The Hidden Markov Model
Hidden Markov models (HMMs) are a modeling framework for time series
data where a sequence of observation is assumed to depend on a latent
state process. The peculiarity is that, instead of the observation
process, the state process cannot be directly observed. However, the
latent states comprise information about the environment the model is
applied on.
The connection between hidden state process and observed
state-dependent process arises by the following: Let \(N\) be the number of possible states. We
assume that for each point in time \(t = 1,
\ldots, T\), an underlying process \((S_t)_{t = 1, \ldots, T}\) selects one of
those \(N\) states. Then, depending on
the active state \(S_t \in \{ 1, \ldots, N
\}\), the observation \(X_t\)
from the state-dependent process \((X_t)_{t =
1, \ldots, T}\) is generated by one of \(N\) distributions \(f^{(1)},\dots,f^{(N)}.\)
Furthermore, we assume \((S_t)_t\)
to be Markovian, i.e. we assume that the actual state only depends on
the previous state. Henceforth, we can identify the process by its
initial distribution \(\delta\) and its
transition probability matrix (t.p.m.) \(\Gamma\). Moreover, by construction, we
force the process \((X_t)_{t = 1, \ldots,
T}\) to satisfy the conditional independence assumption, i.e. the
actual observation \(X_t\) depends on
the current state \(S_t\), but does not
depend on previous observations or states at all. The following graphic
visualizes the dependence structure:
Dependence structure of the HMM.
Referring to financial data, the different states can serve as
proxies for the actual market situation, e.g. calm or nervous. Even
though these moods cannot be observed directly, price changes or trading
volumes, which clearly depend on the current mood of the market, can be
observed. Thereby, using an underlying Markov process, we can detect
which mood is active at any point in time and how the different moods
alternate. Depending on the current mood, a price change is generated by
a different distribution. These distributions characterize the moods in
terms of expected return and volatility.
Following Zucchini, MacDonald, and Langrock
(2016), we assume that the initial
distribution \(\delta\) equals the
stationary distribution \(\pi\), where
\(\pi = \pi \Gamma\), i.e. the
stationary and henceforth the initial distribution is determined by
\(\Gamma\). This is reasonable
from a practical point of view: On the one hand, the hidden state
process has been evolving for some time before we start to observe it
and hence can be assumed to be stationary. On the other hand, setting
\(\delta=\pi\) reduces the number of
parameters that need to be estimated, which is convenient from a
computational perspective.
Adding a Hierarchical Structure
The hierarchical hidden Markov model (HMMM) is a flexible extension
of the HMM that can jointly model data observed on two different time
scales. The two time series, one on a coarser and one on a finer scale,
differ in the number of observations, e.g. monthly observations on the
coarser scale and daily or weekly observations on the finer scale.
Following the concept of HMMs, we can model both state-dependent time
series jointly. First, we treat the time series on the coarser scale as
stemming from an ordinary HMM, which we refer to as the coarse-scale
HMM: At each time point \(t\) of the
coarse-scale time space \(\{1,\dots,T\}\), an underlying process
\((S_t)_t\) selects one state from the
coarse-scale state space \(\{1,\dots,N\}\). We call \((S_t)_t\) the hidden coarse-scale state
process. Depending on which state is active at \(t\), one of \(N\) distributions \(f^{(1)},\dots,f^{(N)}\) realizes the
observation \(X_t\). The process \((X_t)_t\) is called the observed
coarse-scale state-dependent process. The processes \((S_t)_t\) and \((X_t)_t\) have the same properties as
before, namely \((S_t)_t\) is a
first-order Markov process and \((X_t)_t\) satisfies the conditional
independence assumption.
Subsequently, we segment the observations of the fine-scale time
series into \(T\) distinct chunks, each
of which contains all data points that correspond to the \(t\)-th coarse-scale time point. Assuming
that we have \(T^*\) fine-scale
observations on every coarse-scale time point, we face \(T\) chunks comprising of \(T^*\) fine-scale observations each.
The hierarchical structure now evinces itself as we model each of the
chunks by one of \(N\) possible
fine-scale HMMs. Each of the fine-scale HMMs has its own t.p.m. \(\Gamma^{*(i)}\), initial distribution \(\delta^{*(i)}\), stationary distribution
\(\pi^{*(i)}\), and state-dependent
distributions \(f^{*(i,1)},\dots,f^{*(i,N^*)}\). Which
fine-scale HMM is selected to explain the \(t\)-th chunk of fine-scale observations
depends on the hidden coarse-scale state \(S_t\). The \(i\)-th fine-scale HMM explaining the \(t\)-th chunk of fine-scale observations
consists of the following two stochastic processes: At each time point
\(t^*\) of the fine-scale time space
\(\{1,\dots,T^*\}\), the process \((S^*_{t,t^*})_{t^*}\) selects one state
from the fine-scale state space \(\{1,\dots,N^*\}\). We call \((S^*_{t,t^*})_{t^*}\) the hidden fine-scale
state process. Depending on which state is active at \(t^*\), one of \(N^*\) distributions \(f^{*(i,1)},\dots,f^{*(i,N^*)}\) realizes
the observation \(X^*_{t,t^*}\). The
process \((X^*_{t,t^*})_{t^*}\) is
called the observed fine-scale state-dependent process.
The fine-scale processes \((S^*_{1,t^*})_{t^*},\dots,(S^*_{T,t^*})_{t^*}\)
and \((X^*_{1,t^*})_{t^*},\dots,(X^*_{T,t^*})_{t^*}\)
satisfy the Markov property and the conditional independence assumption,
respectively, as well. Furthermore, it is assumed that the fine-scale
HMM explaining \((X^*_{t,t^*})_{t^*}\)
only depends on \(S_t\). This
hierarchical structure is visualized in the following:
Dependence structure of the HHMM.