In veronicagiro/qshift: Distribution factory
knitr::opts_chunk$set(fig.width = 7, fig.height = 7)
Introduction
This vignette will describe the R package, qshift which generates shifted distribution functions for any distribution chosen. The package was realized using a functional programming approach and inspires its computational method to qdist package developed by Andrea Spanò and available in the Github repository qdist.
Shifted probability distribution
Shifted distributions are used daily in many statistical fields.
Given a random variable $X$ with distribution function $F(x)$, a shifted random variable $Y$ from $X$ is a random variable with a constant, $k$, added, defined as $Y = X+k$. $Y$ support changes according to shift value.
- The density function $f(y)$ of $Y$ is $f(x-k)$.
- The probability function $F(y)$ of $Y$ is $F(x-k)$
- The quantile (inverse of probability) function $F^{-1}(p)$ of $Y$ is: $F^{-1}(x) + k$
- Random numbers from the distribution of $Y$ are equal to random numbers generated from $X$ distribution plus $k$
Shifted probability functions in R
R does not provide a specific package for shifted distribution functions, but it provides package stats which contains a wide set of functions for standard distributions: probability, density, quantile and random number generation functions. All them share a stable naming convention both for the names of the functions and in the first argument of these functions. In particular:
- density functions:
d<dist>(x) with x vector of quantiles;
- probability distribution functions:
p<dist>(q) with q vector of quantiles;
- quantile functions:
q<dist>(p) with p vector of probabilities;
- random number generation:
r<dist>(n) with n number of observations.
where <dist> indicates the conventional name of distribution family.
Let us see an example.
Considering the Normal distribution (norm), we have: dnorm(x), pnorm(q), qnorm(p) and rnorm(n).
So, we use to write:
dnorm(x = 6:10, mean = 8, sd = 1)
to get density values at 6:10 from a Normal distribution with parameters mean = 8 and sd = 1.
Going back to shifted distribution, we could easily write a shifted version for dnorm(), as:
shift_dnorm <- function(x, mean = 0, sd = 1, shift = 0, ...){
x_shifted <- x - shift
density <- dnorm(x = x_shifted, mean = mean, sd = sd, ...)
return(density)
}
So we replicate the previous example but shifting the Normal distribution of 1 unit:
shift_dnorm(x = 6:10, mean = 8, sd = 1, shift = 1)
At first view, this approach seems easy and cool but thinking about it, we realize that we would need to write a different function for each probability distribution we would like to work with. Functional programming approach could be a solution, as Andrea Spanò demonstrates in his package (see qdist vignette).
qshift package includes four general functions which compute shifted distribution functions for any distribution chosen:
dshift() for shifted density functions;pshift() for shifted probability distributions;qshift() for shifted quantile functions;rshift() for shifted random numbers generation functions.
The four functions share the same logic: they take as input a probability distribution name, as a character string, and return the equivalent shifted distribution as a function object.
require(qshift)
sdnorm <- dshift("norm") # density function
spnorm <- pshift("norm") # probability function
sqnorm <- qshift("norm") # quantile function
srnorm <- rshift("norm") # random generation function
Talking about the computational process, package qshift is based on two key concepts being part of the functional nature of the R programming language:
- the environment of a function can be used as a placeholder for other objects;
- function formals can be manipulated.
For example, let us consider dshift(). As we seen, it takes as input the distribution name (norm) and proceeds as follows:
- gets the corresponding function
dnorm();
- uses
dnorm() to create a function, say density() that computes shifted density for normal distributions;
- modify the formals of
density() so that it has the same formals as dnorm() plus shift, corresponding to the shift value;
- returns
density().
So, the function sdnorm() has the same formals of the function dnorm(), plus the parameter shift which indicates the shift value to be applied to distribution; by default it is set as 0.
args(sdnorm)
args(dnorm)
Other references are in chapters Environments and Functions of Andrea Spanò weeb Book Ramarro, available on Quantide web site.
Examples
Let us consider the Normal distribution and the previously defined functions sdnorm(), spnorm(), sqnorm() and srnorm().
-
Density function, sdnorm().
