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zJwE>9D@Zb`$&3QHtEbEVQE*y1aB3#%eW+asE@+&i4ym^8&S-~oU`+@A7-v_5LzOUa zblq0DCa}(RWPn!dhSDl|zEcN2)M!&F4(Zn3-IFb`va(k#fOaP2p}N>XG0ObyYfX7s z?EZe$p}?05hm+5TL^-R;Tq!Q)s>!paZnZ%rfdr`Pb2u6-CqJvHN#c-Qkn`=~^fy}W z##n=6<+I~Wbg_O2ppDP%`{`Rd)Rg#;5{i~UyE0+bWa=yPV+HxY0F@4{0kzrda8R2a z{l5PbHvU!}{Cn@32#4!O@MZr#a(^5D9|JY%eU~hWb%*QwZIIaJ+ju0x;oALBs79=`uSOW(smqU`p6er7?UV+j9hA^v}Z|Je552fpq9WB334@RR*t k)&B?n1OI{lKy?uLe=6@1xt@>U*8l(j07*qoM6N<$f-A23j{pDw literal 0 HcmV?d00001 diff --git a/必备数学基础.md b/必备数学基础.md index 711b129..4730356 100644 --- a/必备数学基础.md +++ b/必备数学基础.md @@ -953,4 +953,97 @@ Sigmoid没有负数,都是大于0的,当梯度更新的时候,要么全为 - 置信水平(1-a),区间宽度随置信水平的增大而增大 - 数据的离散程度Se,区间宽度随离程度的增大而增大样本容量 - 区间宽度随样本容量的增大而减小 -- X0与X均值之间的差异,随着差异程度的增大而增大 \ No newline at end of file +- X0与X均值之间的差异,随着差异程度的增大而增大 + +#### 回归直线拟合优度 + +回归直线与各观测点的接近程度称为回归直线对数据的拟合优度 + +- 总平方和(SST): + + ![1604845380041](assets/1604845380041.png) + +- 回归平方和(SSR): + + ![1604845414040](assets/1604845414040.png) + +- 残差平方和(SSE): + + ![1604845448704](assets/1604845448704.png) + +总平方和可以分解为回归平方和、残差平方和两部分:SST = SSR + SSE + +- 总平方和(SST),反映因变量的 n 个观察值与其均值的总离差 +- 回归平方和(SSR),反映了y的总变差中,由于x与y之间的线性关系引起的y的变化部分 +- 残差平方和(SSE),反映了除了x对y的线性影响之外的其他因素对y变差的作用,是不能由回归直线来解释的y的变差部分 + +**判定系数** + +回归平方和占总平方和的比例,用R^2表示,其值在0到1之间 + +- R^2 == 0:说明y的变好与x无关,x完全无助于解释y的变差 +- R^2 == 1:说明残差平方和为0,拟合是完全的,y的变化只与x有关 + +![1604845790717](assets/1604845790717.png) + +![1604845818876](assets/1604845818876.png) + +**显著性校验** + +著性检验的主要目的是根据所建立的估计方程用自变量x来估计或预测因变量y的取值。当建立了估计方程后,还不能马上进行估计或预测,因为该估计方程是根据样本数据得到的,它是否真实的反映了变量x和y之间的关系,则需要通过检验后才能证实根。 + +据样本数据拟合回归方程时,实际上就已经假定变量x与y之间存在着线性关系,并假定误差项是一个服从正态分布的随机变量,且具有相同的方差。但这些假设是否成立需要检验 + +显著性检验包括两方面: + +- 线性关系检验 +- 回归系数检验 + +**线性关系检验** + +线性关系检验是检验自变量x和因变量y之间的线性关系是否显著,或者说,它们之间能否用一个线性模型来表示。 + +将均方回归(MsR)同均方残差(MsE)加以比较,应用F检验来分析二者之间的差别是否显著。 + +- 均方回归:回归平方和(SSR)除以相应的自由度(自变量的个数K) +- 均方残差:残差平方和(SSE)除以相应的自由度(n-k-1) + +H0:β1 = 0所有回归系数与零无显著差异,y与全体x的线性关系不显著 + +计算检验统计量F: + +![1604846120803](assets/1604846120803.png) + +**回归系数的显著性检验** + +回归系数显著性检验的目的是通过检验回归系数β的值与0是否有显著性差异,来判断Y与X之间是否有显著的线性关系若β=0,则总体回归方程中不含X项(即Y不随X变动而变动),因此,变量Y与X之间并不存在线性关系;若β≠0,说明变量Y与X之间存在显著的线性关系 + +![1604846273414](assets/1604846273414.png)是根据最小二乘法求出的样本统计量,服从正态分布;![1604846356404](assets/1604846356404.png) + +![1604846273414](assets/1604846273414.png)的分布具有如下性质数学期望E(![1604846273414](assets/1604846273414.png))=![1604846273414](assets/1604846273414.png) + +标准差:![1604846346057](assets/1604846346057.png) + +由于δ未知,需用其估计量Se来代替得到![1604846273414](assets/1604846273414.png)的估计标准差 + +![1604846442945](assets/1604846442945.png) + +计算检验的统计量:![1604846499957](assets/1604846499957.png) + +**线性关系检验与回归系数检验的区别**: + +线性关系的检验是检验自变量与因变量是否可以用线性来表达,而回归系数的检验是对样本数据计算的回归系数检验总体中回归系数是否为0 + +- 在一元线性回归中,自变量只有一个,线性关系检验与回归系数检验是等价的 +- 多元回归分析中,这两种检验的意义是不同的。线性关系检验只能用来检验总体回归关系的显著性,而回归系数检验可以对各个回归系数分别进行检验 + +**多元线性回归分析** + +经常会遇到某一现象的发展和变化取决于几个影响因素的情况,也就是一个因变量和几个自变量有依存关系的情况,这时需用多元线性回归分析。 + +- 多远线性回归分析预测法,是指通过对两上或两个以上的自变量与变量的相关分析,建立预测模型进行预测和控制的方法 + +- 多元线性回归预测模型一般式为 + +![1604846770738](assets/1604846770738.png) +