# Stochastic Processes and Gauss-Markov Models Part II

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## Outline

• The following topics will be covered in this lecture:
• Markov processes
• Discrete Gauss-Markov models
• Propagation of the mean state
• Propagation of the covariance

## Markov processes

• In our last discussion, we considered stochastic processes generally, with a quintessential process as an example, i.e., Gaussian processes.

• The Gaussian process is at the basis of how we will model a noise-perturbed mechanistic system, but we will also require another important property.

• The additional property we will introduce is the Markov hypothesis;

• in particular, a Markov process is sometimes known as a “memoryless process”.
• A memoryless process is physically intuitive in a variety of situations, in that this is extremely similar to an initial value problem.

• An initial value problem is one in which we have well-defined governing process laws, for which the initial value of a model state will determine the subsequent time-evolution.

• In dynamical systems, this is known as the group action property of the flow map.

• If we satisfy certain conditions with the governing laws, there will exist a flow map,

\begin{align} \boldsymbol{\Phi}(s, \pmb{x}_t): &\mathbb{R} \times \mathbb{R}^{N_x} \rightarrow \mathbb{R}^{N_x}\\ &\pmb{x}_t \rightarrow \pmb{x}_{t+s}, \end{align} which completely describes the time-evolution of any initial state.

• Therefore, in order to model the state at time $$t + s$$, we do not need to know the history of the state before time $$t$$;

• rather, $$\pmb{x}_t$$ provides the complete information necessary to evolve the state to $$\pmb{x}_{t+s}$$.

### Markov processes

• Let us firstly denote the following notations:
Time series slice notation
For a sequence of time points $$\{t_i\}_{i=k}^l\subset T$$, we will denote the “slice” notation for time series of random states as \begin{align} X_{l:k} &:= \{X_{l} , X_{l-1} ,\cdots, X_{k}\} \\ \pmb{x}_{l:k} &:= \{\pmb{x}_{l} , \pmb{x}_{l-1}, \cdots , \pmb{x}_{k}\} \end{align} for either the random variables $$X_i$$ or the random vectors $$\pmb{x}_i$$.
• These notations will often be useful to compactify our subsequent analyses, and we will extend this notation also to observable realizations.
Markov processes
A random process $$\{X_t : t\in T\}$$ where $$T\subset\mathbb{R}$$, is said to be a Markov process if for any increasing collection $$t_0 < t_1 < \cdots < t_n \in T$$ \begin{align} \mathcal{P}\left( X_{n} \leq x_n | X_{n-1:0} = x_{n-1:0}\right) = \mathcal{P}\left( X_n \leq x_n | X_{n-1} = x_{n-1}\right), \end{align} or equivalently in terms of the CDF, or the PDF when it exists.
• We similarly extend the notion to random vectors indexed in time, replacing $$X_k$$ with $$\pmb{x}_k$$.

• The above thus gives precisely the analogy of the inital value problem on the last slide.

• In particular, when we condition on knowledge of the realization of the state at the previous time index, the outcome at the current time is independent of the remaining history at times $$t_{n-2} ,\cdots, t_{0}$$.

• In a sense, $$X_{n-1}=x_{n-1}$$ describes an initial value problem for the probability at the subsequent time index.

### Markov processes

• A special property thus emerges from the Markov hypothesis in how we understand a joint density or cdf of a time series.

• Lets suppose that the joint density, $$p\left(\pmb{x}_{n:0}\right)$$, exists for a time series; we will consider how the Markov hypothesis can be applied recursively:

\begin{align} p\left(\pmb{x}_{n:0}\right) = p\left(\pmb{x}_n | \pmb{x}_{n-1:0} \right)p\left(\pmb{x}_{n-1:0}\right) \end{align} simply as a consequence of conditional probability.

• However we know that, for a Markov process,

\begin{align} p\left(\pmb{x}_n | \pmb{x}_{n-1:1} \right) \equiv p\left(\pmb{x}_n | \pmb{x}_{n-1}\right). \end{align}

• We can therefore simplify the above with the equivalence as

\begin{align} p\left(\pmb{x}_{n:0}\right) &= p\left(\pmb{x}_n | \pmb{x}_{n-1} \right)p\left(\pmb{x}_{n-1:0}\right)\\ &= p\left(\pmb{x}_n | \pmb{x}_{n-1} \right)p\left(\pmb{x}_{n-1}|\pmb{x}_{n-2:0}\right)p\left(\pmb{x}_{n-2:0}\right)\\ &=p\left(\pmb{x}_n | \pmb{x}_{n-1} \right)p\left(\pmb{x}_{n-1}|\pmb{x}_{n-2}\right)p\left(\pmb{x}_{n-2:0}\right). \end{align}

• Notice the pattern extends recursively so that we can identify

\begin{align} p(\pmb{x}_{n:0}) \equiv p(\pmb{x}_0)\prod_{k=1}^{n-1} p(\pmb{x}_{k+1}|\pmb{x}_k). \end{align}

### Markov processes

• Recall the identity in the last slide,

\begin{align} p(\pmb{x}_{n:0}) \equiv p(\pmb{x}_0)\prod_{k=1}^{n-1} p(\pmb{x}_{k+1}|\pmb{x}_k). \end{align}

• In the above, we will typically call $$p(\pmb{x}_0)$$ our “prior” knowledge, assuming that there is no history that we consider before time $$t_0$$.

