Contents

1 개요

2 fitdistrplus Package Install

3 사용방법

4 어떤 분포에 가까운가?

5 정규분포인지 아닌지 검정

6 같은 분포에서 온 샘플인가?

7 이거 하나면 충분

8 참고자료

install.packages("fitdistrplus",dependencies=T)
library(fitdistrplus)

x1 <- c(6.4, 13.3, 4.1, 1.3, 14.1, 10.6, 9.9, 9.6, 15.3, 
22.1,13.4, 13.2, 8.4, 6.3, 8.9, 5.2, 10.9, 14.4)
plotdist(x1)

descdist(x1)

# "norm"대신에 다음과 같은 것들을 넣을 수 있다.
# "lnorm", "exp", "pois", "cauchy", "gamma", "logis", "nbinom", "geom", "beta", "weibull"
f1 <- fitdist(x1, "norm")
summary(f1)
plot(f1)
summary(gofstat(f1))
gofstat(f1)$chisqpvalue

> summary(f1)
Fitting of the distribution ' norm ' by maximum likelihood 
Parameters : 
      estimate Std. Error
mean 10.411111   1.118886
sd    4.747033   0.791172
Loglikelihood:  -53.57625   AIC:  111.1525   BIC:  112.9332 
Correlation matrix:
     mean sd
mean    1  0
sd      0  1 

> gofstat(f1)
Kolmogorov-Smirnov statistic:  0.1104554 
Cramer-von Mises statistic:  0.02848332 
Anderson-Darling statistic:  0.225598 
[1] 0.1847961

• Chi-squared p-value: 0.185
  
• 가설
  
  • 귀무가설: 차이가 없다. -> 정규분포다.
    
  • 대립가설: 차이가 있다. -> 정규분포가 아니다.
    
• p-value는 대립가설이 1종 오류가 나타날 확률이다. p-value=0.185로 유의확률 0.05보다 크므로 대립가설이 틀렸다고 볼 수 있다. 그러므로 귀무가설 채택. 즉, 정규분포라고 볼 수 있다.
  
  

x2<-c(rep(4,1),rep(2,3),rep(1,7),rep(0,12))
gofstat(fitdist(x2,"pois"))$chisqpvalue

Chi-squared statistic:  0.2507318 
[1] 0.6165603 

options(show.error.messages = FALSE)
dist_name <- c()
p_value <- c()
dist <- c("norm", "lnorm", "pois", "exp", "gamma", "nbinom", "geom", "beta", "unif", "logis")

for(i in dist)
{
  print(i)
  f1 <- try(with(trees, fitdist(Height, i, method="mme")))
  if(class(f1) == "try-error") next;
  
  pval <- gofstat(f1)$chisqpvalue
  if (pval >= 0.05)
  {
    print(paste("분포명:", i, ", p-value =", as.character(pval)))
    dist_name <- c(dist_name, i)
    p_value <- c(p_value, pval)
  }
}
print("chisq.test...")
print(data.frame(dist_name, p_value))
#with(trees, hist(Height))
with(trees, descdist(Height))

> print(data.frame(dist_name, p_value))
  dist_name p_value
1      norm   0.595
2     lnorm   0.537
3      pois   0.238
4     gamma   0.558

• 귀무가설: 정규분포와 차이가 없다.
  
• 대립가설: 정규분포와 차이가 있다.
  
  

> source("rftn.f")
> rad.d <-read.table("rad.d", header=T)
> shapiro.test(rad.d$open)

        Shapiro-Wilk normality test

data:  rad.d$open 
W = 0.8292, p-value = 1.977e-05

install.packages("fBasics")
library("fBasics")

tmp <- rnorm(10000)

ksnormTest(tmp)
shapiroTest(tmp)
jarqueberaTest(tmp) #이게 좋구나~~!!!!
dagoTest(tmp)

• http://rss.acs.unt.edu/Rdoc/library/fBasics/html/NormalityTests.html
  
• ksnormTest - Kolmogorov-Smirnov normality test,
  
• shapiroTest - Shapiro-Wilk's test for normality,
  
• jarqueberaTest - Jarque–Bera test for normality,
  
• dagoTest - D'Agostino normality test.
  
  
  

library(nortest)
ad.test(rnorm(10000, mean = 5, sd = 3))
ad.test(runif(10000, min = 2, max = 4))

x <- rnorm(100, mean = 0, sd = 1)
y <- rnorm(100, mean = 1, sd = 1)
ks.test(x,y)

> ks.test(x,y)

	Two-sample Kolmogorov-Smirnov test

data:  x and y 
D = 0.38, p-value = 1.071e-06
alternative hypothesis: two-sided 

• p-value = 1.071e-06 이므로 귀무가설 기각. 즉, 두 샘플은 다른 모집단 분포다.
  
