![Page 1: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/1.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Statistics, Visualization and More Using ”R” (298.916)
Block I+II: Distributions & simulations, loops and functions, linearregression
Ass.-Prof. Dr. Wolfgang Trutschnig
Research group for Stochastics/StatisticsDepartment for Mathematics
University Salzburg
www.trutschnig.net
Salzburg, March 2017
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 2: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/2.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Plan for today:
I Some standard probability distributions (uniform, exponential, normaldistribution) which will be used throughout the seminar
I Generating samples from these distributions
I Loops and if/ifelse
I Writing own R-functions
I Pearson vs. Spearman (rank) correlation
I First steps linear regression
I Exercises
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 3: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/3.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Uniform distribution U(a, b)
Uniform distribution U(a, b)
I Range: Interval [a, b] (with a < b).
I The density f is given by
f (x) =1
b − a1[a,b](x).
I For X ∼ U(a, b) we have
E(X ) =a + b
2, V(X ) =
(b − a)2
12.
I Where does this distributionnaturally appear?
I Generate a sample of sizen = 10.000.
1 n <− 100002 x <− r u n i f ( n , min=−1,max=1)3 h i s t ( x , p r o b a b i l i t y = TRUE)
Den
sity
−1.0 −0.5 0.0 0.5 1.00.
00.
10.
20.
30.
40.
5
Figure: Histogram of a sample of size 10.000 from
X ∼ U(−1, 1)
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 4: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/4.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exponential distribution E(λ)
Exponential distribution E(λ) (λ > 0)
I Range: Interval [0,∞).
I The density f is given by
f (x) = λe−λx1[0,∞)(x).
I For X ∼ E(λ) we have
E(X ) =1
λ, V(X ) =
1
λ2.
I Where does this distributionnaturally appear?
I Generate a sample of sizen = 10.000.
1 n <− 100002 lambda <− 33 x <− r e x p ( n , r=lambda )4 h i s t ( x , p r o b a b i l i t y = TRUE)
Den
sity
0.0 0.5 1.0 1.5 2.0 2.5 3.00.
00.
51.
01.
52.
0
Figure: Histogram of a sample of size 10.000 from X ∼ E(3)
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 5: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/5.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Normal distribution
Normal distribution N (µ, σ2)
I Range: R
I Density of N (µ, σ2)
f (x) =1
√2πσ2
e− (x−µ)2
2σ2 .
I For X ∼ N (µ, σ2) we have
E(X ) = µ, V(X ) = σ2.
I The most important case isX ∼ N (0, 1), for which we haveE(X ) = 0, V(X ) = 1.
I Generate a sample of sizen = 10.000.
1 n <− 100002 mu <− 0 ; s igma <− 13 x <− rnorm ( n , mean=mu, sd=
sigma )4 h i s t ( x , p r o b a b i l i t y = TRUE)
Den
sity
−4 −2 0 2 40.
00.
10.
20.
30.
4
Figure: Histogram of a sample of size 10.000 from
X ∼ N (0, 1)
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 6: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/6.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Normal distribution
I Why ’Normal’ distribution?
Example (Coin tossing)
I A fair coin is tossed n times.
I x1, x2, . . . , xn denote the results (0or 1).
I Calculate the standardized value
z := 2√n (xn−0.5) =
√n
(xn − 0.5)√
0.5 0.5.
I1 n <− 1002 x <− sample ( c ( 0 , 1 ) , n , r e p l a c e
=TRUE)3 z <− s q r t ( n )∗ ( mean ( x )−0.5) /
s q r t ( 0 . 5 ∗ 0 . 5 )
I Repeat R = 20.000 times and plotthe histogram of z1, . . . , zR .
n=30
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
n=100
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
n=500
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
n=5000
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
Figure: Histogram of z1, . . . , zR .
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 7: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/7.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Normal distribution
Example (Rolling a dice)
I A dice is rolled n times.
I x1, x2, . . . , xn denotes the results.
I Calculate the standardized value
z :=√n
(xn − 3.5)√35/12
.
I1 n <− 1002 x <− sample ( 1 : 6 , n , r e p l a c e=
TRUE)3 z <− s q r t ( n )∗ ( mean ( x )−3.5) /
s q r t (35 / 12)
I Repeat R = 20.000 times and plotthe histogram of z1, . . . , zR .
n=30
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
n=100
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
n=500
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
n=5000
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
Figure: Histogram of z1, . . . , zR .
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 8: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/8.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Normal distribution
Example (Uniform distribution)
I Suppose that X ∼ U(0, 1) and thatx1, x2, . . . , xn denote a sample of X .
I Calculate the standardized value
z :=√n
(xn − 1/2)√1/12
.
