Handling Correlated shocks in HMM

Specific Domains Probabilistic programming
1

Is there a way to handle the below in Turing?

y(t) = f(y(t-1), s(t)) + g(y(t-1), s(t)) * e1(t)
s(t) = p(s(t-1)) + q(s(t-1)) * e2(t)

y(t) is the data (Tx1) and s(t) is the (hidden) state variable. e1 & e2 are correlated N(0,1) with correlation coeff rho. f,g,p and q are well behaved functions. Objective is to estimate rho and parameters of these functions.

Any suggestions pse?

Thanks!

2

This model should be straightforward to implement in turing. Try sampling e ~ MvNormal(Sigma) where Sigma is the covariance matrix of the noise terms.

3

Something like this?
e ~ MvNormal(S)
y = (…) + (…) * e[1]

Thanks.

1 Like
4

My Turing implementation of the model:

@model sv_model(r) = begin
    
    N = length(r)
    h = Array{Float64}(undef, N)
        
    σ ~ TruncatedNormal(0.3, 0.1, 0.001, 10)
    ρ ~ Uniform(-1,1)
    ϕ ~ Uniform(-1,1)
    μ ~ Normal(-1.0, 0.3)
    
    h₀ = μ
    r₀ = 0.0
    r[1] ~ Normal(0.0, exp(0.5*μ))
    h[1] ~ Normal(μ, sqrt(1 - ρ*ρ)*σ)
    
    
    for i = 2:N
        r[i] ~ Normal(0.0, exp(0.5*h[i-1]))
        h[i] ~ Normal(μ + ϕ * (h[i-1] - μ) + σ*ρ*exp(-0.5*h[i-1])*r[i-1], sqrt(1 - ρ*ρ)*σ)
    end

end

With HMC() or NUTS() for algorithms, I get the below error.

TypeError: in typeassert, expected Float64, got ForwardDiff.Dual{Nothing,Float64,10}

Stacktrace:
 [1] setindex!(::Array{Float64,1}, ::ForwardDiff.Dual{ForwardDiff.Tag{Turing.Core.var"#f#7"

With PG(), i get occasionally lucky. Otherwise, most of the times i get the following message (though i can’t see why sigma is < 0)

CTaskException:
ArgumentError: Normal: the condition σ >= zero(σ) is not satisfied.
Stacktrace:
 [1] macro expansion at /Users/balaji/.julia/packages/Distributions/RAeyY/src/utils.jl:6 [inlined]
 [2] #Normal#98 at /Users/balaji/.julia/packages/Distributions/RAeyY/src/univariate/continuous/normal.jl:37 [inlined]
 [3] Normal at /Users/balaji/.julia/packages/Distributions/RAeyY/src/univariate/continuous/normal.jl:37 [inlined]
 [4] macro expansion at ./In[133]:18 [inlined]
 [5] ##evaluator#771(::Random._GLOBAL_RNG, ::DynamicPPL.Model{var"###evaluator#771",(:r,),Tuple{Array{Float64,1}},(),

The same exact model runs in Stan (julia interface) though.

5

The performance tips section of the turing manual shows you how to solve that problem.