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growth_dist_medium.R
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wdir <- ""
dataDir <- "data/"
packagesFile <- "packages.txt"
#source(paste(wdir, "functions.R", sep="")) ### this also loads every needed package
#loadDatasets(paste(wdir,dataDir,sep="")) ###USE THIS IF YOU CURRENTLY HAVEN'T DATASETS IN WORKSPACE
source(paste(wdir, "growth_rate_dist.R", sep=""))
small_growth <- groupByGrowth(small_firms)$Growth
medium_growth <- groupByGrowth(medium_firms)$Growth
large_growth <- groupByGrowth(large_firms)$Growth
sample_size <-5000
growth_medium <- sample(medium_growth, sample_size)
############################################################################################################
fit_norm <- fitdist(growth_medium, distr="norm", method="mle")
fit_cauchy <- fitdist(growth_medium, distr="cauchy", method="mle")
fit_laplace1 <-fitdist(growth_medium, distr = "laplace", method = "mle", start=list(location=-0.1, scale=0.001))
est1 <- c()
est2 <- c()
for (i in 1:1000) {
fcauchy<-fitdist(sample(medium_growth, sample_size), method="mle", distr = "cauchy")
est1[i] <- fcauchy$estimate[1]
est2[i] <- fcauchy$estimate[2]
}
plotConfidInterv(est2, fit_cauchy$estimate[2], xtitle = "Scale parameter distribution")
############################################################################################################
################################################################################################
mean(growth_medium)
length(growth_medium)
shifted<- growth_medium + abs(min(growth_medium)) + .000001 # to fit distributions that require values to be positive
min(shifted)
scaled <- (growth_medium - min(growth_medium) +0.000001) /(max(growth_medium) - min(growth_medium)+ 0.000002)
#fit distributions that require values to be within ]0, 1[
#Powerlaw fit using the PowerLaw library
# m_m = conpl$new(shifted)
# est = estimate_xmin(m_m)
# m_m$setXmin(est)
# bs_p <- bootstrap_p(m_m, no_of_sims=1000, threads=8)
# bs_p$p
#Get an idea of the possible distribution for the empirical data
descdist(growth_medium, discrete = FALSE)
#################################################################################################
#Fit distributions using fitdist and mle2
#norm, exp, gamma, beta, weibull, cauchy, pareto, logis, lnorm, laplace
growth <- growth_medium; fit_laplace2 <- mle2(LL2, start=list(m=-0.1, s=0.001), method = "L-BFGS-B", lower = c(m=-0.1, s=0.001))
fit_logis <-fitdist(growth_medium, distr="logis", method="mle")
fit_lnorm <- fitdist(shifted, distr="lnorm", method="mle")
fit_pareto <- mle2(LL3, start = list(m = 1, s = min(shifted)), method = "L-BFGS-B", lower = c(m=0.001), fixed = list(s=(min(shifted))))
fit_exp <- fitdist(shifted, distr="exp", method="mle")
fit_weib <- fitdist(shifted, distr="weibull", method="mle")
fit_gamma <- fitdist(shifted, distr="gamma", method="mle")
fit_beta <- fitdist(scaled, distr="beta", method="mle")
##################################################################################################
#Compute confidence intervals for the estimated parameters using boostrap
# For mle2 objects:
confint(fit_laplace2) # gives the confidence intervals of the estimated parameters (default is 95% c.l.)