I1 n <− 1002 x <− r u n i f ( n , 0 , 1 )3 z <− s q r t ( n )∗ ( mean ( x )−0.5) /
s q r t (1 / 12)
I Repeat R = 20.000 times and plotthe histogram of z1, . . . , zR .
n=30
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
n=100
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
n=500
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
n=5000
x
freq
uenc
y
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
Figure: Histogram of z1, . . . , zR .
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 9: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/9.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Normal distribution
I In each of the considered cases the standardized mean approximately had aN (0, 1)-distribution.
I ...we observed examples for the central limit theorem (CLT).
I The general result is as follows:
Theorem (CLT)
Suppose that (Xn)n∈N is an i.i.d. sequence of random variables with finite varianceV(X1) = σ2 > 0. Set µ := E(X1) and define Zn as
Zn :=
∑ni=1 Xi − nµ√nσ
=√n
Xn − µσ
for every n ∈ N. Then FZn (x) −→ Φ(x) for n→∞ and arbitrary x ∈ R(Φ.... distribution function of N (0, 1)).
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 10: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/10.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Loops
Learning by doing - let’s have a look at an example
I1 R <− 100002 e r g <− r e p ( 0 ,R)3 n <− 5004 f o r ( i i n 1 :R){5 x <− r u n i f ( n , 0 , 1 )6 e r g [ i ] <− s q r t ( n )∗ ( mean ( x )−0.5) / s q r t (1 / 12)7 }8 h i s t ( erg , c o l=” l i g h t b l u e ” , main=”” , x l a b=” x ” , y l a b=” f r e q u e n c y ” ,
p r o b a b i l i t y=TRUE, x l i m=c (−4 ,4) , b r e a k s =35)
I @line 2: construct a vector with name ’erg’ of length R only containing zeros.
I @line 4: repeat the same procedure R times; save the result of the first run inthe 1-st coordinate of ’erg’, the result of the second run in the 2nd coordinate,the result of the third run in the 3rd coordinate, and so on till run number R.
I @line 8: plot a histogram of the resulting values.
Solve exercises 01 - 03 in the R-script R-Codes-R-SVm01.R.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 11: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/11.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
If and ifelse statements
Learning by doing - let’s have a look at two examples of if-statements
I1 x <− rnorm ( 1 )2 i f ( x<0){ p r i n t ( ” N e g a t i v e v a l u e ” )}
I @line 1: sample of size one of X ∼ N (0, 1).
I @line 2: If the value is negative print ’Negative value’.
I1 n <− 10002 x <− rnorm ( n )3 z <− r e p ( 0 , n )4 f o r ( i i n 1 : n ){5 i f ( x [ i ]>=0){z [ i ] <− 1}6 }7 mean ( z )
I What is the code doing?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 12: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/12.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
If and ifelse statements
I If loops can be avoided by using ’ifelse’ instead.
I The subsequent code is significantly faster than the previous snippet:
I1 n <− 10002 x <− rnorm ( n )3 z <− i f e l s e ( x>=0 ,1 ,0)4 mean ( z )
Solve exercises 04 - 07 in the R-script R-Codes-R-SVm01.R.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 13: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/13.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
R-functions
Learning by doing - let’s have a look at the following simple function
I1 #@ f u n c t i o n s :2 my . fun <− f u n c t i o n ( n ){3 x <− r u n i f ( n ,−1 ,1)4 a <− min ( x )5 b <− max ( x )6 r e s <− c ( a , b )7 r e t u r n ( r e s )8 }9
10 my . fun ( 1 0 0 )
I What does the function do?
I How can the function be applied?
I NB: A function takes some arguments/data as input, does some calculations andthen returns a result as output.
I Any structures (vector, data.frame, list, etc.) can serve as input and as output.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 14: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/14.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
R-functions
I Let’s extend the previous function ’my.fun’ in such a way that the user can alsochoose the parameters of the uniform distribution and that the functionautomatically produces a histogram:
I1 my . fun2 <− f u n c t i o n ( n , a=0,b=1){2 x <− r u n i f ( n , min=a , max=b )3 h i s t ( x , p r o b a b i l i t y = TRUE, c o l=” l i g h t b l u e ” )4 a <− min ( x )5 b <− max ( x )6 r e s <− c ( a , b )7 r e t u r n ( r e s )8 }9
10 my . fun2 (1000 , a=3,b=5)
Solve exercises 08 - 09 in the R-script R-Codes-R-SVm01.R.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 15: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/15.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercise 10: A small simulation study concerning GPS bias
Exercise 10: A small simulation study concerning GPS bias
I Ten students of geoinformatics want to test GPS-based distance measurements.
I They (consecutively) record the GPS-coordinates of (the outer track of) the100m starting line in an athletics stadium close by, then (consecutively) walkalong the outer track till the finishing line, and again record the GPS-coordinates.
I Each of them repeats this procedure 50 times.
