hdMTD

Overview

hdMTD is an R programming language package for the estimation of parameters in Mixture Transition Distribution (MTD) models. An MTD is a Markov chain with transition probabilities represented as a convex combination of conditional distributions. Given a sample from an MTD chain, hdMTD can estimate the relevant past set using various methods, such as Bayesian Information Criterion (BIC) and the Forward Stepwise and CUT (FSC) algorithm, which is efficient in estimating the set of relevant lags even in high-dimension, i.e., when dependence extends to pasts so distant that they may be close to the sample size. The package also computes the Maximum Likelihood Estimate (MLE) for transition probabilities, estimates oscillations, determines MTD parameters through the Expectation-Maximization (EM) algorithm, and can also simulate an MTD sample from its invariant distribution using the perfect sample algorithm.

Installation

r remotes::install_github("MaiaraGripp/hdMTD")

Usage

Given a sample from an MTD chain, the hdMTD() function estimates the relevant lag set $\Lambda$ using a specified method and suitable parameters. The available methods include:

• The FSC algorithm developed by Ost and Takahashi (2023).
  
• The first step of FSC, referred to as the FS method.
  
• The second step of FSC, known as the CUT method.
  
• A Bayesian Information Criterion ( BIC ) approach.

Additionally, the package provides the following functionalities:

Model Definition and Sampling

• Create an MTD object with all parameters necessary to define an MTD model using MTDmodel().
  
• Generate a perfect sample from the invariant distribution of an MTD Markov chain using perfectSample().

Estimation and Computation

• Compute the oscillations of an MTD object or estimate them from a chain sample using oscillations().
  
• Obtain the Maximum Likelihood Estimates (MLE) of transition probabilities for a given set of lags using probs().
  
• Compute the absolute frequency of all observed sequences of length $d$ in a sample using countsTab(). Aggregate these frequencies to consider only sequences indexed by subsets of ${1, \dots, d}$ using freqTab().

