statisticstime-seriesjulia

# How to do a Yule-Walker analysis in Julia

Is there a method in Julia to perform a Yule-Walker analysis similar to the equivalent method in Python statsmodels.regression.linear_model.yule_walker ?

Solution

• In the end, as recommended, I have implemented it myself, and the code is like this. I have tested it for some cases, and the results are the same as those of the Python function.

``````function yule_walker(
x::Vector{Float64};
order::Int64=1,
df::Union{Nothing,Int64}=nothing,
inv=false,
demean=true,
)
throw(ArgumentError("ACF estimation method must be 'adjusted' or 'MLE'"))

x = copy(x)
if demean
x .-= mean(x)
end
n = isnothing(df) ? length(x) : df

if ndims(x) > 1 || size(x, 2) != 1
throw(ArgumentError("Expecting a vector to estimate AR parameters"))
end

r = zeros(Float64, order + 1)
r[1] = sum(x .^ 2) / n
for k = 1:order
r[k+1] = sum(x[1:end-k] .* x[k+1:end]) / (n - k * adj_needed)
end
R = Toeplitz(r[1:(end - 1)], conj(r[1:(end - 1)]))

rho = 0
try
rho = R \ r[2:end]
catch err
if occursin("Singular matrix", string(err))
@warn "Matrix is singular. Using pinv."
rho = pinv(R) * r[2:end]
else
throw(err)
end
end

sigmasq = r[1] - dot(r[2:end], rho)
sigma = isnan(sigmasq) || sigmasq <= 0 ? NaN : sqrt(sigmasq)

if inv
return rho, sigma, inv(R)
else
return rho, sigma
end
end
``````

It's tested for these cases,

``````tol = 1e-5
@testset "yule_walker" begin
rho, sigma = yule_walker([1.0, 2, 3]; order=1)
@test rho == [0.0]

rho, sigma = yule_walker([1.0, 2, 3]; order=2)
@test rho == [0.0, -1.5]

x = [0.9901178, -0.74795127, 0.44612542, 1.1362954, -0.04040932]
rho, sigma = yule_walker(x; order=3, method="mle")
@test rho ≈ [-0.9418963, -0.90335955, -0.33267884]
@test sigma ≈ 0.44006365345695164

rho, sigma = yule_walker(x; order=3)
@test isapprox(rho, [0.10959317, 0.05242324, 1.06587676], atol=tol)
@test isapprox(sigma, 0.15860522671108127, atol=tol)

rho, sigma = yule_walker(x; order=5, method="mle")
@test isapprox(
rho, [-1.24209771, -1.56893346, -1.16951484, -0.79844781, -0.27598787], atol=tol
)
@test isapprox(sigma, 0.3679474002175471, atol=tol)

x = [
0.9901178,
-0.74795127,
0.44612542,
1.1362954,
-0.04040932,
0.28625813,
0.88901716,
-0.1079814,
-0.33231995,
0.4607741,
]

rho, sigma = yule_walker(x; order=3, method="mle")
@test isapprox(
rho, [-0.4896151627237206, -0.5724647370433921, 0.09083516892540627], atol=tol
)
@test isapprox(sigma, 0.4249693094713215, atol=tol)

x = [
0.9901178,
-0.74795127,
0.44612542,
1.1362954,
-0.04040932,
0.28625813,
0.88901716,
-0.1079814,
-0.33231995,
0.4607741,
0.7729643,
-1.0998684,
1.098167,
1.0105597,
-1.3370227,
1.239718,
-0.01393661,
-0.4790918,
1.5009186,
-1.1647809,
]

rho, sigma = yule_walker(x; order=3, method="mle")
@test isapprox(rho, [-0.82245705, -0.57029742, 0.12166898], atol=tol)
@test isapprox(sigma, 0.5203501608988023, atol=tol)

rho, sigma = yule_walker(x; order=3)
@test isapprox(rho, [-0.93458149, -0.68653741, 0.10161722], atol=tol)
@test isapprox(sigma, 0.4269012058667671, atol=tol)

rho, sigma = yule_walker(x; order=5, method="mle")
@test isapprox(
rho, [-0.83107755, -0.56407764, 0.20950143, 0.1232321, 0.10249279], atol=tol
)
@test isapprox(sigma, 0.5172269743102993, atol=tol)

rho, sigma = yule_walker(x; order=5)
@test isapprox(
rho, [-0.96481241, -0.65359486, 0.31587079, 0.28403115, 0.1913565], atol=tol
)
@test isapprox(sigma, 0.41677565377507053, atol=tol)
end;
``````