This function has a double use:
- generate probability values from a non-shifted normal distribution by leaving parameter
shift set to its default (shift = 0)
- generate probability values from any shifted normal distribution by setting
shift value
r
x <- seq(4, 14, len = 100)
not_shifted <- sdnorm(x = x, mean = 8, sd = 1.5)
positive_shifted <- sdnorm(x = x, mean = 8, sd = 1.5, shift = 1)
negative_shifted <- sdnorm(x = x, mean = 8, sd = 1.5, shift = -1)
We can visualize these results by plotting them:
```r
require(ggplot2)
df <- data.frame(x=x,
sdnorm_type = factor(c(rep("not_shifted", times=length(x)),
rep("positive_shifted", times=length(x)),
rep("negative_shifted", times=length(x))),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=sdnorm_type)) +
geom_line(size=1) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
xlab("Value") + ylab("Density") +
ggtitle("Density plot for shifted and not-shifted Normal distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
-
Probability function, spnorm():
r
not_shifted <- spnorm(q = x, mean = 8, sd = 1.5)
positive_shifted <- spnorm(q = x, mean = 8, sd = 1.5, shift = 1)
negative_shifted <- spnorm(q = x, mean = 8, sd = 1.5, shift = -1)
We can visualize these results by plotting them:
```r
df <- data.frame(x=x,
spnorm_type = factor(c(rep("not_shifted", times=length(x)),
rep("positive_shifted", times=length(x)),
rep("negative_shifted", times=length(x))),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=spnorm_type)) +
geom_line(size=1) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
xlab("Quantile") + ylab("Probability") +
ggtitle("Probability plot for shifted and not-shifted Normal distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
-
Quantile function, sqnorm():
r
x <- (1:999)/1000
not_shifted <- sqnorm(p = x, mean = 8, sd = 1.5)
positive_shifted <- sqnorm(p = x, mean = 8, sd = 1.5, shift = 1)
negative_shifted <- sqnorm(p = x, mean = 8, sd = 1.5, shift = -1)
We can visualize these results by plotting them:
```r
df <- data.frame(x=x,
sqnorm_type = factor(c(rep("not_shifted", times=length(x)),
rep("positive_shifted", times=length(x)),
rep("negative_shifted", times=length(x))),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=sqnorm_type)) +
geom_line(size=1) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
ylab("Quantile") + xlab("Probability") +
ggtitle("Quantile plot for shifted and not-shifted Normal distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
-
Random generation function, srnorm():
r
x <- 100000
not_shifted <- srnorm(n = x, mean = 8, sd = 1.5)
positive_shifted <- srnorm(n = x, mean = 8, sd = 1.5, shift = 3)
negative_shifted <- srnorm(n = x, mean = 8, sd = 1.5, shift = -3)
srnorm() and all functions generated by rshift() have a further parameter, set_seed, which aim is to fix the seed in random number generation, when set as TRUE. By default it is set as FALSE. In this case it is set as TRUE in order to fix the seed for visualizing the differences between random generated distributions.
We can visualize these results by plotting them:
```r
df <- data.frame(srnorm_type = factor(c(rep("not_shifted", times=x),
rep("positive_shifted", times=x),
rep("negative_shifted", times=x)),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- pl <- ggplot(data = df, mapping = aes(x=value, group=srnorm_type, fill=srnorm_type)) +
geom_density(alpha=0.6) +
scale_fill_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
ggtitle("Density plot for shifted and not-shifted Normal distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
As expected, the Normal shifted distribution is a Normal distribution with changed mean. The change in the mean is equal to the shift.
As we know, Normal distribution support goes from $-\infty$ to $+\infty$. What happens considerig a distribution with a limited support?
Let us consider the Uniform distribution.
First of all we have to define its probability functions:
sdunif <- dshift("unif")
spunif <- pshift("unif")
squnif <- qshift("unif")
srunif <- rshift("unif")
We limited Uniform support between 6 and 12.