• This initial density, instead, represents all knowledge about the process that would provide (uncertain) initial value data for a prediction problem.

• On the other hand, $$p(\pmb{x}_{k+1}| \pmb{x}_{k})$$ is known as a transition probability density, describing the probability for a realization at the next time index given the realization at the previous one.

• Given some initial realization of the random vector $$\pmb{x}_0 = \pmb{x}_0^\ast$$, this chain of transition probabilities thus determines the entire joint density of the time series.

## Discrete Gauss-Markov models

• A Gauss–Markov process is simply a random process that is both Gaussian and Markov.

• These processes play a big role in filtering theory and signal processing for two basic reasons:

• The first is that Markov processes can be completely described by an initial condition and a transition density function, which is a great simplification.
• The second is that the Gaussian is closed under linear (and affine) transformations.
• Because of these properties, a Gauss–Markov process can be represented by the state vector of a multistate linear dynamical system,

\begin{align} \pmb{x}_k := \mathbf{M}_{k} \pmb{x}_{k-1} + \pmb{w}_k \end{align} where

• $$\pmb{x}_k\in \mathbb{R}^{N_x}$$ represents the random vector governed by the process laws encompassed in the linear transformation $$\mathbf{M}_k\in\mathbb{R}^{N_x \times N_x}$$;
• $$\mathbf{M}_k$$ is assumed to have no zero eigenvalues, so that it is an invertible linear transformation;
• $$\pmb{w}_k$$ is known as process noise, representing inadequacies of the deterministic laws encoded in $$\mathbf{M}_k$$;
• particularly, we will typically assume that

\begin{align} \pmb{w}_k \sim N(\pmb{0}, \mathbf{Q}_k) \end{align} so that $$\pmb{w}_k$$ represents an unbiased shock to the deterministic evolution of $$\pmb{x}_{k-1}$$; and

• we assume that there is an initial prior given as $$\pmb{x}_0 \sim N\left(\overline{\pmb{x}}_0, \mathbf{B}_0\right)$$.

### Discrete Gauss-Markov models

• Recall the discrete Gauss-Markov model

\begin{align} \pmb{x}_k := \mathbf{M}_{k} \pmb{x}_{k-1} + \pmb{w}_k \end{align}

• Note that we do not assume to observe $$\pmb{w}_k$$, but we will typically assume to know its statistics, i.e., we know the covariance of the noise $$\mathbf{Q}_k$$.

• We will also assume that the sequence of model noise $$\pmb{w}_{k:1}$$ is white-in-time, i.e.,

\begin{align} \mathbb{E}\left[\pmb{w}_k \pmb{w}_l^\top \right] := \delta_{k,l}\mathbf{Q}_k \end{align} where $$\delta_{k,l}$$ denotes the Kronecker delta, i.e.,

\begin{align} \delta_{k,l} := \begin{cases} 1 & \quad \text{if } k=l\\ 0 &\quad \text{else} \end{cases} \end{align}

• The white-in-time assumption is what keeps the process Markovian, so that there are not correlations between the noise realizations at different times;

• thereby, this prevents the dependence of the transition probability on far-past history.

### Discrete Gauss-Markov models

• When the conditions of the last two slides are satisfied, we arrive at an extremely powerful result:
Discrete Gauss-Markov models
Suppose that we have an initial prior $$\pmb{x}_0 \sim N\left(\overline{\pmb{x}}_0, \mathbf{B}_0\right)$$, and the time evolution of the random vector $$\pmb{x}_k$$ is governed by the discrete Gauss-Markov model \begin{align} \pmb{x}_k := \mathbf{M}_{k} \pmb{x}_{k-1} + \pmb{w}_k \end{align} as already discussed. Then, the joint density for the time series $$\pmb{x}_{k:0}$$ is Gaussian, as are all of the conditional densities and the marginal densities.
• We note, we can easily derive the transition density for an arbitrary state as follows,

\begin{align} \pmb{x}_k - \mathbf{M}_k\pmb{x}_{k-1} = \pmb{w}_k \sim N\left(\pmb{0}, \mathbf{Q}_k\right). \end{align}

• Therefore, we can write

\begin{align} p(\pmb{x}_k | \pmb{x}_{k-1}) = \left(2 \pi\right)^{-\frac{N_x}{2}} \vert\mathbf{Q}_k\vert^{-\frac{1}{2}} \exp\left\{-\frac{1}{2}\left(\pmb{x}_k - \mathbf{M}_k\pmb{x}_{k-1}\right)^\top \mathbf{Q}^{-1}_k\left(\pmb{x}_k - \mathbf{M}_k\pmb{x}_{k-1}\right)\right\} \end{align}

• In the above, we can thus qualitatively consider the transition probability to be given as

• a hyper-exponential penalty function, where the probability density decays at order $$\parallel \pmb{y}\parallel^2$$; where
• $$\parallel \pmb{y}\parallel:= \parallel \pmb{x}_k - \mathbf{M}_k\pmb{x}_{k-1}\parallel_{\mathbf{Q}_k}$$, i.e., the norm of the discrepancy from the deterministic evolution of the last state;
• weighted inverse proportionally to the spread of the model noise (or the model uncertainty).

