R help - Oct 2021 - How to find local minimum between distributions using mixtools?

Hello,
I have two peaks distributions (and some noise values in between), and
I used mixtools to characterize the two overlapping data. How can I
find the minimum between the peak distributions?
The sigma value of the second population is close to that value, but I
am not sure if this is really the cut-off point between the two
distributions (by eye, I would say the value is 0.125 against 0.1 of
the sigma value). Is there a better approach?
Thanks

```
set.seed(10)
negative_mr <- runif(50, 0, 0.099)
negative_fcn <- runif(50, 1, 40)
positive_mr <- c(runif(30, 0.2, 0.5), runif(20, 0.4, 0.5))
positive_fcn <- c(runif(30, 25, 40), runif(20, 10, 25))
uncertain_mr <- runif(10, 0.099, 0.2)
uncertain_fcn <- runif(10, 2, 40)
df <- data.frame(Y=c(negative_mr, uncertain_mr, positive_mr),
X=c(negative_fcn, uncertain_fcn, positive_fcn),
GROUP=c(rep("negative", length(negative_mr)),
rep("uncertain", length(uncertain_mr)),
rep("positive", length(positive_mr))))
library(mixtools)
model = normalmixEM((x = df$Y))
summary(model)
# plot
plot(model, which=2)
Cut_off <- model$sigma[2]
Cut_offE <- 0.125
abline(v=Cut_off, col="blue", lwd=2)
abline(v=Cut_offE, col="blue", lwd=2, lty=2)
plot(df$Y[df$GROUP=="negative"]~df$X[df$GROUP=="negative"],
pch=1, ylim=c(0,0.5), xlim=c(0,41), cex=1.5,
xlab=expression(bold("X")),
ylab=expression(bold("Y")))
points(df$Y[df$GROUP=="positive"]~df$X[df$GROUP=="positive"],
pch=16, cex=1.5)
points(df$Y[df$GROUP=="uncertain"]~df$X[df$GROUP=="uncertain"],
pch=16, cex=1.5, col="grey")
abline(h=Cut_off, col="blue", lwd=2)
abline(h=Cut_offE, col="blue", lwd=2, lty=2)
```


-- 
Best regards,
Luigi

Do you really want the minimum?
It sounds as though your model is a*N(x1,s1) + (1-a)*N(x2,s2) where
you use mixtools to estimate
the parameters.  Finding the derivative of that is fairly
straightforward calculus, and solving for the
derivative being zero gives you extrema (you want the one between x1 and x2).

In practice I might start with a quick and dirty hack like
x <- seq(from min(x1,x2), to=max(x1,x2), length=399)
p <- a*dnorm(x, x1, s1) + (1-a)*dnorm(x, x2, s2)
m <- min(p)
x[x == m]

On Wed, 13 Oct 2021 at 22:12, Luigi Marongiu <marongiu.luigi at gmail.com>
wrote:
>
> Hello,
> I have two peaks distributions (and some noise values in between), and
> I used mixtools to characterize the two overlapping data. How can I
> find the minimum between the peak distributions?
> The sigma value of the second population is close to that value, but I
> am not sure if this is really the cut-off point between the two
> distributions (by eye, I would say the value is 0.125 against 0.1 of
> the sigma value). Is there a better approach?
> Thanks
>
> ```
> set.seed(10)
> negative_mr <- runif(50, 0, 0.099)
> negative_fcn <- runif(50, 1, 40)
> positive_mr <- c(runif(30, 0.2, 0.5), runif(20, 0.4, 0.5))
> positive_fcn <- c(runif(30, 25, 40), runif(20, 10, 25))
> uncertain_mr <- runif(10, 0.099, 0.2)
> uncertain_fcn <- runif(10, 2, 40)
> df <- data.frame(Y=c(negative_mr, uncertain_mr, positive_mr),
> X=c(negative_fcn, uncertain_fcn, positive_fcn),
> GROUP=c(rep("negative", length(negative_mr)),
> rep("uncertain", length(uncertain_mr)),
> rep("positive", length(positive_mr))))
> library(mixtools)
> model = normalmixEM((x = df$Y))
> summary(model)
> # plot
> plot(model, which=2)
> Cut_off <- model$sigma[2]
> Cut_offE <- 0.125
> abline(v=Cut_off, col="blue", lwd=2)
> abline(v=Cut_offE, col="blue", lwd=2, lty=2)
>
plot(df$Y[df$GROUP=="negative"]~df$X[df$GROUP=="negative"],
> pch=1, ylim=c(0,0.5), xlim=c(0,41), cex=1.5,
> xlab=expression(bold("X")),
> ylab=expression(bold("Y")))
>
points(df$Y[df$GROUP=="positive"]~df$X[df$GROUP=="positive"],
> pch=16, cex=1.5)
>
points(df$Y[df$GROUP=="uncertain"]~df$X[df$GROUP=="uncertain"],
> pch=16, cex=1.5, col="grey")
> abline(h=Cut_off, col="blue", lwd=2)
> abline(h=Cut_offE, col="blue", lwd=2, lty=2)
> ```
>
>
> --
> Best regards,
> Luigi
>
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