-
Density function, sdunif():
r
x <- seq(4, 14, len = 100)
not_shifted <- sdunif(x = x, min = 6, max = 12)
positive_shifted <- sdunif(x = x, min = 6, max = 12, shift = 1)
negative_shifted <- sdunif(x = x, min = 6, max = 12, shift = -1)
We can visualize these results by plotting them:
```r
df <- data.frame(x=x,
sdunif_type = factor(c(rep("not_shifted", times=length(x)),
rep("positive_shifted", times=length(x)),
rep("negative_shifted", times=length(x))),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=sdunif_type)) +
geom_line(size=1, alpha =0.8) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
xlab("Value") + ylab("Density") +
ggtitle("Density plot for shifted and not-shifted Uniform distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
-
Probability function, spunif():
r
not_shifted <- spunif(q = x, min = 6, max = 12)
positive_shifted <- spunif(q = x, min = 6, max = 12, shift = 1)
negative_shifted <- spunif(q = x, min = 6, max = 12, shift = -1)
We can visualize these results by plotting them:
```r
df <- data.frame(x=x,
spunif_type = factor(c(rep("not_shifted", times=length(x)),
rep("positive_shifted", times=length(x)),
rep("negative_shifted", times=length(x))),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=spunif_type)) +
geom_line(size=1, alpha =0.8) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
xlab("Quantile") + ylab("Probability") +
ggtitle("Probability plot for shifted and not-shifted Uniform distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
-
Quantile function, squnif():
r
x <- (1:999)/1000
not_shifted <- squnif(p = x, min = 6, max = 12)
positive_shifted <- squnif(p = x, min = 6, max = 12, shift = 1)
negative_shifted <- squnif(p = x, min = 6, max = 12, shift = -1)
We can visualize these results by plotting them:
```r
df <- data.frame(x=x,
squnif_type = factor(c(rep("not_shifted", times=length(x)),
rep("positive_shifted", times=length(x)),
rep("negative_shifted", times=length(x))),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=squnif_type)) +
geom_line(size=1, alpha =0.8) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
ylab("Quantile") + xlab("Probability") +
ggtitle("Quantile plot for shifted and not-shifted Uniform distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
-
Random generation function, srunif():
r
x <- 10000
not_shifted <- srunif(n = x, min = 6, max = 12)
positive_shifted <- srunif(n = x, min = 6, max = 12, shift = 3)
negative_shifted <- srunif(n = x, min = 6, max = 12, shift = -3)
We can visualize these results by plotting them:
```r
df <- data.frame(srunif_type = factor(c(rep("not_shifted", times=x),
rep("positive_shifted", times=x),
rep("negative_shifted", times=x)),
levels = c("not_shifted", "positive_shifted", "negative_shifted")),
value = c(not_shifted, positive_shifted, negative_shifted))
pl <- pl <- ggplot(data = df, mapping = aes(x=value, group=srunif_type, fill=srunif_type)) +
geom_density(alpha=0.6) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
xlab("Value") + ylab("Density") +
ggtitle("Density plot for shifted and not-shifted Uniform distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
As we see, Uniform support changes according to the transformation (the shift) applied to the distribution.
Extending the computation
Further implementations are possible starting from the definition of a shifted distribution.
Suppose we want to define a function for maximum likelihood estimate for the shifted Weibull distribution.
Firstly let us define the shifted density function for the Weibull distribution:
sdweibull <- dshift("weibull")
and, subsequently, by using sdweibull() within an estimator function:
ltweibull <- function(x){
# starting values for parameters (scale, shape) and shift
shift <- min(x) - 0.01
x1 <- x - shift
shape <- (sd(x1)/mean(x1))^(-1.086)
scale <- mean(x1)/gamma(1+1/shape)
# parameters vector definition
theta <- c(shape, scale, shift)
# likelihood function
ll <- function(theta, x){
shape <- theta[1]
scale <- theta[2]
shift <- theta[3]
ld <- sdweibull(x = x, shape = shape, scale = scale, shift = shift, log = TRUE)
-sum(ld)
}
# maximum likelihood estimation
optim(par = theta, fn = ll, x = x)[["par"]]
}
Let us see how it works on a random generated sample from a shifted Weibull distribution:
# generate a random sample from a shifted weibull distribution
srweibull <- rshift("weibull")
x <- srweibull(n = 100000, shape = 1, scale = 5, shift = 1)
# maximum likelihood estimate for the shifted weibull distribution
ltweibull(x = x)
veronicagiro/qshift documentation built on May 3, 2019, 6:10 p.m.
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Introduction
This vignette will describe the R package, qshift which generates shifted distribution functions for any distribution chosen. The package was realized using a functional programming approach and inspires its computational method to qdist package developed by Andrea Spanò and available in the Github repository qdist.
Shifted probability distribution
Shifted distributions are used daily in many statistical fields.
Given a random variable $X$ with distribution function $F(x)$, a shifted random variable $Y$ from $X$ is a random variable with a constant, $k$, added, defined as $Y = X+k$. $Y$ support changes according to shift value.
- The density function $f(y)$ of $Y$ is $f(x-k)$.
