,946.8,852.1,28,68.5 397,0,0,186,0,1040,734,28,36.94 362.6,189,0,164.9,11.6,944.7,755.8,28,71.3 236,157,0,192,0,972.6,749.1,3,13.57 141.3,212,0,203.5,0,971.8,748.5,90,39.66 231.8,0,121.6,174,6.7,1056.4,778.5,28,33.73 349,0,0,192,0,1047,806,180,41.05 167.4,129.9,128.6,175.5,7.8,1006.3,746.6,3,14.94 491,26,123,210,3.9,882,699,3,25.61 213.5,0,174.2,159.2,11.7,1043.6,771.9,28,44.64 393,0,0,192,0,940,758,28,39.58 305.3,203.5,0,203.5,0,965.4,631,7,25.89 298,0,107,210,11,880,744,28,31.87 287,121,94,188,9,904,696,28,41.94 425,106.3,0,151.4,18.6,936,803.7,28,61.8 250.2,166.8,0,203.5,0,977.6,694.1,90,47.71 277.2,97.8,24.5,160.7,11.2,1061.7,782.5,56,66.82 284,119.7,0,168.3,7.2,970.4,794.2,28,40.93 424,22,132,178,8.5,882,750,3,32.01 167.4,129.9,128.6,175.5,7.8,1006.3,746.6,100,53.66 350,0,0,203,0,974,775,180,32.72 202,11,141,206,1.7,942,801,28,21.97 139.6,209.4,0,192,0,1047,806.9,28,28.24 238,0,0,185,0,1118,789,28,17.54 424,22,132,168,8.9,822,750,7,40.29 239.6,359.4,0,185.7,0,941.6,664.3,7,25.42 401.8,94.7,0,147.4,11.4,946.8,852.1,91,75.5 425,106.3,0,153.5,16.5,852.1,887.1,28,60.29 446,24,79,162,11.6,967,712,3,25.02 154,144,112,220,10,923,658,28,16.5 380,0,0,228,0,932,670,365,52.52 178,129.8,118.6,179.9,3.6,1007.3,746.8,14,34.24 164,163,128,197,8,961,641,28,27.23 160,250,0,168,12,1049,688,28,39.45 167.4,129.9,128.6,175.5,7.8,1006.3,746.6,28,41.2 165,0,143.6,163.8,0,1005.6,900.9,100,37.96 233.8,0,94.6,197.9,4.6,947,852.2,28,22.84 316.1,210.7,0,185.7,0,977,689.3,28,38.7 323.7,282.8,0,183.8,10.3,942.7,659.9,56,80.2 250,0,95.7,191.8,5.3,948.9,857.2,28,27.22 387,20,94,157,14.3,938,845,28,50.24 158,0,195,220,11,898,713,28,8.54 288,192,0,192,0,932,717.8,3,16.11 213.5,0,174.2,154.6,11.7,1052.3,775.5,100,59.3 213.7,98.1,24.5,181.7,6.9,1065.8,785.4,28,45.71 190.3,0,125.2,166.6,9.9,1079,798.9,3,12.55 313,0,0,178,8,1000,822,28,25.1 296,0,0,192,0,1085,765,90,29.39 359,19,141,154,10.9,942,801,28,62.94 153.1,145,113,178.5,8,867.2,824,28,26.23 298,0,107,186,6,879,815,28,42.64 140,133,103,200,7,916,753,28,36.44 139.6,209.4,0,192,0,1047,806.9,180,44.21 147,115,89,202,9,860,829,28,19.99 288,192,0,192,0,932,717.8,7,23.52 183.9,122.6,0,203.5,0,959.2,800,7,10.79 203.5,305.3,0,203.5,0,963.4,630,3,9.56 144.8,0,133.6,180.8,11.1,979.5,811.5,28,13.2 141.3,212,0,203.5,0,971.8,748.5,7,10.39 297.2,0,117.5,174.8,9.5,1022.8,753.5,3,21.91 135,0,166,180,10,961,805,28,13.29 531.3,0,0,141.8,28.2,852.1,893.7,3,41.3 276.4,116,90.3,179.6,8.9,870.1,768.3,28,44.28 342,38,0,228,0,932,670,270,55.06 540,0,0,173,0,1125,613,7,52.61 

