\subsection{Reversibility}

Your time series is said to be irreversible (directional) if it has probabilistic properties that depend on the direction of time.  A time series is said to be reversible if it has no such properties.  Directionality is only defined with respect to stationary series and is not concerned with deterministic behavior \citep{Lawrance1991}.

Appropriate plotting of time series can often reveal directionality; there is a lack of directional symmetry in certain characteristic behaviour sequences, such as rises and falls. Figures 1 and 2 give the plots of Cleveland \& McGill (1987) and Cleveland, McGill \& McGill (1986), and additionally their reversed versions, illustrating directionality of the much revered sunspots and Canadian lynx time series, respectively. There has been relatively little written about directionality in time series, and no consolidated account, as is intended here. One reason for this may be that, until recently, time series modelling has been coterminus with standard Gaussian autoregressive moving average models (ARMA models); the Gaussian assumption makes these reversible. Nevertheless, scattered comments on the topic have appeared in the research literature and will presently be cited. Many specialists are aware of the topic but very few books make mention of it; Tong (1990) is an exception.

A time series, modelled by random variables $\{H_t, t = 0, \pm 1, \pm 2, \ldots \}$, is said to be reversible when, for all $r = 1, 2, \ldots$	and $t = 0, \pm 1,	\pm 2, \ldots$,	the joint distribution of 
\begin{equation}
H_t, H_{t+1}, \ldots, H_{t+r}
\end{equation}
is equal to the joint distribution of
\begin{equation}
H_{t+r}, H_{t+r-l},	\ldots,	H_t.
\end{equation}
This definition makes clear that reversibility is a property of dependence in the joint distributions of the series. It will be clear that it is dependence of higher order than second order moments of the form E($H_t, H_{t+r}$), and is not detected by autocorrelations. The first formal definition of reversibility appears to be by Brillinger \& Rosenblatt (1967). Reversibility may seem a strange notion in terms of time, but there are clear possibilities of bi-directionality when spatial distance is used instead.

Detect irreversibility in hurricane counts.  Sunspot series is irreversible.  If the El Nino, SST, and NAO are reversible then the irreversibility in hurricane counts arises from the sun.




http://rss.acs.unt.edu/Rdoc/library/flexmix/html/KLdiv.html

visF = get.visibility(annual$US.1)
edgeF = visF$sm[visF$sm[, 1] < visF$sm[, 2], ]
gF = graph.edgelist(edgeF, directed=FALSE)
degree.distribution(gF)
visB = get.visibility(rev(annual$US.1))
edgeB = visB$sm[visB$sm[, 1] < visB$sm[, 2], ]
gB = graph.edgelist(edgeB, directed=FALSE)
degree.distribution(gB)

edgeB = vis$sm[vis$sm[, 1] > vis$sm[, 2], ]

gF = graph.edgelist(edgeF, directed=FALSE)
gB = graph.edgelist(edgeB, directed=FALSE)
diameter(gF); diameter(gB)