回归分析方法.ppt

,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,第,4,讲,线性回归分析,2012,年,9,月,城市规划数据分析方法,有趣的发现,英国著名的统计学家,F.Galton,及其弟子,K.Pearson,研究了,1078,对夫妇及其一个成年儿子的身高关系。他们以儿子身高作为纵坐标、夫妇平均身高为横坐标作散点图，结果发现二者的关系近似于一条直线。经计算得到了如下方程：,由此方程可以看到：夫妇平均身高增加或减少一个单位，儿子的身高只增加或减少,0.516,个单位。也就是说，子代的身高就不像父辈身高那样分化，而是逐渐向平均身高回归。,Galton,引进“回归”（,regression),一词来表达这种变化关系。不过后来人们研究其它变量间的关系时，并没有发现如上所述的回归现象，但仍沿用“回归”的概念以纪念统计学家,F.Galton,。,“,回归,”,，指在依据大样本数据作出的变量间关系的散点图中，可以找到一条特定的直线或曲线，这样就可以把这条直线或曲线叫做其它测量变量与被预测变量之间关系的回归线，它能够最理想的反映变量间的预测关系。,3.1,回归分析的目的,相关分析揭示了要素之间的相关程度。然而，诸要素之间关系的进一步具体化，譬如某一要素与其它要素之间的关系若能用一定的函数形式予以近似地表达，那么其实用意义将会更大。,回归分析方法，就是研究要素之间具体数量关系的一种强有力的工具，运用这种方法能够建立反映要素之间具体数量关系的数学模型，即回归模型。,3.2,一元线性回归,定义：假设有两个要素（变量）,x,和,y,，,x,为自变量，,y,为因变量。,一元线性回归模型的基本结构形式为,上式代表,x,与,y,之间相关关系的拟合直线，称为回归直线；是,y,的估计值，亦称回归值。,拟合的好坏程度用拟合优度（,R,2,）度量,7,参数估计,:,普通最小二乘法,（,ordinary least squares,OLS,）,:,最小二乘法的基本原则是：最优拟合直线应该使各点到直线的距离的和最小，也可表述为距离的平方和最小。,假定根据这一原理得到的,、,估计值为 、，则直线可表示为,:,参数检验,一元线性回归的步骤,利用散点图或相关分析确定两变量是否存在线性关系；,利用,Excel,、,SPSS,、,Eview,等软件计算回归方程的回归常数和回归系数，得到回归方程；,进行拟合优度检验。,判断回归方程对样本数据的代表程度。检验的方法是使用判定系数,R,2,，取值范围在,0,R,2,1,。当其等于,0,时，,x,与,y,没有任何关系；当其为,1,时，回归线是完全拟合的，即所有散点均落在回归线上；其越接近于,1,，回归线拟合得越好。,序号,面积,A,周长,P,序号,面积,A,周长,P,1,10447.370,625.392,30,232844.300,4282.043,2,15974.730,612.286,31,4054.660,289.307,3,30976.770,775.712,32,30833.840,895.980,4,9442.902,530.202,33,1823.355,205.131,5,10858.920,1906.103,34,26270.300,968.060,6,21532.910,1297.962,35,13573.960,1045.072,7,6891.680,417.058,36,65590.080,2250.435,8,3695.195,243.907,37,157270.400,2407.549,9,2260.180,197.239,38,2086.426,266.541,10,334.332,99.729,39,3109.070,261.818,11,11749.080,558.921,40,2038.617,320.396,12,2372.105,199.667,41,3432.137,253.335,13,8390.633,592.893,42,1600.391,230.030,14,6003.719,459.467,43,3867.586,419.406,15,527620.200,6545.291,44,1946.184,198.661,16,179686.200,2960.475,45,77.305,56.902,17,14196.460,597.993,46,7977.719,715.752,18,22809.180,1103.070,47,19271.820,1011.127,19,71195.940,1154.118,48,8263.480,680.710,20,3064.242,245.049,49,14697.130,1234.114,21,469416.700,8226.009,50,4519.867,326.317,22,5738.953,498.656,51,13157.660,1172.916,23,8359.465,415.151,52,6617.270,609.801,24,6205.016,414.790,53,4064.137,437.355,25,60619.020,1549.871,54,5645.820,432.355,26,14517.740,791.943,55,6993.355,503.784,27,31020.100,1700.965,56,4304.281,267.951,28,26447.160,1246.977,57,6336.383,347.136,29,7985.926,918.312,58,2651.414,292.235,GDP,与人口,3.3,多元线性回归,一个事件往往存在两个或多个要素，它们之间也存在着相互影响、相互关联的关系。