R for Statistics - University of Ottawa€¦ · R for Statistics SSO meeting What is R? I Software...

R for Statistics

Rafa l KulikDepartment of Mathematics and Statistics

University of Ottawa

Statistical Society of Ottawa23 September 2011

Rafa l Kulik

R for Statistics SSO meeting

Plan

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Plan

I What is R?

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Plan

I What is R?I Basic Syntax: Vectors and Functions.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.I Simple Data Analysis.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.I Simple Data Analysis.I Statistical tests.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.I Simple Data Analysis.I Statistical tests.I Linear Regression.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.I Simple Data Analysis.I Statistical tests.I Linear Regression.I Simulation.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.I Simple Data Analysis.I Statistical tests.I Linear Regression.I Simulation.I Bootstrap.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.I Simple Data Analysis.I Statistical tests.I Linear Regression.I Simulation.I Bootstrap.I Random Matrices.

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Plan

I What is R?I Basic Syntax: Vectors and Functions.I Getting data into and out of R.I Simple Data Analysis.I Statistical tests.I Linear Regression.I Simulation.I Bootstrap.I Random Matrices.I Kernel smoothing in time series.

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What is R?

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What is R?

I Software for computation, simulation, data manipulation.

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What is R?

I Software for computation, simulation, data manipulation.I It is open-source and free implementation of S-PLUS.

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What is R?

I Software for computation, simulation, data manipulation.I It is open-source and free implementation of S-PLUS.I Can be used on Windows, Macintosh, Unix, Linux.

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What is R?

I Software for computation, simulation, data manipulation.I It is open-source and free implementation of S-PLUS.I Can be used on Windows, Macintosh, Unix, Linux.I Started by R. Gentleman and R. Ihaka from the University of Auckland.

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What is R?

I Software for computation, simulation, data manipulation.I It is open-source and free implementation of S-PLUS.I Can be used on Windows, Macintosh, Unix, Linux.I Started by R. Gentleman and R. Ihaka from the University of Auckland.I R is maintained by R development team. Its webpage is

www.r-project.org

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What is R?


www.r-project.org

I You may want to download Tinn-R - a free and simple replacement forthe code editor provided by R-Gui (graphical user interface).

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What is R?


www.r-project.org

I You may want to download Tinn-R - a free and simple replacement forthe code editor provided by R-Gui (graphical user interface).I The first issue of R journal appeared last year.

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Basic Syntax: Vectors and Functions

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I Writing data manually as a vector:

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

I Elements of the vector can be printed as a[3], a[2:4].

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

I Elements of the vector can be printed as a[3], a[2:4].I Basic operations on vectors.

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

I Elements of the vector can be printed as a[3], a[2:4].I Basic operations on vectors.I You may write a vector as the argument of a function. e.g. log(a).

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

I Elements of the vector can be printed as a[3], a[2:4].I Basic operations on vectors.I You may write a vector as the argument of a function. e.g. log(a).I Operation a+b does not work. Extract e.g. b=b[2:6]. Now it works.

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

I Elements of the vector can be printed as a[3], a[2:4].I Basic operations on vectors.I You may write a vector as the argument of a function. e.g. log(a).I Operation a+b does not work. Extract e.g. b=b[2:6]. Now it works.I Note that a*b, a/b are coordinatewise.

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

I Elements of the vector can be printed as a[3], a[2:4].I Basic operations on vectors.I You may write a vector as the argument of a function. e.g. log(a).I Operation a+b does not work. Extract e.g. b=b[2:6]. Now it works.I Note that a*b, a/b are coordinatewise.I However, max(a,b) is not coordinatewise. You have to use pmax(a,b).

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a=c(2,4,7,6,5) b=c(1,3,4,6,8,10)

I Elements of the vector can be printed as a[3], a[2:4].I Basic operations on vectors.I You may write a vector as the argument of a function. e.g. log(a).I Operation a+b does not work. Extract e.g. b=b[2:6]. Now it works.I Note that a*b, a/b are coordinatewise.I However, max(a,b) is not coordinatewise. You have to use pmax(a,b).I Type help(name) to learn how to use a function name.

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Getting data into and out of R

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I Write longer sequences using scan() function. Type f=scan() and starttyping. Hit ENTER once to change a line, hit twice to finish.

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I Write longer sequences using scan() function. Type f=scan() and starttyping. Hit ENTER once to change a line, hit twice to finish.I You may read data from a file: yuan=scan("C:/docs/yuan.txt") or

swiss=scan("C:/docs/swiss.txt",nlines=500).