I For each of the 500 pairs they calculate the distance in meters.
I Given the sample size of n = 500 they expect the mean distance to be prettyclose to 100m (why?).
I All the bigger the surprise when the mean distance turns out to be roughly 102m.
I What went wrong - just bad luck?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 16: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/16.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercise 10: A small simulation study concerning GPS bias
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
−40
−20
0
20
40
0 50 100 150x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 17: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/17.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercise 10: A small simulation study concerning GPS bias
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
−40
−20
0
20
40
0 50 100 150x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 18: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/18.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercise 10: A small simulation study concerning GPS bias
I What went wrong - just bad luck?
I We answer the question by means of simulations.
I Assume that the starting point S and the end point Z have the following exactcoordinates: S = (0, 0),Z = (100, 0)
I S ′,Z ′ will denote the measured coordinates; F = (X1,Y1) denotes themeasurement error in S, G = (X2,Y2) the measurement error in Z .
I In other words
S ′ = S + (X1,Y1) = (X1,Y1)
Z ′ = Z + (X2,Y2) = (100 + X2,Y2)
I The measured distance d therefore given by
d =√
(100 + X2 − X1)2 + (Y2 − Y1)2
I To simplify matters we assume that the errors follow a normal distribution, i.e.X1,X2,Y2,Y2 ∼ N (0, σ2).
I Consider the case σ = 15.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 19: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/19.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercise 10: A small simulation study concerning GPS bias
Exercise 10:
I Simulate n = 100.000 (or more) distance measurements.
I Calculate the corresponding distances distances d1, . . . , dn.
I Calculate the mean distance dn - is it greater or smaller than 100m?
I Produce a boxplot of the calculated distances.
I Analyze what happens if σ2 is increased or reduced.
I Find a possible explanation of the observation made.
I Write a function with sample size n as input parameter which produces a boxplotof the distances and returns the mean dn.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 20: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/20.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Quick Reminder: Pearson correlation coefficient ρ
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−5
0
5
10
−2 0 2 4x
y
Figure: What is the correlation coefficient of the drawn
sample?
I The graphic depicts a sample(x1, y1), . . . , (xn, yn).
I Give a rough estimate of thecorrelation coefficient ρ of the sample
I How can ρ be calculated?
I Let sx (resp. sy ) denote the standarddeviation of the x-coordinates(y -coordinates) of the sample, i.e.
sx =
√√√√ 1
n − 1
n∑i=1
(xi − xn)2
sy =
√√√√ 1
n − 1
n∑i=1
(yi − yn)2
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 21: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/21.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Quick Reminder: Pearson correlation coefficient ρ
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−5
0
5
10
−2 0 2 4x
y
Figure: What is the correlation coefficient of the drawn
sample?
I Let sxy denote the (empirical)covariance of the sample, i.e.
sxy =1
n − 1
n∑i=1
(xi − xn)(yi − yn)
I The (Pearson) correlation coefficientρxy is defined as
ρxy =sxy
sx sy
if sx , sy > 0.
I In our case we get ρxy = 0.97464.
I How can this value be interpreted?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 22: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/22.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Quick Reminder: Pearson correlation coefficient ρ
Properties of ρ
I Whenever ρxy exists (i.e. whenever sx , sy > 0) we have −1 ≤ ρxy ≤ 1.
I We have ρxy = ρyx . As a consequence we will simply write ρ in the sequel.
I ρ = 1 if and only if (x1, y1), . . . , (xn, yn) lie on a straight line with positive slope.
I ρ = −1 if and only if (x1, y1), . . . , (xn, yn) lie on a straight line with negativeslope.
I In case of ρ = 0 we call the sample (x1, y1), . . . , (xn, yn) uncorrelated.
I ρ = 0 is not a measure of dependence - it only measures linear dependence.
I ρ = 0 means that there is no linear dependence.
I If instead of (x1, y1), . . . , (xn, yn) we consider (2x1, 3y1), . . . , (2xn, 3yn), whathappens to ρ?
I If instead of (x1, y1), . . . , (xn, yn) we consider (−2x1,−3y1), . . . , (−2xn,−3yn),what happens to ρ?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 23: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/23.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Quick Reminder: Pearson correlation coefficient ρ
I If instead of (x1, y1), . . . , (xn, yn) we consider (−2x1,−3y1), . . . , (−2xn,−3yn),what happens to ρ?
1 f i l e <− u r l ( ” h t t p : //www. t r u t s c h n i g . n e t / geo r e g 1 . RData” )2 l o a d ( f i l e )3 A<−geo r e g 14 head ( geo r e g 1 )5
6 c o r (A$x , A$ y )7 c o r (2∗A$x , 3∗A$ y )8 c o r (−2∗A$x ,−3∗A$ y )9 c o r (−2∗A$x , 3∗A$ y )
I ρ does not change under linear transformations with the same sign.
