#Solution_Practice_3
> cc<-c(1,2,3,4,5,2,1,2,3,4,3,2,1,2,3,4,3,2,1,2,5,4,3,2,1)
> A<-matrix(cc,5,5)
> A
[,1] [,2] [,3] [,4] [,5]
[1,] 1 2 3 4 5
[2,] 2 1 2 3 4
[3,] 3 2 1 2 3
[4,] 4 3 2 1 2
[5,] 5 4 3 2 1
> b<-matrix(c(7,-1,-3,5,17),5,1)
> b
[,1]
[1,] 7
[2,] -1
[3,] -3
[4,] 5
[5,] 17
> x<-solve(A)%*%b
> x
[,1]
[1,] -2
[2,] 3
[3,] 5
[4,] 2
[5,] -4
#factor_function
فرم کلی این تابع به صورت :
factor(x = character(), levels, labels = levels,
exclude = NA, ordered = is.ordered(x))
که از یک رشته شی محتويات انها را خارج و در
Levels
قرار میدهد و میتوان به انها برچسب نیز نسبت داد ، و ordered ترتیب را برای ما مشخص میکند
ترتیب اعضای فاکتور را مشخص میکند
@R_Experts
فرم کلی این تابع به صورت :
factor(x = character(), levels, labels = levels,
exclude = NA, ordered = is.ordered(x))
که از یک رشته شی محتويات انها را خارج و در
Levels
قرار میدهد و میتوان به انها برچسب نیز نسبت داد ، و ordered ترتیب را برای ما مشخص میکند
> mons = c("March","April","January","November","January",
+ "September","October","September","November","August",
+ "January","November","November","February","May","August",
+ "July","December","August","August","September","November",
+ "February","April")
> mons = factor(mons)
> table(mons)
mons
April August December February January July
2 4 1 2 3 1
March May November October September
1 1 5 1 3
ordered=TRUEترتیب اعضای فاکتور را مشخص میکند
> fert = c(10,20,20,50,10,20,10,50,20)
> fert = factor(fert,levels=c(10,20,50),ordered=TRUE)
> fert
[1] 10 20 20 50 10 20 10 50 20
Levels: 10 < 20 < 50
@R_Experts
#abs
x: Numeric value, array or vector
@R_Experts
abs(x)
x: Numeric value, array or vector
> abs(-1)
[1] 1
> abs(20)
[1] 20
> abs(0)
[1] 0
> x <- c(-2,4,0,45,9,-4)
> abs(x)
[1] 2 4 0 45 9 4
> x <- matrix(c(-3,5,-7,1,-9,4),nrow=3,ncol=2,byrow=TRUE)
> abs(x[1,])
[1] 3 5
> abs(x[,1])
[1] 3 7 9
@R_Experts
#atan
atan() function returns the radian arctangent of number data.
x: Numeric value, array or vector
@R_Experts
atan() function returns the radian arctangent of number data.
atan(x)
x: Numeric value, array or vector
> atan(1)
[1] 0.7853982
> atan(0)
[1] 0
> atan(0.5)
[1] 0.4636476
> x <- c(1, 0, 0.5)
> atan(x)
[1] 0.7853982 0.0000000 0.4636476
@R_Experts
#beta
beta() function return the beta function and the natural logarithm of the beta function.
B(a,b) = Γ(a)Γ(b)/Γ(a+b)
beta(a, b)
a,b: non-negative numeric vectors
@R_Experts
beta() function return the beta function and the natural logarithm of the beta function.
B(a,b) = Γ(a)Γ(b)/Γ(a+b)
beta(a, b)
a,b: non-negative numeric vectors
> beta(4,9)
[1] 0.0005050505
> x <- c(3,6, 4)
> y <- c(7,4, 12)
> beta(x,y)
[1] 0.0039682540 0.0019841270 0.0001831502
@R_Experts
#choose()
choose(n,r)
n: n elements
r: r subset elements
...
nCr = n!/(r! * (n-r)!)
@R_Experts
choose(n,r)
n: n elements
r: r subset elements
...
nCr = n!/(r! * (n-r)!)
> choose(5,2)
[1] 10
> choose(2,1)
[1] 2
@R_Experts
#log( )
function computes natural logarithms (Ln) for a number or vector. log10 computes common logarithms (Lg).log2 computes binary logarithms (Log2). log(x,b) computes logarithms with base b.