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I Write longer sequences using scan() function. Type f=scan() and starttyping. Hit ENTER once to change a line, hit twice to finish.I You may read data from a file: yuan=scan("C:/docs/yuan.txt") or

swiss=scan("C:/docs/swiss.txt",nlines=500).I Export data: swiss=write(data,"C:/docs/data.txt").

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Example:

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Example:

Merge yuan and swiss together, write them as file currency.1

1See Tables for matrix operations.

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Example:

Merge yuan and swiss together, write them as file currency.1

currency=matrix(0,length(yuan),2)currency[,1]=yuan;currency[,2]=swiss ;write(currency,"C:/currency.txt"); # NOT GOODcurrency=t(currency);write(currency,"C:/currency.txt"); # NOT GOODwrite(currency,"C:/currency.txt",ncolumns=2) # GOOD !!!

1See Tables for matrix operations.

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Simple Data Analysis

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We have already data yuan and swiss. They describe exchange rates ofChinese Yuan and Swiss Frank, respectively, vs. US Dollar.

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We have already data yuan and swiss. They describe exchange rates ofChinese Yuan and Swiss Frank, respectively, vs. US Dollar. Before you startwith statistical analysis, note that:

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We have already data yuan and swiss. They describe exchange rates ofChinese Yuan and Swiss Frank, respectively, vs. US Dollar. Before you startwith statistical analysis, note that: financial data are not stationary.

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We have already data yuan and swiss. They describe exchange rates ofChinese Yuan and Swiss Frank, respectively, vs. US Dollar. Before you startwith statistical analysis, note that: financial data are not stationary. If St,t = 1, . . . , n, describes a value of financial instrument at time t, the typicaltransformation is

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Rt = log(

St

St−1

), t = 2, . . . , n.

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Rt = log(

St

St−1

), t = 2, . . . , n.

We will write a script which computes log-returns for yuan and swiss,then performs a basic statistical analysis. This will be stored in the fileReturns.R.

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Script Returns.R

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Script Returns.R

yuanreturns=yuan[2:500]/yuan[1:499];yuanreturns=log(yuanreturns);swissreturns=log(swiss[2:500]/swiss[1:499]);print(summary(yuanreturns)); # Print summary statisticsprint(summary(swissreturns));plot(yuanreturns); # Point plotplot(swissreturns);

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Script Returns.R

yuanreturns=yuan[2:500]/yuan[1:499];yuanreturns=log(yuanreturns);swissreturns=log(swiss[2:500]/swiss[1:499]);print(summary(yuanreturns)); # Print summary statisticsprint(summary(swissreturns));plot(yuanreturns); # Point plotplot(swissreturns);

This is not a good script, since the first plot is invisible. You may add thecommand par(). You may also make the pictures much nicer.

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yuanreturns=yuan[2:500]/yuan[1:499];yuanreturns=log(yuanreturns);swissreturns=log(swiss[2:500]/swiss[1:499]);print("Summary statistics for YUAN");print(summary(yuanreturns));print("Summary statistics for SWISS");print(summary(swissreturns));par(mfrow=c(1,2));plot(yuanreturns,xlab="Time",type="l",col="blue");plot(swissreturns,main="SWISS vs. USD",ylab="SWISS log-returns");

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The script has still a lot of disadvantages: data sets names have to replacedmanually, everything is printed out at the same time.

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The script has still a lot of disadvantages: data sets names have to replacedmanually, everything is printed out at the same time. We will write afunction which will perform a simple data analysis for a given data set.

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Function DataAnalysis

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Function DataAnalysis

DataAnalysis<−function(data,gr=TRUE){ print(summary(data))

if(gr){ readline(prompt = "Press <Enter> to continue...")par(mfrow=c(1,1))plot(data,type="l",col="blue")abline(mean(data),0,col="red")readline(prompt = "Press <Enter> to continue...")boxplot(data)readline(prompt = "Press <Enter> to continue...")hist(data,breaks=floor(sqrt(length(data))),prob=TRUE)curve(dnorm(x,mean(data),sd(data)),add=TRUE,col="red")readline(prompt = "Press <Enter> to continue...")qqnorm(data); qqline(data,col="red")} }

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Statistical tests

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Statistical tests

I There is just one command with many possible specifications: t.test(x,y).

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Statistical tests

I There is just one command with many possible specifications: t.test(x,y).I Variables:

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Statistical tests


• x-data set, y-second data set, default=NULL;

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Statistical tests


• x-data set, y-second data set, default=NULL;• alternative=c(”two.sided”,”less”,”greater”);

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Statistical tests


• x-data set, y-second data set, default=NULL;• alternative=c(”two.sided”,”less”,”greater”);• paired=FALSE;

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Statistical tests


• x-data set, y-second data set, default=NULL;• alternative=c(”two.sided”,”less”,”greater”);• paired=FALSE;• conf.level=0.95.

