mirror of
https://asciireactor.com/otho/cs-5821.git
synced 2024-11-21 20:25:07 +00:00
lab2 added
This commit is contained in:
caes
parent
8c20745057
commit
eb36ab4234
BIN
lab2/.RData
Normal file
BIN
lab2/.RData
Normal file
Binary file not shown.
9
lab2/.Rhistory
Normal file
9
lab2/.Rhistory
Normal file
@ -0,0 +1,9 @@
|
||||
x <- c(1,3,2,5)
|
||||
x
|
||||
x <- c(1,3,2,5)
|
||||
q()
|
||||
x = matrix ( data = c (1,2,3,4), nrow=2, ncol=2)
|
||||
x
|
||||
matrix ( c(1,2,3,4),2,2,byrow=TRUE)
|
||||
ls()
|
||||
q()
|
||||
BIN
lab2/Figure.pdf
Normal file
BIN
lab2/Figure.pdf
Normal file
Binary file not shown.
65
lab2/lab.r
Normal file
65
lab2/lab.r
Normal file
@ -0,0 +1,65 @@
|
||||
x <- c (1,3,2,5)
|
||||
x
|
||||
|
||||
y = c(5,4,7)
|
||||
|
||||
length(x)
|
||||
|
||||
length(y)
|
||||
|
||||
ls()
|
||||
|
||||
|
||||
x+y
|
||||
|
||||
x = matrix ( data = c (1,2,3,4), nrow=2, ncol=2)
|
||||
|
||||
matrix ( c(1,2,3,4),2,2,byrow=TRUE)
|
||||
|
||||
sqrt( x )
|
||||
|
||||
|
||||
x = rnorm(50)
|
||||
y = x + rnorm(50,mean=50,sd=.1)
|
||||
cor(x,y)
|
||||
|
||||
|
||||
set.seed(1303)
|
||||
rnorm(50)
|
||||
|
||||
set.seed(3)
|
||||
y = rnorm(100)
|
||||
mean(y)
|
||||
|
||||
var(y)
|
||||
sqrt(var(y))
|
||||
sd (y)
|
||||
|
||||
|
||||
x = rnorm(100)
|
||||
y = rnorm(100)
|
||||
plot(x,y)
|
||||
|
||||
|
||||
pdf("Figure.pdf")
|
||||
plot(x,y,col="green")
|
||||
dev.off()
|
||||
|
||||
x = seq (1 ,10)
|
||||
x =1:10
|
||||
x = seq ( - pi , pi , length =50)
|
||||
|
||||
|
||||
y=x
|
||||
f = outer(x,y,function(x,y) cos(y)/(1+x^2))
|
||||
contour(x,y,f)
|
||||
contour(x,y,f,nlevels=45,add=T)
|
||||
fa=(f-t(f))/2
|
||||
contour(x,y,fa,nlevels=15)
|
||||
|
||||
image (x ,y , fa )
|
||||
persp (x ,y , fa )
|
||||
persp (x ,y , fa , theta =30)
|
||||
persp (x ,y , fa , theta =30 , phi =20)
|
||||
persp (x ,y , fa , theta =30 , phi =70)
|
||||
persp (x ,y , fa , theta =30 , phi =40)
|
||||
611
lab2/program
Normal file
611
lab2/program
Normal file
@ -0,0 +1,611 @@
|
||||
|
||||
|
||||
|
||||
> x <- c (1 ,3 ,2 ,5)
|
||||
> x
|
||||
[1] 1 3 2 5
|
||||
Note that the > is not part of the command; rather, it is printed by R to
|
||||
indicate that it is ready for another command to be entered. We can also
|
||||
save things using = rather than <- :
|
||||
> x = c (1 ,6 ,2)
|
||||
> x
|
||||
[1] 1 6 2
|
||||
> y = c (1 ,4 ,3)
|
||||
Hitting the up arrow multiple times will display the previous commands,
|
||||
which can then be edited. This is useful since one often wishes to repeat
|
||||
a similar command. In addition, typing ?funcname will always cause R to
|
||||
open a new help file window with additional information about the function
|
||||
funcname .
|
||||
We can tell R to add two sets of numbers together. It will then add the
|
||||
first number from x to the first number from y , and so on. However, x and
|
||||
y should be the same length. We can check their length using the length()
|
||||
length()
|
||||
function.
|
||||
> length ( x )
|
||||
[1] 3
|
||||
> length ( y )
|
||||
[1] 3
|
||||
> x+y
|
||||
[1] 2 10 5
|
||||
The ls() function allows us to look at a list of all of the objects, such
|
||||
ls()
|
||||
as data and functions, that we have saved so far. The rm() function cSan be
|
||||
rm()
|
||||
used to delete any that we don’t want.
|
||||
> ls ()
|
||||
[1] " x " " y "
|
||||
> rm (x , y )
|
||||
> ls ()
|
||||
character (0)
|
||||
It’s also possible to remove all objects at once:
|
||||
> rm ( list = ls () )44
|
||||
2. Statistical Learning
|
||||
The matrix() function can be used to create a matrix of numbers. Before
|
||||
matrix()
|
||||
we use the matrix() function, we can learn more about it:
|
||||
> ? matrix
|
||||
The help file reveals that the matrix() function takes a number of inputs,
|
||||
but for now we focus on the first three: the data (the entries in the matrix),
|
||||
the number of rows, and the number of columns. First, we create a simple
|
||||
matrix.
|
||||
> x = matrix ( data = c (1 ,2 ,3 ,4) , nrow =2 , ncol =2)
|
||||
> x
|
||||
[ ,1] [ ,2]
|
||||
[1 ,]
|
||||
1
|
||||
3
|
||||
[2 ,]
|
||||
2
|
||||
4
|
||||
Note that we could just as well omit typing data= , nrow= , and ncol= in the
|
||||
matrix() command above: that is, we could just type
|
||||
> x = matrix ( c (1 ,2 ,3 ,4) ,2 ,2)
|
||||
and this would have the same effect. However, it can sometimes be useful to
|
||||
specify the names of the arguments passed in, since otherwise R will assume
|
||||
that the function arguments are passed into the function in the same order
|
||||
that is given in the function’s help file. As this example illustrates, by
|
||||
default R creates matrices by successively filling in columns. Alternatively,
|
||||
the byrow=TRUE option can be used to populate the matrix in order of the
|
||||
rows.