plot(profile(fit_laplace2)) # plots profile (confidence intervals) of the parameters
# For fitdistr objects:
boot <- bootdist(fit_cauchy, bootmethod = "param", niter = 1000) #uses parametric bootsrap to generate
# 1000 samples and compute their parameters according to the given distribution
boot$CI[,-1] # returns the 95% bootstrap CIs for all parameters
##################################################################################################
summary(fit_norm)
summary(fit_cauchy)
summary(fit_laplace1)
summary(fit_laplace2)
summary(fit_logis)
summary(fit_lnorm)
summary(fit_pareto)
summary(fit_exp)
summary(fit_weib)
summary(fit_gamma)
summary(fit_beta)
#KS test
ks.test(growth_medium, "pnorm", fit_norm$estimate[1], fit_norm$estimate[2])
ks.test(growth_medium, "pcauchy", fit_cauchy$estimate[1], fit_cauchy$estimate[2])
ks.test(growth_medium, "plaplace", fit_laplace1$estimate[1], fit_laplace1$estimate[2])
ks.test(growth_medium, "plaplace", coef(fit_laplace2)[1], coef(fit_laplace2)[2])
ks.test(growth_medium, "plogis",fit_logis$estimate[1], fit_logis$estimate[2])
ks.test(shifted, "plnorm", fit_lnorm$estimate[1] , fit_lnorm$estimate[2])
ks.test(shifted, "ppareto", coef(fit_pareto)[1], coef(fit_pareto)[2])
ks.test(shifted, "pexp", fit_exp$estimate[1])
ks.test(shifted, "pweibull", fit_weib$estimate[1], fit_weib$estimate[2])
ks.test(shifted, "pgamma", fit_gamma$estimate[1], fit_gamma$estimate[2])
ks.test(scaled, "pbeta",fit_beta$estimate[1], fit_beta$estimate[2])
#############################################################################
# Generates 1000 ks test statistics using parametric bootstrap and returns
# a p-value based on a logspline distribution fit
n.sims <- 1000
stats <- replicate(n.sims, {
r <- rcauchy(length(growth_medium)
, fit_cauchy$estimate[1]
, fit_cauchy$estimate[2])
as.numeric(ks.test(r
, "pcauchy"
, fit_cauchy$estimate[1]
, fit_cauchy$estimate[2])$statistic)})
fit_log <- logspline(stats)
1-plogspline(ks.test(growth_medium
, "pcauchy"
, fit_cauchy$estimate[1]
, fit_cauchy$estimate[2])$statistic
, fit_log
)
##############################################################################
# Take a look at the AICs and BICs among other gof statistics
gof_original <- gofstat(list(fit_norm, fit_cauchy,fit_logis, fit_laplace1))
gof_shifted <- gofstat(list(fit_lnorm, fit_exp, fit_weib, fit_gamma))
gof_scaled <-gofstat(fit_beta)
gof_laplace<-c(AIC(fit_laplace2), BIC(fit_laplace2)) # BIC is NA. It can be retrieved
gof_pareto<-c(AIC(fit_pareto), BIC(fit_pareto)) # using mle of {stats4} instead of mle2
gof_original
gof_original$chisqpvalue # returns the Chi-square p-values for the given fits. (They are surprisingly low)
gof_laplace
gof_pareto
gof_shifted
gof_scaled
#####################################################################################
breaks <- -Inf
breaks <- append (breaks, c(seqlast(min(growth_medium),max(growth_medium),0.2), Inf))
growth_medium.cut<-cut(growth_medium,breaks=breaks, include.lowest = TRUE) #binning data
#table(growth_medium.cut) # frequencies of empirical data in the respective bins
p<-c()
for (i in 1:(length(breaks)-1)) {
p[i] <- pcauchy(breaks[i+1],fit_cauchy$estimate[1], fit_cauchy$estimate[2])-pcauchy(breaks[i],fit_cauchy$estimate[1], fit_cauchy$estimate[2])
}
f.os<-c()
for (j in 1:(length(breaks)-1)) {
f.os[j]<- table(growth_medium.cut)[[j]]
