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Given this data set:
y<-c(-13,16,35,40,28,36,43,33,40,33,22,-5,-27,-31,-29,-25,-26,-31,-26,-24,-25,-29,-23,4)
t<-1:24
My goal is to calculate two areas. The first area would integrate only data from the first part of the curve found above the Zero line. The second area would integrate data from the second part of the curve found below the zero line.
First I would like to fit a sine wave to this data. Using this excellent answer:
https://stats.stackexchange.com/questions/60994/fit-a-sinusoidal-term-to-data
I was able to fit a sine wave (I will be using the periodic with second harmonic which looks to have a better fit)
ssp <- spectrum(y)
per <- 1/ssp$freq[ssp$spec==max(ssp$spec)]
reslm <- lm(y ~ sin(2*pi/per*t)+cos(2*pi/per*t))
summary(reslm)
rg <- diff(range(y))
plot(y~t,ylim=c(min(y)-0.1*rg,max(y)+0.1*rg))
lines(fitted(reslm)~t,col=4,lty=2) # dashed blue line is sin fit
# including 2nd harmonic really improves the fit
reslm2 <- lm(y ~ sin(2*pi/per*t)+cos(2*pi/per*t)+sin(4*pi/per*t)+cos(4*pi/per*t))
summary(reslm2)
lines(fitted(reslm2)~t,col=3) # solid green line is periodic with second harmonic
abline(h=0,lty=2)
Next I would like to calculate the area under the curve that is only positive, as well as the area under the curve that is exclusively negative. I've had luck looking at similar answers using the AUC functions in the Bolstad2 and Mess packages. But my data points do not fall neatly on zero line, and I do not know how to break up the sine function into areas only above the Zero line and only below the Zero line.
First things first. To get an exact calculation, you will need to work with the exact function of the 2nd harmonic fourier. Secondly, the beauty of harmonics functions is that they are repetitive. So if you want to find where your function reaches 0, you merely need to expand your interval to so you can be sure to find more than 2 roots.
First we get the exact function from the regression model
fourierfnct <- function(t){
fnct <- reslm2$coeff[1]+
reslm2$coeff[2]*sin(2*pi/per*t)+
reslm2$coeff[3]*cos(2*pi/per*t)+
reslm2$coeff[4]*sin(4*pi/per*t)+
reslm2$coeff[5]*cos(4*pi/per*t)
return(fnct)
}
secondly,you can write a function which can find the roots (where the function is 0). R provides a uniroot function which you can use to find multiple roots in a loop.
manyroots <- function(f,inter,period){
roots <- array(NA, inter)
for(i in 1:(length(inter)-1)){
roots[i] <- tryCatch({
return_value <- uniroot(f,c(inter[i],inter[i+1]))$root
}, error = function(err) {
return_value <- -1
})
}
retroots <- roots[-which(roots==-1)]
return(retroots)
}
then you simply calculate the roots, and use them to integrate the function across those boundaries.
roots <- manyroots(fourierfnct,seq(0,25),per)
integrate(fourierfnct, roots[1],roots[2])
#300.6378 with absolute error < 3.3e-12
integrate(fourierfnct, roots[2],roots[3])
#-284.6378 with absolute error < 3.2e-12
This may not be the solution you are looking for, but you could try this:
# Create a new t vector but with more subdivisions
t2 = seq(1,24,length.out = 10000)
# Evaluate your model on this t2
y2 = predict(reslm2, newdata = data.frame(t = t2))
lines(t2[y2>=0],y2[y2>=0],col="red")
# Estimate the area where the curve is greater than 0
sum(diff(t2)[1]*y2[y2>0])
# Estimate the area where the curve is less than 0
sum(diff(t2)[1]*y2[y2<0])
I'm trying to get a 95% confidence interval around some predicted values, but am not capable of achieving this.