|
||||
> matrix ( c (1 ,2 ,3 ,4) ,2 ,2 , byrow = TRUE )
|
||||
[ ,1] [ ,2]
|
||||
[1 ,]
|
||||
1
|
||||
2
|
||||
[2 ,]
|
||||
3
|
||||
4
|
||||
Notice that in the above command we did not assign the matrix to a value
|
||||
such as x . In this case the matrix is printed to the screen but is not saved
|
||||
for future calculations. The sqrt() function returns the square root of each
|
||||
sqrt()
|
||||
element of a vector or matrix. The command x^2 raises each element of x
|
||||
to the power 2 ; any powers are possible, including fractional or negative
|
||||
powers.
|
||||
> sqrt ( x )
|
||||
[ ,1]
|
||||
[1 ,] 1.00
|
||||
[2 ,] 1.41
|
||||
> x ^2
|
||||
[ ,1]
|
||||
[1 ,]
|
||||
1
|
||||
[2 ,]
|
||||
4
|
||||
[ ,2]
|
||||
1.73
|
||||
2.00
|
||||
[ ,2]
|
||||
9
|
||||
16
|
||||
The rnorm() function generates a vector of random normal variables,
|
||||
rnorm()
|
||||
with first argument n the sample size. Each time we call this function, we
|
||||
will get a different answer. Here we create two correlated sets of numbers,
|
||||
x and y , and use the cor() function to compute the correlation between
|
||||
cor()
|
||||
them.2.3 Lab: Introduction to R
|
||||
45
|
||||
> x = rnorm (50)
|
||||
> y = x + rnorm (50 , mean =50 , sd =.1)
|
||||
> cor ( x , y )
|
||||
[1] 0.995
|
||||
By default, rnorm() creates standard normal random variables with a mean
|
||||
of 0 and a standard deviation of 1. However, the mean and standard devi-
|
||||
ation can be altered using the mean and sd arguments, as illustrated above.
|
||||
Sometimes we want our code to reproduce the exact same set of random
|
||||
numbers; we can use the set.seed() function to do this. The set.seed()
|
||||
set.seed()
|
||||
function takes an (arbitrary) integer argument.
|
||||
> set . seed (1303)
|
||||
> rnorm (50)
|
||||
[1] -1.1440
|
||||
1.3421
|
||||
. . .
|
||||
2.1854
|
||||
0.5364
|
||||
0.0632
|
||||
0.5022 -0.0004
|
||||
We use set.seed() throughout the labs whenever we perform calculations
|
||||
involving random quantities. In general this should allow the user to re-
|
||||
produce our results. However, it should be noted that as new versions of
|
||||
R become available it is possible that some small discrepancies may form
|
||||
between the book and the output from R .
|
||||
The mean() and var() functions can be used to compute the mean and
|
||||
mean()
|
||||
variance of a vector of numbers. Applying sqrt() to the output of var() var()
|
||||
will give the standard deviation. Or we can simply use the sd() function.
|
||||
sd()
|
||||
> set . seed (3)
|
||||
> y = rnorm (100)
|
||||
> mean ( y )
|
||||
[1] 0.0110
|
||||
> var ( y )
|
||||
[1] 0.7329
|
||||
> sqrt ( var ( y ) )
|
||||
[1] 0.8561
|
||||
> sd ( y )
|
||||
[1] 0.8561
|
||||
2.3.2 Graphics
|
||||
The plot() function is the primary way to plot data in R . For instance,
|
||||
plot()
|
||||
plot(x,y) produces a scatterplot of the numbers in x versus the numbers
|
||||
in y . There are many additional options that can be passed in to the plot()
|
||||
function. For example, passing in the argument xlab will result in a label
|
||||
on the x-axis. To find out more information about the plot() function,
|
||||
type ?plot .
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
x = rnorm (100)
|
||||
y = rnorm (100)
|
||||
plot (x , y )
|
||||
plot (x ,y , xlab =" this is the x - axis " , ylab =" this is the y - axis " ,
|
||||
main =" Plot of X vs Y ")46
|
||||
2. Statistical Learning
|
||||
We will often want to save the output of an R plot. The command that we
|
||||
use to do this will depend on the file type that we would like to create. For
|
||||
instance, to create a pdf, we use the pdf() function, and to create a jpeg,
|
||||
pdf()
|
||||
we use the jpeg() function.
|
||||
jpeg()
|
||||
> pdf (" Figure . pdf ")
|
||||
> plot (x ,y , col =" green ")
|
||||
> dev . off ()
|
||||
null device
|
||||
1
|
||||
The function dev.off() indicates to R that we are done creating the plot.
|
||||
dev.off()
|
||||
Alternatively, we can simply copy the plot window and paste it into an
|
||||
appropriate file type, such as a Word document.
|
||||
The function seq() can be used to create a sequence of numbers. For
|
||||
seq()
|
||||
instance, seq(a,b) makes a vector of integers between a and b . There are
|
||||
many other options: for instance, seq(0,1,length=10) makes a sequence of
|
||||
10 numbers that are equally spaced between 0 and 1 . Typing 3:11 is a
|
||||
shorthand for seq(3,11) for integer arguments.
|
||||
> x = seq (1 ,10)
|
||||
> x
|
||||
[1] 1 2 3 4 5 6 7
|
||||
> x =1:10
|
||||
> x
|
||||
[1] 1 2 3 4 5 6 7
|
||||
> x = seq ( - pi , pi , length =50)
|
||||
8 9 10
|
||||
8 9 10
|
||||
We will now create some more sophisticated plots. The contour() func-
|
||||
contour()
|
||||
tion produces a contour plot in order to represent three-dimensional data; contour plot
|
||||
it is like a topographical map. It takes three arguments:
|
||||
1. A vector of the x values (the first dimension),
|
||||
2. A vector of the y values (the second dimension), and
|
||||
3. A matrix whose elements correspond to the z value (the third dimen-
|
||||
sion) for each pair of ( x , y ) coordinates.
|
||||
As with the plot() function, there are many other inputs that can be used
|
||||
to fine-tune the output of the contour() function. To learn more about
|
||||
these, take a look at the help file by typing ?contour .