- The probability function $F(y)$ of $Y$ is $F(x-k)$
- The quantile (inverse of probability) function $F^{-1}(p)$ of $Y$ is: $F^{-1}(x) + k$
- Random numbers from the distribution of $Y$ are equal to random numbers generated from $X$ distribution plus $k$
Shifted probability functions in R
R does not provide a specific package for shifted distribution functions, but it provides package stats which contains a wide set of functions for standard distributions: probability, density, quantile and random number generation functions. All them share a stable naming convention both for the names of the functions and in the first argument of these functions. In particular:
- density functions:
d<dist>(x)withxvector of quantiles; - probability distribution functions:
p<dist>(q)withqvector of quantiles; - quantile functions:
q<dist>(p)withpvector of probabilities; - random number generation:
r<dist>(n)withnnumber of observations.
where <dist> indicates the conventional name of distribution family.
Let us see an example.
Considering the Normal distribution (norm), we have: dnorm(x), pnorm(q), qnorm(p) and rnorm(n).
So, we use to write:
dnorm(x = 6:10, mean = 8, sd = 1)
to get density values at 6:10 from a Normal distribution with parameters mean = 8 and sd = 1.
Going back to shifted distribution, we could easily write a shifted version for dnorm(), as:
shift_dnorm <- function(x, mean = 0, sd = 1, shift = 0, ...){ x_shifted <- x - shift density <- dnorm(x = x_shifted, mean = mean, sd = sd, ...) return(density) }
So we replicate the previous example but shifting the Normal distribution of 1 unit:
shift_dnorm(x = 6:10, mean = 8, sd = 1, shift = 1)
At first view, this approach seems easy and cool but thinking about it, we realize that we would need to write a different function for each probability distribution we would like to work with. Functional programming approach could be a solution, as Andrea Spanò demonstrates in his package (see qdist vignette).
qshift package includes four general functions which compute shifted distribution functions for any distribution chosen:
dshift()for shifted density functions;pshift()for shifted probability distributions;qshift()for shifted quantile functions;rshift()for shifted random numbers generation functions.
The four functions share the same logic: they take as input a probability distribution name, as a character string, and return the equivalent shifted distribution as a function object.
require(qshift) sdnorm <- dshift("norm") # density function spnorm <- pshift("norm") # probability function sqnorm <- qshift("norm") # quantile function srnorm <- rshift("norm") # random generation function
Talking about the computational process, package qshift is based on two key concepts being part of the functional nature of the R programming language:
- the environment of a function can be used as a placeholder for other objects;
- function formals can be manipulated.
For example, let us consider dshift(). As we seen, it takes as input the distribution name (norm) and proceeds as follows:
- gets the corresponding function
dnorm(); - uses
dnorm()to create a function, saydensity()that computes shifted density for normal distributions; - modify the formals of
density()so that it has the same formals asdnorm()plusshift, corresponding to the shift value; - returns
density().
So, the function sdnorm() has the same formals of the function dnorm(), plus the parameter shift which indicates the shift value to be applied to distribution; by default it is set as 0.
args(sdnorm) args(dnorm)
Other references are in chapters Environments and Functions of Andrea Spanò weeb Book Ramarro, available on Quantide web site.
Examples
Let us consider the Normal distribution and the previously defined functions sdnorm(), spnorm(), sqnorm() and srnorm().
-
Density function,
sdnorm().
This function has a double use:- generate probability values from a non-shifted normal distribution by leaving parameter
shiftset to its default (shift = 0) - generate probability values from any shifted normal distribution by setting
shiftvalue
r x <- seq(4, 14, len = 100) not_shifted <- sdnorm(x = x, mean = 8, sd = 1.5) positive_shifted <- sdnorm(x = x, mean = 8, sd = 1.5, shift = 1) negative_shifted <- sdnorm(x = x, mean = 8, sd = 1.5, shift = -1)We can visualize these results by plotting them:```r require(ggplot2) df <- data.frame(x=x, sdnorm_type = factor(c(rep("not_shifted", times=length(x)), rep("positive_shifted", times=length(x)), rep("negative_shifted", times=length(x))), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
