Simple Linear Regression and CorrelationPPT

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Linear Regression and Correlation

It is the square of the coefficient of correlation.
It ranges from 0 to 1. It does not give any information on the
direction of the relationship between the variables.
It can range from -1.00 to 1.00. Values of -1.00 or 1.00 indicate perfect and
strong correlation. Values close to 0.0 indicate weak correlation. Negative values indicate an inverse
relationship and positive values indicate a direct relationship.
13-9
Perfect Correlation
13-10
Minitab Scatter Plots
13-11
Correlation Coefficient - Interpretation
intervals for the dependent variable.
13-2
Regression Analysis - Introduction
Recall in Chapter 4 the idea of showing the relationship between two variables with a scatter diagram was introduced.
13-12
Hale Waihona Puke Correlation Coefficient - Formula

学习笔记：伍德里奇《计量经济学》第五版-第二章简单回归模型

~除了x 以外影响y 的因素？~y 和x 的函数关系？~何以确定在其他条件不变的情况下刻画了y 和x 的关系由以上得简单线性模型（simple linear regression model ）：y = b0+ b1x + u （2.1）y ：因变量x ：自变量u ：误差项（干扰项），即“观测不到的”因素（该模型没有限制x 和u 的关系，因此不能说明x 对y 的影响2.4节是如何解决x 的初始值不同时，同样变化量对y 的影响的？E(u) = 0 (2.5)（代价：方程中要包含截距b0 因为这样可以通过微调截距项来使第一个假定一定成立对u 做的第一个假定：E(u|x) = E(u)(2.6)（前提：u 和x 是随机变量均值独立假定（任何给定x 下u 的平均值都一样）：E(u|x)= 0 （2.7）结合均值独立与均值为0，得零条件期望假定：E(y|x) = b0 + b1x （2.8）（E(y|x)称为总体回归函数（population regression function ，PRF ），说明了y 的均值是如何随着x 的变动而变动的结合方程（2.1）和假定（2.7）得条件均值函数：一、y 和x关系的起点随机变量：具有数值特征并由一个实验决定其结果的变量•（是为了解决协方差受度量单位影响的问题，是协方差的改进）（u 和x 不相关，u 也能和x ²相关，对于大部分回归不行）相关系数（仅衡量线性相关程度）：•yi = b0 + b1xi + ui （2.9)抽取一个容量为n 的随机样本E(u)=0 （2.10）利用Cov(x,u)=E(xu)=0 (2.11)和假定（2.6）得：E(y –b0 –b1x) = 0 （2.12）E[x(y –b0 –b1x)] = 0 （2.13）因此方程（2.10）和（2.11）可写为在样本中就对应和（2.14）（2.15）结合（2.9）的均值形式（2.16）可以解出参变量（实际上就是矩法估计）（）（前提：分母大于0，即样本中所有x 不完全相等（含义：若样本中x 和y 正相关，则斜率系数为正二、普通最小二乘法（如何估计参变量）协方差：•不相关和协方差=0可互推，但不一定独立，独立一定不相关•矩法估计：利用要估计的参数与某种均值的关系，用样本矩代替总体矩u 的解法。

各种线性回归模型原理

各种线性回归模型原理线性回归是一种广泛应用于统计学和机器学习领域的方法，用于建立自变量和因变量之间线性关系的模型。

在这里，我将介绍一些常见的线性回归模型及其原理。

1. 简单线性回归模型（Simple Linear Regression）简单线性回归模型是最简单的线性回归模型，用来描述一个自变量和一个因变量之间的线性关系。

模型方程为：Y=α+βX+ε其中，Y是因变量，X是自变量，α是截距，β是斜率，ε是误差。

模型的目标是找到最优的α和β，使得模型的残差平方和最小。

这可以通过最小二乘法来实现，即求解最小化残差平方和的估计值。

2. 多元线性回归模型（Multiple Linear Regression）多元线性回归模型是简单线性回归模型的扩展，用来描述多个自变量和一个因变量之间的线性关系。