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
y=x
|
||||
f = outer (x ,y , function (x , y ) cos ( y ) /(1+ x ^2) )
|
||||
contour (x ,y , f )
|
||||
contour (x ,y ,f , nlevels =45 , add = T )
|
||||
fa =( f - t ( f ) ) /2
|
||||
contour (x ,y , fa , nlevels =15)
|
||||
The image() function works the same way as contour() , except that it
|
||||
image()
|
||||
produces a color-coded plot whose colors depend on the z value. This is2.3 Lab: Introduction to R
|
||||
47
|
||||
known as a heatmap, and is sometimes used to plot temperature in weather heatmap
|
||||
forecasts. Alternatively, persp() can be used to produce a three-dimensional
|
||||
persp()
|
||||
plot. The arguments theta and phi control the angles at which the plot is
|
||||
viewed.
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
image (x ,y , fa )
|
||||
persp (x ,y , fa )
|
||||
persp (x ,y , fa , theta =30)
|
||||
persp (x ,y , fa , theta =30 , phi =20)
|
||||
persp (x ,y , fa , theta =30 , phi =70)
|
||||
persp (x ,y , fa , theta =30 , phi =40)
|
||||
2.3.3 Indexing Data
|
||||
We often wish to examine part of a set of data. Suppose that our data is
|
||||
stored in the matrix A .
|
||||
> A = matrix (1:16 ,4 ,4)
|
||||
> A
|
||||
[ ,1] [ ,2] [ ,3] [ ,4]
|
||||
[1 ,]
|
||||
1
|
||||
5
|
||||
9
|
||||
13
|
||||
[2 ,]
|
||||
2
|
||||
6
|
||||
10
|
||||
14
|
||||
[3 ,]
|
||||
3
|
||||
7
|
||||
11
|
||||
15
|
||||
[4 ,]
|
||||
4
|
||||
8
|
||||
12
|
||||
16
|
||||
Then, typing
|
||||
> A [2 ,3]
|
||||
[1] 10
|
||||
will select the element corresponding to the second row and the third col-
|
||||
umn. The first number after the open-bracket symbol [ always refers to
|
||||
the row, and the second number always refers to the column. We can also
|
||||
select multiple rows and columns at a time, by providing vectors as the
|
||||
indices.
|
||||
> A [ c (1 ,3) , c (2 ,4) ]
|
||||
[ ,1] [ ,2]
|
||||
[1 ,]
|
||||
5
|
||||
13
|
||||
[2 ,]
|
||||
7
|
||||
15
|
||||
> A [1:3 ,2:4]
|
||||
[ ,1] [ ,2] [ ,3]
|
||||
[1 ,]
|
||||
5
|
||||
9
|
||||
13
|
||||
[2 ,]
|
||||
6
|
||||
10
|
||||
14
|
||||
[3 ,]
|
||||
7
|
||||
11
|
||||
15
|
||||
> A [1:2 ,]
|
||||
[ ,1] [ ,2] [ ,3] [ ,4]
|
||||
[1 ,]
|
||||
1
|
||||
5
|
||||
9
|
||||
13
|
||||
[2 ,]
|
||||
2
|
||||
6
|
||||
10
|
||||
14
|
||||
> A [ ,1:2]
|
||||
[ ,1] [ ,2]
|
||||
[1 ,]
|
||||
1
|
||||
5
|
||||
[2 ,]
|
||||
2
|
||||
648
|
||||
2. Statistical Learning
|
||||
[3 ,]
|
||||
[4 ,]
|
||||
3
|
||||
4
|
||||
7
|
||||
8
|
||||
The last two examples include either no index for the columns or no index
|
||||
for the rows. These indicate that R should include all columns or all rows,
|
||||
respectively. R treats a single row or column of a matrix as a vector.
|
||||
> A [1 ,]
|
||||
[1] 1 5
|
||||
9 13
|
||||
The use of a negative sign - in the index tells R to keep all rows or columns
|
||||
except those indicated in the index.
|
||||
> A [ - c (1 ,3) ,]
|
||||
[ ,1] [ ,2] [ ,3] [ ,4]
|
||||
[1 ,]
|
||||
2
|
||||
6
|
||||
10
|
||||
14
|
||||
[2 ,]
|
||||
4
|
||||
8
|
||||
12
|
||||
16
|
||||
> A [ - c (1 ,3) ,-c (1 ,3 ,4) ]
|
||||
[1] 6 8
|
||||
The dim() function outputs the number of rows followed by the number of
|
||||
dim()
|
||||
columns of a given matrix.
|
||||
> dim ( A )
|
||||
[1] 4 4
|
||||
2.3.4 Loading Data
|
||||
For most analyses, the first step involves importing a data set into R . The
|
||||
read.table() function is one of the primary ways to do this. The help file
|
||||
read.table()
|
||||
contains details about how to use this function. We can use the function
|
||||
write.table() to export data.
|
||||
write.
|
||||
Before attempting to load a data set, we must make sure that R knows table()
|
||||
to search for the data in the proper directory. For example on a Windows
|
||||
system one could select the directory using the Change dir. . . option under
|
||||
the File menu. However, the details of how to do this depend on the op-
|
||||
erating system (e.g. Windows, Mac, Unix) that is being used, and so we
|
||||
do not give further details here. We begin by loading in the Auto data set.
|
||||
This data is part of the ISLR library (we discuss libraries in Chapter 3) but
|
||||
to illustrate the read.table() function we load it now from a text file. The
|
||||
following command will load the Auto.data file into R and store it as an
|
||||
object called Auto , in a format referred to as a data frame. (The text file data frame
|
||||
can be obtained from this book’s website.) Once the data has been loaded,
|
||||
the fix() function can be used to view it in a spreadsheet like window.
|
||||
However, the window must be closed before further R commands can be
|
||||
entered.
|
||||
> Auto = read . table (" Auto . data ")
|
||||
> fix ( Auto )2.3 Lab: Introduction to R
|
||||
49
|
||||
Note that Auto.data is simply a text file, which you could alternatively
|
||||
open on your computer using a standard text editor. It is often a good idea
|
||||
to view a data set using a text editor or other software such as Excel before
|
||||
loading it into R .