- generate probability values from a non-shifted normal distribution by leaving parameter
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=sdnorm_type)) + geom_line(size=1) + scale_color_manual(values = c("not_shifted" = "cyan3", "positive_shifted" = "#89FF99", "negative_shifted" = "#FF89B4")) + xlab("Value") + ylab("Density") + ggtitle("Density plot for shifted and not-shifted Normal distributions") + theme(legend.title=element_blank(), legend.text= element_text(size=10), plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl) ```
-
Probability function,
spnorm():r not_shifted <- spnorm(q = x, mean = 8, sd = 1.5) positive_shifted <- spnorm(q = x, mean = 8, sd = 1.5, shift = 1) negative_shifted <- spnorm(q = x, mean = 8, sd = 1.5, shift = -1)We can visualize these results by plotting them:```r df <- data.frame(x=x, spnorm_type = factor(c(rep("not_shifted", times=length(x)), rep("positive_shifted", times=length(x)), rep("negative_shifted", times=length(x))), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=spnorm_type)) + geom_line(size=1) + scale_color_manual(values = c("not_shifted" = "cyan3", "positive_shifted" = "#89FF99", "negative_shifted" = "#FF89B4")) + xlab("Quantile") + ylab("Probability") + ggtitle("Probability plot for shifted and not-shifted Normal distributions") + theme(legend.title=element_blank(), legend.text= element_text(size=10), plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl) ```
-
Quantile function,
sqnorm():r x <- (1:999)/1000 not_shifted <- sqnorm(p = x, mean = 8, sd = 1.5) positive_shifted <- sqnorm(p = x, mean = 8, sd = 1.5, shift = 1) negative_shifted <- sqnorm(p = x, mean = 8, sd = 1.5, shift = -1)We can visualize these results by plotting them:```r df <- data.frame(x=x, sqnorm_type = factor(c(rep("not_shifted", times=length(x)), rep("positive_shifted", times=length(x)), rep("negative_shifted", times=length(x))), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=sqnorm_type)) + geom_line(size=1) + scale_color_manual(values = c("not_shifted" = "cyan3", "positive_shifted" = "#89FF99", "negative_shifted" = "#FF89B4")) + ylab("Quantile") + xlab("Probability") + ggtitle("Quantile plot for shifted and not-shifted Normal distributions") + theme(legend.title=element_blank(), legend.text= element_text(size=10), plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl) ```
-
Random generation function,
srnorm():r x <- 100000 not_shifted <- srnorm(n = x, mean = 8, sd = 1.5) positive_shifted <- srnorm(n = x, mean = 8, sd = 1.5, shift = 3) negative_shifted <- srnorm(n = x, mean = 8, sd = 1.5, shift = -3)srnorm()and all functions generated byrshift()have a further parameter,set_seed, which aim is to fix the seed in random number generation, when set asTRUE. By default it is set asFALSE. In this case it is set asTRUEin order to fix the seed for visualizing the differences between random generated distributions.
We can visualize these results by plotting them:```r df <- data.frame(srnorm_type = factor(c(rep("not_shifted", times=x), rep("positive_shifted", times=x), rep("negative_shifted", times=x)), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
pl <- pl <- ggplot(data = df, mapping = aes(x=value, group=srnorm_type, fill=srnorm_type)) + geom_density(alpha=0.6) + scale_fill_manual(values = c("not_shifted" = "cyan3", "positive_shifted" = "#89FF99", "negative_shifted" = "#FF89B4")) + ggtitle("Density plot for shifted and not-shifted Normal distributions") + theme(legend.title=element_blank(), legend.text= element_text(size=10), plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl) ```
As expected, the Normal shifted distribution is a Normal distribution with changed mean. The change in the mean is equal to the shift.
As we know, Normal distribution support goes from $-\infty$ to $+\infty$. What happens considerig a distribution with a limited support?
Let us consider the Uniform distribution.
First of all we have to define its probability functions:
sdunif <- dshift("unif") spunif <- pshift("unif") squnif <- qshift("unif") srunif <- rshift("unif")
We limited Uniform support between 6 and 12.
