I am trying to use Julia to estimate a continuous univariate distribution using N observed data points (stored as an array of Float64 numbers), and then sample from this estimated distribution. I have no prior knowledge restricting attention to some family of distributions.
I was thinking of using the KernelDensity package to estimate the distribution, but I'm not sure how to sample from the resulting output.
Any help/tips would be much appreciated.
Without any restrictions on the estimated distribution, a natural candidate would be the empirical distribution function (see Wikipedia). For this distribution there are very nice theorems about convergence to actual distribution (see Dvoretzky–Kiefer–Wolfowitz inequality).
With this choice, sampling is especially simple. If dataset is a list of current samples, then dataset[rand(1:length(dataset),sample_size)] is a set of new samples from the empirical distribution. With the Distributions package, it could be more readable, like so:
using Distributions
new_sample = sample(dataset,sample_size)
Finally, Kernel density estimation is also good, but might need a parameter to be chosen (the kernel and its width). This shows a preference for a certain family of distributions. Sampling from a kernel distribution is surprisingly similar to sampling from the empirical distribution: 1. choose a sample from the empirical distributions; 2. perturb each sample using a sample from the kernal function.
For example, if the kernel function is a Normal distribution of width w, then the perturbed sample could be calculated as:
new_sample = dataset[rand(1:length(dataset),sample_size)]+w*randn(sample_size)
Related
I'm trying to replicate the precipitation mixture model from this paper: http://dx.doi.org/10.1029/2006WR005308
f(r) is the gamma PDF, g(r) is the generalized Pareto PDF, and w(r) is the weighting function, which depends on the value r being considered. I've looked at R packages like distr and mixtools that handle mixture models, but I only see examples where w is a constant, and I haven't found any implementations where the mixture is a function of the value. I'm struggling to create valid custom functions to represent h(r) so if someone could point me to a package that would be super helpful.
I have a data set where observations come from highly distinct groups. Each group may have a wildly different distribution, so I am trying to find the best distribution using fitdist from fitdistrplus, then use gamlssML from the gamlss package to find the best parameters.
My issue is with transforming the data after this step. For some of the distributions, like the Box-Cox t, I can find the equation for normalizing the data using the BCT coefficients, but for many of these distributions I cannot.
Does gamlss have a function that normalizes the data after fitting? Their documentation only provides the transformations for a small number of distributions https://www.gamlss.com/wp-content/uploads/2018/01/DistributionsForModellingLocationScaleandShape.pdf
Thanks a lot
The normalised data values (for any distribution) are exactly equal to the residuals from a gamlss fit,
m1 <- gamlss()
which can be accessed by
residuals(m1) or
m1$residuals
When using the GAMLSS package in R, there are many different ways to fit a distribution to a set of data. My data is a single vector of values, and I am fitting a distribution over these values.
My question is this: what is the main difference between using fitDist() and gamlss() since they give similar but different answers for parameter values, and different worm plots?
Also, using the function confint() works for gamlss() fitted objects but not for objects fitted with fitDist(). Is there any way to produce confidence intervals for parameters fitted with the fitDist() function? Is there an accuracy difference between the two procedures? Thanks!
m1 <- fitDist()
fits many distributions and chooses the best according to a
generalized Akaike information criterion, GAIC(k), wit penalty k for each
fitted parameter in the distribution, where k is specified by the user,
e.g. k=2 for AIC,
k = log(n) for BIC,
k=4 for a Chi-squared test (rounded from 3.84, the 5% critical value of a Chi-squared distribution with 1 degree of fereedom), which is my preference.
m1$fits
gives the full results from the best to worst distribution according to GAIC(k).
hybrid Gibbs models are flexible for fitting spatial pattern data, however, I am confused on how to get the confidence interval for the fitted model's estimate. for instance, I fitted a hybrid geyer model including a hardcore and a geyer saturation components, got the estimates:
Mo.hybrid<-Hybrid(H=Hardcore(), G=Geyer(81,1))
my.hybrid<-ppm(my.X~1,Mo.hybrid, correction="bord")
#beta = 1.629279e-06
#Hard core distance: 31.85573
#Fitted G interaction parameter gamma: 10.241487
what I interested is the gamma, which present the aggregation of points. obviously, the data X is a sample, i.e., of cells in a anatomical image. in order to report statistical result, a confidence interval for gamma is needed. however, i do not have replicates for the image data.
can i simlate 10 time of the fitted hybrid model, then refitted them to get confidence interval of the estimate? something like:
mo.Y<-rmhmodel(cif=c("hardcore","geyer"),
par=list(list(beta=1.629279e-06,hc=31.85573),
list(beta=1, gamma=10.241487,r=81,sat=1)), w=my.X)
Y1<-rmh(model=mo.Y, control = list(nrep=1e6,p=1, fixall=TRUE),
start=list(n.start=c(npoint(my.X))))
Y1.fit<-ppm(Y1~1, Mo.hybrid,rbord=0.1)
# simulate and fit Y2,Y3,...Y10 in same way
or:
Y10<-simulate(my.hybrid,nsim=10)
Y1.fit<-ppm(Y10[1]~1, Mo.hybrid,rbord=0.1)
# fit Y2,Y3,...Y10 in same way
certainly, the algorithms is different, the rmh() can control simulated intensity while the simulate() does not.
now the questions are:
is it right to use simualtion to get confidence interval of estimate?
or the fitted model can provide estimate interval that could be extracted?
if simulation is ok, which algorithm is better in my case?
The function confint calculates confidence intervals for the canonical parameters of a statistical model. It is defined in the standard stats package. You can apply it to fitted point process models in spatstat: in your example just type confint(my.hybrid).
You wanted a confidence interval for the non-canonical parameter gamma. The canonical parameter is theta = log(gamma) so if you do exp(confint(my.hybrid) you can read off the confidence interval for gamma.
Confidence intervals and other forms of inference for fitted point process models are discussed in detail in the spatstat book chapters 9, 10 and 13.
The confidence intervals described above are the asymptotic ones (based on the asymptotic variance matrix using the central limit theorem).
If you really wanted to estimate the variance-covariance matrix by simulation, it would be safer and easier to fit the model using method='ho' (which performs the simulation) and then apply confint as before (which would then use the variance of the simulations rather than the asymptotic variance).
rmh.ppm and simulate.ppm are essentially the same algorithm, apart from some book-keeping. The differences observed in your example occur because you passed different arguments. You could have passed the same arguments to either of these functions.
I want to fit a distribution to my data. I use fitdistrplus package in r to find the distribution. I can compare the goodness of fit results for different distributions to see which one is more fitted to my data but I don't know how to check the pvalue for goodness of fit test for each of the distributions. The results might show that among gamma, lognormal and exponential, exponential distribution has the lower statistics for anderson darling test but I don't know how to check if pvalue for these tests does not reject the null hypothesis. Is there any built in function in R which gives the pvalues?
Here is a piece of code I used as an example:
d <- sample(100,50)
library(fitdistrplus)
descdist(d)
fitg <- fitdist(d,"gamma")
fitg2 <- fitdist(d,"exp")
gofstat(list(fitg,fitg2))
This code makes 50 random numbers from 0 to 100 and tries to find best fitted model to these data. If descdist(d) shows that gamma and exponential are the two candidates as the best fitted model, fitg and fitg2 finds their related models. the last line compares Ks and anderson darling statistics to show which distribution is most fitted. Distribution with lower value for these tests is the best. However, I dont know how to find p-values for fitg and fitg2 before comparying them. If pvalues show that none of these distributions are not fitted to these data, there is no point to comparing their goodness of fit statistics to my knowledge.
Any help is appreciated.
Thanks