|
||||
This particular data set has not been loaded correctly, because R has
|
||||
assumed that the variable names are part of the data and so has included
|
||||
them in the first row. The data set also includes a number of missing
|
||||
observations, indicated by a question mark ? . Missing values are a common
|
||||
occurrence in real data sets. Using the option header=T (or header=TRUE ) in
|
||||
the read.table() function tells R that the first line of the file contains the
|
||||
variable names, and using the option na.strings tells R that any time it
|
||||
sees a particular character or set of characters (such as a question mark),
|
||||
it should be treated as a missing element of the data matrix.
|
||||
> Auto = read . table (" Auto . data " , header =T , na . strings ="?")
|
||||
> fix ( Auto )
|
||||
Excel is a common-format data storage program. An easy way to load such
|
||||
data into R is to save it as a csv (comma separated value) file and then use
|
||||
the read.csv() function to load it in.
|
||||
> Auto = read . csv (" Auto . csv " , header =T , na . strings ="?")
|
||||
> fix ( Auto )
|
||||
> dim ( Auto )
|
||||
[1] 397 9
|
||||
> Auto [1:4 ,]
|
||||
The dim() function tells us that the data has 397 observations, or rows, and
|
||||
dim()
|
||||
nine variables, or columns. There are various ways to deal with the missing
|
||||
data. In this case, only five of the rows contain missing observations, and
|
||||
so we choose to use the na.omit() function to simply remove these rows.
|
||||
na.omit()
|
||||
> Auto = na . omit ( Auto )
|
||||
> dim ( Auto )
|
||||
[1] 392
|
||||
9
|
||||
Once the data are loaded correctly, we can use names() to check the
|
||||
names()
|
||||
variable names.
|
||||
> names ( Auto )
|
||||
[1] " mpg "
|
||||
[5] " weight "
|
||||
[9] " name "
|
||||
" cylinders "
|
||||
" d i s p l a c e m e n t " " horsepower "
|
||||
" a c c e l e r a t i o n " " year "
|
||||
" origin "
|
||||
2.3.5 Additional Graphical and Numerical Summaries
|
||||
We can use the plot() function to produce scatterplots of the quantitative
|
||||
variables. However, simply typing the variable names will produce an error
|
||||
message, because R does not know to look in the Auto data set for those
|
||||
variables.
|
||||
scatterplot50
|
||||
2. Statistical Learning
|
||||
> plot ( cylinders , mpg )
|
||||
Error in plot ( cylinders , mpg ) : object ’ cylinders ’ not found
|
||||
To refer to a variable, we must type the data set and the variable name
|
||||
joined with a $ symbol. Alternatively, we can use the attach() function in
|
||||
attach()
|
||||
order to tell R to make the variables in this data frame available by name.
|
||||
> plot ( Auto$cylinders , Auto$mpg )
|
||||
> attach ( Auto )
|
||||
> plot ( cylinders , mpg )
|
||||
The cylinders variable is stored as a numeric vector, so R has treated it
|
||||
as quantitative. However, since there are only a small number of possible
|
||||
values for cylinders , one may prefer to treat it as a qualitative variable.
|
||||
The as.factor() function converts quantitative variables into qualitative
|
||||
as.factor()
|
||||
variables.
|
||||
> cylinders = as . factor ( cylinders )
|
||||
If the variable plotted on the x-axis is categorial, then boxplots will
|
||||
automatically be produced by the plot() function. As usual, a number
|
||||
of options can be specified in order to customize the plots.
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
ylab =" MPG ")
|
||||
mpg )
|
||||
mpg ,
|
||||
mpg ,
|
||||
mpg ,
|
||||
mpg ,
|
||||
boxplot
|
||||
col =" red ")
|
||||
col =" red " , varwidth = T )
|
||||
col =" red " , varwidth =T , horizontal = T )
|
||||
col =" red " , varwidth =T , xlab =" cylinders " ,
|
||||
The hist() function can be used to plot a histogram. Note that col=2
|
||||
hist()
|
||||
has the same effect as col="red" .
|
||||
histogram
|
||||
> hist ( mpg )
|
||||
> hist ( mpg , col =2)
|
||||
> hist ( mpg , col =2 , breaks =15)
|
||||
The pairs() function creates a scatterplot matrix i.e. a scatterplot for every
|
||||
pair of variables for any given data set. We can also produce scatterplots
|
||||
for just a subset of the variables.
|
||||
scatterplot
|
||||
matrix
|
||||
> pairs ( Auto )
|
||||
> pairs (∼ mpg + d i s p l a c e m e n t + horsepowe r + weight +
|
||||
acceleration , Auto )
|
||||
In conjunction with the plot() function, identify() provides a useful
|
||||
identify()
|
||||
interactive method for identifying the value for a particular variable for
|
||||
points on a plot. We pass in three arguments to identify() : the x-axis
|
||||
variable, the y-axis variable, and the variable whose values we would like
|
||||
to see printed for each point. Then clicking on a given point in the plot
|
||||
will cause R to print the value of the variable of interest. Right-clicking on
|
||||
the plot will exit the identify() function (control-click on a Mac). The
|
||||
numbers printed under the identify() function correspond to the rows for
|
||||
the selected points.2.3 Lab: Introduction to R
|
||||
51
|
||||
> plot ( horsepower , mpg )
|
||||
> identify ( horsepower , mpg , name )
|
||||
The summary() function produces a numerical summary of each variable in
|
||||
summary()
|
||||
a particular data set.
|
||||
> summary ( Auto )
|
||||
mpg
|
||||
Min .
|
||||
: 9.00
|
||||
1 st Qu .:17.00
|
||||
Median :22.75
|
||||
Mean
|
||||
:23.45
|
||||
3 rd Qu .:29.00
|
||||
Max .
|
||||
:46.60
|
||||
cylinders
|
||||
Min .
|
||||
:3.000
|
||||
1 st Qu .:4.000
|
||||
Median :4.000
|
||||
Mean
|
||||
:5.472
|
||||
3 rd Qu .:8.000
|
||||
Max .
|
||||
:8.000
|
||||
horsepower
|
||||
Min .
|
||||
: 46.0
|
||||
1 st Qu .: 75.0
|
||||
Median : 93.5
|
||||
Mean
|
||||
:104.5
|
||||
3 rd Qu .:126.0
|
||||
Max .
|
||||
:230.0 weight
|
||||
Min .
|
||||
:1613
|
||||
1 st Qu .:2225
|
||||
Median :2804
|
||||
Mean
|
||||
:2978
|
||||
3 rd Qu .:3615
|
||||
Max .