I ρ changes, however, under non-linear transformations:
I If instead of (x1, y1), . . . , (xn, yn) we consider (x31 , y
31 ), . . . , (x3
n , y3n ) then we get
ρ = 0.9.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 24: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/24.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Spearman rank correlation ρS
I Assume we want to have a measure quantifying if there is a monotonicrelationship between the x- and the y -coordinates of a sample(x1, y1), . . . , (xn, yn).
I ’Monotonic relationship’ (or concordance) in the sense that if the x-coordinatesincrease then also the y -coordinates (grow or fall together).
I There is no need for the relationship to be linear.
I One natural idea is to work with ranks - best explained by some simple examples:
1 x1 <− c ( 3 , 1 , 4 , 15 , 13)2 r 1 <− rank ( x1 )3 x14 #[ 1 ] 3 1 4 15 135 r 16 #[ 1 ] 2 1 3 5 4
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 25: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/25.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Spearman rank correlation ρS
1 x1 <− c ( 3 , 1 , 3 , 15 , 13)2 r 1 <− rank ( x1 )3 x14 #[ 1 ] 3 1 3 15 135 r 16 #[ 1 ] 2 . 5 1 . 0 2 . 5 5 . 0 4 . 0
I The values are sorted - the rank rk(xi ) of observation xi is the position after theranking.
I In case of ties averages of the ranks will be calculated (other choices are optionalin the function).
I From (x1, y1), . . . , (xn, yn) we get the sample ranks(rkx (x1), rky (y1)), . . . , (rkx (xn), rky (yn)).
I rkx (xi ) is the rank of observation xi among x1, . . . , xn.
I rky (yi ) is the rank of observation yi among y1, . . . , yn.
I The Spearman rank correlation is defined as the Pearson correlation of theseranks.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 26: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/26.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Spearman rank correlation ρS
Example
I Considering the following sample of size n = 5
x y3.05 10.211.38 2.194.32 19.31
15.51 241.087.08 50.81
x y rk.x rk.y3.05 10.21 2.00 2.001.38 2.19 1.00 1.004.32 19.31 3.00 3.00
15.51 241.08 5.00 5.007.08 50.81 4.00 4.00
I What can be seen?
I For ρS we get ρs = 1
1 c o r ( rank (E$ x ) , rank (E$ y ) )2 c o r (E$x , E$y , method=” spearman ” )
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 27: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/27.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Spearman rank correlation ρS
Properties of ρS :
I Whenever ρS exists we have −1 ≤ ρxy ≤ 1.
I ρS is symmetric too.
I ρS = 1 if and only if: for each pair (xi , yi ), (xj , yj ) we have xi ≤ xj and only ifyi ≤ yj .
I ρS = −1 if and only if: for each pair (xi , yi ), (xj , yj ) we have xi ≤ xj if and onlyif yi ≥ yj .
I ρS = 0 is not a measure of dependence - it only measures monotonicdependence (aka concordance).
I ρS = 0 means that there is no monotonic relationship dependence.
I If instead of (x1, y1), . . . , (xn, yn) we consider (2x1, 3y1), . . . , (2xn, 3yn), whathappens to ρS?
I If instead of (x1, y1), . . . , (xn, yn) we consider (−2x1,−3y1), . . . , (−2xn,−3yn),what happens to ρS?
I If instead of (x1, y1), . . . , (xn, yn) we consider (x31 , y
31 ), . . . , (x3
n , y3n ), what
happens to ρS?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 28: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/28.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Spearman rank correlation ρS
1 f i l e <− u r l ( ” h t t p : //www. t r u t s c h n i g . n e t / geo r e g 1 . RData” )2 l o a d ( f i l e )3 A<−geo r e g 14 head ( geo r e g 1 )5
6 c o r (A$x , A$y , method = ” spearman ” )7 c o r (2∗A$x , 3∗A$y , method = ” spearman ” )8 c o r (−2∗A$x ,−3∗A$y , method = ” spearman ” )9 c o r (A$ x ˆ3 ,A$ y ˆ3 , method = ” spearman ” )
I For all four cases we get ρS = 0.9633945.
I Easy to verify: ρS is invariant under monotonic transformations (both increasingor both decreasing).
I Let’s add two outliers to A and see how ρ and ρS change.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 29: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/29.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Spearman rank correlation ρS
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
−2 0 2 4 6 8 10
−5
05
10
x
y
1 Dazu<−data . f rame ( x=c ( 1 0 , 1 0 . 3 ) , y=c( 2 , 2 . 4 ) )
2 A1<−r b i n d (A, Dazu )3 p l o t (A1)4 c o r (A1$x , A1$ y )5 c o r (A1$x , A1$y , method = ” spearman ” )
I Which is more influenced by the twonew points?