因此，多元线性回归模型更带有普遍性的意义。,为常数,称为回归系数,F,检验回归方程拟合效果,GDP,与人口,3.4,多项式回归,（二次多项式回归）,假设有两个要素（变量）,x,和,y,，,x,为自变量，,y,为因变量。,一元线性回归模型的基本结构形式为。,例：太阳光的紫外线随一日中时间的变化而变化，今测得某地,56,月份晴天一日内不同时间的紫外线强度如下，试作回归分析。,画散点图，可以看出，紫外线强度与一日内的时间大致呈抛物线关系，便可得到拟合方程。,时间,t,6,8,10,12,14,16,18,紫外线强度,I,0.6,1,1.07,1.17,1.09,0.89,0.48,3.5,多项式回归,（三次多项式回归）,假设有两个要素（变量）,x,和,y,，,x,为自变量，,y,为因变量。,一元线性回归模型的基本结构形式为。,研究饲料的含磷量与饲料系数的关系，得如下数据，试进行回归分析,含磷量,x,%0.35 0.77 1.04 1.36 1.70,饲料系数,y,2.65 2.01 1.77 2.25 4.27,配置抛物线方程，得方程,1,：,增加高次方项，并进行优化筛选，得方程,2,：,显然，方程,2,要好于方程,1,，两者的预测值也表明方程,2,要更接近于实测值：,多项式回归,（二次多项式回归）,假设有两个要素（变量）,x,和,y,，,x,为自变量，,y,为因变量。,一元线性回归模型的基本结构形式为。,3.5,非线性回归模型,非线性关系线性化的几种情况,:,对于指数曲线 ，,令,可以将其转化为直线形式：，,其中，；,对于对数曲线 ，,令 ，,可以将其转化为直线形式：；,对于幂函数曲线 ，,令 ，,可以将其转化为直线形式：,其中，；,对于双曲线 ，,令 ，,转化为直线形式：,对于,S,型曲线 ，可,转化为直线形式：,对于幂乘积：,，,只要令 ，就可以将其转化为线性形式：,其中，；,对于对数函数和,只要令 ，就可以将其化为线性形式：,例,:,下表给出了某地区林地景观斑块面积（,Area,）与周长（,Perimeter,）的数据。,下面我们建立林地景观斑块面积,A,与周长,P,之间的非线性回归模型,。,序号,面积,A,周长,P,序号,面积,A,周长,P,1,10447.370,625.392,42,232844.300,4282.043,2,15974.730,612.286,43,4054.660,289.307,3,30976.770,775.712,44,30833.840,895.980,4,9442.902,530.202,45,1823.355,205.131,5,10858.920,1906.103,46,26270.300,968.060,6,21532.910,1297.962,47,13573.960,1045.072,7,6891.680,417.058,48,65590.080,2250.435,8,3695.195,243.907,49,157270.400,2407.549,9,2260.180,197.239,50,2086.426,266.541,10,334.332,99.729,51,3109.070,261.818,11,11749.080,558.921,52,2038.617,320.396,12,2372.105,199.667,53,3432.137,253.335,13,8390.633,592.893,54,1600.391,230.030,14,6003.719,459.467,55,3867.586,419.406,15,527620.200,6545.291,56,1946.184,198.661,16,179686.200,2960.475,57,77.305,56.902,17,14196.460,597.993,58,7977.719,715.752,18,22809.180,1103.070,59,19271.820,1011.127,19,71195.940,1154.118,60,8263.480,680.710,20,3064.242,245.049,61,14697.130,1234.114,21,469416.700,8226.009,62,4519.867,326.317,22,5738.953,498.656,63,13157.660,1172.916,23,8359.465,415.151,64,6617.270,609.801,24,6205.016,414.790,65,4064.137,437.355,25,60619.020,1549.871,66,5645.820,432.355,26,14517.740,791.943,67,6993.355,503.784,27,31020.100,1700.965,68,4304.281,267.951,28,26447.160,1246.977,69,6336.383,347.136,29,7985.926,918.312,70,2651.414,292.235,30,3638.766,399.725,71,2656.824,298.473,31,585425.100,11474.770,72,1846.988,179.866,32,35220.640,1877.476,73,1616.684,172.808,33,10067.820,497.394,74,1730.563,172.143,34,27422.570,1934.596,75,11303.970,881.042,35,43071.550,1171.413,76,14019.790,638.176,36,57585.940,2275.389,77,9277.172,862.088,37,28254.130,1322.795,78,13684.750,712.787,38,497261.000,9581.298,79,1949.164,228.403,39,24255.030,994.906,80,4846.016,324.481,40,1837.699,229.401,81,521457.400,7393.938,41,1608.625,225.842,82,564370.800,12212.410,（,1,）作变量替换，令：，将上表中的原始数据进行对数变换，变换后得到的各新变量对应的观测数据如下表所示。