模型方程为：Y=α+β1X1+β2X2+...+βnXn+ε其中，Y是因变量，X1,X2,...,Xn是自变量，α是截距，β1,β2,...,βn是自变量的系数，ε是误差。

多元线性回归模型的参数估计同样可以通过最小二乘法来实现，找到使残差平方和最小的系数估计值。

3. 岭回归（Ridge Regression）岭回归是一种用于处理多重共线性问题的线性回归方法。

在多元线性回归中，如果自变量之间存在高度相关性，会导致参数估计不稳定性。

岭回归加入一个正则化项，通过调节正则化参数λ来调整模型的复杂度，从而降低模型的过拟合风险。

模型方程为：Y=α+β1X1+β2X2+...+βnXn+ε+λ∑βi^2其中，λ是正则化参数，∑βi^2是所有参数的平方和。

岭回归通过最小化残差平方和和正则化项之和来估计参数。

当λ=0时，岭回归变为多元线性回归，当λ→∞时，参数估计值将趋近于0。

4. Lasso回归（Lasso Regression）Lasso回归是另一种用于处理多重共线性问题的线性回归方法，与岭回归不同的是，Lasso回归使用L1正则化，可以使得一些参数估计为0，从而实现特征选择。

7_simple linear regression and correlation

8
Least squares
Least square
2 2 2 2 2 ˆ ˆ ˆ ˆ ˆ LS minimizes i 1 2 3 4 i 1 n
Y
^ 2 ^ 1
ˆX ˆ ˆ2 Y2 2
^ 4 ^ 3
ˆX ˆi ˆ ˆi Y i
Simple linear regression and correlation
1
Regression overview
Simple linear regression
Y X
2
Regression overview
Brief history y
• Auguste Bravais -- Correlation in 1846 • Sir Francis Galton developed the procedure d of f regression i and d correlation l ti during 1875-1885 • Karl Pearson – correlation coefficient in 1895
Not!
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Assumptions
Actual relationship is linear
Not!
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Assumptions
L Log-transforming f i the h data d may improve i the h situation i i
250 200 150 100 50 0
5
Regression overview
Data
BMI (Kg/m2) 20 30 50 45 10 30 40 25 50 20 10 55 60 50 35 Birth-weight (Kg) 2.7 2 7 2.9 3.4 3.0 2.2 31 3.1 3.3 2.3 3.5 2.5 1.5 3.8 37 3.7 3.1 2.8

回归分析

Regression Analysis 回归分析
y

x
5
Regression Analysis
变量间的关系
（函数关系）
函数关系的例子
回归分析
某种商品的销售额 (y) 与销售量 (x) 之间的关系可表示为 y = p x (p 为单价) 圆的面积(S)与半径之间的关系可表示为S = r2
样本相关系数的定义公式是：
r
( X X )(Y Y ) ( X X ) (Y Y )
t t 2 t t
2
上式中， X 和 Y 分别是Ｘ和Ｙ的样本平均数。样本相关系数是根据样本观测值计算的，抽取的样本不同，其具体的数值也会有所差异。容易证明，样本相关系数是总体相关系数的一致估计量。
r的取值相关程度
|r|＜0.3 不线性相关
0.3≤|r|<0.5 0.5≤|r|<0.8
|r|≥0.8
低度线性相中度线性相高度线性关关相关
23
Regression Analysis 回归分析
•
3.如果|ｒ|=1，则表明Ｘ与Ｙ完全线性相关，当ｒ=1时，称为完全正相关，而ｒ=-1时，称为完全负相关。
相关分析（Correlation Analysis）是用于度量两个
数值变量间的关联程度
3
Regression Analysis 回归分析
一、函数关系与相关关系
1.函数关系
当一个或几个变量取一定的值时，另一个变量有确定值与之相对应，我们称这种关系为确定性的函数关系。
4
（函数关系）
（1）是一一对应的确定关系（2）设有两个变量 x 和 y ，变量 y 随变量 x 一起变化，并完全依赖于 x ，当变量 x 取某个数值时， y 依确定的关系取相应的值，则称 y 是 x 的函数，记为 y = f (x)，其中 x 称为自变量，y 称为因变量（3）各观测点落在一条线上

