Fitting Probability distribution in R
Discrete and Continuous random Variables
Eralda Gjika (Dhamo)
18 January 2021
Fitting Probability distributions
DATASET CYBER DATA
data used for this session are imported from: https://osf.io/aj7vr/
# read the data and save it as "cyber.data"
head(cyber.data,10)Because the data are weekly observations we can have as a first graph a time series plot.
library(forecast)
# ts.plot() will create a time series plot
ts.plot(cyber.data[,2],main="Event count", col="red",xlab="week", ylab="number of attacs",lwd=2)
text(280,120,"Maximum",col="blue")# will add text to our graph
ts.plot(cyber.data[,3],main="Average length", col="blue",xlab="week", ylab="Average time of attacs",lwd=2)
text(110,1700,"Maximum",col="red")
summary(cyber.data[,c(2,3)])# descriptive statistics of our dataset event.count avg.report.length
Min. : 0.00 Min. : 0.0
1st Qu.: 10.00 1st Qu.: 394.0
Median : 21.00 Median : 509.0
Mean : 25.42 Mean : 533.7
3rd Qu.: 35.00 3rd Qu.: 654.0
Max. :126.00 Max. :1700.0 Analyse the data by observing boxplot for any outliers.
boxplot(cyber.data[,2],col="green",main="BoxPlot attacs",xlab="Count")
boxplot(cyber.data[,3],col="orange",main="BoxPlot av.length",xlab="time")
#
par(mfrow=c(1,2))
hist(cyber.data[,2],col="green",main="Histogram attacs",xlab="number of attacs",ylab="Freq")
hist(cyber.data[,3],col="orange",main="Histogram av.length",xlab="time")
we observe some outliers in both variables, especially in the upper region. This was also observed in the time series plot. For now we are not dealing with this outliers . The aim of this material is just to show some basic functions from the library(fitdistr) and library(fitdistrplus) which are used for fitting probability distributions to a vector of numerical data.
FITTING DISTRIBUTIONS IN R
Packages needed for this part are:
library(fitdistrplus)
library(stats4)
library(MASS)
# for other necessary test or graphical tools
library(survival)
library(actuar)
library(distrMod)Probability density function (p.d.f) and Cummulative density function (c.d.f)
We can use the function plotdist(data) to obtain the histogram and the cummulative distribution graph of teh data.
plotdist(cyber.data$event.count, histo = TRUE, demp = TRUE)
plotdist(cyber.data$avg.report.length, histo = TRUE, demp = TRUE)
Exercise: Try to simulate 10^5 observations from the most known probability distributions you know and plot their Empirical density and Cummulative distribution. Are these graphs as you expected?
Skewness-kurtosis graph for the choice of distributions (Cullen and Frey, 1999)
The function descdist() provides a skewness-kurtosis graph to help to choose the best candidate(s) to fit a given dataset. If we want to use it for discrete distributions we may use argument discrete=TRUE.
descdist(cyber.data[,3])summary statistics
------
min: 0 max: 1700
median: 509
mean: 533.7213
estimated sd: 211.2284
estimated skewness: 1.014491
estimated kurtosis: 6.198751 
descdist(cyber.data[,3], boot = 1000)# bootstrapped values 1000summary statistics
------
min: 0 max: 1700
median: 509
mean: 533.7213
estimated sd: 211.2284
estimated skewness: 1.014491
estimated kurtosis: 6.198751 
descdist(cyber.data[,3], boot = 10000)# bootstrapped values 10^4summary statistics
------
min: 0 max: 1700
median: 509
mean: 533.7213
estimated sd: 211.2284
estimated skewness: 1.014491
estimated kurtosis: 6.198751 
# observe the concentration of the simulations
descdist(cyber.data[,3], boot = 1000,discrete=TRUE)summary statistics
------
min: 0 max: 1700
median: 509
mean: 533.7213
estimated sd: 211.2284
estimated skewness: 1.014491
estimated kurtosis: 6.198751 
Trying to fit probability distributions to the number of cyber attacs.
# Negative Binomial
nbinom.f<- fitdist(cyber.data$event.count, "nbinom") NaNs producedNaNs producedNaNs producedpois.f<- fitdist(cyber.data$event.count, "pois")# PoissonNaNs producedNaNs producednorm.f <- fitdist(cyber.data$event.count, "norm") # NormalNaNs producedNaNs producedexp.f <- fitdist(cyber.data$event.count, "exp") # ExponentialNaNs producedNaNs producedInformation about the parameter estimation
summary(nbinom.f)Fitting of the distribution ' nbinom ' by maximum likelihood
