JHU Coursera Regression Model Quizes

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JHU Coursera Regression Model Quizes.

JHU Coursera Regression Model Quizes 回归模型问题集

JHU DataScience Specialization/Cousers Reproducible Data/Week1-4/Regression Model Quizes

主要练习手工计算回归模型的基础方法

Week 2

Quiz 1

手算均值

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x <- c(0.18, -1.54, 0.42, 0.95)
w <- c(2, 1, 3, 1)
mu.y <- sum(w * x) / sum(w)
sprintf("mean of y is : %f",mu.y)
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## [1] "mean of y is : 0.147143"

Quiz 2

线性回归

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x <- c(0.8, 0.47, 0.51, 0.73, 0.36, 0.58, 0.57, 0.85, 0.44, 0.42)
y <- c(1.39, 0.72, 1.55, 0.48, 1.19, -1.59, 1.23, -0.65, 1.49, 0.05)
pander(lm(y~x)) #THROUGH THE ORIGIN
Fitting linear model: y ~ x
  Estimate Std. Error t value Pr(>
(Intercept) 1.567 1.252 1.252 0.246
x -1.713 2.105 -0.8136 0.4394 
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pander(lm(y~x-1)) #去除截距
Fitting linear model: y ~ x - 1
  Estimate Std. Error t value Pr(>
x 0.8263 0.5817 1.421 0.1892

Quiz 3

mtcars 回归系数

(Intercept) wt
37.29 -5.344

Quiz 4

练习求b1

\[\begin{align} Cor(Y,X) &= 0.5 \qquad Sd(Y) = 1 \qquad Sd(X) = 0.5 \\ \beta_1 &= Cor(Y,X) * \frac{Sd(Y)}{Sd(X)} \end{align}\]

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B1 = 0.5 * 1 / 0.5

Quiz 5

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corr <- .4; emean <- 0; varr1 <- 1
varr2 <- 1; b0 <- 0; x <- 1.5
b1 <- corr * sqrt(varr1) / sqrt(varr2)
(y <- b0 + b1 * x)
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## [1] 0.6

Quiz 6

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x <- c(8.58, 10.46, 9.01, 9.64, 8.86)
(x - mean(x)) / sd(x) # Choose No.1
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## [1] -0.9718658  1.5310215 -0.3993969  0.4393366 -0.5990954

Quiz 7

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x <- c(0.8, 0.47, 0.51, 0.73, 0.36, 0.58, 0.57, 0.85, 0.44, 0.42)
y <- c(1.39, 0.72, 1.55, 0.48, 1.19, -1.59, 1.23, -0.65, 1.49, 0.05)
pander(lm(y~x))
Fitting linear model: y ~ x
  Estimate Std. Error t value Pr(>
(Intercept) 1.567 1.252 1.252 0.246
x -1.713 2.105 -0.8136 0.4394

Quiz 8

It must be identically 0.

Quiz 9

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x <- c(0.8, 0.47, 0.51, 0.73, 0.36, 0.58, 0.57, 0.85, 0.44, 0.42)
mean(x)
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## [1] 0.573

Quiz 10

\[\begin{align} \beta_1 &= Cor(Y,X)*Sd(Y)/Sd(X) \\ Y_1 &= Cor(Y,X)*Sd(X)/Sd(Y) \\ \beta_1/Y_1 &= Sd(Y)^2/Sd(X)^2 \notag \\ &= Var(Y)/Var(X) \end{align}\]

Week 3

Quiz 1

求系数

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x <- c(0.61, 0.93, 0.83, 0.35, 0.54, 0.16, 0.91, 0.62, 0.62)
y <- c(0.67, 0.84, 0.6, 0.18, 0.85, 0.47, 1.1, 0.65, 0.36)
fit <- lm(y~x)
pander(fit)
Fitting linear model: y ~ x
  Estimate Std. Error t value Pr(>
(Intercept) 0.1885 0.2061 0.9143 0.391
x 0.7224 0.3107 2.325 0.05296

Quiz 2

总平方和与回归与残差关系

\[\begin{align} y_i &= \beta_0 + \beta_1 x_i + e_i \\ \hat y_i &= \beta_0 + \beta_1 x_i\\ e_i &= y_i - \hat y_i \\ SS_{total} &= \|y_i-\bar y \mathbf{1}\|^2 = \| \hat y_i-\bar y \mathbf{1}\|^2 + \|\hat \epsilon\| \notag \\ &= \sum_{i=1}^n (y_i-\bar y)^2 = \sum_{i=1}^n (\hat y_i-\bar y)^2 + \sum_{i=1}^n (y_i-\hat y)^2 \notag \\ &= SS_{regression} + SS_{residual} \\ \mathbf{1} &= (1,1,\ldots,1)^T \end{align}\]

