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An important Lemma.
Note: Unless otherwise stated, the algorithms and the corresponding screenshots are adopted from Robert and Casella (2005).
Box-Muller Algorithm
function boxmuller()
    x = ones(2)
    u = rand(2)
    logu = sqrt(-2*log(u[1]))
    x[1] = logu * cos(2*pi*u[2])
    x[2] = logu * sin(2*pi*u[2])
    return(x)
end
​
## example
boxmuller()
Accept-Reject Method
Example:
f(x)∝exp⁡(−x2/2)(sin⁡(6x)2+3cos⁡(x)2sin⁡(4x)2+1)f(x)\propto \exp(-x^2/2)(\sin(6x)^2+3\cos(x)^2\sin(4x)^2+1)f(x)∝exp(−x2/2)(sin(6x)2+3cos(x)2sin(4x)2+1)
The Julia code is as follows:
function AccRej(f::Function, M)
    ## use normal distribution N(0, 1) as sampling function g
    while true
        x = randn()
        u = rand()
        cutpoint = f(x)/(M*g(x))
        if u <= cutpoint
            return([x, f(x)])
        end
    end
end
​
## density function of N(0, 1)
function g(x)
    return(exp(-0.5*x^2)/sqrt(2*pi))
end
​
## example function and ignore the normalized constant
function f(x)
    return(exp(-x^2/2)*(sin(6*x)^2 + 3*cos(x)^2*sin(4*x)^2 + 1))
end
​
## example
N = 500;
data = ones(N, 2);
for i = 1:500
    data[i,:] = AccRej(f, sqrt(2*pi)*5)
end
Envelope Accept-Reject
It is easy to write the Julia code:
function EnvAccRej(f::function, M, gl::function)
    while true
        x = randn() # still assume gm is N(0,1)
        u = rand()
        cutpoint1 = gl(x)/(M*g(x))
        cutpoint2 = f(x)/(M*g(x))
        if u <= cutpoint1
            return(x)
        elseif u <= cutpoint2
            return(x)
        end
    end
end
Atkinson's Poisson Simulation
It is necessary to note that the parameters in the algorithm are not same with those in the density function. In other words, the corresponding density function of the algorithm should be
f(x)=βexp⁡(α−βx)[1+exp⁡(α−βx)]2.f(x) = \beta \frac{\exp(\alpha-\beta x)}{[1+\exp(\alpha-\beta x)]^2}.f(x)=β[1+exp(α−βx)]2exp(α−βx)​.
The following Julia code can generate the poisson random variable with respect to λ\lambdaλ
.
function AtkinsonPois(lambda)
    # parameters
    beta = pi/sqrt(3*lambda)
    alpha = lambda*beta
    c = 0.767 - 3.36/lambda
    k = log(c) - lambda - log(beta)
    # step 1: propose new x
    u1 = rand()
    while true
        global x
        x = (alpha - log((1-u1)/u1))/beta
        x > -0.5 && break
    end
    while true
        # step 2: transform to N
        N = floor(Int, x)
        u2 = rand()
        # step 3: accept or not
        lhs = alpha - beta*x + log(u2/(1+exp(alpha-beta*x))^2)
        rhs = k + N*log(lambda) - log(factorial(lambda))
        if lhs <= rhs
            return(N)
        end
    end
end
​
## example
N = 100;
res = ones(Int, N);
for i = 1:N
    res[i] = AtkinsonPois(10)
end
# ans: 8 9 13 10 12 .......
As mentioned in the above exercise, another poisson generation method can be derived from the following exercise.
We can write the following Julia code to implement this generation procedure.
function SimplePois(lambda)
    s = 0
    k = 0
    while true
        u = rand()
        x = -log(u)/lambda
        s = s + x
        if s > 1
            return(k)
        end
        k = k + 1
    end
end
​
## example for simple poisson
res2 = ones(Int, N);
for i = 1:N
    res2[i] = SimplePois(10)
end
# ans: 5 7 16 7 10 .......
ARS Algorithm
ARS is based on the construction of an envelope and the derivation of a corresponding Accept-Reject algorithm. It provides a sequential evaluation of lower and upper envelopes of the density fff
 when h=log⁡fh=\log fh=logf
 is concave.
Let Sn{\cal S}_nSn​
 be a set of points xi,i=0,1,…,n+1x_i,i=0,1,\ldots,n+1xi​,i=0,1,…,n+1
, in the support of fff
 such that h(xi)=log⁡f(xi)h(x_i)=\log f(x_i)h(xi​)=logf(xi​)
 is known up to the same constant. Given the concavity of hhh
, the line Li,i+1L_{i,i+1}Li,i+1​
 through (xi,h(xi))(x_i,h(x_i))(xi​,h(xi​))
 and (xi+1,h(xi+1))(x_{i+1},h(x_{i+1}))(xi+1​,h(xi+1​))
 is below the graph of hhh
 in [xi,xi+1][x_i,x_{i+1}][xi​,xi+1​]
 and is above this graph outside this interval.