Parameter Estimation

• Estimate the parameters of an MTD model from a sample using MTDest(), which implements an Expectation-Maximization (EM) algorithm based on Lèbre and Bourguinon (2008).

``` r

library(hdMTD) set.seed(1234)

1. Simulating an MTD Markov Chain:

Lambda <- c(1, 5, 10) # Set of relevant lags Λ = {-10, -5, -1} A <- c(0, 1) # State space

Create an MTD model

MTD <- MTDmodel(Lambda = Lambda, A = A) MTD # An MTD class object

> $P

> 0 1

> 000 0.3796866 0.6203134

> 001 0.4474859 0.5525141

> 010 0.2728461 0.7271539

> 011 0.3406455 0.6593545

> 100 0.4389599 0.5610401

> 101 0.5067593 0.4932407

> 110 0.3321195 0.6678805

> 111 0.3999188 0.6000812

>

> $lambdas

> lam0 lam-1 lam-5 lam-10

> 0.2228608 0.2280200 0.3149060 0.2342131

>

> $pj

> $pj$p-1

> 0 1

> 0 0.01405572 0.9859443

> 1 0.31139528 0.6886047

>

> $pj$p-5

> 0 1

> 0 0.7104103 0.2895897

> 1 0.3711330 0.6288670

>

> $pj$p-10

> 0 1

> 0 0.5052655 0.4947345

> 1 0.7583400 0.2416600

>

>

> $p0

> p0(0) p0(1)

> 0.1544877 0.8455123

>

> $Lambda

> [1] 1 5 10

>

> $A

> [1] 0 1

Compute oscillations for the MTD model

oscillation(MTD)

> -1 -5 -10

> 0.06779938 0.10684048 0.05927337

Generate a sample from the MTD

X <- perfectSample(MTD = MTD, N = 1000) X[1:10] # Display the last 10 sampled states (X[1] is the latest sampled state).

> [1] 1 0 0 0 1 0 1 0 0 1

2. Inference on MTD Markov Chains

oscillation(X, S = c(1, 10)) # Estimated oscillations for lags -10 and -1

> -1 -10

> 0.04735179 0.06575918

Estimate the set of relevant lags using the "Forward Stepwise" (FS) method

S1 <- hdMTD(X = X, d = 15, method = "FS", l = 3) S1 # Estimated lags (size l=3)

> [1] 5 10 1

Alternative equivalent function:

S1 <- hdMTD_FS(X = X, d = 15, l = 3)

Refining the estimated lags using "BIC"

S2 <- hdMTD(X, d = max(S1), method = "BIC", S = S1, minl = 1, maxl = 3) S2 # Estimated set of relevant lags with the "BIC" method using "FS" output.

> [1] 5

Alternative equivalent function:

S2 <- hdMTD_BIC(X, d = max(S1), S = S1, minl = 1, maxl = 3)

"BIC" estimation for sets of relevant lags

S3 <- hdMTD(X, d = 12, method = "BIC", minl = 1, maxl = 3, byl = TRUE, BICvalue = TRUE) S3 # Sets of relevant lags (subsets of 1:12) with sizes from size 1 to 3 with lowest BIC.

> 5 5,10 5,7,10 smallest: 5

> 668.7065 675.4400 682.0869 668.7065

"CUT" estimation for sets of relevant lags given S1

S4 <- hdMTD(X, d = 20, method = "CUT", S = S1, alpha = 0.1) S4 # Estimated set of relevant lags with the "CUT" method using "FS" output.

> [1] 10 5

# "FSC" method combining FS and CUT

S5 <- hdMTD(X, d = 20, method = "FSC", l=3, alpha = 0.01); S5 # Estimated set of relevant lags with the "FSC".

> [1] 5

Validation: S5 should match the result of running "FS" on the first half of the sample

and "CUT" on the second half

all.equal(S5, hdMTDCUT(X[501:1000], d = 20, S = hdMTDFS(X[1:500], d = 20, l=3), alpha = 0.01))

> [1] TRUE

3. Estimating Transition Probabilities

Estimate the transition probability matrix given a sample and set of relevant lags

p <- probs(X, S = S4, matrixform = TRUE) p # MLE given the CUT output as set of relevant lags

> 0 1

> 00 0.3779070 0.6220930

> 01 0.3254717 0.6745283

> 10 0.5070423 0.4929577

> 11 0.3638677 0.6361323

4. Estimating MTD Parameters with the Expectation-Maximization (EM) Algorithm

Initial parameter values for the EM algorithm

init <- list( 'lambdas'= c(0.05,0.3,0.3,0.35), 'p0' = c(0.5,0.5), 'pj' = list( matrix(c(0.5,0.5, 0.5,0.5),ncol=2,nrow = 2), matrix(c(0.5,0.5, 0.5,0.5),ncol=2,nrow = 2), matrix(c(0.5,0.5, 0.5,0.5),ncol=2,nrow = 2) ) )

Estimate the MTD model parameters

estParam <- MTDest(X,S=c(1,5,10),init=init, iter = TRUE); estParam

> $lambdas

> lam-0 lam-1 lam-5 lam-10

> 0.04889442 0.29599651 0.30684622 0.34826285

>

> $pj

> $pj$p_-1

> 0 1

> 0 0.2986150 0.7013850

> 1 0.4432366 0.5567634

>

> $pj$p_-5

> 0 1

> 0 0.5928177 0.4071823

> 1 0.2647329 0.7352671

>

> $pj$p_-10

> 0 1

> 0 0.2658844 0.7341156

> 1 0.4675134 0.5324866

>

>

> $p0

> p0(0) p0(1)

> 0.3845184 0.6154816

>

> $iterations

> [1] 9

>

> $distlogL

> [1] 28.863219344 2.133782102 1.082393935 0.549555520 0.279169153

> [6] 0.141869452 0.072120503 0.036675669 0.018657448 0.009494794

Build the estimated MTD model using the inferred parameters

estMTD <- MTDmodel(Lambda, A, lam0 = estParam$lambdas[1], lamj = estParam$lambdas[-1], p0 = estParam$p0, pj = estParam$pj)

Display the estimated transition probability matrix

estMTD$P

> 0 1

> 000 0.3816913 0.6183087

> 001 0.4244988 0.5755012

> 010 0.2810198 0.7189802

> 011 0.3238273 0.6761727

> 100 0.4519112 0.5480888

> 101 0.4947187 0.5052813

> 110 0.3512397 0.6487603

> 111 0.3940471 0.6059529

```

Data sets

This package includes three real-world data sets acquired from external sources and one simulated data set (testChains), which was generated using the perfectSample function.

• raindata: This dataset was obtained from Kaggle, but its original source is the Australian Bureau of Meteorology
  (Data Source, Climate Data).
  Copyright: Commonwealth of Australia 2010, Bureau of Meteorology.

• sleepscoring: This dataset was collected at the Haaglanden Medisch Centrum (HMC, The Netherlands) Sleep Center and is publicly available on PhysioNet
  (DOI: 10.13026/t79q-fr32).

• tempdata: This dataset was obtained from INMET (National Institute of Meteorology, Brazil) and is available at INMET Data Portal.

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References

The methods implemented in this package are based on the following works:

• “FS”, “CUT”, and “FSC” methods:
  These functions are based on: Ost G, Takahashi DY (2023). Sparse Markov Models for High-Dimensional Inference.
  Journal of Machine Learning Research, 24(279), 1–54.
  URL: http://jmlr.org/papers/v24/22-0266.html

• “BIC” method:
  This function is based on the well-known Bayesian Information Criterion (BIC) method for model selection:
  Imre Csiszár & Paul C. Shields (2000). The Consistency of the BIC Markov Order Estimator.
  The Annals of Statistics, 28(6), 1601-1619.
  DOI: 10.1214/aos/1015957472

• MTDest function (Expectation-Maximization method):
  The MTDest function applies the Expectation-Maximization (EM) algorithm for parameter estimation in Mixture Transition Distribution (MTD) models, based on the following paper:
  Lebre, Sophie & Bourguignon, Pierre-Yves (2008). An EM Algorithm for Estimation in the Mixture Transition Distribution Model.
  Journal of Statistical Computation and Simulation, 78.
  DOI: 10.1080/00949650701266666