-
Density function,
sdunif():r x <- seq(4, 14, len = 100) not_shifted <- sdunif(x = x, min = 6, max = 12) positive_shifted <- sdunif(x = x, min = 6, max = 12, shift = 1) negative_shifted <- sdunif(x = x, min = 6, max = 12, shift = -1)We can visualize these results by plotting them:```r df <- data.frame(x=x, sdunif_type = factor(c(rep("not_shifted", times=length(x)), rep("positive_shifted", times=length(x)), rep("negative_shifted", times=length(x))), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=sdunif_type)) + geom_line(size=1, alpha =0.8) + scale_color_manual(values = c("not_shifted" = "cyan3", "positive_shifted" = "#89FF99", "negative_shifted" = "#FF89B4")) + xlab("Value") + ylab("Density") + ggtitle("Density plot for shifted and not-shifted Uniform distributions") + theme(legend.title=element_blank(), legend.text= element_text(size=10), plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl) ```
-
Probability function,
spunif():r not_shifted <- spunif(q = x, min = 6, max = 12) positive_shifted <- spunif(q = x, min = 6, max = 12, shift = 1) negative_shifted <- spunif(q = x, min = 6, max = 12, shift = -1)We can visualize these results by plotting them:```r df <- data.frame(x=x, spunif_type = factor(c(rep("not_shifted", times=length(x)), rep("positive_shifted", times=length(x)), rep("negative_shifted", times=length(x))), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=spunif_type)) + geom_line(size=1, alpha =0.8) + scale_color_manual(values = c("not_shifted" = "cyan3", "positive_shifted" = "#89FF99", "negative_shifted" = "#FF89B4")) + xlab("Quantile") + ylab("Probability") + ggtitle("Probability plot for shifted and not-shifted Uniform distributions") + theme(legend.title=element_blank(), legend.text= element_text(size=10), plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl) ```
-
Quantile function,
squnif():r x <- (1:999)/1000 not_shifted <- squnif(p = x, min = 6, max = 12) positive_shifted <- squnif(p = x, min = 6, max = 12, shift = 1) negative_shifted <- squnif(p = x, min = 6, max = 12, shift = -1)We can visualize these results by plotting them:```r df <- data.frame(x=x, squnif_type = factor(c(rep("not_shifted", times=length(x)), rep("positive_shifted", times=length(x)), rep("negative_shifted", times=length(x))), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
pl <- ggplot(data = df, mapping = aes(x=x, y=value, col=squnif_type)) +
geom_line(size=1, alpha =0.8) +
scale_color_manual(values = c("not_shifted" = "cyan3",
"positive_shifted" = "#89FF99",
"negative_shifted" = "#FF89B4")) +
ylab("Quantile") + xlab("Probability") +
ggtitle("Quantile plot for shifted and not-shifted Uniform distributions") +
theme(legend.title=element_blank(), legend.text= element_text(size=10),
plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl) ```
-
Random generation function,
srunif():r x <- 10000 not_shifted <- srunif(n = x, min = 6, max = 12) positive_shifted <- srunif(n = x, min = 6, max = 12, shift = 3) negative_shifted <- srunif(n = x, min = 6, max = 12, shift = -3)We can visualize these results by plotting them:```r df <- data.frame(srunif_type = factor(c(rep("not_shifted", times=x), rep("positive_shifted", times=x), rep("negative_shifted", times=x)), levels = c("not_shifted", "positive_shifted", "negative_shifted")), value = c(not_shifted, positive_shifted, negative_shifted))
pl <- pl <- ggplot(data = df, mapping = aes(x=value, group=srunif_type, fill=srunif_type)) + geom_density(alpha=0.6) + scale_color_manual(values = c("not_shifted" = "cyan3", "positive_shifted" = "#89FF99", "negative_shifted" = "#FF89B4")) + xlab("Value") + ylab("Density") + ggtitle("Density plot for shifted and not-shifted Uniform distributions") + theme(legend.title=element_blank(), legend.text= element_text(size=10), plot.title = element_text(size=14), axis.title=element_text(size=12))
print(pl)
```
As we see, Uniform support changes according to the transformation (the shift) applied to the distribution.
Extending the computation
Further implementations are possible starting from the definition of a shifted distribution.
Suppose we want to define a function for maximum likelihood estimate for the shifted Weibull distribution.
Firstly let us define the shifted density function for the Weibull distribution:
sdweibull <- dshift("weibull")
and, subsequently, by using sdweibull() within an estimator function:
ltweibull <- function(x){ # starting values for parameters (scale, shape) and shift shift <- min(x) - 0.01 x1 <- x - shift shape <- (sd(x1)/mean(x1))^(-1.086) scale <- mean(x1)/gamma(1+1/shape) # parameters vector definition theta <- c(shape, scale, shift) # likelihood function ll <- function(theta, x){ shape <- theta[1] scale <- theta[2] shift <- theta[3] ld <- sdweibull(x = x, shape = shape, scale = scale, shift = shift, log = TRUE) -sum(ld) } # maximum likelihood estimation optim(par = theta, fn = ll, x = x)[["par"]] }
Let us see how it works on a random generated sample from a shifted Weibull distribution:
# generate a random sample from a shifted weibull distribution srweibull <- rshift("weibull") x <- srweibull(n = 100000, shape = 1, scale = 5, shift = 1) # maximum likelihood estimate for the shifted weibull distribution ltweibull(x = x)
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