Basically, I estimated a growth curve like this:
set.seed(123)
dat=data.frame(size=rnorm(50,10,3),age=rnorm(50,5,2))
S <- function(t,ts,C,K) ((C*K)/(2*pi))*sin(2*pi*(t-ts))
sommers <- function(t,Linf,K,t0,ts,C)
Linf*(1-exp(-K*(t-t0)-S(t,ts,C,K)+S(t0,ts,C,K)))
model <- nls(size~sommers(age,Linf,K,t0,ts,C),data=dat,
start=list(Linf=10,K=4.7,t0=2.2,C=0.9,ts=0.1))
I have independent size measurements, for which I would like to predict the age. Therefore, the inverse of the function, which is not very straightforward, I calculated like this:
model.out=coef(model)
S.out <- function(t)
((model.out[[4]]*model.out[[2]])/(2*pi))*sin(2*pi*(t-model.out[[5]]))
sommers.out <- function(t)
model.out[[1]]*(1-exp(-model.out[[2]]*(t-model.out[[3]])-S.out(t)+S.out(model.out[[3]])))
inverse = function (f, lower = -100, upper = 100) {
function (y) uniroot((function (x) f(x) - y), lower = lower, upper = upper)[1]
}
sommers.inverse = inverse(sommers.out, 0, 25)
x= sommers.inverse(10) #this works with my complete dataset, but not with this fake one
Although this works fine, I need to know the confidence interval (95%) around this estimate (x). For linear models there is for example "predict(... confidence=)". I could also bootstrap the function somehow to get the quantiles associated with the parameters (didn't find how), to then use the extremes of those to calculate the maximum and minimum values predictable. But that doesn't really look like the good way of doing this....
Any help would be greatly appreciated.
EDIT after answer:
So this worked (explained in the book of Ben Bolker, see answer):
vmat = mvrnorm(1000, mu = coef(mfit), Sigma = vcov(mfit))
dist = numeric(1000)
for (i in 1:1000) {dist[i] = sommers_inverse(9.938,vmat[i,])}
quantile(dist, c(0.025, 0.975))
On the rather bad fake data I gave, this works of course rather horrible. But on the real data (which I have a problem recreating), this is ok!
Unless I'm mistaken, you're going to have to use either regular (parametric) bootstrapping or a method called either "population predictive intervals" (e.g., see section 5 of chapter 7 of Bolker 2008), which assumes that the sampling distributions of your parameters are multivariate Normal. However, I think you may have bigger problems, unless I've somehow messed up your model in adapting it ...
Generate data (note that random data may actually bad for testing your model - see below ...)
set.seed(123)
dat <- data.frame(size=rnorm(50,10,3),age=rnorm(50,5,2))
S <- function(t,ts,C,K) ((C*K)/(2*pi))*sin(2*pi*(t-ts))
sommers <- function(t,Linf,K,t0,ts,C)
Linf*(1-exp(-K*(t-t0)-S(t,ts,C,K)+S(t0,ts,C,K)))
Plot the data and the initial curve estimate:
plot(size~age,data=dat,ylim=c(0,16))
agevec <- seq(0,10,length=1001)
lines(agevec,sommers(agevec,Linf=10,K=4.7,t0=2.2,ts=0.1,C=0.9))
I had trouble with nls so I used minpack.lm::nls.lm, which is slightly more robust. (There are other options here, e.g. calculating the derivatives and providing the gradient function, or using AD Model Builder or Template Model Builder, or using the nls2 package.)
For nls.lm we need a function that returns the residuals:
sommers_fn <- function(par,dat) {
with(c(as.list(par),dat),size-sommers(age,Linf,K,t0,ts,C))
}
library(minpack.lm)
mfit <- nls.lm(fn=sommers_fn,
par=list(Linf=10,K=4.7,t0=2.2,C=0.9,ts=0.1),
dat=dat)
coef(mfit)
## Linf K t0 C ts
## 10.6540185 0.3466328 2.1675244 136.7164179 0.3627371
Here's our problem:
plot(size~age,data=dat,ylim=c(0,16))
lines(agevec,sommers(agevec,Linf=10,K=4.7,t0=2.2,ts=0.1,C=0.9))
with(as.list(coef(mfit)), {
lines(agevec,sommers(agevec,Linf,K,t0,ts,C),col=2)
abline(v=t0,lty=2)
abline(h=c(0,Linf),lty=2)
})
With this kind of fit, the results of the inverse function are going to be extremely unstable, as the inverse function is many-to-one, with the number of inverse values depending sensitively on the parameter values ...
sommers_pred <- function(x,pars) {
with(as.list(pars),sommers(x,Linf,K,t0,ts,C))
}
sommers_pred(6,coef(mfit)) ## s(6)=9.93
sommers_inverse <- function (y, pars, lower = -100, upper = 100) {
uniroot(function(x) sommers_pred(x,pars) -y, c(lower, upper))$root
}
sommers_inverse(9.938, coef(mfit)) ## 0.28
If I pick my interval very carefully I can get back the correct answer ...
sommers_inverse(9.938, coef(mfit), 5.5, 6.2)
Maybe your model will be better behaved with more realistic data. I hope so ...