Lecture 8 Simple Linear Regression (W)

26
13
• The mean response μ y has a straight-line
relationship with x:
μy =α + βx
The slope β and intercept α are unknown
parameters.
• The standard deviation of y (call it σ ) is the same for all values of x. The value of σ
10
• Plot and interpret. As always, we first examine the data. Figure 3 is a scatterplot of the crying data. Plot the explanatory variable (crying intensity at birth) horizontally and the response variable (IQ at age 3) vertically. Look for the form, direction, and strength of the relationship as well as for outliers or other deviations. There is a moderate positive linear relationship, with no extreme outliers or potentially influential observations.
18
9
Example 2: Crying and IQ
• Infants who cry easily may be a sign of higher IQ. Crying intensity and IQ data on 38 infants:

SIMPLE lINEAR rEGRESSION英文版完整六西格玛方案

function).
Y = f(X1, X2, X3, etc.)
response variable predictor variables
Sounds like correlation.. DOE..
Overview of Regression
Correlation vs Regression
Regression and correlation are closely related and
• Common Regression Mistakes
Overview of Regression
General Use of Regression Analysis
A primary goal in many studies is to understand and quantify relationships between variables. Regression analysis is used when…
R = 0.9
R = 0.1 R = -0.7
Overview of Regression
Correlation vs Regression
Brief Review of Correlation
R =
S XY S XX S YY
S XY =
S XX =
SYY =
i=1 n
i=1

n
i=1 n
Simple Linear Regression
Learning Objectives
• Overview of Regression
Correlation vs Regression
• Linear Regression

英文文献回归模型r语言

英文文献回归模型r语言回归模型在统计学和机器学习中被广泛应用，而R语言作为一种流行的统计分析工具，也被用于实现各种回归模型。

在英文文献中，关于回归模型和R语言的结合有很多相关的研究和资料。

这些文献涵盖了从基础到高级的各种回归模型在R语言中的实现和应用。

首先，让我们从基础开始。

有一些文献专门介绍了如何在R语言中实现简单线性回归（simple linear regression）和多元线性回归（multiple linear regression）。

这些文献通常会讲解如何使用R中的lm()函数来拟合回归模型，以及如何解释和评估模型的结果。

一些经典的参考书籍如《An Introduction to Statistical Learning》和《Applied Regression Analysis》提供了丰富的案例和代码，可以帮助读者深入理解回归模型在R中的实现。

其次，针对特定领域的研究，有许多文献探讨了高级的回归模型在R语言中的应用。

比如，关于时间序列分析的文献会介绍如何使用R中的arima()函数来构建自回归（autoregressive）、移动平均（moving average）和ARIMA模型。

另外，关于广义线性模型（generalized linear model）和混合效应模型（mixed effects model）的文献也有很多，这些模型在R语言中有丰富的包和函数来支持。