Parameters : Loglikelihood: -1536.555 AIC: 3077.11 BIC: 3084.916
Correlation matrix:
size mu
size 1.000000e+00 7.175825e-06
mu 7.175825e-06 1.000000e+00summary(pois.f)Fitting of the distribution ' pois ' by maximum likelihood
Parameters : Loglikelihood: -3522.192 AIC: 7046.384 BIC: 7050.287 summary(norm.f)Fitting of the distribution ' norm ' by maximum likelihood
Parameters : Loglikelihood: -1612.522 AIC: 3229.044 BIC: 3236.849
Correlation matrix:
mean sd
mean 1 0
sd 0 1summary(exp.f)Fitting of the distribution ' exp ' by maximum likelihood
Parameters : Loglikelihood: -1550.139 AIC: 3102.277 BIC: 3106.18 Compare the distributions
we can use these graphical test to compare the fitted distribiutions with the real observations. We expect the distributions to follow the patterns of teh real observations in the histogram, QQ-plot, CDF plot and also PP-plot.
par(mfrow = c(2, 2))
plot.legend <- c("Binomial neg", "Poisson","Normal","Exponential")
denscomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)
qqcomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)cdfcomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)
ppcomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)
Simulate based on the parameter estimated
Scatterplot view. we expect a high linear relationship.We use now the parameter estimated above to simulate random observations from these distributions. And we are using the scatterplot to understand the fitting. We are looking for scatterplots which show a linear pattern.
set.seed(512)
Ex=rexp(length(cyber.data$event.count),0.03934638)# estimated rate= 0.03934638
plot(cyber.data$event.count,Ex)# scatterplot observed vs fitted
No.r=rnorm(length(cyber.data$event.count),25.41530,19.82332) # estimated parameters
plot(cyber.data$event.count,No.r)# scatterplot observed vs fitted
Po=rpois(length(cyber.data$event.count),25.4153)# estimated parameters
plot(cyber.data$event.count,Po)# scatterplot observed vs fitted
neg.bin=rnbinom(length(cyber.data$event.count),size=1.658441, mu=25.415193)# estimated parameters
plot(cyber.data$event.count,neg.bin)# scatterplot observed vs fitted
Two fitted distribution
To observe more clear the fitting between the two most accurate fitting we can construct graphics above only for these two distributions.
par(mfrow = c(2, 2))
plot.legend <- c("Normal","Exponencial")
denscomp(list(norm.f,exp.f), legendtext = plot.legend)
qqcomp(list(norm.f,exp.f), legendtext = plot.legend)cdfcomp(list(norm.f,exp.f), legendtext = plot.legend)
ppcomp(list(norm.f,exp.f), legendtext = plot.legend)
Compare the Negative Binomial fit and the normal fit.
nbinom.f <- fitdist(cyber.data$event.count, "nbinom") # NaNs producedNaNs producedNaNs producednorm.f <- fitdist(cyber.data$event.count, "norm") # NaNs producedNaNs produced#
par(mfrow = c(2, 2))
plot.legend <- c("Binomial neg", "Normal")
denscomp(list(nbinom.f,norm.f), legendtext = plot.legend)
qqcomp(list(nbinom.f,norm.f), legendtext = plot.legend)cdfcomp(list(nbinom.f,norm.f), legendtext = plot.legend)
ppcomp(list(nbinom.f,norm.f), legendtext = plot.legend)
Average time of attacks
First we substitute the 0 values with values very close to 0 (example 0.1) because for some probability distributions there are conditions of values > zero.
avg.report.length=ifelse(cyber.data$avg.report.length==0,0.01,cyber.data$avg.report.length)
expo.f <- fitdist(avg.report.length, "exp",method="mme")# NaNs producedNaNs producedlnorm.f<- fitdist(avg.report.length, "lnorm",method="mme")# NaNs producedNaNs producednorma.f<- fitdist(avg.report.length, "norm")# NaNs producedNaNs producedweib <- fitdist(avg.report.length, "weibull", start = list(shape = 1, scale = 500))# NaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedlogis.f <- fitdist(avg.report.length, "llogis",start=NULL, fix.arg=NULL, discrete=FALSE, keepdata = TRUE)NaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedNaNs producedCullen and Frey graph
descdist(avg.report.length)summary statistics
------
min: 0.01 max: 1700
median: 509
mean: 533.7214
estimated sd: 211.2282
estimated skewness: 1.0145
estimated kurtosis: 6.198748
