1、Simple linear regression,一、Concept and maths model： The simplest functional relationship of one variable to another in a population is the simple linear regression. X and Y model y=a+bx,independent variable,dependent variable,x,y,regression coefficient or slope,b,intercept,a,二、Basis thought 尋找一條最佳直

2、線反映x與y的線性依 存規(guī)律。,y x,Simple linear regression,Scatter plot,所求直線使得理論 值與實(shí)際觀察值最接近 用直線去描述這些觀察 點(diǎn)時(shí)，使得error Least !,三、 Basis theory 1.data format i x y 1 x1 y1 2 x2 y2 . n xn yn,Simple linear regression,2.error question 第i個(gè)個(gè)體的誤差為： i =(a+bx i )-yi 所有n 個(gè)個(gè)體的誤差為： |i | i2=I 從而，所求直線 即使得I 為最小,b=Lxy /Lxx a=y-bx 求最小

3、值點(diǎn),Simple linear regression,Lxx -Recall the sum of squares of deviations from the mean,Lxy -Recall the sum of the cross products of deviations from the mean,Simple linear regression,三、 Basis method： method of least squares(最小二乘法)，即 在線性回歸分析中，估計(jì)模型的參數(shù)時(shí)，保證 the error sum of squares最小的原則。,Simple linear re

4、gression,四、Testing the Significance of a regression,1. t-Testing ： tb=(b-0)/sb 其中，sb2=ss剩/(（n-2）Lxx)，v=n-2 residual ss剩= Lyy- b Lxy 2.coefficient of determination R2= regression ss回/total ss總 =b Lxy/Lyy 3. F-Testing ： F= MS回/MS剩,五、Confidence intervals in regression 1. Confidence intervals for an Estimated y(預(yù)測(cè))： 已知自變量x去估計(jì)反應(yīng)變量y。 (1) 的估計(jì)(條件平均值的100(1)%可信區(qū) 間 ):,Simple linear regression,(2)個(gè)體Y值的100(1)%容許區(qū)間:,Simple linear regression,2. prediction interval(預(yù)報(bào))： 已知反應(yīng)變量y去估計(jì)自變量x。,Simple linear regression,Simple Linear Correlation,一、The correlation co