此外，还有一些文献专门讨论了回归诊断（regression diagnostics）和模型选择（model selection）在R语言中的实现。

这些内容涉及到如何检验回归模型的假设、识别异常值和影响点，以及利用交叉验证等方法选择最佳的模型。

最后，关于回归模型和R语言的文献还包括了一些实际案例和研究论文，这些文献通过具体的数据集和分析过程展示了回归模型在R中的应用。

这些案例可以帮助读者更好地理解如何将理论知识转化为实际研究中的解决方案。

Simple

Simple Linear Regression 笔记Simple Linear RegressionCreating the Regression LineCalculating b1 & b0, creating the lineand testing its significance with a t-test.DEFINITIONS:b1 - This is the SLOPE of the regression line. Thus this is the amount that the Y variable (dependent) will change for each 1 unit change in the X variable.b0 - This is the intercept of the regression line with the y-axis. In otherwords it is the value of Y if the value of X = 0.Y-hat = b0 + b1(x) - This is the sample regression line. You must calculate b0 & b1 to create this line. Y-hat stands for the predicted value of Y, and it can be obtained by plugging an individual value of x into the equation and calculating y-hat. EXAMPLE:A firm wants to see if there is sales is explained by the number of hours overtime that their salespeople work. Using a spreadsheet containing 25 months of sales & overtime figures, the following calculations are made; SSx = 85, SSy = 997 and SSxy = 2,765, X-bar = 13 and Y-bar = 67,987, also s(b1) = 21.87. Create the regression line.(1) find b1 - One method of caluating b1 is b1 = SSxy/SSx = 2765/85 = 32.53. This is the slope of the line - for every unit change in X, y will increase by 32.53. It is a positive number, thus its a direct relationship - as X goes up, so does Y. However, if b1 = -32.53, then we would know the relationship between X & Y is an inverse relationship - as X goes up, y goes down)(2) find b0 - again the formula is on pg. 420 and is b0 = Y-bar - b1(x-bar) = 67,987 - 32.53(13) = 67,987 - 422.89 = 67,564, this is the intercept of the line and the Y-axis, and can be interpreted as the value of Y if zero hours of overtime (x=0) are worked.(3) Create Line - Y-hat = b0 + b1(x) or Y-hat = 67,564 + 32.53(x), This line quantifies the relationship between X & Y. Under the normal error model, b1 is unbiased for Beta1 with:Sy/x is the residual standard deviation:But is this Relationship "Significant"Since it is based on a sample and we wish to generalize to a population, it must be tested to see if it is "significant," meaning would the relationship we found actually exist in the population or is the result due to sampling error (our sample did not represent the true population). The specific test we use is a t-test to test to see if b1 is different from 0. Since B1 would be the slope of the regression line in the population, it makes sense to test to see if it is different from zero. If it is zero, then our slope is 0, meaning if we graphed the relationship between X & Y we would end up with a horizontal (flat) line. And if this line is flat then we know that no matter what value the X variable takes on, the Y variable's value will not change. This means there is no linear relationship between the two variables. This also means that the regression line we calculated is useless for explaining or predicting the dependent variable.TESTING B1 We use our standard five step hypothesis testing procedure.Hypotheses: H0: B1 = 0, H1: B1 not = 0Critical value: a t-value based on n-2 degrees of freedom. Also divide alpha by 2 because it is a 2-tailed test. In this case n = 25 (25 months data used) thus n-2 = 23. With alpha = .05 we have alpha/2 = .025 and then t = 2.069 (from t-table inside front cover of book).Calculated Value: The formula is on page 442 and is simply t = b1/s(b1) = 32.53/21.87 = 1.49. s(b1) is the standard error of b1 and is given in the problem)Compare: t-calc < t-crit and thus accept H0.Conclusion: B1 = 0, the population slope of our regression is a flat line, thus there is no linear relationship between sales and overtime worked, and the sample regression line we calculated is not useful for explaining or predicting sales dollars from overtime worked.CorrelationCorrelation is a measure of the degree of linear association between two variables. The value of a correlation can range from -1, thru 0, to +1. A correlation = 0 means there is no LINEAR association between the two variables, a value of -1 or +1 means there is a perfect linear association between the two variables, the difference being that -1 indicates a perfect inverse relationship and +1 a perfect positive relationship. The sample notation for a correlation is "r" while the population correlation coefficient is represented by the greek letter "Rho" (which looks like a small "p").We often want to find out if a calculated sample correlation would be "significant." Again this would mean we would test to see if Rho = 0 or not. If Rho=0 then there would be no linear relationship between the two variables in the population.AN EXAMPLE:Based on a sample of 42 days, the correlation between sales and number of sunny hours in the day is calculated for the Sunglass Hut store in Meridian Mall. The r = .56. Is this a "significant" correlation?This is a basic hypothesis test.....Hypotheses: H0: Rho = 0, H1: Rho not = 0.Critical Value: The t-test for the significance of Rho has n-2 degrees of freedom, and alpha will need to be divided by 2, thus n-2 = 40 and alpha (.05/2) = .025 ... from the table we find: 2.021.Calculated Value: The formula on page 438 is t = r / sqr root of (1-r-sqrd)/(n-2). In this case that equals .56 / the square root of (1-.56-squared)/(40) = .56/.131 = 4.27Compare: The t-calc is larger than the t-crit thus we REJECT Ho.Conclusion: Rho does not equal zero and thus there is evidence of a linear association between the two variables in the population.The F-test in RegressionEXAMPLEUsing the information given, construct the ANOVA table and determine whether there is a regression relationship between years of car ownership (Y) and salary (X). n= 47, SSR = 458 and SSE = 1281.ANOVA Table: The anova table is on page 451, and is basically the same as a one-way ANOVA table. The first thing we need is the df, and by definition the df for the regression = 1, the df for the error = n-2 or 45, and the total df = n-1 or 46. Next we need the MS calculations. MSR = SSR/df for the regression = SSR/1 = SSR or 458. MSE = SSE/n-2 = 1281/45 = 28.47. Finally, the F-calc = MSR/MSE or 458/28.47 = 16.09.Hypotheses: H0: There is no regression relationship,i.e, B1 =0. H1: There is a regression relationship, i.e, B1 is not = 0. Critical Value: F(num. df, den. df) = F(1, 45) at alpha = .05 = 4.08Calculated Value: from above ANOVA table = 16.09Compare: F-calc larger than F-crit thus REJECTConclusion: There is a regression (linear) relationship between years of car ownership and salary.The Coefficient of Determination - r-sqrdWe can also test the significance of the regression coefficient using an F-test. Since we only have one coefficient in simple linear regression, this test is analagous to the t-test. However, when we proceed to multiple regression, the F-test will be atest of ALL of the regression coefficients jointly being 0. (Note: b0 is not a coefficient and we generally do not test its significance although we could do so with a t-test just as we did ofr b1.r-sqrd is always a number between 0 and 1. The closer it is to 1.0 the better the X-Y relationship predicts or explains the variance in Y. Unfortunately there are no set values that allow you to say that is a "good" r-sqrd or "bad" r-sqrd. Such a determination is subjective and is determined by the research you are conducting. If nobody has ever explained more that 15% of the variance in some Y variable before, and you design a study that explains 25% of variance, then this might be considered good r-sqrd, even though the actual number, 25%, is not very high.EXAMPLE:What is the r-sqrd if SSR = 345 and SSE = 123?r-sqrd = SSR/SST. We don't have SST, but we know that SSR + SSE = SST, thus SST = 345 + 123 = 468, thus r-sqrd = 345/468 = .737. This means that the regression relationship between X & Y explains 73.7% of the variance in the Y variable. Under most circumstances this would be a high amount, but again we would have to know more about our research varaibles.。