I We get ρ = 0.8187617 (beforeρ = 0.97464)
I Moreover ρS = 0.9349794 (beforeρS = 0.9633945)
I ρ is less robust against outliers than ρS
I Rank-based quantities are generallyrobust
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 30: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/30.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●●
● ●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−4 −2 0 2 4
−3
−2
−1
01
23
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 31: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/31.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●●
● ●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−4 −2 0 2 4
−3
−2
−1
01
23
rho=0.0477
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
rho_S=0.0469
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 32: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/32.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
● ●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●● ●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●● ●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
−4 −2 0 2 4
−4
−2
02
4
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 33: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/33.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
● ●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●● ●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●● ●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
−4 −2 0 2 4
−4
−2
02
4
rho=0.7266
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
rho_S=0.7013
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 34: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/34.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●●
●
●
●
● ●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●●
●
●
●●
●●
●
●●
●
●
●
● ●●
●
●
●
●
●
● ●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●●
●
●
●●
●
●
●
● ●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●● ●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
−4 −2 0 2 4
−20
−10
010
20
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 35: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/35.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●●
●
●
●
● ●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●●
●
●
●●
●●
●
●●
●
●
●
● ●●
●
●
●
●
●
● ●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●●
●
●
●●
●
●
●
● ●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●● ●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
−4 −2 0 2 4
−20
−10
010
20
rho=−0.9175
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
rho_S=−0.9652
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 36: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/36.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 37: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/37.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
rho=0.0143
x
y
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
rho_S=0.0251
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 38: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/38.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●●
●
●●
●●
●
●
●●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●●
●●
● ●●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
● ●
●
●●●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●● ●
●●
●
●
●●
●●
●●
● ●
●
●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ● ●
● ●
●
● ●
●
● ●●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●●
●
●
●
● ●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
● ●
●●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
● ●
●
●
●
●
●
● ●●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
● ●
●
● ●
●
●●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
● ●
●
●
● ●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●●
●●
●
●
● ●
●
● ●
●
●●
●
●
●●
●●
●●
●
●
●
●
●
●
●●
●
●
● ●●
●
●
−3 −2 −1 0 1 2 3
02
46
8
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
● ●●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 39: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/39.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●●
●
●●
●●
●
●
●●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●●
●●
● ●●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
● ●
●
●●●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●● ●
●●
●
●
●●
●●
●●
● ●
●
●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ● ●
● ●
●
● ●
●
● ●●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●●
●
●
●
● ●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
● ●
●●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
● ●
●
●
●
●
●
● ●●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●●
●
●
●
●●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
● ●
●
● ●
●
●●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
● ●
●
●
● ●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●●
●●
●
●
● ●
●
● ●
●
●●
●
●
●●
●●
●●
●
●
●
●
●
●
●●
●
●
● ●●
●
●
−3 −2 −1 0 1 2 3
02
46
8
rho=0.8576
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
● ●●
●
●
0 200 400 600 800 1000
020
040
060
080
010
00
rho_S=0.9422
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 40: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/40.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Some examples and exercises
Solve Exercise 11 and Exercise 12 in the R-script R-Codes-R-SVm01.R.
Exercise 13: Can you find a sample (x1, y1), . . . , (xn, yn) for which the Pearsoncorrelation ρ and the Spearman correlation ρS have different sign?Hint: Running simulations is never a bad idea; simulate five x-coordinates and fivey -coordinates from U(0, 1) and calculate ρ and ρS ; repeat several times
Solve Exercise 14 in the R-script R-Codes-R-SVm01.R.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 41: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/41.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
What is regression all about - a general perspective
Known:
I We know that there is a relationship between quantities X and Y of thefollowing form:
Y = r(X ) + ε (1)
I r is an unknown function and ε is a random error fulfilling E(ε) = 0.
I Usually we also assume that ε is not influenced by X (might be a too restrictivecondition in various situations).
I We call X the predictor and Y the response.
Wanted:
I Based on observations (x1, y1), (x2, y2), . . . , (xn, yn) from (1) we want todetermine/estimate the function r (why?).
I If we have a good estimator r of r then we can predict Y for arbitrary values ofX by considering r(X ).
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 42: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/42.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
A real-life example
Example (Offer optimization in supermarkets)
I A supermarket chain wants to optimize their offers.
I If the price is only reduced by 5% then the sales numbers will only go up a bit.
I If the price is reduced by 50% then the sales numbers will go up a lot but thecompany might earn less because the margin is too small.
I Objective: Determine the optimal price reduction in the sense that thesupermarket’s profit is maximal.
I X ...price reduction (absolute or percentage) of a certain product.
I Y ...net earnings (based on this product).
I Y = r(X ) + ε.
I What do you think: Is the model solely based on price reduction as predictorgood?
