S-shaped data: Smoothing with quasibinomial distribution

January 16, 2016
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Figure 1: Synthetic data and fitted curves.

S-shaped distributed data can be found in many applications. Such data can be approximated with logistic distribution function [1].  Cumulative distribution function of logistic distribution function is a logistic function, i.e., logit.

To demonstrate this, in this short example, after generating a synthetic data, we will fit quasibinomial regression model to different observations.

ggplot [2], an implementation of grammar of graphics [3], provides capability to apply regression or customised smoothing onto a raw data during plotting.

Generating Synthetic Data

Let generate set of $n$ observation over time $t$, denoted, $X_{1}, X_{2}, …, X_{n}$ for $k$ observation $X=(x_{1}, x_{2}, …, x_{k})$. We will use cumulative function for logistic distribution [4],
$$F(x;\mu,s) = \frac{1}{2} + \frac{1}{2} \tanh((x-\mu)/2s)$$, adding some random noise to make
it realistic.

Let’s say there are $k=6$ observations with the following parameter sets, $\mu = \{9,2,3,5,7,5\}$  and $s=\{2,2,4,3,4,2\}$, we will utilise `mapply` [5] in generating a syntetic data frame.

generate_logit_cdf <- function(mu, s, 
sigma_y=0.1,
x=seq(-5,20,0.1)) {
x_ms <- (x-mu)/s
y <- 0.5 + 0.5 * tanh(x_ms)
y <- abs(y + rnorm(length(x), 0, sigma_y))
ix <- which(y>=1.0)
if(length(ix)>=1) {
y[ix] <- 1.0
}
return(y)
}
set.seed(424242)
x <- seq(-5,20,0.025) # 1001 observation
mu_vec <- c(1,2,3,5,7,8) # 6 variables
s_vec <- c(2,2,4,3,4,2)
# Syntetic variables
observations_df<- mapply(generate_logit_cdf,
mu_vec,
s_vec,
MoreArgs = list(x=x))
# Give them names
colnames(observations_df) <- c("Var1", "Var2", "Var3", "Var4", "Var5", "Var6")
head(observations_df)

Smoothing of observations

Using the syntetic data we have generated, `observations_df`,
we can noq use `ggplot` and `quasibinomial` `glm` to visualise
and smooth the variables.

library(ggplot2)
library(reshape2)
df_all <- reshape2:::melt(observations_df)
colnames(df_all) <- c("x", "observation", "y")
df_all$observation <- as.factor(df_all$observation)
p1<-ggplot(df_all, aes(x=x, y=y, colour=observation)) + geom_point() +
scale_color_brewer(palette = "Reds") +
theme(
panel.background = element_blank(),
axis.text.x = element_text(face="bold", color="#000000", size=11),
axis.text.y = element_text(face="bold", color="#000000", size=11),
axis.title.x = element_text(face="bold", color="#000000", size=11),
axis.title.y = element_text(face="bold", color="#000000", size=11)
# legend.position = "none"
)
l1<-ggplot(df_all, aes(x=x, y=y, colour=observation)) +
geom_point(size=3) + scale_color_brewer(palette = "Reds") +
scale_color_brewer(palette = "Reds") +
#geom_smooth(method="loess", se = FALSE, size=1.5) +
geom_smooth(aes(group=observation),method="glm", family=quasibinomial(), formula="y~x",
se = FALSE, size=1.5) +
xlab("x") +
ylab("y") +
#scale_y_continuous(breaks=seq(0.0,1,0.1)) +
#scale_x_continuous(breaks=seq(0.0,230,20)) +
#ggtitle("") +
theme(
panel.background = element_blank(),
axis.text.x = element_text(face="bold", color="#000000", size=11),
axis.text.y = element_text(face="bold", color="#000000", size=11),
axis.title.x = element_text(face="bold", color="#000000", size=11),
axis.title.y = element_text(face="bold", color="#000000", size=11)
)
library(gridExtra)
gridExtra:::grid.arrange(p1,l1)

References
[1] https://en.wikipedia.org/wiki/Logistic_distribution#Applications
[2] http://www.ggplot.org
[3] The Grammar of Graphics, L. Wilkinson, http://www.amzn.com/038724544
[4] http://en.wikipedia.org/wiki/Logistic_distribution#Cumulative_distribution_function.
[5] https://stat.ethz.ch/R-manual/R-devel/library/base/html/mapply.html

Notes:
* Quasibinomial distribution is as a term used in R’s GLM implementation context, see here.

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