oscillation {hdMTD} | R Documentation |
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:
|
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\}\setminus
S
. 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))