|
||||
:5140
|
||||
year
|
||||
Min .
|
||||
:70.00
|
||||
1 st Qu .:73.00
|
||||
Median :76.00
|
||||
Mean
|
||||
:75.98
|
||||
3 rd Qu .:79.00
|
||||
Max .
|
||||
:82.00 origin
|
||||
Min .
|
||||
:1.000
|
||||
1 st Qu .:1.000
|
||||
Median :1.000
|
||||
Mean
|
||||
:1.577
|
||||
3 rd Qu .:2.000
|
||||
Max .
|
||||
:3.000
|
||||
displacement
|
||||
Min .
|
||||
: 68.0
|
||||
1 st Qu .:105.0
|
||||
Median :151.0
|
||||
Mean
|
||||
:194.4
|
||||
3 rd Qu .:275.8
|
||||
Max .
|
||||
:455.0
|
||||
acceleration
|
||||
Min .
|
||||
: 8.00
|
||||
1 st Qu .:13.78
|
||||
Median :15.50
|
||||
Mean
|
||||
:15.54
|
||||
3 rd Qu .:17.02
|
||||
Max .
|
||||
:24.80
|
||||
name
|
||||
amc matador
|
||||
: 5
|
||||
ford pinto
|
||||
: 5
|
||||
toyota corolla
|
||||
: 5
|
||||
amc gremlin
|
||||
: 4
|
||||
amc hornet
|
||||
: 4
|
||||
chevrolet chevette : 4
|
||||
( Other )
|
||||
:365
|
||||
For qualitative variables such as name , R will list the number of observations
|
||||
that fall in each category. We can also produce a summary of just a single
|
||||
variable.
|
||||
> summary ( mpg )
|
||||
Min . 1 st Qu .
|
||||
9.00
|
||||
17.00
|
||||
Median
|
||||
22.75
|
||||
Mean 3 rd Qu .
|
||||
23.45
|
||||
29.00
|
||||
Max .
|
||||
46.60
|
||||
Once we have finished using R , we type q() in order to shut it down, or
|
||||
q()
|
||||
quit. When exiting R , we have the option to save the current workspace so
|
||||
workspace
|
||||
that all objects (such as data sets) that we have created in this R session
|
||||
will be available next time. Before exiting R , we may want to save a record
|
||||
of all of the commands that we typed in the most recent session; this can
|
||||
be accomplished using the savehistory() function. Next time
|
||||
516
lab2/text
Normal file
516
lab2/text
Normal file
@ -0,0 +1,516 @@
|
||||
"
|
||||
R uses functions to perform operations. To run a function called funcname, we type funcname(input1, input2) , where the inputs (or arguments) input1 function and input2 tell R how to run the function. A function can have any number of inputs. For example, to create a vector of numbers, we use the function c() (for concatenate). Any numbers inside the parentheses are joined together. The following command instructs R to join together the numbers 1, 3, 2, and 5, and to save
|
||||
"
|
||||
> x <- c (1 ,3 ,2 ,5)
|
||||
> x
|
||||
[1] 1 3 2 5
|
||||
|
||||
"
|
||||
Note that the > is not part of the command; rather, it is printed by R to indicate that it is ready for another command to be entered. We can also save things using = rather than <- :
|
||||
"
|
||||
|
||||
> x = c (1 ,6 ,2)
|
||||
> x
|
||||
[1] 1 6 2
|
||||
> y = c (1 ,4 ,3)
|
||||
|
||||
"
|
||||
Hitting the up arrow multiple times will display the previous commands, which can then be edited. This is useful since one often wishes to repeat a similar command. In addition, typing ?funcname will always cause R to open a new help file window with additional information about the function funcname . We can tell R to add two sets of numbers together. It will then add the first number from x to the first number from y , and so on. However, x and y should be the same length. We can check their length using the length() function.
|
||||
"
|
||||
|
||||
> length ( x )
|
||||
[1] 3
|
||||
> length ( y )
|
||||
[1] 3
|
||||
> x+y
|
||||
[1] 2 10 5
|
||||
|
||||
"
|
||||
The ls() function allows us to look at a list of all of the objects, such as data and functions, that we have saved so far. The rm() function can be used to delete any that we don’t want.
|
||||
"
|
||||
|
||||
> ls ()
|
||||
[1] " x " " y "
|
||||
> rm (x , y )
|
||||
> ls ()
|
||||
character (0)
|
||||
|
||||
"
|
||||
It’s also possible to remove all objects at once:
|
||||
"
|
||||
|
||||
> rm ( list = ls () )
|
||||
|
||||
"
|
||||
The matrix() function can be used to create a matrix of numbers. Before we use the matrix() function, we can learn more about it:
|
||||
"
|
||||
|
||||
> ? matrix
|
||||
|
||||
"
|
||||
The help file reveals that the matrix() function takes a number of inputs, but for now we focus on the first three: the data (the entries in the matrix), the number of rows, and the number of columns. First, we create a simple matrix.
|
||||
"
|
||||
|
||||
> x = matrix ( data = c (1 ,2 ,3 ,4) , nrow =2 , ncol =2)
|
||||
|
||||
"
|
||||
Note that we could just as well omit typing data= , nrow= , and ncol= in the matrix() command above: that is, we could just type
|
||||
> x = matrix ( c (1 ,2 ,3 ,4) ,2 ,2) and this would have the same effect. However, it can sometimes be useful to specify the names of the arguments passed in, since otherwise R will assume that the function arguments are passed into the function in the same order that is given in the function’s help file. As this example illustrates, by default R creates matrices by successively filling in columns. Alternatively, the byrow=TRUE option can be used to populate the matrix in order of the rows.
|
||||
"
|
||||
|
||||
> matrix ( c (1 ,2 ,3 ,4) ,2 ,2 , byrow = TRUE )
|
||||
|
||||
"
|
||||
Notice that in the above command we did not assign the matrix to a value such as x. In this case the matrix is printed to the screen but is not saved for future calculations. The sqrt() function returns the square root of each sqrt() element of a vector or matrix. The command x^2 raises each element of x to the power 2 ; any powers are possible, including fractional or negative powers.