@R_Experts
function computes natural logarithms (Ln) for a number or vector. log10 computes common logarithms (Lg).log2 computes binary logarithms (Log2). log(x,b) computes logarithms with base b.
>log(5) #ln5
[1] 1.609438
>log10(5) #lg5
[1] 0.69897
>log2(5) #log25
[1] 2.321928
>log(9,base=3) #log39 = 2
[1] 2
>x <- rep(1:12)
>x
[1] 1 2 3 4 5 6 7 8 9 10 11 12
>log(x)
[1] 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595 1.9459101
[8] 2.0794415 2.1972246 2.3025851 2.3978953 2.4849066
>log(x,6)
[1] 0.0000000 0.3868528 0.6131472 0.7737056 0.8982444 1.0000000 1.0860331
[8] 1.1605584 1.2262944 1.2850972 1.3382908 1.3868528
@R_Experts
log10() function computes base 10 logarithm.
x: numeric vector
@R_Experts
log10(x)
x: numeric vector
> log10(100)
[1] 2
> x <- c(100,1000, 10000)
> log10(x)
[1] 2 3 4
@R_Experts
#exp
exp(x) function compute the exponential value of a number or number vector, ex.
expm1() function computes exp() minus 1:»>حاصل منهای 1
@R_Experts
exp(x) function compute the exponential value of a number or number vector, ex.
> x <- 5
> exp(x)
[1] 148.4132
> y <- rep(1:20)
> exp(y)
[,1] [,2] [,3] [,4] [,5]
[1,] 2.718282 20.08554 148.4132 1096.633 8103.084
[2,] 7.389056 54.59815 403.4288 2980.958 22026.466
expm1() function computes exp() minus 1:»>حاصل منهای 1
> expm1(5)
[1] 147.4132
> expm1(rep(1:20))
[,1] [,2] [,3] [,4] [,5]
[1,] 1.718282 19.08554 147.4132 1095.633 8102.084
[2,] 6.389056 53.59815 402.4288 2979.958 22025.466
@R_Experts
#factorial
factorial() function computes the factorial of a number.
factorial(x)
x: numeric vector
@R_Experts
factorial() function computes the factorial of a number.
factorial(x)
x: numeric vector
> factorial(2) #2 × 1
[1] 2
> factorial(1) #1 × 1
[1] 1
> factorial(3) #3 × 2 × 1
[1] 6
> factorial(4) #4 × 3 × 2 × 1
[1] 24
> factorial(c(4,3,2))
[1] 24 6 2
@R_Experts
#max_min
max() function computes the maximun value of a vector. min() function computes the minimum value of a vector.
max(x,na.rm=FALSE)
min(x,na.rm=FALSE)
• x: number vector
• na.rm: whether NA should be removed, if not, NA will be returned
...
Missing value affect the results:
After define na.rm=TRUE, result is meaningful:
Compare more than 1 vectors:
@R_Experts
max() function computes the maximun value of a vector. min() function computes the minimum value of a vector.
max(x,na.rm=FALSE)
min(x,na.rm=FALSE)
• x: number vector
• na.rm: whether NA should be removed, if not, NA will be returned
...
> x <- c(1,2.3,2,3,4,8,12,43,-4,-1)
> max(x)
[1] 43
> min(x)
[1] -4
Missing value affect the results:
> y<- c(x,NA)
> y
[1] 1.0 2.3 2.0 3.0 4.0 8.0 12.0 43.0 -4.0 -1.0 NA
> max(y)
[1] NA
> min(y)
[1] NA
After define na.rm=TRUE, result is meaningful:
> max(y,na.rm=TRUE)
[1] 43
Compare more than 1 vectors:
> x2 <- c(-100,-43,0,3,1,-3)
> min(x,x2)
[1] -100
@R_Experts
#atan2
atan2(y, x) function returns the radian arctangent between the x-axis and the vector from the origin to (x, y).
در مختصات دکارتی این نقطه نسبت به مبدا زاویه ای تولید میکند را تانژانت معکوس رادیانش را بر میگرداند
atans(y, x)
x, y: Numeric value, array or vector
@R_Experts
atan2(y, x) function returns the radian arctangent between the x-axis and the vector from the origin to (x, y).
