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https://github.com/dmetivie/smoothperiodicstatsmodels.jl

A Julia package for time series and other models with smoothly varying periodic parameters

https://github.com/dmetivie/smoothperiodicstatsmodels.jl

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Keywords

autoregressive-models hidden-markov-model seasonality
Last synced: 5 months ago · JSON representation

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A Julia package for time series and other models with smoothly varying periodic parameters

Basic Info
  • Host: GitHub
  • Owner: dmetivie
  • License: mit
  • Language: Julia
  • Default Branch: master
  • Homepage:
  • Size: 321 KB
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Topics
autoregressive-models hidden-markov-model seasonality
Created over 2 years ago · Last pushed 6 months ago
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Readme License

README.md

SmoothPeriodicStatsModels

Build Status

This package is primarily built to be an extension of the StochasticWeatherGenerators package. See the tutorial for an example of usage.

It builds several smooth versions of statistical models for time series, currently including:

  • AR(1) Auto-Regressive model of order 1
  • Mixture Models
  • Auto-Regressive Periodic Hidden Markov Model. For now, it only has Bernoulli mixture emission distributions, i.e., conditionally independent distributions.

Smooth means that temporal coefficients of the models e.g., $p(t)\in [0,1]$ for Bernoulli parameter and $t\in [1,T]$ have a periodic functional form to enforce smoothness i.e., $p(t)\simeq p(t+1)$. Note that the package PeriodicHiddenMarkovModels.jl also deals with Periodic HMM, but right now, it does not enforce smoothness; hence if you don't have many observations, you can have $p(t)$ very different from $p(t+1)$. Moreover, this causes identifiability issues of the hidden states.

The HMM part of this package could be moved at some point to PeriodicHiddenMarkovModels.jl, making use under the hood of the very nice and flexible HiddenMarkovModels.jl package.

This is inspired by seasonal Hidden Markov Models; see A. Touron (2019) and used in the paper Interpretable Seasonal Multisite Hidden Markov Model for stochastic rain generation in France.

Example

This example shows how to use the ARPeriodicHMM model, which is a periodic HMM with multivariate Bernoulli emission distributions of dimension $D=6$ and autoregressive dependence on past observations. This model assumes conditional independence, i.e., given the hidden states and the past observations, the multivariate Bernoulli distribution is independent (conditional independence),

math \mathbb{P}(Y_n = y_t \mid Z_n = k, Y_{n-1} = y_{n-1}) = \prod_{j=1}^D \mathbb{P}\left(Y_{n}^{(j)} = y_{n}^{(j)}\mid Z_n = k, Y_{n-1}^{(j)} = y_{n-1}^{(j)}\right).

Under the hood, to deal with arbitrary autoregressive orders, the past observations are encoded in a variable $H$ of dimension $D \times (2^p)$, where $p$ is the autoregressive order (see the paper for a more precise definition).

Set up

julia using SmoothPeriodicStatsModels using Random

```julia Random.seed!(2020) K = 3 # Number of hidden states T = 11 # Period N = 49586 # Length of observation D = 6 # Dimension of observed variables autoregressiveorder = 1

sizeorder = 2^autoregressiveorder degreeofP = 1 sizedegreeofP = 2*degreeof_P + 1

transθ = 4*(rand(K, K - 1, sizedegreeofP) .- 1/2) Bernoulliθ = 2*(rand(K, D, sizeorder, sizedegreeofP) .- 1/2) hmm = Trig2ARPeriodicHMM([1/3, 1/6, 1/2], transθ, Bernoulli_θ, T) ```

Simulations

julia z_ini = 1 y_past = rand(Bool, autoregressive_order, D) n2t = SmoothPeriodicStatsModels.n_to_t(N,T) z, y = rand(hmm, n2t; y_ini=y_past, z_ini=z_ini, seq=true)

Fit

```julia transθguess = rand(K, K-1, sizedegreeofP) Bernoulliθguess = zeros(K, D, sizeorder, sizedegreeofP) transθguess[:,:,1] .= transθ[:,:,1] # cheating on initial guess to recover very good MLE maxima Bernoulliθguess[:,:,:,1] = Bernoulliθ[:,:,:,1] hmmguess = Trig2ARPeriodicHMM([1/4, 1/4, 1/2], transθguess, Bernoulliθguess, T)