I Which other predictors would you choose?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 43: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/43.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−5
0
5
10
−2 0 2 4x
y
Figure: Prediction at the point x = 1.5?
I The graphic depicts measurements(x1, y1), . . . , (xn, yn).
I It is known that the data comes fromthe following linear model
Y = aX + b︸ ︷︷ ︸r(X )
+ε.
I In other words: yi = axi + b + εi fori ∈ {1, . . . , n}.
I εi ...samples of the random error εfulfilling E(ε) = 0 that do not influenceeach other and are not influenced by xi .
I Wanted: Forecast the y -value at thepoint x = 1.5.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 44: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/44.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−5
0
5
10
−2 0 2 4x
y
Figure: Prediction at the point x = 1.5?
I How would you predict the value atthe point x = 1.5?
I Problem: We do not know theparameters a and b.
I Choose a and b in such a way thatthe straight line y = ax + b fits thedata in the best possible way.
I Denote the optimal values by a andb.
I Given a and b, predicty = a 1.5 + b.
I Which of the following straightlines fits best?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 45: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/45.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−5
0
5
10
−2 0 2 4x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 46: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/46.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
I Choose those values for a and b that minimize the prediction errors at the pointsin the sample.
I Choosing a and b as parameters we would forecast axi + b for xi .
I The error ri we make is ri = yi − (axi + b) = yi − axi − b. Plot ri
I The sum of all squared errors is given by
F (a, b) :=n∑
i=1
(yi − axi − b
)2(2)
I Choose a and b in such a way that F (a, b) is minimal.
I Analytic calculation yields the following optimal values
a =
∑ni=1(xi − xn)(yi − yn)∑n
i=1(xi − xn)2=
sxy
s2x
(3)
b = yn − a xn. (4)
I For our given sample we get a = 2.010 and b = 0.897.
I The forecast at the point x = 1.5 therefore is y = 2.01 · 1.5 + 0.897 = 3.912
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 47: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/47.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
Back
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−3 −2 −1 0 1 2 3 4
−5
05
10
x
y
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 48: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/48.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
I Before fitting linear models in R some additional observations:
I The estimate slope a =sxys2x
looks a bit like the Pearson correlation ρ =sxysx sy
.
I Using both expressions we get
a = ρsy
sx
I Increasing x by one standard deviation sx increases y by ρ standard deviationssy , in fact
r(x + sx ) = a(x + sx ) + b = ax + b︸ ︷︷ ︸y
+asx = y + ρsy
sxsx = y + ρsy .
I How do we quantify if our optimal model offers a good explanation of the model?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 49: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/49.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
I A natural idea is the coefficient of determination R2
I Easy to show:
n∑i=1
(yi − yn)2 =n∑
i=1
(yi − yi )2︸ ︷︷ ︸
r2i
+n∑
i=1
(yi − yn)2
I Variance of y1, . . . , yn equals the variance of the residuals plus the variance ofthe forecasts y1, . . . , yn.
I Calculate
R2 = 1−∑n
i=1(yi − yi )2∑n
i=1(yi − yn)2=
∑ni=1(yi − yn)2∑ni=1(yi − yn)2
(5)
I R2 is the portion of y -variance explained by the model.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 50: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/50.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−5
0
5
10
−2 0 2 4x
y
R^2=0.9499
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
0.0
2.5
5.0
7.5
−2 0 2 4x
y
R^2=0.1044
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 51: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/51.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
Properties of R2:
I We have 0 ≤ R2 ≤ 1.
I The higher R2 the higher the percentage of variance explained by the model.
I If R2 is close to 1 then the model explains the data very well.
I If R2 is close to 0 the model does not help much to explain the data.
I There should be a strong interrelation between R2 and the correlation ρ of theoriginal sample (x1, y1), . . . , (xn, yn)...
I Calculations in R will make this clear.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 52: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/52.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
1 f i l e <− u r l ( ” h t t p : //www. t r u t s c h n i g . n e t / geo r e g 1 . RData” )2 l o a d ( f i l e )3 head ( geo r e g 1 )4 A<−geo r e g 15
6 model<−lm ( data=A, y ˜ x ) #use what eve r name you want i n s t e a d o fmodel
7 summary ( model )
I yields
1
2 C a l l :3 lm ( f o r m u l a = y ˜ x , data = A)4
5 R e s i d u a l s :6 Min 1Q Median 3Q Max7 −3.07477 −0.63681 −0.03544 0.70030 1.95308
I and
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 53: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/53.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Linear regression
1 C o e f f i c i e n t s :2 E s t i m a t e Std . E r r o r t v a l u e Pr (>| t | )3 ( I n t e r c e p t ) 0 .89704 0.11406 7 . 8 6 5 1 . 1 3 e−11 ∗∗∗4 x 2 .00965 0.05035 39 .917 < 2e−16 ∗∗∗5 −−−6 S i g n i f . codes : 0 ∗∗∗ 0 . 0 0 1 ∗∗ 0 . 0 1 ∗ 0 . 0 5 .