平方和与残差

\[\begin{align} SS_x &= \sum_{i=1}^n {(x_i - \bar x)^2} \\ e_i &= y_i - (\beta_1x_i + \beta_0) \\ \hat \sigma^2 &= \frac{1}{n-2} \sum_{i=1}^n e_i^2 \\ &= \frac{1}{n-2} SS_{residual} \sim \chi_{(n-2)}^2 \end{align}\]

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# init
n <- length(x)
beta1 <- cor(y, x) * sd(y) / sd(x)
beta0 <- mean(y) - beta1 * mean(x)
ssx <- sum((x - mean(x))^2)
e <- y - (beta1 * x + beta0)
sigma <- sqrt(sum(e^2) / (n-2))

方差与系数方差关系

\[\begin{align} \sigma^2_{\hat\beta_1}&=Var(\hat\beta_1)=\frac{\sigma^2}{SS_x}\\ \sigma^2_{\hat\beta_0}&=Var(\hat\beta_0)=(\frac{1}{n}+\frac{\bar X^2}{SS_x})\sigma^2 \end{align}\]

\[\begin{align} R^2 = \frac{SS_{regression}}{SS_{total}} = \frac{\sum_{i=1}^n (\hat y_i-\bar y)^2}{\sum_{i=1}^n (y_i-\bar y)^2} \end{align}\]

t分布与方差

\[\begin{align} \frac{\hat\beta_j-\beta_j}{\sigma_{\hat\beta_j}} \sim t_{\beta_j}(n-p) \end{align}\]

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# 计算 Beta1
seBeta0 <- sqrt(1/n + mean(x)^2/ssx)*sigma
seBeta1 <- sigma / sqrt(ssx)
tBeta0 <- beta0 / seBeta0
tBeta1 <- beta1 / seBeta1
pBeta0 <- 2 * pt(tBeta0, df = n-2, lower.tail = FALSE)
pBeta1 <- 2 * pt(tBeta1, df = n-2, lower.tail = FALSE)
  Estimate Std. Error t value P(>
(Intercept) 0.1885 0.2061 0.9143 0.391
x 0.7224 0.3107 2.325 0.05296

Quiz 3

Dataset: mtcars

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# 构造模型
fit <- lm(mpg~wt,mtcars)
pander(summary(fit))
  Estimate Std. Error t value Pr(>
(Intercept) 37.29 1.878 19.86 8.242e-19
wt -5.344 0.5591 -9.559 1.294e-10
Fitting linear model: mpg ~ wt
Observations Residual Std. Error \(R^2\) Adjusted \(R^2\)
32 3.046 0.7528 0.7446 
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sumCoef <- summary(fit)$coefficients
b0 <- sumCoef[1,1]
b1 <- sumCoef[2,1]
seb0 <- sumCoef[1, 2]
seb1 <- sumCoef[2, 2]
b0i <- b0 + c(-1, 1) * qt(.9, df = fit$df) * seb0
b1i <- b1 + c(-1, 1) * qt(.9, df = fit$df) * seb1 
x0 <- mean(mtcars$wt)
  fit lwr upr
confidence 20.09 18.99 21.19
prediction 20.09 13.77 26.41
manual 20.09 15.27 24.91

BBBBB

Quiz 4

The estimated expected change in mpg per 1,000 lb increase in weight.

Quiz 5

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pander(predict(fit,newdata = data.frame(wt=3000/1000),interval = ("prediction")))
fit lwr upr
21.25 14.93 27.57

Quiz 6

fit lwr upr
20.09 12.58 27.6
fit lwr upr
20.09 18.99 21.19
fit lwr upr
20.09 13.77 26.41

Quiz 7

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fit3<-lm(mpg~I(wt/100),mtcars)
pander(summary(fit3)$coefficients)
  Estimate Std. Error t value Pr(>
(Intercept) 37.29 1.878 19.86 8.242e-19
I(wt/100) -534.4 55.91 -9.559 1.294e-10 
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#It would get multiplied by 100.