For x∈[xi,xi+1]x\in [x_i,x_{i+1}]x∈[xi​,xi+1​]
, define
hˉn(x)=min⁡{Li−1,i(x),Li+1,i+2(x)}andh‾n(x)=Li,i+1(x) ,\bar h_n(x)=\min\{L_{i-1,i}(x),L_{i+1,i+2}(x)\}\quad\text{and}\quad \underline h_n(x)=L_{i,i+1}(x)\,,hˉn​(x)=min{Li−1,i​(x),Li+1,i+2​(x)}andh​n​(x)=Li,i+1​(x),
the envelopes are
h‾n(x)≤h(x)≤hˉn(x)\underline h_n(x)\le h(x)\le \bar h_n(x)h​n​(x)≤h(x)≤hˉn​(x)
uniformly on the support of fff
. Thus,
exp⁡h‾n(x)=f‾n(x)≤f(x)≤fˉn(x)=exp⁡hˉn(x)=wˉngn(x) .\exp \underline h_n(x) = \underline f_n(x) \le f(x)\le \bar f_n(x)=\exp\bar h_n(x) = \bar w_ng_n(x)\,.exph​n​(x)=f​n​(x)≤f(x)≤fˉ​n​(x)=exphˉn​(x)=wˉn​gn​(x).
​Davison (2008) provides another version of ARS
and gives an illustration example.
Let's consider the slightly simple verison in which we do not need to consider h−(y)h_-(y)h−​(y)
. It is obvious that
Consider the CDF of g+=exp⁡(h+)g_+=\exp(h_+)g+​=exp(h+​)
:
G+(y)=∫−∞yexp⁡(h+(x))dx=∫−∞min⁡{z1,y}exp⁡(h+(x))dx+∫z1min⁡{z2,max⁡{y,z1}}exp⁡(h+(x))dx+⋯+∫zk−1min⁡{zk,max⁡{y,zk−1}}exp⁡(h+(x))dx+∫zkmax⁡{zk,y}exp⁡(h+(x))dx\begin{aligned}
    G_+(y) & = \int_{-\infty}^y \exp(h_+(x)) dx \\
           & = \int_{-\infty}^{\min\{z_1,y\}} \exp(h_+(x)) dx \\
           & \qquad + \int_{z_1}^{\min\{z_2,\max\{y, z_1\}\}} \exp(h_+(x)) dx \\
           & \qquad + \cdots \\
           & \qquad + \int_{z_{k-1}}^{\min\{z_k,\max\{y, z_{k-1}\}\}} \exp(h_+(x)) dx \\
           & \qquad + \int_{z_k}^{\max\{z_k,y\}} \exp(h_+(x)) dx
\end{aligned}G+​(y)​=∫−∞y​exp(h+​(x))dx=∫−∞min{z1​,y}​exp(h+​(x))dx+∫z1​min{z2​,max{y,z1​}}​exp(h+​(x))dx+⋯+∫zk−1​min{zk​,max{y,zk−1​}}​exp(h+​(x))dx+∫zk​max{zk​,y}​exp(h+​(x))dx​
and
∫zjzj+1exp⁡(h+(x))dx=exp⁡(h(yj+1))⋅1h′(yj+1)exp⁡((y−yj+1)h′(yj+1))∣zjzj+1,  j=1,…,k−1\int_{z_{j}}^{z_{j+1}}\exp(h_+(x))dx = \exp({h(y_{j+1})})\cdot \frac{1}{h'(y_{j+1})}\exp((y-y_{j+1})h'(y_{j+1}))\mid_{z_j}^{z_{j+1}},\; j=1,\ldots,k-1∫zj​zj+1​​exp(h+​(x))dx=exp(h(yj+1​))⋅h′(yj+1​)1​exp((y−yj+1​)h′(yj+1​))∣zj​zj+1​​,j=1,…,k−1
Then use the inverse of G+(y)G_+(y)G+​(y)
 to sample random variable whose density function is g+(x)g_+(x)g+​(x)
. So we can use the following Julia program to sample from g+(x)g_+(x)g+​(x)
.