}
chisq.test(x=f.os, p=p)# p-value is very low, probably due to binning choice
chisq.test(x=f.os, simulate.p.value = T) # test with Monte Carlo simulated p-values
#################################################################################
#################################################################################
#Visualise the obtained distributions
par(mfrow=c(1,1))
grid(5,5)
x<-growth_medium #(normal, cauchy, laplace, logis)
hist(x, prob=T, breaks=c(seqlast(min(growth_medium),max(growth_medium),0.1)), xlab="Growth",xlim = c(-3, 4), col="grey", main="Fit for medium firm growth distribution")
curve(dnorm(x, fit_norm$estimate[1], fit_norm$estimate[2]), add=T,col = 1, lwd=2)
curve(dcauchy(x, fit_cauchy$estimate[1], fit_cauchy$estimate[2]), add=T,col = 2, lwd=2)
curve(dlaplace(x, fit_laplace1$estimate[1] , fit_laplace1$estimate[2]), add=T, col=4, lwd=2)
#curve(dlaplace(x, coef(fit_laplace2)[1] , coef(fit_laplace2)[2]), add=T, col=5, lwd=2)
curve(dlogis(x, fit_logis$estimate[1], fit_logis$estimate[2]), add=T,col = 7, lwd=2)
legend("topright", c("Normal", "Cauchy", "Laplace", "Logistic"), col=c(1,2,4,7), lwd=3)
x<-shifted #(lnorm, pareto, exp, weib, gamma)
hist(x, add = F, prob=T, breaks=c(seqlast(min(shifted),max(shifted),0.2)),xlim=c(0, 20), xlab="Growth rate", col="light grey", main="Empirical growth rate distribution in medium firms")
curve(dlnorm(x, fit_lnorm$estimate[1], fit_lnorm$estimate[2]), add=T,col =6, lwd=2)
curve(dpareto(x, coef(fit_pareto)[1] , coef(fit_pareto)[2]), add=T,col =7 , lwd=2)
curve(dexp(x, fit_exp$estimate[1]), add=T,col = 8, lwd=2)
curve(dweibull(x, fit_weib$estimate[1], fit_weib$estimate[2]), add=T,col = 9, lwd=2)
curve(dgamma(x, fit_gamma$estimate[1], fit_gamma$estimate[2]), add=T,col = 10, lwd=2)
x<-scaled #(beta)
hist(x, add = F, prob=T, breaks=c(seqlast(min(scaled),max(scaled),0.02)),xlim=c(0, 1), xlab="Growth rate", col="light grey", main="Empirical growth rate distribution in medium firms")
curve(dbeta(x, fit_beta$estimate[1], fit_beta$estimate[2]), add=T,col = 11, lwd=2)
kurtosis(medium_growth)
skewness(medium_growth)
mean(medium_growth)
var(medium_growth)
# Visualise some Q-Q and P-P plots
# For fitdistr objects:
# QQ plot
qqcomp(fit_norm)
ppcomp(fit_norm)
# PP plot
qqcomp(fit_cauchy)
ppcomp(fit_cauchy)
#For other objects:
#QQ plot
lengr <- length(growth_medium)
y <- rlaplace(lengr, coef(fit_laplace2)[1], coef(fit_laplace2)[2])
qqplot(y, growth_medium, xlab="Theoretical Quantiles", ylab = "Empirical Quantiles")
qqline(rlaplace(lengr), col = 2,lwd=2,lty=2, distribution = qlaplace)
#qnorm(0.5) # for a given surface under the std normal dist (50% in this case) give me the corresponding z score
#CDF plot:
#lines(ecdf(rcauchy(lengr, fit_cauchy$estimate[1], fit_cauchy$estimate[2])), col="blue", cex=0.5, xlim=c(-1,1))
#plot(ecdf(rlaplace(lengr, 0.0315464, 0.0571342)), col="blue", cex=0.5)
plot(ecdf(growth_medium),cex=0.5, col="red")
curve(pcauchy(x,fit_cauchy$estimate[1], fit_cauchy$estimate[2]), add = T , cex=0.5, col="blue")
curve(plaplace(x,fit_laplace1$estimate[1], fit_laplace1$estimate[2]), add = T , cex=0.5, col="green")
curve(plaplace(x,coef(fit_laplace2)[1], coef(fit_laplace2)[2]), add = T , cex=0.5,col="purple")
###########################################################################################################