در مختصات دکارتی این نقطه نسبت به مبدا زاویه ای تولید میکند را تانژانت معکوس رادیانش را بر میگرداند
atans(y, x)
x, y: Numeric value, array or vector
> atan2(2,1)
[1] 1.107149
> y <- c(2,3)
> x <- c(5,6)
> atan2(y,x)
[1] 0.3805064 0.4636476
@R_Experts
#cos
cos() function computes the cosine value of numeric value.
cos(x)
x: Numeric value, array or vector
@R_Experts
cos() function computes the cosine value of numeric value.
cos(x)
x: Numeric value, array or vector
> cos(pi)
[1] -1
> cos(-pi)
[1] -1
> cos(pi/3)
[1] 0.5
> cos(0)
[1] 1
> x <- c(pi, pi/4, pi/3)
> cos(x)
[1] -1.0000000 0.7071068 0.5000000
@R_Experts
#acos
acos() function returns the radian arccosine of number data.
acos(x)
x: Numeric value, array or vector
@R_Experts
acos() function returns the radian arccosine of number data.
acos(x)
x: Numeric value, array or vector
> acos(1)
[1] 0
> acos(0)
[1] 1.570796
> x <- c(-1,-0.866025,-0.707107,-0.5,0,0.5,0.707107,0.866025,1)
> acos(x)
[1] 3.1415926 2.6179931 2.3561948 2.0943951 1.5707963 1.0471976 0.7853979
[8] 0.5235996 0.0000000
@R_Experts
#sqrt
sqrt() function computes the square root of a numeric vector.
sqrt(x)
x: numeric or complex vector, array
sqrt() function computes the square root of a numeric vector.
sqrt(x)
x: numeric or complex vector, array
> sqrt(9)
[1] 3
> sqrt(-1)
[1] NaN
Warning message:
In sqrt(-1) : NaNs produced
> sqrt(3+5i)
[1] 2.101303+1.189738i
> sqrt(c(4,9,16))
[1] 2 3 4
#gamma
gamma() function returns the gamma function Γx.
gamma(x): numeric vectors
gamma() function returns the gamma function Γx.
gamma(x): numeric vectors
> gamma(2)
[1] 1
> gamma(3)
[1] 2
> gamma(4)
[1] 6
> gamma(5)
[1] 24
> gamma(6)
[1] 120
> gamma(c(4,5,6))
[1] 6 24 120
#outer
outer() function applies a function to two arrays.
x,y: arrays
FUN: function to use on the outer products, default is multiply
...
Calculate logarithm value of array x elements using array y as bases:
Add array x elements with array y elements:
Multiply array x elements with array y elements:
Concatenate characters to the array elements:
@R_Experts
outer() function applies a function to two arrays.
outer(x, y, FUN="*", ...)
x %o% y
x,y: arrays
FUN: function to use on the outer products, default is multiply
...
>x <- c(1,2.3,2,3,4,8,12,43)
>y<- c(2,4)
Calculate logarithm value of array x elements using array y as bases:
>outer(x,y,"log")
[,1] [,2]
[1,] 0.000000 0.0000000
[2,] 1.201634 0.6008169
[3,] 1.000000 0.5000000
[4,] 1.584963 0.7924813
[5,] 2.000000 1.0000000
[6,] 3.000000 1.5000000
[7,] 3.584963 1.7924813
[8,] 5.426265 2.7131324
Add array x elements with array y elements:
> outer(x,y,"+")
[,1] [,2]
[1,] 3.0 5.0
[2,] 4.3 6.3
[3,] 4.0 6.0
[4,] 5.0 7.0
[5,] 6.0 8.0
[6,] 10.0 12.0
[7,] 14.0 16.0
[8,] 45.0 47.0
Multiply array x elements with array y elements:
> x %o% y #equal to outer(x,y,"*")
[,1] [,2]
[1,] 2.0 4.0
[2,] 4.6 9.2
[3,] 4.0 8.0
[4,] 6.0 12.0
[5,] 8.0 16.0
[6,] 16.0 32.0
[7,] 24.0 48.0
[8,] 86.0 172.0
Concatenate characters to the array elements:
>z <- c("a","b")
>outer(x,z,"paste")
[,1] [,2]
[1,] "1 a" "1 b"
[2,] "2.3 a" "2.3 b"
[3,] "2 a" "2 b"
[4,] "3 a" "3 b"
[5,] "4 a" "4 b"
[6,] "8 a" "8 b"
[7,] "12 a" "12 b"
[8,] "43 a" "43 b" @R_Experts
#svd
svd() function computes the singular-value decomposition of a rectangular matrix.