I wish to numerically find the maximum of the function multiplied by Beta 3 shown on p346 of the following link when tau=30:
http://www.ssc.upenn.edu/~fdiebold/papers/paper49/Diebold-Li.pdf
They give the answer on p347 as 0.0609.
I would like to confirm this numerically in R. I.e. to take the derivative and find the value where it reaches zero.
library(numDeriv)
x <- 30
testh <- function(lambda){ ((1-exp(-lambda*30))/(lambda*30)) - exp(-lambda*30) }
grad_h <- function(lambda){
val <- grad(testh, lambda)
return(val^2)
}
OptLam <- optimize(f=grad_h, interval=c(0.0001,120), tol=0.0000000000001)
I take the square of the gradient as I want the minimum to be at zero.
Unfortunately, the answer comes back as Lambda=120!! With lambda at 120 the value of the objective function is 5.36e-12.
By working by hand I can func a lower value of the numerical derivative that is closer to zero (it is also close to the analytical value given above):
grad_h(0.05977604)
## [1] 4.24494e-12
Why is the function above not finding this lower value? I have set the tolerance very high so it should be able to find such this optimal value?
Is it possible to correct the existing method so that it gives the correct answer?
Is there a better way to find the maximum gradient of a function numerically in R?
For example is there an optimizer that looks for zero rather than trying to find a minimum of maximum?
You can use uniroot to find where the derivative is 0. This might work for you,
grad_h <- function(lambda){
val=grad(testh,lambda)
return(val)
}
## The root
res <- uniroot(grad_h, c(0,120), tol=1e-10)
## see it
ls <- seq(0.001, 1, length=1000)
plot(ls, testh(ls), col="salmon")
abline(v=res$root, col="steelblue", lwd=2, lty=2)
text(x=res$root, y=testh(res$root),
labels=sprintf("(%f, %s)", res$root,
format(testh(res$root), scientific = T)), adj=-0.1)
I have frequency values changing with the time (x axis units), as presented on the picture below. After some normalization these values may be seen as data points of a density function for some distribution.
Q: Assuming that these frequency points are from Weibull distribution T, how can I fit best Weibull density function to the points so as to infer the distribution T parameters from it?
sample <- c(7787,3056,2359,1759,1819,1189,1077,1080,985,622,648,518,
611,1037,727,489,432,371,1125,69,595,624)
plot(1:length(sample), sample, type = "l")
points(1:length(sample), sample)
Update.
To prevent from being misunderstood, I would like to add little more explanation. By saying I have frequency values changing with the time (x axis units) I mean I have data which says that I have:
7787 realizations of value 1
3056 realizations of value 2
2359 realizations of value 3 ... etc.
Some way towards my goal (incorrect one, as I think) would be to create a set of these realizations:
# Loop to simulate values
set.values <- c()
for(i in 1:length(sample)){
set.values <<- c(set.values, rep(i, times = sample[i]))
}
hist(set.values)
lines(1:length(sample), sample)
points(1:length(sample), sample)
and use fitdistr on the set.values:
f2 <- fitdistr(set.values, 'weibull')
f2
Why I think it is incorrect way and why I am looking for a better solution in R?
in the distribution fitting approach presented above it is assumed that set.values is a complete set of my realisations from the distribution T
in my original question I know the points from the first part of the density curve - I do not know its tail and I want to estimate the tail (and the whole density function)
Here is a better attempt, like before it uses optim to find the best value constrained to a set of values in a box (defined by the lower and upper vectors in the optim call). Notice it scales x and y as part of the optimization in addition to the Weibull distribution shape parameter, so we have 3 parameters to optimize over.
Unfortunately when using all the points it pretty much always finds something on the edges of the constraining box which indicates to me that maybe Weibull is maybe not a good fit for all of the data. The problem is the two points - they ares just too large. You see the attempted fit to all data in the first plot.