|
||||
"
|
||||
|
||||
> sqrt ( x )
|
||||
|
||||
> x^2
|
||||
|
||||
"
|
||||
The rnorm() function generates a vector of random normal variables,
|
||||
rnorm() with first argument n the sample size. Each time we call this function, we will get a different answer. Here we create two correlated sets of numbers, x and y , and use the cor() function to compute the correlation between
|
||||
"
|
||||
|
||||
> x = rnorm (50)
|
||||
> y = x + rnorm (50 , mean =50 , sd =.1)
|
||||
> cor ( x , y )
|
||||
[1] 0.995
|
||||
|
||||
"
|
||||
By default, rnorm() creates standard normal random variables with a mean of 0 and a standard deviation of 1. However, the mean and standard deviation can be altered using the mean and sd arguments, as illustrated above. Sometimes we want our code to reproduce the exact same set of random numbers; we can use the set.seed() function to do this. The set.seed()
|
||||
"
|
||||
|
||||
> set . seed (1303)
|
||||
> rnorm (50)
|
||||
|
||||
"
|
||||
We use set.seed() throughout the labs whenever we perform calculations involving random quantities. In general this should allow the user to reproduce our results. However, it should be noted that as new versions of R become available it is possible that some small discrepancies may form between the book and the output from R. The mean() and var() functions can be used to compute the mean and variance of a vector of numbers. Applying sqrt() to the output of var() will give the standard deviation. Or we can simply use the sd() function.
|
||||
"
|
||||
|
||||
> set . seed (3)
|
||||
> y = rnorm (100)
|
||||
> mean ( y )
|
||||
[1] 0.0110
|
||||
> var ( y )
|
||||
[1] 0.7329
|
||||
> sqrt ( var ( y ) )
|
||||
[1] 0.8561
|
||||
> sd ( y )
|
||||
[1] 0.8561
|
||||
|
||||
"
|
||||
2.3.2 Graphics
|
||||
|
||||
The plot() function is the primary way to plot data in R . For instance, plot(x,y) produces a scatterplot of the numbers in x versus the numbers in y. There are many additional options that can be passed in to the plot() function. For example, passing in the argument xlab will result in a label on the x-axis. To find out more information about the plot() function, type ?plot .
|
||||
"
|
||||
|
||||
x = rnorm (100)
|
||||
y = rnorm (100)
|
||||
plot (x , y )
|
||||
plot (x ,y , xlab =" this is the x - axis " , ylab =" this is the y - axis " , main =" Plot of X vs Y ")
|
||||
|
||||
"
|
||||
We will often want to save the output of an R plot. The command that we use to do this will depend on the file type that we would like to create. For instance, to create a pdf, we use the pdf() function, and to create a jpeg, we use the jpeg() function.
|
||||
"
|
||||
|
||||
> pdf (" Figure . pdf ")
|
||||
> plot (x ,y , col =" green ")
|
||||
> dev . off ()
|
||||
|
||||
"
|
||||
The function dev.off() indicates to R that we are done creating the plot. Alternatively, we can simply copy the plot window and paste it into an appropriate file type, such as a Word document. The function seq() can be used to create a sequence of numbers. For instance, seq(a,b) makes a vector of integers between a and b . There are many other options: for instance, seq(0,1,length=10) makes a sequence of 10 numbers that are equally spaced between 0 and 1 . Typing 3:11 is a shorthand for seq(3,11) for integer arguments.
|
||||
"
|
||||
|
||||
> x = seq (1 ,10)
|
||||
> x
|
||||
[1] 1 2 3 4 5 6 7
|
||||
> x =1:10
|
||||
> x
|
||||
[1] 1 2 3 4 5 6 7
|
||||
> x = seq ( - pi , pi , length =50)
|
||||
|
||||
"
|
||||
We will now create some more sophisticated plots. The contour() function produces a contour plot in order to represent three-dimensional data; it is like a topographical map. It takes three arguments:
|
||||
1. A vector of the x values (the first dimension),
|
||||
2. A vector of the y values (the second dimension), and
|
||||
3. A matrix whose elements correspond to the z value (the third dimen
|
||||
sion) for each pair of ( x , y ) coordinates.
|
||||
|
||||
As with the plot() function, there are many other inputs that can be used to fine-tune the output of the contour() function. To learn more about these, take a look at the help file by typing ?contour .
|
||||
"
|
||||
|
||||
y=x
|
||||
f = outer (x ,y , function (x , y ) cos ( y ) /(1+ x ^2) )
|
||||
contour (x ,y , f )
|
||||
contour (x ,y ,f , nlevels =45 , add = T )
|
||||
fa =( f - t ( f ) ) /2
|
||||
contour (x ,y , fa , nlevels =15)
|
||||
|
||||
"
|
||||
The image() function works the same way as contour() , except that it produces a color-coded plot whose colors depend on the z value. This is known as a heatmap, and is sometimes used to plot temperature in weather heatmap forecasts. Alternatively, persp() can be used to produce a three-dimensional plot. The arguments theta and phi control the angles at which the plot is viewed.
|
||||
"
|
||||
|
||||
> image (x ,y , fa )
|
||||
> persp (x ,y , fa )
|
||||
> persp (x ,y , fa , theta =30)
|
||||
> persp (x ,y , fa , theta =30 , phi =20)
|
||||
> persp (x ,y , fa , theta =30 , phi =70)
|
||||
> persp (x ,y , fa , theta =30 , phi =40)
|
||||
|
||||
|
||||
"
|
||||
2.3.3 Indexing Data
|
||||
|
||||
We often wish to examine part of a set of data. Suppose that our data is stored in the matrix A.
|
||||
"
|
||||
> A = matrix (1:16 ,4 ,4)
|
||||
> A
|
||||
|
||||
"Then, typing"
|
||||
|
||||
> A [2 ,3]
|
||||
[1] 10
|
||||
will select the element corresponding to the second row and the third col-
|
||||
umn. The first number after the open-bracket symbol [ always refers to
|
||||
the row, and the second number always refers to the column. We can also
|
||||
select multiple rows and columns at a time, by providing vectors as the
|
||||
indices.