descdist(avg.report.length, boot = 1000)# bootstrapped values 1000summary statistics
------
min: 0.01 max: 1700
median: 509
mean: 533.7214
estimated sd: 211.2282
estimated skewness: 1.0145
estimated kurtosis: 6.198748 
descdist(avg.report.length, boot = 10000)# bootstrapped values 10^4summary statistics
------
min: 0.01 max: 1700
median: 509
mean: 533.7214
estimated sd: 211.2282
estimated skewness: 1.0145
estimated kurtosis: 6.198748 
# observe the concentration of the simulations
descdist(avg.report.length, boot = 1000,discrete=TRUE)summary statistics
------
min: 0.01 max: 1700
median: 509
mean: 533.7214
estimated sd: 211.2282
estimated skewness: 1.0145
estimated kurtosis: 6.198748 
Parameter estimations:
summary(expo.f)Fitting of the distribution ' exp ' by matching moments
Parameters : Loglikelihood: -2664.434 AIC: 5330.868 BIC: 5334.77 summary(lnorm.f)Fitting of the distribution ' lnorm ' by matching moments
Parameters : Loglikelihood: -3620.588 AIC: 7245.177 BIC: 7252.982 summary(norma.f)Fitting of the distribution ' norm ' by maximum likelihood
Parameters : Loglikelihood: -2478.007 AIC: 4960.013 BIC: 4967.818
Correlation matrix:
mean sd
mean 1 0
sd 0 1summary(weib)Fitting of the distribution ' weibull ' by maximum likelihood
Parameters : Loglikelihood: -2510.2 AIC: 5024.4 BIC: 5032.205
Correlation matrix:
shape scale
shape 1.0000000 0.2850167
scale 0.2850167 1.0000000summary(logis.f)Fitting of the distribution ' llogis ' by maximum likelihood
Parameters : Loglikelihood: -2538.638 AIC: 5081.277 BIC: 5089.082
Correlation matrix:
shape scale
shape 1.00000000 0.02488679
scale 0.02488679 1.00000000Graphic test check:
par(mfrow = c(2, 2))
plot.legend <- c("Exponential","Normal", "log-Normal","Weibull","Logis")
denscomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)
qqcomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)cdfcomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)
ppcomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)
#
gofstat(list(expo.f,norma.f,lnorm.f,weib,logis.f),fitnames = c("Exponential","Normal", "log-Normal","Weibull","Logis"))Goodness-of-fit statistics
Exponential Normal log-Normal
Kolmogorov-Smirnov statistic 0.3433618 0.06189011 0.03403799
Cramer-von Mises statistic 13.4476900 0.42673770 0.14139236
Anderson-Darling statistic 66.2372441 2.74835073 11.17728575
Weibull Logis
Kolmogorov-Smirnov statistic 0.09875598 0.08651839
Cramer-von Mises statistic 0.76792533 0.51785182
Anderson-Darling statistic 5.80145260 4.89002332
Goodness-of-fit criteria
Exponential Normal log-Normal
Akaike's Information Criterion 5330.868 4960.013 7245.177
Bayesian Information Criterion 5334.770 4967.818 7252.982
Weibull Logis
Akaike's Information Criterion 5024.400 5081.277
Bayesian Information Criterion 5032.205 5089.082We are looking for the lowest value of the test statistics and also the lowest value of the information criteria. The fitting which has the lowest value is supposed to be the most accurate among those compared. In this situation the “best” fitting of the data is from the Normal or Log-normal distributions. (observe the values of test statistics and AIC/BIC).
Log-normal and Normal distributions
par(mfrow = c(2, 2))
plot.legend <- c("Normal","Log-normal")
denscomp(list(norma.f,logis.f), legendtext = plot.legend)
qqcomp(list(norma.f,logis.f), legendtext = plot.legend)cdfcomp(list(norma.f,logis.f), legendtext = plot.legend)
ppcomp(list(norma.f,logis.f), legendtext = plot.legend)
Simulation Normal and Log normal
Log.nor=rlnorm(length(cyber.data$avg.report.length),6.2073341,0.3809332)
plot(cyber.data$avg.report.length,Log.nor)# 
Nor=rnorm(length(cyber.data$avg.report.length),533.7214 ,210.9395)
plot(cyber.data$avg.report.length,Nor)#
#
par(mfrow=c(1,3))
hist(cyber.data$avg.report.length)
hist(Log.nor)hist(Nor)
Eralda Gjika (Dhamo)
January 2021
Reference:
https://www.r-project.org/conferences/useR-2009/slides/Delignette-Muller+Pouillot+Denis.pdf
rdocumentation.org/packages/fitdistrplus/versions/1.1-3/topics/descdist
---
title: "Fitting Probability distribution in R"
subtitle: "Discrete and Continuous random Variables"
author: "Eralda Gjika (Dhamo)"
date: "18 January 2021"
output: html_notebook
---