• 비모수 검정으로 모집단에 대한 어떤 가정도 필요없다.
  
  

#install.packages("propagate")
library(propagate)
y <- c(0.0941,0.2372,0.2923,0.1750,0.0863,0.0419,0.0207,0.0128,0.0142,0.0071,0.0041,0.0031,0.0032,0.0057,0.0022,0.0001) 
fit.result <- fitDistr(y)
fit.result

#도움말을 찾아보면 함수의 내용이 무엇인지 알 수 있다.
print(propagate:::dJSB)

> fit.result <- fitDistr(y)
Fitting Normal distribution...Done.
Fitting Skewed-normal distribution...Done.
Fitting Generalized normal distribution...........10.........20.......Done.
Fitting Log-normal distribution...Done.
Fitting Scaled/shifted t- distribution...Done.
Fitting Logistic distribution...Done.
Fitting Uniform distribution...Done.
Fitting Triangular distribution...Done.
Fitting Trapezoidal distribution...Done.
Fitting Curvilinear Trapezoidal distribution...Done.
Fitting Generalized Trapezoidal distribution...Done.
Fitting Gamma distribution...Done.
Fitting Cauchy distribution...Done.
Fitting Laplace distribution...Done.
Fitting Gumbel distribution...Done.
Fitting Johnson SU distribution...........10.........20.........30.........40.........50
.........60.........70.........80.Done.
Fitting Johnson SB distribution...........10.........20.........30.........40.........50
.........60.........70.........80.Done.
Fitting 3P Weibull distribution...........10.........20.......Done.
Fitting 4P Beta distribution...Done.
Fitting Arcsine distribution...Done.
Fitting von Mises distribution...Done.
> fit.result
$aic
              Distribution      AIC
17              Johnson SB 613.7082
18              3P Weibull 624.7589
3       Generalized normal 624.7920
16              Johnson SU 626.7920
4               Log-normal 627.1804
12                   Gamma 634.9023
14                 Laplace 639.8102
13                  Cauchy 640.2875
5        Scaled/shifted t- 642.1013
15                  Gumbel 642.8420
2            Skewed-normal 646.8412
6                 Logistic 647.8476
1                   Normal 652.1788
9              Trapezoidal 654.7066
20                 Arcsine 670.4624
19                 4P Beta 705.6771
11 Generalized Trapezoidal 836.8772
21               von Mises 882.3921
7                  Uniform 897.6248
8               Triangular 905.4962
10 Curvilinear Trapezoidal 911.1446

$fit
$fit$Normal
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.000717984960447366, 0.00725890392042258 
residual sum-of-squares: 197174
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Skewed-normal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.00583038510142194, 0.0106443425236555, 8.50327745471171 
residual sum-of-squares: 174116
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Generalized normal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.0818148807363509, 0.0385147298611958, -2.26477841096039 
residual sum-of-squares: 119822
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Log-normal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -4.42473398922409, 2.05820723941306 
residual sum-of-squares: 129074
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Scaled/shifted t-`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.002960556811455, 0.00543082394110837, 0.847275538361585 
residual sum-of-squares: 160675
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$Logistic
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00106198716159507, 0.00448320432790849 
residual sum-of-squares: 183218
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$Uniform
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00249994429141502, 0.289899967178183 
residual sum-of-squares: 12634542
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$Triangular
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.58442884603643, 0.276474920439136, 0.00750016053696837 
residual sum-of-squares: 13956564
reason terminated: Relative error between `par' and the solution is at most `ptol'.

$fit$Trapezoidal
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.267697695351802, 0.162169769399358, -0.118546092194162, 0.0169541854752057 
residual sum-of-squares: 192315
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Curvilinear Trapezoidal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -3.51380272100041, 0.781019642275179, 0.778518731596065 
residual sum-of-squares: 15358740
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Generalized Trapezoidal`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.316014522465303, -0.0122579620659835, 0.00250001098860818, -0.0569154565231172, 16.9883350504257, -13.687268319418, 1.38700131982915 
residual sum-of-squares: 3808834
reason terminated: Relative error between `par' and the solution is at most `ptol'.

$fit$Gamma
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.365709818937871, 25.9401735353146 
residual sum-of-squares: 147123
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$Cauchy
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.0027034462361882, 0.00566077284092268 
residual sum-of-squares: 161183
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$Laplace
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00208489337704117, 0.0115719008947262 
residual sum-of-squares: 159884
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$Gumbel
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: -0.000564496234569831, 0.00585195632924723 
residual sum-of-squares: 168315
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Johnson SU`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00238945145777224, 2.41428652017113e-08, -6.58408257452447, 0.441518979946343 
residual sum-of-squares: 119822
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`Johnson SB`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00249961631424682, 0.290585603944236, 0.633747903543896, 0.35472053982219 
residual sum-of-squares: 95990
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`3P Weibull`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00206926024634554, 0.60287651598625, 0.0650136696113521 
residual sum-of-squares: 119755
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`4P Beta`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 243.96530522719, 244.003863012748, -0.219805373171359, 0.221019706262809 
residual sum-of-squares: 456251
reason terminated: Relative error between `par' and the solution is at most `ptol'.