|
||||
> A [ c (1 ,3) , c (2 ,4) ]
|
||||
[ ,1] [ ,2]
|
||||
[1 ,]
|
||||
5
|
||||
13
|
||||
[2 ,]
|
||||
7
|
||||
15
|
||||
> A [1:3 ,2:4]
|
||||
[ ,1] [ ,2] [ ,3]
|
||||
[1 ,]
|
||||
5
|
||||
9
|
||||
13
|
||||
[2 ,]
|
||||
6
|
||||
10
|
||||
14
|
||||
[3 ,]
|
||||
7
|
||||
11
|
||||
15
|
||||
> A [1:2 ,]
|
||||
[ ,1] [ ,2] [ ,3] [ ,4]
|
||||
[1 ,]
|
||||
1
|
||||
5
|
||||
9
|
||||
13
|
||||
[2 ,]
|
||||
2
|
||||
6
|
||||
10
|
||||
14
|
||||
> A [ ,1:2]
|
||||
[ ,1] [ ,2]
|
||||
[1 ,]
|
||||
1
|
||||
5
|
||||
[2 ,]
|
||||
2
|
||||
648
|
||||
2. Statistical Learning
|
||||
[3 ,]
|
||||
[4 ,]
|
||||
3
|
||||
4
|
||||
7
|
||||
8
|
||||
The last two examples include either no index for the columns or no index
|
||||
for the rows. These indicate that R should include all columns or all rows,
|
||||
respectively. R treats a single row or column of a matrix as a vector.
|
||||
> A [1 ,]
|
||||
[1] 1 5
|
||||
9 13
|
||||
The use of a negative sign - in the index tells R to keep all rows or columns
|
||||
except those indicated in the index.
|
||||
> A [ - c (1 ,3) ,]
|
||||
[ ,1] [ ,2] [ ,3] [ ,4]
|
||||
[1 ,]
|
||||
2
|
||||
6
|
||||
10
|
||||
14
|
||||
[2 ,]
|
||||
4
|
||||
8
|
||||
12
|
||||
16
|
||||
> A [ - c (1 ,3) ,-c (1 ,3 ,4) ]
|
||||
[1] 6 8
|
||||
The dim() function outputs the number of rows followed by the number of
|
||||
dim()
|
||||
columns of a given matrix.
|
||||
> dim ( A )
|
||||
[1] 4 4
|
||||
2.3.4 Loading Data
|
||||
For most analyses, the first step involves importing a data set into R . The
|
||||
read.table() function is one of the primary ways to do this. The help file
|
||||
read.table()
|
||||
contains details about how to use this function. We can use the function
|
||||
write.table() to export data.
|
||||
write.
|
||||
Before attempting to load a data set, we must make sure that R knows table()
|
||||
to search for the data in the proper directory. For example on a Windows
|
||||
system one could select the directory using the Change dir. . . option under
|
||||
the File menu. However, the details of how to do this depend on the op-
|
||||
erating system (e.g. Windows, Mac, Unix) that is being used, and so we
|
||||
do not give further details here. We begin by loading in the Auto data set.
|
||||
This data is part of the ISLR library (we discuss libraries in Chapter 3) but
|
||||
to illustrate the read.table() function we load it now from a text file. The
|
||||
following command will load the Auto.data file into R and store it as an
|
||||
object called Auto , in a format referred to as a data frame. (The text file data frame
|
||||
can be obtained from this book’s website.) Once the data has been loaded,
|
||||
the fix() function can be used to view it in a spreadsheet like window.
|
||||
However, the window must be closed before further R commands can be
|
||||
entered.
|
||||
> Auto = read . table (" Auto . data ")
|
||||
> fix ( Auto )2.3 Lab: Introduction to R
|
||||
49
|
||||
Note that Auto.data is simply a text file, which you could alternatively
|
||||
open on your computer using a standard text editor. It is often a good idea
|
||||
to view a data set using a text editor or other software such as Excel before
|
||||
loading it into R .
|
||||
This particular data set has not been loaded correctly, because R has
|
||||
assumed that the variable names are part of the data and so has included
|
||||
them in the first row. The data set also includes a number of missing
|
||||
observations, indicated by a question mark ? . Missing values are a common
|
||||
occurrence in real data sets. Using the option header=T (or header=TRUE ) in
|
||||
the read.table() function tells R that the first line of the file contains the
|
||||
variable names, and using the option na.strings tells R that any time it
|
||||
sees a particular character or set of characters (such as a question mark),
|
||||
it should be treated as a missing element of the data matrix.
|
||||
> Auto = read . table (" Auto . data " , header =T , na . strings ="?")
|
||||
> fix ( Auto )
|
||||
Excel is a common-format data storage program. An easy way to load such
|
||||
data into R is to save it as a csv (comma separated value) file and then use
|
||||
the read.csv() function to load it in.
|
||||
> Auto = read . csv (" Auto . csv " , header =T , na . strings ="?")
|
||||
> fix ( Auto )
|
||||
> dim ( Auto )
|
||||
[1] 397 9
|
||||
> Auto [1:4 ,]
|
||||
The dim() function tells us that the data has 397 observations, or rows, and
|
||||
dim()
|
||||
nine variables, or columns. There are various ways to deal with the missing
|
||||
data. In this case, only five of the rows contain missing observations, and
|
||||
so we choose to use the na.omit() function to simply remove these rows.
|
||||
na.omit()
|
||||
> Auto = na . omit ( Auto )
|
||||
> dim ( Auto )
|
||||
[1] 392
|
||||
9
|
||||
Once the data are loaded correctly, we can use names() to check the
|
||||
names()
|
||||
variable names.
|
||||
> names ( Auto )
|
||||
[1] " mpg "
|
||||
[5] " weight "
|
||||
[9] " name "
|
||||
" cylinders "
|
||||
" d i s p l a c e m e n t " " horsepower "
|
||||
" a c c e l e r a t i o n " " year "
|
||||
" origin "
|
||||
2.3.5 Additional Graphical and Numerical Summaries
|
||||
We can use the plot() function to produce scatterplots of the quantitative
|
||||
variables. However, simply typing the variable names will produce an error
|
||||
message, because R does not know to look in the Auto data set for those
|
||||
variables.
|
||||
scatterplot50
|
||||
2. Statistical Learning
|
||||
> plot ( cylinders , mpg )
|
||||
Error in plot ( cylinders , mpg ) : object ’ cylinders ’ not found
|
||||
To refer to a variable, we must type the data set and the variable name
|
||||
joined with a $ symbol. Alternatively, we can use the attach() function in
|
||||
attach()
|
||||
order to tell R to make the variables in this data frame available by name.
|
||||
> plot ( Auto$cylinders , Auto$mpg )
|
||||
> attach ( Auto )
|
||||
> plot ( cylinders , mpg )
|
||||
The cylinders variable is stored as a numeric vector, so R has treated it
|
||||
as quantitative. However, since there are only a small number of possible
|
||||
values for cylinders , one may prefer to treat it as a qualitative variable.
|
||||
The as.factor() function converts quantitative variables into qualitative
|
||||
as.factor()
|
||||
variables.
|
||||
> cylinders = as . factor ( cylinders )
|
||||
If the variable plotted on the x-axis is categorial, then boxplots will
|
||||
automatically be produced by the plot() function. As usual, a number
|
||||
of options can be specified in order to customize the plots.