If I drop those first two points and just fit the rest, we get a much better fit. You see this in the second plot. I think this is a good fit, it is in any case a local minimum in the interior of the constraining box.
library(optimx)
sample <- c(60953,7787,3056,2359,1759,1819,1189,1077,1080,985,622,648,518,
611,1037,727,489,432,371,1125,69,595,624)
t.sample <- 0:22
s.fit <- sample[3:23]
t.fit <- t.sample[3:23]
wx <- function(param) {
res <- param[2]*dweibull(t.fit*param[3],shape=param[1])
return(res)
}
minwx <- function(param){
v <- s.fit-wx(param)
sqrt(sum(v*v))
}
p0 <- c(1,200,1/20)
paramopt <- optim(p0,minwx,gr=NULL,lower=c(0.1,100,0.01),upper=c(1.1,5000,1))
popt <- paramopt$par
popt
rms <- paramopt$value
tit <- sprintf("Weibull - Shape:%.3f xscale:%.1f yscale:%.5f rms:%.1f",popt[1],popt[2],popt[3],rms)
plot(t.sample[2:23], sample[2:23], type = "p",col="darkred")
lines(t.fit, wx(popt),col="blue")
title(main=tit)
You can directly calculate the maximum likelihood parameters, as described here.
# Defining the error of the implicit function
k.diff <- function(k, vec){
x2 <- seq(length(vec))
abs(k^-1+weighted.mean(log(x2), w = sample)-weighted.mean(log(x2),
w = x2^k*sample))
}
# Setting the error to "quite zero", fulfilling the equation
k <- optimize(k.diff, vec=sample, interval=c(0.1,5), tol=10^-7)$min
# Calculate lambda, given k
l <- weighted.mean(seq(length(sample))^k, w = sample)
# Plot
plot(density(rep(seq(length(sample)),sample)))
x <- 1:25
lines(x, dweibull(x, shape=k, scale= l))
Assuming the data are from a Weibull distribution, you can get an estimate of the shape and scale parameter like this:
sample <- c(7787,3056,2359,1759,1819,1189,1077,1080,985,622,648,518,
611,1037,727,489,432,371,1125,69,595,624)
f<-fitdistr(sample, 'weibull')
f
If you are not sure whether it is distributed Weibull, I would recommend using the ks.test. This tests whether your data is from a hypothesised distribution. Given your knowledge of the nature of the data, you could test for a few selected distributions and see which one works best.
For your example this would look like this:
ks = ks.test(sample, "pweibull", shape=f$estimate[1], scale=f$estimate[2])
ks
The p-value is insignificant, hence you do not reject the hypothesis that the data is from a Weibull distribution.
Update: The histograms of either the Weibull or exponential look like a good match to your data. I think the exponential distribution gives you a better fit. Pareto distribution is another option.
f<-fitdistr(sample, 'weibull')
z<-rweibull(10000, shape= f$estimate[1],scale= f$estimate[2])
hist(z)
f<-fitdistr(sample, 'exponential')
z = rexp(10000, f$estimate[1])
hist(z)
I am trying to find the best PDF of a continuous data that has unknown distribution, using the "density" function in R. Now, given a new data point, I want to find the probability density of this data point based on the kernel density estimator that I have from the "density" function result.
How can I do that?
If your new point will be within the range of values produced by density, it's fairly easy to do -- I'd suggest using approx (or approxfun if you need it as a function) to handle the interpolation between the grid-values.
Here's an example:
set.seed(2937107)
x <- rnorm(10,30,3)
dx <- density(x)
xnew <- 32.137
approx(dx$x,dx$y,xout=xnew)
If we plot the density and the new point we can see it's doing what you need:
This will return NA if the new value would need to be extrapolated. If you want to handle extrapolation, I'd suggest direct computation of the KDE for that point (using the bandwidth from the KDE you have).
This is one year old, but nevertheless, here is a complete solution. Let's call
d <- density(xs)
and define h = d$bw. Your KDE estimation is completely determined by
the elements of xs,
the bandwidth h,
the type of kernel functions.
Given a new value t, you can compute the corresponding y(t), using the following function, which assumes you have used Gaussian kernels for estimation.
myKDE <- function(t){
kernelValues <- rep(0,length(xs))
for(i in 1:length(xs)){
transformed = (t - xs[i]) / h
kernelValues[i] <- dnorm(transformed, mean = 0, sd = 1) / h
}
return(sum(kernelValues) / length(xs))
}
What myKDE does is it computes y(t) by the definition.
See: docs
dnorm(data_point, its_mean, its_stdev)