svd(x, nu = min(n, p), nv = min(n, p), LINPACK = FALSE)
La.svd(x, nu = min(n, p), nv = min(n, p))
x: a numeric, logical or complex matrix
nu: the number of left singular vectors to be computed. This must between 0 and n = nrow(x)
nv: the number of right singular vectors to be computed. This must be between 0 and p = ncol(x)
LINPACK: logical. Should LINPACK be used (for compatibility with R < 1.7.0)? In this case nu must be 0, nrow(x) or ncol(x)
@R_Experts
svd() function computes the singular-value decomposition of a rectangular matrix.
svd(x, nu = min(n, p), nv = min(n, p), LINPACK = FALSE)
La.svd(x, nu = min(n, p), nv = min(n, p))
x: a numeric, logical or complex matrix
nu: the number of left singular vectors to be computed. This must between 0 and n = nrow(x)
nv: the number of right singular vectors to be computed. This must be between 0 and p = ncol(x)
LINPACK: logical. Should LINPACK be used (for compatibility with R < 1.7.0)? In this case nu must be 0, nrow(x) or ncol(x)
> x <- matrix(1:16,4,4)
> x
[,1] [,2] [,3] [,4]
[1,] 1 5 9 13
[2,] 2 6 10 14
[3,] 3 7 11 15
[4,] 4 8 12 16
> svd(x)
$d
[1] 3.862266e+01 2.071323e+00 2.076990e-15 4.119458e-16
$u
[,1] [,2] [,3] [,4]
[1,] -0.4284124 -0.7186535 0.43803202 0.3288281
[2,] -0.4743725 -0.2738078 -0.82913672 -0.1119477
[3,] -0.5203326 0.1710379 0.34417739 -0.7625890
[4,] -0.5662928 0.6158835 0.04692732 0.5457086
$v
[,1] [,2] [,3] [,4]
[1,] -0.1347221 0.82574206 0.5322301 -0.1293488
[2,] -0.3407577 0.42881720 -0.6132292 0.5691660
[3,] -0.5467933 0.03189234 -0.3702319 -0.7502855
[4,] -0.7528288 -0.36503251 0.4512310 0.3104683
@R_Experts
#diag
diag() function extracts or replaces the diagonal of a matrix, or constructs a diagonal matrix.
x: matrix, vector
nrow, ncol: Optional dimensions for the result when x is not a matrix
: either a single value or a vector of length equal to that of the current diagonal. Should be of a mode which can be coerced to that of x
...
@R_Experts
diag() function extracts or replaces the diagonal of a matrix, or constructs a diagonal matrix.
diag(x = 1, nrow, ncol)
diag(x) <- value
x: matrix, vector
nrow, ncol: Optional dimensions for the result when x is not a matrix
: either a single value or a vector of length equal to that of the current diagonal. Should be of a mode which can be coerced to that of x
...
> diag(10,3,4)
[,1] [,2] [,3] [,4]
[1,] 10 0 0 0
[2,] 0 10 0 0
[3,] 0 0 10 0
> diag(3)
[,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 1 0
[3,] 0 0 1
@R_Experts
#dim
dim() function gets or sets the dimension of a matrix, array or data frame.
dim(x)
x: array, matrix or data frame.
>BOD #R Biochemical Oxygen Demand Dataset
@R_Experts
dim() function gets or sets the dimension of a matrix, array or data frame.
dim(x)
x: array, matrix or data frame.
>BOD #R Biochemical Oxygen Demand Dataset
Time demand
1 1 8.3
2 2 10.3
3 3 19.0
4 4 16.0
5 5 15.6
6 7 19.8
>class(BOD)
[1] "data.frame"
>dim(BOD) #get dimension
[1] 6 2
Set dimension of a matrix:
>x <- rep(1:20)
>x
[1] 1 2 3 4 5 6 7 8 9 10
Set dimension to 2 × 5:
>dim(x) <- c(2,5)
>x
[,1] [,2] [,3] [,4] [,5]
[1,] 1 3 5 7 9
[2,] 2 4 6 8 10
@R_Experts