# example 3.22 in Davison(2008)
r = 2; m = 10; mu = 0; sig2 = 1;
# range of y
yl = -2; yu = 2;
# function of h
function h(y)
    return(r * y - m * log(1 + exp(y)) - (y - mu)^2 / (2 * sig2))
end
​
function h(y::Array)
    return(r * y - m * log.(1 .+ exp.(y)) .- (y .- mu) .^2 ./ (2 * sig2))
end
​
# derivative of h
function dh(y)
    return(r - m * exp(y) / (1 + exp(y)) - (y - mu) / sig2)
end
​
function dh(y::Array)
    return(r .- m * exp.(y) ./ (1 .+ exp.(y)) .- (y .- mu)./sig2)
end
​
# intersection point
function zfix(yfixed::Array)
    yf0 = yfixed[1:end-1]
    yf1 = yfixed[2:end]
    zfixed = yf0 .+ (h(yf0) .- h(yf1) .+ (yf1 .- yf0) .* dh(yf1)) ./ (dh(yf1) .- dh(yf0))
    return(zfixed)
end
​
# evaluate log-density (not necessary)
function hplus(y::Float64, yfixed::Array)
    zfixed = zfix(yfixed)
    n = size(zfixed, 1)
    for i = 1:n
        if i == 1 && y < zfixed[i]
            return(h(yfixed[i]) + (y - yfixed[i]) * dh(yfixed[i]))
        elseif i < n && y >= zfixed[i] && y < zfixed[i+1]
            return(h(yfixed[i+1]) + (y - yfixed[i+1]) * dh(yfixed[i+1]))
        elseif i == n && y >= zfixed[n]
            return(h(yfixed[i]) + (y - yfixed[i]) * dh(yfixed[i]))
        end
    end
end
​
# calculate G_+(z_i)
function gplus_cdf(yfixed::Array, zfixed::Array)
    n = size(zfixed, 1)
    s = zeros(n+1)
    pr = zeros(n+1)
    for i = 1:(n+1)
        ## integral from -infty to zi
        if i == 1
        #    s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])) - )
            s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])) - exp((yl-yfixed[i]) * dh(yfixed[i])))
        elseif i == n+1
            s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((yu-yfixed[i]) * dh(yfixed[i])) - exp((zfixed[n]-yfixed[i]) * dh(yfixed[i])))
        else
            s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])) - exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])))
        end
    end
    pr = s / sum(s)
    return cumsum(pr), sum(s)
end
​
# sample from gplus density
function gplus_sample(yfixed)
    zfixed = zfix(yfixed)
    gp = gplus_cdf(yfixed, zfixed)
    zpr = gp[1]
    norm_const = gp[2]
    n = size(zfixed, 1)
    u = rand()
    # Invert the gplus pdf
    for i = 1:n
        if i == 1 && u < zpr[i]
            ey = u * dh(yfixed[i]) * norm_const / exp(h(yfixed[i])) + exp((yl-yfixed[i])*dh(yfixed[i]))
            return(yfixed[i] + log(ey)/dh(yfixed[i]))    
        elseif i == n && u >= zpr[i]
            ey = (u - zpr[i]) * dh(yfixed[i+1]) * norm_const / exp(h(yfixed[i+1])) + exp((zfixed[i]-yfixed[i+1])*dh(yfixed[i+1]))
            return(yfixed[i+1] + log(ey)/dh(yfixed[i+1]))
        elseif i < n && u >= zpr[i] && u < zpr[i+1]
            ey = (u - zpr[i]) * dh(yfixed[i+1]) * norm_const / exp(h(yfixed[i+1])) + exp((zfixed[i]-yfixed[i+1])*dh(yfixed[i+1]))
            return(yfixed[i+1] + log(ey)/dh(yfixed[i+1]))
        end
    end
end
Back to the main sampling algorithm, we can implement the procedure as follows:
## adaptive rejection sampling
function ars(yfixed::Array)
    x = gplus_sample(yfixed)
    u = rand()
    if u <= exp(h(x)-hplus(x, yfixed))
        return(x)
    else
        return(ars(append!(yfixed, x)))
    end
end
​
## example
N = 100
res = ones(N);
for i = 1:N
    res[i] = ars([-1.8,-1.1,-0.5,-0.2])
end
Based on the ARS algorithm, we can also get the Supplemental ARS algorithm:
Exponential Random Variable
The inverse transform sampling from a uniform distribution can be easily used to sample an exponential random variable. The CDF is
F(x)=1−exp⁡(−λx) ,F(x) = 1-\exp(-\lambda x)\,,F(x)=1−exp(−λx),
whose inverse function is
F−1(y)=−log⁡(1−y)λ .F^{-1}(y) = -\frac{\log(1-y)}{\lambda}\,.F−1(y)=−λlog(1−y)​.
Thus, we can sample U∼U(0,1)U\sim U(0, 1)U∼U(0,1)
, and then perform the transfomation
−log⁡Uλ .-\frac{\log U}{\lambda}\,.−λlogU​.
Today, I came across another method from Xi'an's blog, which points to the question on StackExchange.
Still confused about the details, as commented in the code.
The theoretical part is as follows,
I rewrite the provided C code in Julia, and compare the distribution with samples from inverse CDF and the package Distributions
a = [exp_rand() for i=1:1000]
b = [-log(rand()) for i=1:1000]
c = rand(Exponential(1), 1000)
histogram(a, bins=40, label = "sexp")
histogram!(b, bins=40, alpha=0.5, label = "invF")
histogram!(c, bins=40, alpha=0.5, label = "rand")
AIS
Special Distribution
Last modified 1yr ago
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Outline
Box-Muller Algorithm
Accept-Reject Method
Envelope Accept-Reject
Atkinson's Poisson Simulation
ARS Algorithm
Exponential Random Variable