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
>
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
plot ( cylinders ,
|
||||
ylab =" MPG ")
|
||||
mpg )
|
||||
mpg ,
|
||||
mpg ,
|
||||
mpg ,
|
||||
mpg ,
|
||||
boxplot
|
||||
col =" red ")
|
||||
col =" red " , varwidth = T )
|
||||
col =" red " , varwidth =T , horizontal = T )
|
||||
col =" red " , varwidth =T , xlab =" cylinders " ,
|
||||
The hist() function can be used to plot a histogram. Note that col=2
|
||||
hist()
|
||||
has the same effect as col="red" .
|
||||
histogram
|
||||
> hist ( mpg )
|
||||
> hist ( mpg , col =2)
|
||||
> hist ( mpg , col =2 , breaks =15)
|
||||
The pairs() function creates a scatterplot matrix i.e. a scatterplot for every
|
||||
pair of variables for any given data set. We can also produce scatterplots
|
||||
for just a subset of the variables.
|
||||
scatterplot
|
||||
matrix
|
||||
> pairs ( Auto )
|
||||
> pairs (∼ mpg + d i s p l a c e m e n t + horsepowe r + weight +
|
||||
acceleration , Auto )
|
||||
In conjunction with the plot() function, identify() provides a useful
|
||||
identify()
|
||||
interactive method for identifying the value for a particular variable for
|
||||
points on a plot. We pass in three arguments to identify() : the x-axis
|
||||
variable, the y-axis variable, and the variable whose values we would like
|
||||
to see printed for each point. Then clicking on a given point in the plot
|
||||
will cause R to print the value of the variable of interest. Right-clicking on
|
||||
the plot will exit the identify() function (control-click on a Mac). The
|
||||
numbers printed under the identify() function correspond to the rows for
|
||||
the selected points.2.3 Lab: Introduction to R
|
||||
51
|
||||
> plot ( horsepower , mpg )
|
||||
> identify ( horsepower , mpg , name )
|
||||
The summary() function produces a numerical summary of each variable in
|
||||
summary()
|
||||
a particular data set.
|
||||
> summary ( Auto )
|
||||
mpg
|
||||
Min .
|
||||
: 9.00
|
||||
1 st Qu .:17.00
|
||||
Median :22.75
|
||||
Mean
|
||||
:23.45
|
||||
3 rd Qu .:29.00
|
||||
Max .
|
||||
:46.60
|
||||
cylinders
|
||||
Min .
|
||||
:3.000
|
||||
1 st Qu .:4.000
|
||||
Median :4.000
|
||||
Mean
|
||||
:5.472
|
||||
3 rd Qu .:8.000
|
||||
Max .
|
||||
:8.000
|
||||
horsepower
|
||||
Min .
|
||||
: 46.0
|
||||
1 st Qu .: 75.0
|
||||
Median : 93.5
|
||||
Mean
|
||||
:104.5
|
||||
3 rd Qu .:126.0
|
||||
Max .
|
||||
:230.0 weight
|
||||
Min .
|
||||
:1613
|
||||
1 st Qu .:2225
|
||||
Median :2804
|
||||
Mean
|
||||
:2978
|
||||
3 rd Qu .:3615
|
||||
Max .
|
||||
:5140
|
||||
year
|
||||
Min .
|
||||
:70.00
|
||||
1 st Qu .:73.00
|
||||
Median :76.00
|
||||
Mean
|
||||
:75.98
|
||||
3 rd Qu .:79.00
|
||||
Max .
|
||||
:82.00 origin
|
||||
Min .
|
||||
:1.000
|
||||
1 st Qu .:1.000
|
||||
Median :1.000
|
||||
Mean
|
||||
:1.577
|
||||
3 rd Qu .:2.000
|
||||
Max .
|
||||
:3.000
|
||||
displacement
|
||||
Min .
|
||||
: 68.0
|
||||
1 st Qu .:105.0
|
||||
Median :151.0
|
||||
Mean
|
||||
:194.4
|
||||
3 rd Qu .:275.8
|
||||
Max .
|
||||
:455.0
|
||||
acceleration
|
||||
Min .
|
||||
: 8.00
|
||||
1 st Qu .:13.78
|
||||
Median :15.50
|
||||
Mean
|
||||
:15.54
|
||||
3 rd Qu .:17.02
|
||||
Max .
|
||||
:24.80
|
||||
name
|
||||
amc matador
|
||||
: 5
|
||||
ford pinto
|
||||
: 5
|
||||
toyota corolla
|
||||
: 5
|
||||
amc gremlin
|
||||
: 4
|
||||
amc hornet
|
||||
: 4
|
||||
chevrolet chevette : 4
|
||||
( Other )
|
||||
:365
|
||||
For qualitative variables such as name , R will list the number of observations
|
||||
that fall in each category. We can also produce a summary of just a single
|
||||
variable.
|
||||
> summary ( mpg )
|
||||
Min . 1 st Qu .
|
||||
9.00
|
||||
17.00
|
||||
Median
|
||||
22.75
|
||||
Mean 3 rd Qu .
|
||||
23.45
|
||||
29.00
|
||||
Max .
|
||||
46.60
|
||||
Once we have finished using R , we type q() in order to shut it down, or
|
||||
q()
|
||||
quit. When exiting R , we have the option to save the current workspace so
|
||||
workspace
|
||||
that all objects (such as data sets) that we have created in this R session
|
||||
will be available next time. Before exiting R , we may want to save a record
|
||||
of all of the commands that we typed in the most recent session; this can
|
||||
be accomplished using the savehistory() function. Next time
|
||||
Loading…
Reference in New Issue
Block a user