$fit$Arcsine
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00241801362975452, 0.295326975917706 
residual sum-of-squares: 268803
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fit$`von Mises`
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00235128799831663, 314.160915522474 
residual sum-of-squares: 9759620
reason terminated: Number of iterations has reached `maxiter' == 1024.


$bestfit
Nonlinear regression via the Levenberg-Marquardt algorithm
parameter estimates: 0.00249961631424682, 0.290585603944236, 0.633747903543896, 0.35472053982219 
residual sum-of-squares: 95990
reason terminated: Relative error in the sum of squares is at most `ftol'.

$fitted
 [1] 62.219673 20.891649 12.605071  9.187138  7.285103  6.065145  5.213740  4.585130  4.101920  3.719054  3.408463
[12]  3.151718  2.936210  2.753014  2.595632  2.459222  2.340100  2.235416  2.142929  2.060858  1.987768  1.922492
[23]  1.864074  1.811724  1.764788  1.722718  1.685056  1.651422  1.621494  1.595006  1.571740  1.551516  1.534193
[34]  1.519663  1.507852  1.498715  1.492237  1.488439  1.487372  1.489126  1.493832  1.501671  1.512883  1.527777
[45]  1.546754  1.570328  1.599166  1.634138  1.676391  1.727475  1.789524  1.865572  1.960094  2.080031  2.236881
[56]  2.451540  2.767639  3.300214  4.355798

$residuals
 [1]  0.2803273  4.1083505 12.3949289 -9.1871379  5.2148973 -6.0651453 -5.2137405 -4.5851299  8.3980803 -3.7190542
[11] -3.4084626 -3.1517175 -2.9362096 -2.7530136 -2.5956321 -2.4592222 -2.3401002 10.2645843 10.3570709 -2.0608580
[21] -1.9877677 -1.9224916 -1.8640737 -1.8117243 -1.7647878 -1.7227176 -1.6850565 -1.6514218 -1.6214937 -1.5950062
[31] -1.5717396 -1.5515155 -1.5341925 -1.5196632 10.9921481 -1.4987145 -1.4922374 -1.4884391 -1.4873721 -1.4891258
[41] -1.4938321 -1.5016714 -1.5128828 -1.5277768 -1.5467535 -1.5703281 -1.5991664 10.8658625 -1.6763910 -1.7274750
[51] -1.7895244 -1.8655720 -1.9600941 -2.0800311 -2.2368812 -2.4515397 -2.7676387 -3.3002136  8.1442017

############### distribution fitting ###################
## skew-normal distribution, taken from package 'VGAM'
dsn <- function (x, location = 0, scale = 1, shape = 0, log = FALSE) 
{
  zedd <- (x - location)/scale
  loglik <- log(2) + dnorm(zedd, log = TRUE) + pnorm(shape * zedd, log.p = TRUE) - log(scale)
  if (log) loglik else exp(loglik)  
}

## generalized normal distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dgnorm <- function(x, alpha = 1, xi = 1, kappa = -0.1) 
{
  1/(exp(log(1 - (kappa * (x - xi))/alpha)^2/(2 * kappa^2)) * (sqrt(2 * pi) * (alpha - x * kappa + kappa * xi))) 
}

## scaled and shifted t-distribution,
dst <- function (x, mean = 0, sd = 1, df = 2) 
{
  dt((x - mean)/sd, df = df)/sd
}

## Gumbel distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dgumbel <- function(x, location = 0, scale = 1) {
  z <- (x - location)/scale
  (1/scale) * exp(-z - exp(-z))
}

## Johnson SU distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dJSU <- function (x, xi = 0, lambda = 1, gamma = 1, delta = 1) 
{
  z <- (x - xi)/lambda
  delta/(lambda * sqrt(2 * pi) * sqrt(z^2 + 1)) * exp(-0.5 * (gamma + delta * log(z + sqrt(z^2 + 1)))^2)
}

## Johnson SB distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dJSB <- function (x, xi = 0, lambda = 1, gamma = 1, delta = 1) 
{
  z <- (x - xi)/lambda
  delta/(lambda * sqrt(2 * pi) * z * (1 - z)) * exp(-0.5 * (gamma + delta * log(z/(1 - z)))^2)
}

