advanced-algorithmic-trading

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• Smoothing - Estimating the past values of the state given the observations
Filtering and smoothing are similar, but not the same. Perhaps the best way to think of the
difference is that with smoothing we are really wanting to understand what has happened to
states in the past given our current knowledge, whereas with filtering we really want to know
what is happening with the state right now.
In this chapter we are going to discuss the theory of the state space model and how we can
use the Kalman Filter to carry out the various types of inference described above. We will then
apply it to trading situations such as cointegrated pairs later in the book.
A Bayesian approach will be utilised for the problem. This is the statistical framework
that allows updates of beliefs in light of new information, which is precisely the desired behaviour
sought from the Kalman Filter.
I would like to warn you that state-space models and Kalman Filters suffer from an abundance
of mathematical notation, even if the conceptual ideas behind them are relatively straightforward.
I will try and explain all of this notation in depth, as it can be confusing for those new to
engineering control problems or state-space models in general. Fortunately we will be letting
Python do the heavy lifting of solving the model for us, so the verbose notation will not be a
problem in practice.
13.1 Linear State-Space Model
Let us begin by discussing all of the elements of the linear state-space model.
Since the states of the system are time-dependent we need to subscript them with t. We will
use θ t to represent a column vector of the states.
In a linear state-space model we say that these states are a linear combination of the prior
state at time t − 1 as well as system noise (random variation). In order to simplify the analysis
we are going to suggest that this noise is drawn from a multivariate normal distribution. Other
distributions can be used for alternative models but they will not be considered here.
The linear dependence of θ t on the previous state θ t−1 is given by the matrix G t , which can
also be time-varying (hence the subscript t). The multivariate time-dependent noise is given by
w t . The relationship is summarised below in what is often called the state equation:
θ t = G t θ t−1 + w t (13.1)
This equation is only half of the story. We also need to discuss the observations–what we
actually see–since the states are hidden to us.
We can denote the time-dependent observations by y t . The observations are a linear combination
of the current state and some additional random variation known as measurement noise,
which is also drawn from a multivariate normal distribution.
If we denote the linear dependence matrix of θ t on y t by F t (also time-dependent) and the
measurement noise by v t we have the observation equation:
y t = F T t θ t + v t (13.2)