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Statistical tests



I Usage:

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Statistical tests



I Usage:• t.test(x,alternative="less");

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Statistical tests



I Usage:• t.test(x,alternative="less");• results=t.test(x,y,paired=TRUE,conf.level=0.97).

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Statistical tests




I Results of performing t.test function are stored in the object results,which is a t.test object.

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Statistical tests




I Results of performing t.test function are stored in the object results,which is a t.test object.Ia=t.test(yuanreturns); a$statistics; a$statistic[[1]]

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Linear Regression

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Linear Regression

Having data (Xt, Yt), t = 1, . . . , n from the model

Yt = β0 + β1Xt + εt, t = 1, . . . , n.

we want to estimate parameters β0 and β1 using mean squared errorcriterion.

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Linear Regression


Yt = β0 + β1Xt + εt, t = 1, . . . , n.

we want to estimate parameters β0 and β1 using mean squared errorcriterion.I The basic function for linear regression model is lm (y~ x).

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Linear Regression


Yt = β0 + β1Xt + εt, t = 1, . . . , n.

we want to estimate parameters β0 and β1 using mean squared errorcriterion.I The basic function for linear regression model is lm (y~ x).I Write lm (y~ x-1) to fit regression line without intercept.

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Linear Regression


Yt = β0 + β1Xt + εt, t = 1, . . . , n.

we want to estimate parameters β0 and β1 using mean squared errorcriterion.I The basic function for linear regression model is lm (y~ x).I Write lm (y~ x-1) to fit regression line without intercept.I The output is a linear model object. If lmresults=lm (y~ x), then:

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Linear Regression


Yt = β0 + β1Xt + εt, t = 1, . . . , n.


• lmresults$coefficients to get intercept and slope;

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Linear Regression


Yt = β0 + β1Xt + εt, t = 1, . . . , n.


• lmresults$coefficients to get intercept and slope;• lmresults$residuals to get residuals;

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Linear Regression


Yt = β0 + β1Xt + εt, t = 1, . . . , n.


• lmresults$coefficients to get intercept and slope;• lmresults$residuals to get residuals;• lmresults$fitted.values to get fitted values.

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Function Linear Regression

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Function Linear Regression

LinearRegression<−function(x,y) {cat("Would you to fit with intercept? (Y/N) ");answer <- readline(); pr=switch(answer,y=,Y=TRUE,FALSE);pr1=switch(answer,n=,N=TRUE,FALSE);if(pr){lmresults= lm (y~ x) } ;if(pr1){lmresults= lm (y~ x-1) } ;par(mfrow=c(1,2)) ;plot(x,y,xlab=""); abline(lmresults) ;plot(x,lmresults$residuals,ylab="Residuals") ;readline(prompt = "Press <Enter> to continue...") ;print(lmresults) ;

}

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Simulation

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Simulation

I Simulation of basic random variables: b=rnorm(6).2

2See Tables for other simulation functions.

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Simulation

I Simulation of basic random variables: b=rnorm(6).2

BrownianMotion<−function(n=1000,m=10) {print("Simulate one path of BM");par(mfrow=c(1,1))l=3*sqrt(n)plot(cumsum(rnorm(n)),type="l",col="red",ylim=c(-l,l),xlab="")readline(prompt = "Press <Enter> to continue...")print("Simulation of"); print(m); print("paths of BM")for (s in 1:m){ points(cumsum(rnorm(n)),type="l" )}

}2See Tables for other simulation functions.

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Bootstrap

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Bootstrap

I Given a sample X1, . . . , Xn, let T (X1, . . . , Xn) = Θn be an estimator ofa quantity θ.

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Bootstrap

I Given a sample X1, . . . , Xn, let T (X1, . . . , Xn) = Θn be an estimator ofa quantity θ.I A confidence interval for θ has typically the following form

[Θn − zα/2se(Θn), Θn + zα/2se(Θn)],

where zα/2 is 1− α/2 quantile of N (0, 1).

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Bootstrap




I Example 1: Xi ∼ N (µ, σ2) and Θn = Xn, then se(Θn) = σ/√

n and wecan easily estimate σ by its sample version.

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Bootstrap





n and wecan easily estimate σ by its sample version.I Example 2: Xi ∼ F and Θn = med(X1, . . . , Xn), then se(Θn) =???.

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Bootstrap






I Solution: use bootstrap to get se(Θn), an estimate of se(Θn).

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Bootstrap






I Solution: use bootstrap to get se(Θn), an estimate of se(Θn).I Such constructed bootstrap confidence interval works well when thedistribution of Θn is approximately normal.