0 . 1 17
8 R e s i d u a l s t a n d a r d e r r o r : 1 . 0 3 on 84 d e g r e e s o f f reedom9 M u l t i p l e R−s q u a r e d : 0 . 9 4 9 9 , A d j u s t e d R−s q u a r e d : 0 .9493
10 F−s t a t i s t i c : 1593 on 1 and 84 DF, p−v a l u e : < 2 . 2 e−16
I Calculate the prediction for x = 1.5
1 ND<−data . f rame ( x=c ( 1 . 5 ) )2 p<−p r e d i c t ( model , new=ND)3 p4 3.91152
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 54: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/54.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercises
Exercise 15:
I Load the dataset geo reg1.RData (see R-Code, end of part 01 in linearregression).
I Produce a scatterplot of the data including the regression line.
I Add the values of the estimated parameters a and b in the title of the plot.
I Produce a boxplots of the residuals r1, . . . , rn.
I Calculate ρ and ρS of the data.
I Forecast r(x) for x ∈ {0, 0.1, 0.2, . . . , 0.9, 1}.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 55: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/55.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercises
Exercise 16:
I The datset ’brainhead.txt’ (see R-Code )contains Brain weight (grams) and headsize (cm3) for 237 adults.
I Fit a linear regression with ’weight’ as response and ’cm3’ as explanatoryvariable.
I Plot the data together with the regression line.
I Calculate the corresponding R2.
I Calculate the biggest ten residuals (’biggest in the sense of absolute value’) -how many man and how many woman are in the ’top-ten’?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 56: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/56.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
@Performance
Summary @univariate linear regression
I (x1, y1), . . . , (xn, yn) are observations from the model Y = aX + b + ε.
I Thereby ε was a random error fulfilling E(ε) = 0; set σ2 = V(ε).
I In other words: yi = axi + b + εi for every i ∈ {1, . . . , n} .
I Using least squares we got the following estimators a of a and b of b
a =
∑ni=1(xi − xn)(yi − yn)∑n
i=1(xi − xn)2=
sxy
s2x
(6)
b = yn − a xn. (7)
I We hope to get a ≈ a and b ≈ b, i.e. we hope that the estimates are close tothe true values.
I Will this always be the case?
I When can we expect to get good estimates?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 57: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/57.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
@Performance
1 #one s i m u l a t i o n2 a<−2 ; b<−13 n<−1004 x<−r u n i f ( n ,−3 ,4) #g e n e r a t e random x v a l u e s5 e r r o r<−rnorm ( n , 0 , 1 ) #e r r o r from normal d i s t r i b u t i o n N( 0 , 1 )6 y<−a∗x+b+e r r o r7 A<−data . f rame ( x=x , y=y )8 p l o t (A)9 model<−lm ( data=A, y ˜ x )
10 a b l i n e ( model )11 summary ( model )
I yields
1
2 C a l l :3 lm ( f o r m u l a = y ˜ x , data = A)4
5 R e s i d u a l s :6 Min 1Q Median 3Q Max7 −2.20647 −0.66814 −0.09888 0.77627 1.95348
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 58: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/58.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
@Performance
1 C o e f f i c i e n t s :2 E s t i m a t e Std . E r r o r t v a l u e Pr (>| t | )3 ( I n t e r c e p t ) 0 .76146 0.10149 7 . 5 0 3 2 . 8 7 e−11 ∗∗∗4 x 2 .09902 0.04686 44 .790 < 2e−16 ∗∗∗5 −−−6 S i g n i f . codes : 0 ∗∗∗ 0 . 0 0 1 ∗∗ 0 . 0 1 ∗ 0 . 0 5 . 0 . 1 17
8 R e s i d u a l s t a n d a r d e r r o r : 0 .9631 on 98 d e g r e e s o f f reedom9 M u l t i p l e R−s q u a r e d : 0 . 9 5 3 4 , A d j u s t e d R−s q u a r e d : 0 .9529
10 F−s t a t i s t i c : 2006 on 1 and 98 DF, p−v a l u e : < 2 . 2 e−16
1 sum ( model $ r e s i d u a l s ˆ2) / ( n−2)
I yields
1
2 [ 1 ] 0 .9275251
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 59: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/59.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
@Performance
1 #s e v e r a l r u n s2 R<−10003 E<−data . f rame ( a=r e p ( 0 ,R) , b=r e p ( 0 ,R) )4
5 a<−2 ; b<−16 n<−1007 f o r ( i i n 1 :R){8 x<−r u n i f ( n ,−3 ,4) #g e n e r a t e random x v a l u e s9 e r r o r<−rnorm ( n , 0 , 1 )
10 y<−a∗x+b+e r r o r11 A<−data . f rame ( x=x , y=y )12 model<−lm ( data=A, y ˜ x )13 E [ i , ]<−as . numer ic ( c o e f f i c i e n t s ( model ) ) [ 2 : 1 ]14 }
I yields
1 a b
2 1 1.994841 0.8079434
3 2 1.987354 1.0237531
4 3 1.951075 0.9251133
5 4 1.999110 1.0721703
6 5 1.996653 0.8200005
7 6 1.968700 1.0383586
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 60: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/60.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
@Performance
1 a b2 Min . : 1 . 8 5 8 Min . : 0 . 6 8 4 13 1 s t Qu . : 1 . 9 6 6 1 s t Qu . : 0 . 9 2 8 04 Median : 2 . 0 0 1 Median : 0 . 9 9 9 45 Mean : 2 . 0 0 2 Mean : 0 . 9 9 8 96 3 rd Qu . : 2 . 0 3 5 3 rd Qu . : 1 . 0 7 2 27 Max . : 2 . 1 6 2 Max . : 1 . 3 0 3 1
I What does the table tell us?
