Original link: http://tecdat.cn/?p=26578
The exponential distribution is a probability distribution of the time between events in a Poisson process, so it is used to predict the waiting time until the next event, for example, the time you need to wait at a bus stop until the next bus arrives(< /strong>Click “Read the original text” at the end of the article to get the completecode data).
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In this article we will use the exponential distribution assuming that its parameter λ , the mean time between events, changes at some time point k , i.e.:
Our main goal is to estimate the parameters λ, α and k using a Gibbs sampler given n samples of observations from this distribution.
Gibbs Sampler
The Gibbs sampler is a special case of the Metropolis-Hastings sampler and is typically used when the target is a multivariate distribution. With this approach, chains are generated by sampling from the marginal distribution of the target distribution, so every candidate point is accepted.
The Gibbs sampler generates a Markov chain as follows:
-
Let
be a random vector in Rd at time t=0 Initialize X(0). -
Repeat for each iteration t=1,2,3,…:
-
Set x1=X1(t-1).
-
For each j=1,…,d:
-
Generate X?j(t) from
, where =”” src=”//i2.wp.com/csdnimg.cn/release/phoenix/outside_default.png” alt=”outside_default.png”> is the univariate conditional density of Xj given X(-j). -
Update
. -
When each candidate point is accepted, set
. -
increase t.
Bayesian formula
A simple formulation of the change point problem assumes that f and g have known densities:
where k is unknown and k=1,2,…,n. Let Yi be the elapsed time (in minutes) between bus arrival at the bus stop. Assume that the change point occurs at the kth minute, that is:
When Y=(Y1,Y2,…,Yn), the likelihood L(Y|k) is given by the following formula:
A Bayesian model assuming independent priors is given by:
The joint distribution of data and parameters is:
in,
As I mentioned before, the implementation of the Gibbs sampler requires sampling from the marginal distribution of the target distribution, so we need to find the complete conditional distribution of λ, α, and k. How can you do this? Simply put, you have to select terms from the connectivity distribution introduced above that only depend on the parameters of interest and ignore the rest.
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The complete conditional distribution of λ is given by:
The complete conditional distribution of α is given by:
The complete conditional distribution of k is given by:
Calculation method
Here you will learn how to estimate the parameters λ, α, and k using the Gibbs sampler using R.
Data
First, we generate data from the next exponential distribution with change points:
set.seed(98712) y <- c(rexp(25, rate = 2), rexp(35, rate = 10))
Considering the situation at the bus station, buses arrive at the bus station every 2 minutes on average at the beginning, but starting from time i=26, buses start arriving at the bus station every 10 minutes on average.
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Bayesian simple linear regression simulation analysis using Gibbs sampling in R language
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Implementation of Gibbs sampler
First, we need to initialize k, λ and α.
n <- length(y) # Number of observations in the sample lci <- 10000 # chain size aba <- alpha <- k <- numeric(lcan) k\[1\] <- sample(1:n,
Now, for each iteration of the algorithm, we need to generate λ(t), α(t) and k(t) as follows (remember there are no change points if k + 1>n):
for (i in 2:lcan){
kt <- k\[i-1\]
# Generate lambda
lambda\[i\] <- rgamma
# Generate α
# generate k
for (j in 1:n) {
L\[j\] <- ((lambda\[i\] / alpha\[i
# Delete the first 9000 values in the chain
bunIn <- 9000
Results
In this section we present the chain generated by the Gibbs sampler and the distribution of its parameters λ, α and k. The true value of the parameter is shown by the red line.
The following table shows the actual values of the parameters and the average of the estimated values obtained using the Gibbs sampler:
res <- c(mean(k\[-(1:bun)\]), mean(lmba\[-(1:burn)\]), mean(apa\[-(1:buI)\ ])) resfil
Conclusion
From the results, we can conclude that the mean of the estimates of the parameters k, λ and α from the exponential distribution with change points obtained using the Gibbs sampler in R is close to the actual values of the parameters, but we expect better estimate. This may be due to the choice of the initial value of the chain or the choice of the prior distribution of λ and α.
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This article is selected from “R Language Bayesian METROPOLIS-HASTINGS GIBBS Gibbs Sampler Estimation of Change Point Exponential Distribution Analysis Poisson Process Station Waiting Time”.
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