# Fitting Probability distributions


# DATASET CYBER DATA
 data used for this session are imported from: https://osf.io/aj7vr/
  
```{r}
# read the data and save it as "cyber.data"
head(cyber.data,10)
```

Because the data are weekly observations we can have as a first graph a time series plot.
```{r}
library(forecast)
# ts.plot() will create a time series plot 
ts.plot(cyber.data[,2],main="Event count", col="red",xlab="week", ylab="number of attacs",lwd=2) 
text(280,120,"Maximum",col="blue")# will add text to our graph
ts.plot(cyber.data[,3],main="Average length", col="blue",xlab="week", ylab="Average time of attacs",lwd=2)
text(110,1700,"Maximum",col="red")
summary(cyber.data[,c(2,3)])# descriptive statistics of our dataset
```
Analyse the data by observing boxplot for any outliers.
```{r}
boxplot(cyber.data[,2],col="green",main="BoxPlot attacs",xlab="Count")
boxplot(cyber.data[,3],col="orange",main="BoxPlot av.length",xlab="time")
#
par(mfrow=c(1,2))
hist(cyber.data[,2],col="green",main="Histogram attacs",xlab="number of attacs",ylab="Freq")
hist(cyber.data[,3],col="orange",main="Histogram av.length",xlab="time")
```
we observe some outliers in both variables, especially in the upper region. This was also observed in the time series plot. 
For now we are not dealing with this outliers . The aim of this material is just to show some basic functions from the library(fitdistr) and library(fitdistrplus) which are used for fitting probability distributions to a vector of numerical data. 