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Bootstrap - ctd.

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Bootstrap - ctd.

n=1000; m=700; k=500;X=rnorm(n,1,1);TX=median(X);TXboot=1:k;for (s in 1:k)

{;Xstar=sample(X,m,replace=TRUE);TXboot[s]=median(Xstar);};

print(sd(TXboot));qqnorm(TXboot);

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Random Matrices

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Random Matrices

Let A be a matrix n × n with the entries N (0, 1). Then the matrixB = A + AT is called Gaussian Unitary Ensemble. Its eigenvalues follow aparticular pattern.

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Random Matrices


n = 5000;A = array(rnorm(n*n),c(n,n));B = (A+t(A))/sqrt(2*n);Eigens <- eigen(B, symmetric=T);Eigenvalues <- Eigens$values;hist(Eigenvalues,xlab="Eigenvalues",freq=T)

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Random Matrices


n = 5000;A = array(rnorm(n*n),c(n,n));B = (A+t(A))/sqrt(2*n);Eigens <- eigen(B, symmetric=T);Eigenvalues <- Eigens$values;hist(Eigenvalues,xlab="Eigenvalues",freq=T)

Note: RMTstat package deals with different aspects of random matrices.

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Kernel Smoothing in Time Series

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I Recall swiss data Xt, t = 1, . . . , n. We could think about a model

Xt = f(t/n) + εt, t = 1, . . . , n.

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I Recall swiss data Xt, t = 1, . . . , n. We could think about a model

Xt = f(t/n) + εt, t = 1, . . . , n.

I We can estimate f via a kernel estimator, e.g.

fh(x) =n∑

t=1

`t(x)Xt,

where

`t(x) =K

(x−t/n

h

)∑n

s=1 K(

x−s/nh

).

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my.data=swiss; n=length(my.data); ggrid=1:n/n;H=c(0.025,0.05,0.1);K<−function(x) { dnorm(x)};fhat<− function(x){sum(data*K((x-ggrid)/h))/sum(K((x-ggrid)/h)) };par(mfrow=c(1,3))fhatvec=1:nfor (h in H){ temp=1;for (i in ggrid)

{fhatvec[temp]=fhat(i); temp=temp+1;};plot(ggrid,my.data,xlab="Time",ylab="Data",type="p")points(ggrid,fhatvec,type="l",col="blue") }

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my.data=swiss; n=length(my.data); ggrid=1:n/n;H=c(0.025,0.05,0.1);K<−function(x) { dnorm(x)};fhat<− function(x){sum(data*K((x-ggrid)/h))/sum(K((x-ggrid)/h)) };par(mfrow=c(1,3))fhatvec=1:nfor (h in H){ temp=1;for (i in ggrid)

{fhatvec[temp]=fhat(i); temp=temp+1;};plot(ggrid,my.data,xlab="Time",ylab="Data",type="p")points(ggrid,fhatvec,type="l",col="blue") }

3

3For other smoothing techniques, e.g. wavelets, see waved, wavelets.

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Some commands

Table 1: Basic functionssqrt() Square root.

abs() Absolut value.

sin(), cos(), tan() Trigonometric functions.

exp() Exponential function.

log(), log2(), log10(), logb() Logarithmic functions.

Table 2: Basic vector functionssum(), prod() Sum and product of all elements.

cumsum(), cumprod() Cumulative sums and products.

length() Length of a vector.

sort(), rev(sort()) Sort a vector in increasing and decreasing order.

Table 3: Sequences

seq(a,b,by=x) Sequence from a to b with the step x. Default x = 1.

seq(a,b,length.out=n) Sequence from a to b with the size n.

rep(a,n) Repeat the value a n times.

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Table 4: Matrix operations

dim() Dimension.

% ∗% Matrix multiplication.

t() Matrix transpose.

det() Matrix determinant.

solve() Inverse.

eigen() Eigenvalues.

Table 5: Graphics

curve(expr) expr - written as e.g. x2

abline(a,b) Adds a line with intercept a and slope b.

points() Overlaying plots.

Table 6: Distributionsdname() Density (probability) function.

pname() Distribution function P (X ≤ x).

qname() Quantiles.

rname() Simulation.

”name” binom, pois, unif, exp, gamma, t, norm

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Table 7: Basic Statisticsmean(), median()

sd(), var()

min(), max()

summary()

quantile()

cor(), cov()

hist() Histogram.

boxplot() Boxplot.

stem() Stem-and-leaf plot.

ecdf Empirical distribution function.

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R for Statistics - University of Ottawa€¦ · R for Statistics SSO meeting What is R? I Software...

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