oscillation: Oscillations of an MTD Markov chain

In hdMTD: Inference for High-Dimensional Mixture Transition Distribution Models

Oscillations of an MTD Markov chain

Description

Calculates the oscillations of an MTD model object or estimates the oscillations of a chain sample.

Usage

oscillation(x, ...)

Arguments

x

Must be an MTD object or a chain sample.

...

Additional parameters that might be required. Such as: S: If x is a chain sample the user should provide a set of lags for which he wishes to estimate the oscillations. It must be labeled as S, an in this scenario the function takes an upper bound for the order as d=max{S}. A: If x is a chain sample, and there may be elements in A that did not appear in x, the state space should be specified, and it must be labeled as A.

Details

The oscillation for a certain lag j of an MTD model ( \{ \delta_j:\ j \in \Lambda \} ), is the product of the weight \lambda_j multiplied by the maximum of the total variation distance between the distributions in a stochastic matrix p_j.

\delta_j = \lambda_j\max_{b,c \in \mathcal{A}} d_{TV}(p_j(\cdot | b), p_j(\cdot | c)).

So, if x is an MTD object, the parameters \Lambda, \mathcal{A}, \lambda_j, and p_j are inputted through, respectively, the entries Lambda, A, lambdas and the list pj of stochastic matrices. Hence, an oscillation \delta_j may be calculated for all j \in \Lambda.

If we wish to estimate the oscillations from a sample, then x must be a chain, and S, a vector representing a set of lags, must be informed. This way the transition probabilities can be estimated. Let \hat{p}(\cdot| x_S) symbolize an estimated distribution in \mathcal{A} given a certain past x_S ( which is a sequence of elements of \mathcal{A} where each element occurred at a lag in S), and \hat{p}(\cdot|b_jx_S) an estimated distribution given past x_S and that the symbol b\in\mathcal{A} occurred at lag j. If N is the sample size, d=max(S) and N(x_S) is the number of times the sequence x_S appeared in the sample, then

\delta_j = \max_{c_j,b_j \in \mathcal{A}} \frac{1}{N-d}\sum_{x_{S} \in \mathcal{A}^{S}} N(x_S)d_{TV}(\hat{p}(. | b_jx_S), \hat{p}(. | c_jx_S) )

is the estimated oscillation for a lag j \in \{1,\dots,d\}\setminusS. Note that \mathcal{A}^S is the space of sequences of \mathcal{A} indexed by S.

Value

If the x parameter is an MTD object, it will provide the oscillations for each element in Lambda. In case x is a chain sample, it estimates the oscillations for a user-inputted set S of lags.

Examples

oscillation( MTDmodel(Lambda=c(1,4),A=c(2,3) ) )
oscillation(MTDmodel(Lambda=c(1,4),A=c(2,3),lam0=0.01,lamj=c(0.49,0.5),
pj=list(matrix(c(0.1,0.9,0.9,0.1),ncol=2)), single_matrix=TRUE))

hdMTD documentation built on June 8, 2025, 10:30 a.m.