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local statistics = {}
local ROOT_2 = math.sqrt(2.0)
-- Approximations for erf(x) and erfInv(x) from
-- https://en.wikipedia.org/wiki/Error_function
local erf
local erf_inv
local A = 8 * (math.pi - 3.0)/(3.0 * math.pi * (4.0 - math.pi))
local B = 4.0 / math.pi
local C = 2.0/(math.pi * A)
local D = 1.0 / A
erf = function(x)
if x == 0 then return 0; end
local xSq = x * x
local aXSq = A * xSq
local v = math.sqrt(1.0 - math.exp(-xSq * (B + aXSq) / (1.0 + aXSq)))
return (x > 0 and v) or -v
end
erf_inv = function(x)
if x == 0 then return 0; end
if x <= -1 or x >= 1 then return nil; end
local y = math.log(1 - x * x)
local u = C + 0.5 * y
local v = math.sqrt(math.sqrt(u * u - D * y) - u)
return (x > 0 and v) or -v
end
local function std_normal(u)
return ROOT_2 * erf_inv(2.0 * u - 1.0)
end
local poisson
local cdf_table = {}
local function generate_cdf(lambda_index, lambda)
local max = math.ceil(4 * lambda)
local pdf = math.exp(-lambda)
local cdf = pdf
local t = { [0] = pdf }
for i = 1, max - 1 do
pdf = pdf * lambda / i
cdf = cdf + pdf
t[i] = cdf
end
return t
end
for li = 1, 100 do
cdf_table[li] = generate_cdf(li, 0.25 * li)
end
poisson = function(lambda, max)
if max < 2 then
return (math.random() < math.exp(-lambda) and 0) or 1
elseif lambda >= 2 * max then
return max
end
local u = math.random()
local lambda_index = math.floor(4 * lambda + 0.5)
local cdfs = cdf_table[lambda_index]
if cdfs then
lambda = 0.25 * lambda_index
if u < cdfs[0] then return 0; end
if max > #cdfs then max = #cdfs + 1 else max = math.floor(max); end
if u >= cdfs[max - 1] then return max; end
if max > 4 then -- Binary search
local s = 0
while s + 1 < max do
local m = math.floor(0.5 * (s + max))
if u < cdfs[m] then max = m; else s = m; end
end
else
for i = 1, max - 1 do
if u < cdfs[i] then return i; end
end
end
return max
else
local x = lambda + math.sqrt(lambda) * std_normal(u)
return (x < 0.5 and 0) or (x >= max - 0.5 and max) or math.floor(x + 0.5)
end
end
-- Error function.
statistics.erf = erf
-- Inverse error function.
statistics.erf_inv = erf_inv
--- Standard normal distribution function (mean 0, standard deviation 1).
--
-- @return
-- Any real number (actually between -3.0 and 3.0).
statistics.std_normal = function()
local u = math.random()
if u < 0.001 then
return -3.0
elseif u > 0.999 then
return 3.0
end
return std_normal(u)
end
--- Standard normal distribution function (mean 0, standard deviation 1).
--
-- @param mu
-- The distribution mean.
-- @param sigma
-- The distribution standard deviation.
-- @return
-- Any real number (actually between -3*sigma and 3*sigma).
statistics.normal = function(mu, sigma)
local u = math.random()
if u < 0.001 then
return mu - 3.0 * sigma
elseif u > 0.999 then
return mu + 3.0 * sigma
end
return mu + sigma * std_normal(u)
end
--- Poisson distribution function.
--
-- @param lambda
-- The distribution mean and variance.
-- @param max
-- The distribution maximum.
-- @return
-- An integer between 0 and max (both inclusive).
statistics.poisson = function(lambda, max)
lambda, max = tonumber(lambda), tonumber(max)
if not lambda or not max or lambda <= 0 or max < 1 then return 0; end
return poisson(lambda, max)
end
return statistics
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