#   FITTING DISTRIBUTIONS IN R
Packages needed for this part are:

```{r}
library(fitdistrplus)
library(stats4)
library(MASS)
# for other necessary test or graphical tools
library(survival)
library(actuar)
library(distrMod)
```

### Probability density function (p.d.f) and Cummulative density function (c.d.f) 

We can use the function plotdist(data) to obtain the histogram and the cummulative distribution graph of teh data.
```{r}
plotdist(cyber.data$event.count, histo = TRUE, demp = TRUE)
plotdist(cyber.data$avg.report.length, histo = TRUE, demp = TRUE)
```
Exercise: Try to simulate 10^5 observations from the most known probability distributions you know and plot their Empirical density and Cummulative distribution. Are these graphs as you expected?

## Skewness-kurtosis graph for the choice of distributions (Cullen and Frey, 1999)

The function descdist() provides a skewness-kurtosis graph to help to
choose the best candidate(s) to fit a given dataset. 
If we want to use it for discrete distributions we may use argument discrete=TRUE.

```{r}
descdist(cyber.data[,3])
descdist(cyber.data[,3], boot = 1000)# bootstrapped values 1000
descdist(cyber.data[,3], boot = 10000)# bootstrapped values 10^4
# observe the concentration of the simulations
descdist(cyber.data[,3], boot = 1000,discrete=TRUE)
```
Trying to fit probability distributions to the number of cyber attacs. 
```{r}
# Negative Binomial
nbinom.f<- fitdist(cyber.data$event.count, "nbinom") 
pois.f<- fitdist(cyber.data$event.count, "pois")# Poisson
norm.f <- fitdist(cyber.data$event.count, "norm") # Normal
exp.f <- fitdist(cyber.data$event.count, "exp") # Exponential
```
Information about the parameter estimation
```{r}
summary(nbinom.f)
summary(pois.f)
summary(norm.f)
summary(exp.f)
```
###  Compare the distributions

we can use these graphical test to compare the fitted distribiutions with the real observations. We expect the distributions to follow the patterns of teh real observations in the histogram, QQ-plot, CDF plot and also PP-plot. 
```{r}
par(mfrow = c(2, 2))
plot.legend <- c("Binomial neg", "Poisson","Normal","Exponential")
denscomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)
qqcomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)
cdfcomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)
ppcomp(list(nbinom.f,pois.f,norm.f,exp.f), legendtext = plot.legend)
```

### Simulate based on the parameter estimated

Scatterplot view. we expect a high linear relationship.We use now the parameter estimated above to simulate random observations from these distributions. And we are using the scatterplot to understand the fitting. We are looking for scatterplots which show a linear pattern.

```{r}
set.seed(512)
Ex=rexp(length(cyber.data$event.count),0.03934638)# estimated rate= 0.03934638
plot(cyber.data$event.count,Ex)# scatterplot observed vs fitted

No.r=rnorm(length(cyber.data$event.count),25.41530,19.82332) # estimated parameters
plot(cyber.data$event.count,No.r)#  scatterplot observed vs fitted

Po=rpois(length(cyber.data$event.count),25.4153)# estimated parameters
plot(cyber.data$event.count,Po)#  scatterplot observed vs fitted

neg.bin=rnbinom(length(cyber.data$event.count),size=1.658441, mu=25.415193)# estimated parameters
plot(cyber.data$event.count,neg.bin)#  scatterplot observed vs fitted
```