## three-parameter weibull distribution with location gamma,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dweibull2 <- function(x, location = 0, shape = 1, scale = 1) {
  (shape/scale) * ((x - location)/scale)^(shape - 1) * exp(-((x - location)/scale)^shape)  
}

## four-parameter beta distribution with boundary parameters,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dbeta2 <- function(x, alpha1 = 1, alpha2 = 1, a = 0, b = 0) {
  (1/beta(alpha1, alpha2)) * ((x - a)^(alpha1 - 1) * (b - x)^(alpha2 - 1))/(b - a)^(alpha1 + alpha2 - 1)
}

## triangular distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dtriang <- function(x, a = 0, b = 1, c = 2) {
  y <- numeric(length(x))
  y[x < a] <- 0
  y[a <= x & x <= b] <- 2 * (x[a <= x & x <= b] - a)/((c - a) * (b - a))
  y[b < x & x <= c] <- 2 * (c - x[b < x & x <= c])/((c - a) * (c - b))
  y[c < x] <- 0  
  return(y)
}

## trapezoidal distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dtrap <- function(x, a = 0, b = 1, c = 2, d = 3) {
  y <- numeric(length(x))
  u <- 2/(d + c - b - a)
  y[x < a] <- 0
  y[a <= x & x < b] <- u * (x[a <= x & x < b] - a)/(b - a)
  y[b <= x & x < c] <- u
  y[c <= x & x < d] <- u * (d - x[c <= x & x < d])/(d - c)  
  y[d <= x] <- 0
  return(y)
}

## curvilinear trapezoidal distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dctrap <- function(x, a = 0, b = 1, d = 0.1) {
  y <- numeric(length(x))
  mp <- (a + b)/2
  sw <- (b - a)/2
  y[x < a - d] <- 0
  y[a - d <= x & x <= a + d] <- log((sw + d)/(mp - x[a - d <= x & x <= a + d]))
  y[a + d < x & x <= b - d] <- log((sw + d)/(sw - d))
  y[b - d <= x & x <= b + d] <- log((sw + d)/(x[b - d <= x & x <= b + d] - mp))
  y[b + d < x] <- 0  
  return(y)
}

## Generalized Trapezoidal distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dgtrap <- function (x, min = 0, mode1 = 1, mode2 = 2, max = 3, n1 = 2, n3 = 2, alpha = 1, log = FALSE) 
{
  y <- numeric(length(x))
  nc <- (2 * n1 * n3) / ((2 * alpha * (mode1 - min) * n3) + ((alpha + 1) * (mode2 - mode1) * n1 * n3) + 
                        (2 * (max - mode2) * n1))
  y[min <= x & x < mode1] <- nc * alpha * ((x[min <= x & x < mode1] - min) / (mode1 - min))^(n1 - 1)
  y[mode1 <= x & x < mode2] <-  nc * (((1 - alpha) * ((x[mode1 <= x & x < mode2] - mode1) / (mode2 - mode1))) + alpha)
  y[mode2 <= x & x <= max] <- nc * ((max - x[mode2 <= x & x <= max]) / (max - mode2))^(n3 - 1)
    
  if (log)  y <- log(y)
  if (any(is.nan(y))) {
    warning("NaN in dtrapezoid")
  }
  else if (any(is.na(y))) {
    warning("NA in dtrapezoid")
  }
  return(y)
}

## Laplacian distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dlaplace <- function(x, mean = 0, sigma = 1) {
  1/(sqrt(2) * sigma) * exp(-(sqrt(2) * abs(x - mean))/sigma)
}

## Arcsine distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
darcsin <- function(x, a = 0, b = 1) { 
  y <- numeric(length(x))  
  y[x <= a] <- 0
  y[a < x & x < b] <- 1/(pi * sqrt((b - x[a < x & x < b]) * (x[a < x & x < b] - a)))
  y[b <= x] <- 0
  return(y)
}

## von Mises distribution,
## taken from PDF in Mathematica's "Ultimate Univariate Probability Distribution Explorer"
dmises <- function(x, mu = 0, kappa = 1) {  
  y <- numeric(length(x))  
  y[x < -pi + mu] <- 0
  y[-pi + mu <= x & x <= pi + mu] <- exp(kappa * cos(x[-pi + mu <= x & x <= pi + mu] - mu))/(2 * pi * besselI(kappa, 0))
  y[pi + mu < x] <- 0
  return(y)
}

• Fitting Distributions
  
• R Distributions
  
• http://blog.naver.com/enjoystat?Redirect=Log&logNo=140109260238
  
• fitdistrplus.pdf
  
• FITTING_DISTRIBUTIONS_WITH_R.pdf
  
• intro2fitdistrplus.pdf