I A graphical overview also helps to interpret the results.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 61: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/61.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
@Performance
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
0.8
1.0
1.2
1.9 2.0 2.1a
b
sample size n= 100
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 62: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/62.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
Natural related questions:
I What happens if the sample size n is increased?
I The more info the better the estimates should (on average) be!
I What other parameter in the simulation could have an influence on the quality ofthe estimates?
I Answer: The variance σ2 of ε is important.
I The higher the variance the poorer the estimates.
I Repeat the simulation (several runs) for higher and lower sample size and varythe variance of the error.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 63: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/63.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
0.8
1.0
1.2
1.9 2.0 2.1a
b
sample size n= 100
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 64: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/64.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.85
0.90
0.95
1.00
1.05
1.10
1.96 2.00 2.04 2.08a
b
sample size n= 500
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 65: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/65.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
0.90
0.95
1.00
1.05
1.10
1.950 1.975 2.000 2.025 2.050a
b
sample size n= 1000
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 66: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/66.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.96
0.98
1.00
1.02
1.98 1.99 2.00 2.01a
b
sample size n= 10000
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 67: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/67.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
0.8
1.0
1.2
1.9 2.0 2.1a
b
sample size n= 100
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 68: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/68.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
●●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.4
0.8
1.2
1.6
1.8 2.0 2.2 2.4a
b
sample size n= 100, sigma^2=4
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 69: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/69.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Influence of the parameters at stake
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
0
1
2
1.6 2.0 2.4a
b
sample size n= 100, sigma^2=16
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 70: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/70.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercises
Exercise 17:Modify the last 30 lines of the R-Code R-Codes-R-SVm01.R to do the following:
I Simulate a sample of size n = 100 from the model Y = 0.5X − 1 + ε wherebyε ∼ N (0, 0.5).
I Include a scatterplot of the data including the regression line; include theestimated parameters a and b in the title of the scatterplot.
I Produce a boxplots of the residuals r1, . . . , rn.
I Calculate ρ and ρS of the data.
I Forecast r(x) for x ∈ {0, 0.1, 0.2, . . . , 0.9, 1}.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 71: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/71.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercises
Exercise 18:Modify the last 30 lines of the R-Code R-Codes-R-SVm01.R to do the following:
I Simulate a sample of size n = 100 from the model Y = 0.5X − 1 + ε withε ∼ N (0, 0.5).
I Save the estimated parameters a and b in a data.frame A.
I Repeat the previous two steps R = 1000 times.
I Produce a boxplots of the estimates a1, . . . , aR and a boxplot of the estimatesb1, . . . , bR .
I Calculate the biggest, the smallest and the median value of a1, . . . , aR .
I Calculate the biggest, the smallest and the median value of b1, . . . , bR .
I Repeat the previous steps for bigger sample size and/or for bigger variance ofthe errors.
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)
![Page 72: Statistics, Visualization and More Using 'R' (298.916 ... · Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression Statistics,](https://reader030.vdocuments.mx/reader030/viewer/2022041110/5f0f1c0e7e708231d44289be/html5/thumbnails/72.jpg)
Standard probability distributions Loops, if/ifelse, R-functions Correlation Regression in general Linear regression
Exercises
Exercise 19:
I In the literature and in bad courses one frequently sees that regression onlyworks in case the errors have normal distribution.
I Consider U(−1, 1)-distributed errors using the command error=runif(n,-1,1) andrepeat the tasks in Exercise 18 and Exercise 19 for this situation.
I Do we also get good results in this setting?
Wolfgang Trutschnig
Statistics, Visualization and More Using ”R” (298.916)