# Two fitted distribution 
To observe more clear the fitting between the two most accurate fitting we can construct graphics above only for these two distributions.
```{r}
par(mfrow = c(2, 2))
plot.legend <- c("Normal","Exponencial")
denscomp(list(norm.f,exp.f), legendtext = plot.legend)
qqcomp(list(norm.f,exp.f), legendtext = plot.legend)
cdfcomp(list(norm.f,exp.f), legendtext = plot.legend)
ppcomp(list(norm.f,exp.f), legendtext = plot.legend)
```
Compare the Negative Binomial fit and the normal fit.
```{r}
nbinom.f <- fitdist(cyber.data$event.count, "nbinom") # 
norm.f <- fitdist(cyber.data$event.count, "norm") # 
#
par(mfrow = c(2, 2))
plot.legend <- c("Binomial neg", "Normal")
denscomp(list(nbinom.f,norm.f), legendtext = plot.legend)
qqcomp(list(nbinom.f,norm.f), legendtext = plot.legend)
cdfcomp(list(nbinom.f,norm.f), legendtext = plot.legend)
ppcomp(list(nbinom.f,norm.f), legendtext = plot.legend)
```


# Average time of attacks

First we substitute the 0 values with values very close to 0 (example 0.1) because for some probability distributions there are conditions of values > zero.

```{r}
avg.report.length=ifelse(cyber.data$avg.report.length==0,0.01,cyber.data$avg.report.length)
expo.f <- fitdist(avg.report.length, "exp",method="mme")# 
lnorm.f<- fitdist(avg.report.length, "lnorm",method="mme")# 
norma.f<- fitdist(avg.report.length, "norm")#  
weib <- fitdist(avg.report.length, "weibull", start = list(shape = 1, scale = 500))# 
logis.f <- fitdist(avg.report.length, "llogis",start=NULL, fix.arg=NULL, discrete=FALSE, keepdata = TRUE)
```
Cullen and Frey graph 
```{r}
descdist(avg.report.length)
descdist(avg.report.length, boot = 1000)# bootstrapped values 1000
descdist(avg.report.length, boot = 10000)# bootstrapped values 10^4
# observe the concentration of the simulations
descdist(avg.report.length, boot = 1000,discrete=TRUE)
```
 
 Parameter estimations:
```{r}
summary(expo.f)
summary(lnorm.f)
summary(norma.f)
summary(weib)
summary(logis.f)
```
Graphic test check:
```{r}
par(mfrow = c(2, 2))
plot.legend <- c("Exponential","Normal", "log-Normal","Weibull","Logis")
denscomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)
qqcomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)
cdfcomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)
ppcomp(list(expo.f,norma.f,lnorm.f,weib,logis.f), legendtext = plot.legend)
#
gofstat(list(expo.f,norma.f,lnorm.f,weib,logis.f),fitnames = c("Exponential","Normal", "log-Normal","Weibull","Logis"))
```
We are looking for the lowest value of the test statistics and also the lowest value of the information criteria. The fitting which has the lowest value is supposed to be the most accurate among those compared. In this situation the "best" fitting of the data is from the Normal or Log-normal distributions. (observe the values of test statistics and AIC/BIC). 

# Log-normal and Normal distributions
```{r}
par(mfrow = c(2, 2))
plot.legend <- c("Normal","Log-normal")
denscomp(list(norma.f,logis.f), legendtext = plot.legend)
qqcomp(list(norma.f,logis.f), legendtext = plot.legend)
cdfcomp(list(norma.f,logis.f), legendtext = plot.legend)
ppcomp(list(norma.f,logis.f), legendtext = plot.legend)
```


# Simulation Normal and Log normal 
```{r}
Log.nor=rlnorm(length(cyber.data$avg.report.length),6.2073341,0.3809332)
plot(cyber.data$avg.report.length,Log.nor)# 
Nor=rnorm(length(cyber.data$avg.report.length),533.7214	,210.9395)
plot(cyber.data$avg.report.length,Nor)# 
# 
par(mfrow=c(1,3))
hist(cyber.data$avg.report.length)
hist(Log.nor)
hist(Nor)
```


### Eralda Gjika (Dhamo)
## January 2021


Reference:

https://www.r-project.org/conferences/useR-2009/slides/Delignette-Muller+Pouillot+Denis.pdf

rdocumentation.org/packages/fitdistrplus/versions/1.1-3/topics/descdist