@time "FitMLE SHMM (Baum Welch)" hmmfit, θqfit, θyfit, hist, histoA, histoB = fitmle(hmmguess, transθguess, Bernoulliθguess, y, ypast, maxiter=10000, robust=true; display=:iter, silence=true, tol=1e-3, θ_iters=true, n2t=n2t);

EM converged in 317 iterations, logtot = -194299.4177103428

FitMLE SHMM (Baum Welch): 72.532794 seconds (344.65 M allocations: 27.641 GiB, 3.59% gc time)

```

Plots

Code for the plot ```julia using LaTeXStrings, Plots using Distributions: succprob default(fontfamily="Computer Modern", linewidth=2, label=nothing, grid=true, framestyle=:default, legendfontsize=14, foreground_color_legend=nothing, background_color_legend=nothing, tickfontsize=14, xlabelfontsize=14, ylabelfontsize=14) ``` ```julia begin     pA = [plot() for k in 1:K]     for k in 1:K         [plot!(pA[k], hmm.A[k, l, :], c=l, label=L"Q_{%$(k)\to %$(l)}", legend=:topleft) for l in 1:K]         [plot!(pA[k], hmm_fit.A[k, l, :], c=l, label=:none, legend=:topleft, s = :dash) for l in 1:K]         hline!(pA[k], [0.5], c=:black, label=:none, s=:dot)         ylims!(0,1) xlabel!(pA[k], L"t")     end     pallA = plot(pA..., size=(1000, 500), bottom_margin=3Plots.mm) end ``` ```julia begin     mm = 1 # H = 1 <=> {Y_{t-1} = 0}     pB = [plot() for j in 1:D]     for j in 1:D         [plot!(pB[j], succprob.(hmm.B[k, :, j, mm]), c=k, label=:none) for k in 1:K]         [plot!(pB[j], succprob.(hmm_fit.B[k, :, j, mm]), c=k, label=:none, s = :dash) for k in 1:K]         hline!(pB[j], [0.5], c=:black, label=:none, s=:dot)         ylims!(pB[j], (0, 1)) xlabel!(pB[j], L"t") annotate!(pB[j], (0.5, 0.93), L"j=%$(j)") j==2 ? title!(pB[j], L"\mathbb{P}(Y_t^{(j)}=1|Z=k, Y_{t-1}^{(j)} = 0)") : nothing     end     pallB = plot(pB..., size=(1000, 500), bottom_margin=3Plots.mm) end begin     mm = 2 # H = 2 <=> {Y_{t-1} = 1}     pB = [plot() for j in 1:D]     for j in 1:D         [plot!(pB[j], succprob.(hmm.B[k, :, j, mm]), c=k, label=:none) for k in 1:K]         [plot!(pB[j], succprob.(hmm_fit.B[k, :, j, mm]), c=k, label=:none, s = :dash) for k in 1:K]         hline!(pB[j], [0.5], c=:black, label=:none, s=:dot)         ylims!(pB[j], (0, 1)) xlabel!(pB[j], L"t") j==2 ? title!(pB[j], L"\mathbb{P}(Y_t^{(j)}=1|Z=k, Y_{t-1}^{(j)} = 1)") : nothing annotate!(pB[j], (0.5, 0.93), L"j=%$(j)")     end     pallB = plot(pB..., size=(1000, 500), bottom_margin=3Plots.mm) end ```

Transition matrix: Transition matrix

Emission distributions: Emission distribution Emission distribution

Owner

  • Name: David Métivier
  • Login: dmetivie
  • Kind: user
  • Location: Montpellier, France
  • Company: INRAe, MISTEA

I am a research scientist with a physics background. Now, I do statistics to tackle environmental, and climate change problems. Julia enthusiast!

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juliahub.com: SmoothPeriodicStatsModels

A Julia package for time series and other models with smoothly varying periodic parameters

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