【源版】医学统计学第十四章--直线回归与相关(课堂)同

第九章双变量回归与相关 simple linear regression
and correlation
2019/11/12
回归分析与相关分析
变量间关系：年龄～身高、肺活量～体重、药物剂量与动物死亡率、体温与脉搏、糖尿病人的血糖和胰岛素水平、年龄与血压等。
一.散点图：为了考察随机变量X与Y的
寻找使S(残差i)2 最小的直线
Yi （Y的估计值） = a + bXi
估计值 Yˆi
Yi
Yˆ 残差i = Yi – 估计值 i
n
n
Q S(Y Yˆ)2 (Yi Yˆi )2 Yi (a bXi 2
i 1
i 1
b

( X X )(Y Y ) (X X )2
Yˆ 1.6671 0.1392 X
回归参数a、b的解释
1. 斜率 (b)
当X每增加1个单位时， Y改变 b个单位
本例b = 0.1392，表明在所研究的年龄范围内，年龄每增加1岁，尿肌酐含量平均增加0.1392
mmol/24h
2. Y的截距 (a) X = 0时Y的平均值本例a＝1.6617，表示年龄为0时，尿肌酐含量的均值为1.6617mmol/24h（注意有时这种解释无实际意义）
Y的离均差平方和的分解
Y Y （Y Yˆ）（Yˆ Y）
可由数学证明：
（Y Y）2 （Yˆ Y）2 （Y Yˆ）2
即 SS总 SS回 SS残
V总 V回 V残
几个平方和的意义
SS总＝ (Y Y )2 ， Y的离均差平方和(total
sum of squares)，未考虑X 与Y 的回归关系时Y 的总变异。