,序号,y=lnA,x=LnP,序号,y=lnA,x=LnP,1,9.254106,6.438379,42,12.35813,8.362186,2,9.678763,6.4172,43,8.307622,5.667487,3,10.34099,6.653782,44,10.33637,6.797918,4,9.153019,6.273258,45,7.508433,5.32365,5,9.292742,7.552816,46,10.17619,6.875294,6,9.977338,7.168551,47,9.515909,6.951841,7,8.83807,6.033226,48,11.09118,7.718879,8,8.214789,5.496789,49,11.96572,7.786364,9,7.7232,5.284414,50,7.643208,5.585528,10,5.812135,4.602457,51,8.042079,5.567651,11,9.37153,6.326008,52,7.620027,5.769558,12,7.771533,5.296653,53,8.140938,5.534711,13,9.034871,6.385013,54,7.378003,5.438211,14,8.700134,6.130066,55,8.260386,6.038839,15,13.17613,8.786501,56,7.573626,5.291597,16,12.09897,7.993105,57,4.347755,4.041328,17,9.560748,6.393579,58,8.984408,6.573334,18,10.03492,7.005852,59,9.866399,6.918821,19,11.17319,7.051092,60,9.019601,6.523136,20,8.027556,5.501457,61,9.595408,7.118109,21,13.05925,9.015056,62,8.416238,5.787871,22,8.655032,6.211917,63,9.484759,7.067248,23,9.03115,6.028643,64,8.797438,6.413133,24,8.733113,6.027773,65,8.309957,6.080744,25,11.01236,7.345927,66,8.638671,6.069247,26,9.583127,6.67449,67,8.852716,6.222147,27,10.34239,7.438951,68,8.367365,5.590806,28,10.1829,7.128478,69,8.754063,5.849717,29,8.985436,6.822537,70,7.882848,5.67756,30,8.1994,5.990776,71,7.884887,5.698678,31,13.28009,9.347906,72,7.521311,5.192213,32,10.46939,7.537684,73,7.388132,5.152181,33,9.217099,6.209381,74,7.456202,5.148326,34,10.21912,7.567654,75,9.332909,6.781105,35,10.67062,7.065966,76,9.548225,6.458614,36,10.96103,7.729906,77,9.135312,6.759358,37,10.24899,7.187502,78,9.524037,6.569182,38,13.11687,9.167568,79,7.575156,5.431112,39,10.09638,6.902648,80,8.485912,5.782227,40,7.51627,5.435471,81,13.16438,8.908416,41,7.383135,5.419837,82,13.24347,9.410208,（,2,）,以,x,为横坐标、,y,为纵坐标，在平面直角坐标系中作出散点图。很明显，,y,与,x,呈线性关系。,（,3,）根据所得表中的数据，运用建立线性回归模型的方法，建立,y,与,x,之间的线性回归模型，得到：,x,与,y,的相关系数 高达,0.9665,。,（,4,）,将上式还原成双对数曲线，即,事实上，有关研究表明（,Li,，,2000,；徐建华等，,2001,），景观形态具有分形特征，景观斑块的面积（,Area,）与周长（,Perimeter,）之间的数量关系都可以用双对数关系拟合：,根据分维数的定义，,D=2/a,即为景观形态的面积,-,周长维数。在上例中，某地区林地景观形态的分维数,D,2/1.505=1.3289,。对于某种景观而言，分维数,D,的大小代表了该景观形态的复杂性。,D,值越大，就表示该景观越复杂。,