讯享网

4.2 程序

讯享网 concrete <- read.csv("concrete.csv") str(concrete) normalize <- function(x){ return ((x - min(x)) / (max(x) - min(x))) } concrete_norm <- as.data.frame(lapply(concrete, normalize)) summary(concrete_norm$strength) summary(concrete$strength) concrete_train <- concrete_norm[1:773,] concrete_test <- concrete_norm[774:1030,] concrete_model <- neuralnet(strength ~ cement + slag + ash + water + superplastic + coarseagg + fineagg + age, data=concrete_train) plot(concrete_model) concrete_test[1:8] model_results <- compute(concrete_model, concrete_test[1:8]) predicted_strength <- model_results$net.result cor(predicted_strength, concrete_test$strength) concrete_model2 <- neuralnet(strength ~ cement + slag + ash + water + superplastic + coarseagg + fineagg + age, data=concrete_train, hidden=5) plot(concrete_model2) model_results2 <- compute(concrete_model2, concrete_test[1:8]) predicted_strength2 <- model_results2$net.result cor(predicted_strength2, concrete_test$strength) 

4.3 程序释义

4.3.1 加载数据

concrete <- read.csv("concrete.csv")

 执行效果：

4.3.2 打印数据列信息

讯享网str(concrete)

执行效果：

4.3.3 定义归一化函数

normalize <- function(x){ return ((x - min(x)) / (max(x) - min(x))) }

        我们定义了一个归一化函数normalize，操那个函数体可以看出，这个函数是将一个数据集中每一个数据减去其中最小数，再除以最大数和最小数的差值。后面我们会将这个函数应用至已有数据的每一列中。

4.3.4 给数据集应用归一化函数

concrete_norm <- as.data.frame(lapply(concrete, normalize))

 从上方代码可以看出，我们将归一化函数normalize使用lapply应用至了concrete数据的每一列的每一项数据中。然后再将归一化后的数据调用as.data.frame函数封装为数据表返回给concrete_norm。

执行完毕后，我查看concrete_norm，会发现如下图所有数据都变成了0--1之间的数据。

4.3.5 调用summary函数统计归一化前后水泥强度信息

summary(concrete_norm$strength) summary(concrete$strength)

执行结果：

4.3.6 将数据分为测试集和训练集

concrete_train <- concrete_norm[1:773,] concrete_test <- concrete_norm[774:1030,]

如上方所示，我们将数据的前773行设置为训练集，将774至1030行设置为训练集。并分别传给两个变量concrete_train、concrete_test。

4.3.7  调用第三方包的函数neuralnet生成训练集的神经网络模型（第三方包准备）

因为neuralnet是一个第三方的包，所以调用前我们需要先引入这个包。

1、如下图操作

2、使用命令：library("neuralnet")命令将neuralnet函数导入。

4.3.8  调用第三方包的函数neuralnet生成训练集的神经网络模型

concrete_model <- neuralnet(strength ~ cement + slag + ash + water + superplastic + coarseagg + fineagg + age, data=concrete_train)

可以看到上方代码我们使用了neuralnet函数，其中参数说明如下：

strength：水泥强度，放在~之前表示其是输出，即Y

cement + slag + ash + water + superplastic + coarseagg + fineagg + age：这是会影响水泥强度的变量X1  X2 ... Xn

data=concrete_train ： 说明要应用的训练集是concrete_train

训练出的模型如下图：

4.3.9 打印神经网络模型

plot(concrete_model)

执行结果如下图

其中我们可以看到有两个值：Error:5.077823,Steps:3676。其中Error代表误差，Steps代表计算的步数。

4.3.10 改进模型

concrete_model2 <- neuralnet(strength ~ cement + slag + ash + water + superplastic + coarseagg + fineagg + age, data=concrete_train, hidden=5)

上方代码表示我们有建立了一个新的神经网络模型concrete_model2，其与concrete_model不同之处在于concrete_model2被声明了要使用5个隐藏层。

模型计算完成后，在执行下方代码打印模型：

plot(concrete_model2)

结果如下图：

我们可以明显的看出，当隐藏层数量上升之后，模型的Error会急剧下降，但是Steps会升高。

4.3.11 将测试集应用至模型，验证模型正确性

1、使用隐藏层数量为1的模型

concrete_test[1:8] model_results <- compute(concrete_model, concrete_test[1:8]) predicted_strength <- model_results$net.result cor(predicted_strength, concrete_test$strength)

其中，第二行是使用concrete_model模型计算测试集，计算结果是model_results。

第三行是获取模型计算结果的水泥强度。

第四行是生成计算出强度和实际强度的相关系数。相关系数越趋近于1，就说么模型越准确。

这里我们计算出的结果如下图：

2、使用隐藏层数量5的模型

model_results2 <- compute(concrete_model2, concrete_test[1:8]) predicted_strength2 <- model_results2$net.result cor(predicted_strength2, concrete_test$strength)

这里我们计算出的结果如下图： 

3、对比两个模型计算结果准确性

通过对比两个模型计算结果与实际结果相关系数。发现隐藏层为5的模型计算结果是0.。隐藏层为1的模型计算结果是0.。显而易见，隐藏层数量越多，模型约准确。

4.4 神经网络与逻辑回归

虽然神经网络很强大，但是很多情况下我们预测依然使用逻辑回归。