Quiz 8

The new intercept would be bhat0???cbhat1

Quiz 9

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fit1 <- lm(mpg ~ wt, data = mtcars)
fit2 <- lm(mpg ~ 1, data = mtcars)
num <- sum((predict(fit1)-mtcars$mpg)^2)
den <- sum((predict(fit2)-mtcars$mpg)^2)
num/den
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## [1] 0.2471672
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1 - summary(fit1)$r.squared#options
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## [1] 0.2471672

Quiz 10

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x <- c(0.61, 0.93, 0.83, 0.35, 0.54, 0.16, 0.91, 0.62, 0.62)
y <- c(0.67, 0.84, 0.6, 0.18, 0.85, 0.47, 1.1, 0.65, 0.36)
sum(resid(lm(y ~ x)))
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## [1] 1.110223e-16
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sum(resid(lm(y ~ 1)))
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## [1] -3.816392e-17
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sum(resid(lm(y ~ x - 1)))
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## [1] 0.220565
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#If an intercept is included, then they will sum to 0.

Week 4

Quiz 1

  Estimate Std. Error t value Pr(>
(Intercept) 29.74 2.541 11.7 2.688e-12
I(1 * (cyl == 8)) -1.815 1.358 -1.337 0.192
I(1 * (cyl == 4)) 4.256 1.386 3.07 0.004718
wt -3.206 0.7539 -4.252 0.000213 
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## [1] -6.07086

Quiz 2

  Estimate Std. Error t value Pr(>
(Intercept) 19.74 1.218 16.21 4.493e-16
I(1 * (cyl == 8)) -4.643 1.492 -3.112 0.004152
I(1 * (cyl == 4)) 6.921 1.558 4.441 0.0001195

Quiz 3

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## 
## Call:
## lm(formula = mpg ~ factor(cyl) + wt, data = mtcars)
## 
## Coefficients:
##  (Intercept)  factor(cyl)6  factor(cyl)8            wt  
##       33.991        -4.256        -6.071        -3.206
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## 
## Call:
## lm(formula = mpg ~ factor(cyl), data = mtcars)
## 
## Coefficients:
##  (Intercept)  factor(cyl)6  factor(cyl)8  
##       26.664        -6.921       -11.564

Quiz 4

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##               Estimate Std. Error   t value     Pr(>|t|)
## (Intercept)  33.990794   1.887793 18.005569 6.257246e-17
## I(wt * 0.5)  -6.411227   1.507791 -4.252065 2.130435e-04
## factor(cyl)6 -4.255582   1.386073 -3.070244 4.717834e-03
## factor(cyl)8 -6.070860   1.652288 -3.674214 9.991893e-04

函数step可以实现基于AIC准则的模型选择，最优模型为AIC值最小的

Fitting linear model: mpg ~ wt + qsec + am
  Estimate Std. Error t value Pr(>
(Intercept) 9.618 6.96 1.382 0.1779
wt -3.917 0.7112 -5.507 6.953e-06
qsec 1.226 0.2887 4.247 0.0002162
am 2.936 1.411 2.081 0.04672

Quiz 5

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x <- c(0.586, 0.166, -0.042, -0.614, 11.72)
y <- c(0.549, -0.026, -0.127, -0.751, 1.344)
fit <- lm(y ~ x)
round(hatvalues(fit),4)
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##      1      2      3      4      5 
## 0.2287 0.2438 0.2525 0.2804 0.9946

Quiz 6

  • 异常值

  • 杠杆点

  • 杠杆点对回归系数没有影响，但是会影响决定系数，可以通过观察帽子矩阵来识别，杠杆作用的平均值为h=p/n，p为自变量个数，n为样本量，如果一个观测值的杠杆值>2h，则应考虑为杠杆点，考虑剔除或采取措施。R中可以通过hatvalues函数计算杠杆值。

  • 影响点

  • 影响点有将回归线拉向它的趋势，因此会影响回归系数的值，可以通过COOK距离来判断，R中可以通过cooks.distance函数计算每个观测值的COOK距离。根据经验，如果距离大于1，则说明观测点为影响点。

  • 此外，还可以通过dffits函数和dfbetas函数计算相应的值，如果dffits>2/根号p/n，那么可认为是影响点，如果dfbetas>2/根号n，也可认为是影响点，p为自变量个数，n为样本量

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x <- c(0.586, 0.166, -0.042, -0.614, 11.72)
y <- c(0.549, -0.026, -0.127, -0.751, 1.344)
fit <- lm(y ~ x)
round(dfbetas(fit),4)
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##   (Intercept)         x
## 1      1.0621   -0.3781
## 2      0.0675   -0.0286
## 3     -0.0174    0.0079
## 4     -1.2496    0.6725
## 5      0.2043 -133.8226
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round(hatvalues(fit),4)
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##      1      2      3      4      5 
## 0.2287 0.2438 0.2525 0.2804 0.9946
THE END