## Propagation of the mean state

• The last result is an extremely important result, that we will utilized heavily in forming Bayesian inferences.

• However, it should be noted that there is another classical approach to handling the propagation of the initial Gaussian $$N(\overline{\pmb{x}}_0,\mathbf{B}_0)$$.

• In particular, a Gaussian distribution is entirely parameterized in terms of the mean and covariance;

• therefore, knowledge of these values completely describes the distribution in question at any time.
• We will consider this approach in the following when we look at how the mean is propagated in time.

• Consider,

\begin{align} \mathbb{E}\left[ \pmb{x}_k \right] &= \mathbb{E}\left[\mathbf{M}_k \pmb{x}_{k-1} + \pmb{w}_k \right] \\ &= \mathbb{E}\left[ \mathbf{M}_k \pmb{x}_{k-1}\right] +\mathbb{E}\left[ \pmb{w}_k \right]\\ &= \mathbf{M}_k \overline{\pmb{x}}_{k-1} + \pmb{0}\\ &\equiv \overline{\pmb{x}}_k \end{align}

• Therefore, if the model noise is unbiased as above, the mean at a subsequent time is given by the deterministic evolution of the mean at the last time.

• Using this relationship recursively, we then say

\begin{align} \overline{\pmb{x}}_k &= \mathbf{M}_k \cdots \mathbf{M}_1 \overline{\pmb{x}}_0 \\ &:= \mathbf{M}_{k:1} \overline{\pmb{x}}_0 . \end{align}

## Propagation of the covariance

• The previous, inductive relationship for the mean state then gives us the ability to compute the covariance.

• Consider,

\begin{align} \mathrm{cov}\left(\pmb{x}_k\right) &:= \mathbb{E}\left[\left(\pmb{x}_k - \overline{\pmb{x}}_k\right)\left(\pmb{x}_k - \overline{\pmb{x}}_k\right)^\top\right]\\ &=\mathbb{E}\left[\left(\mathbf{M}_k \pmb{x}_{k-1} + \pmb{w}_k - \mathbf{M}_k\overline{\pmb{x}}_{k-1} \right)\left(\mathbf{M}_k \pmb{x}_{k-1} + \pmb{w}_k - \mathbf{M}_k\overline{\pmb{x}}_{k-1} \right)^\top\right]\\ &=\mathbb{E}\left[\left(\mathbf{M}_k \pmb{x}_{k-1} - \mathbf{M}_k \overline{\pmb{x}}_{k-1}\right)\left(\mathbf{M}_k \pmb{x}_{k-1} - \mathbf{M}_k \overline{\pmb{x}}_{k-1}\right)^\top\right] \\ & \quad + \mathbb{E}\left[\left(\mathbf{M}_k \pmb{x}_{k-1} - \mathbf{M}_k \overline{\pmb{x}}_{k-1}\right)\pmb{w}_k^\top \right] + \mathbb{E}\left[\pmb{w}_k\left(\mathbf{M}_k \pmb{x}_{k-1} - \mathbf{M}_k \overline{\pmb{x}}_{k-1}\right)^\top \right]\\ &\quad + \mathbb{E}\left[ \pmb{w}_k \pmb{w}^\top_k \right] \end{align}

• Having assumed that the model noise is white-in-time, $$\pmb{x}_{k-1}$$ is independent of $$\pmb{w}_k$$, so that all cross terms have zero covariance.

• Furthermore, we recognize

\begin{align} \left(\mathbf{M}_k \pmb{x}_{k-1} - \mathbf{M}_k \overline{\pmb{x}}_{k-1}\right)=\mathbf{M}_k\left(\pmb{x}_{k-1} -\overline{\pmb{x}}_{k-1}\right) \end{align} such that we write

\begin{align} \mathrm{cov}\left(\pmb{x}_k\right) = \mathbf{M}_k \mathbf{B}_{k-1}\mathbf{M}_k^\top + \mathbf{Q}_k. \end{align}

### Propagation of the covariance

• The recursive form for the mean and covariance provides a complete (recursive) description of the probability distribution of $$\pmb{x}_k$$ in time.

• This is given inductively, with knowledge of the first prior and the model noise statistics, as

\begin{align} \pmb{x}_0 &\sim N(\overline{x}_0, \mathbf{B}_0); \\ \pmb{x}_k & \sim N\left(\mathbf{M}_k\overline{\pmb{x}}_{k-1} , \mathbf{M}_k\mathbf{B}_{k-1}\mathbf{M}_k^\top + \mathbf{Q}_k\right); \end{align} where the second line provides the inductive step, describing all future distributions.

• This is an exact equation, which fully describes the discrete Gauss-Markov model, in the absence of observations of the process.

• That is to say, this form of the discrete Gauss-Markov model only assumes that we use our prior knowledge, but that we do not update this with additional conditional knowledge of the evolution.

• Shortly, we will introduce the technology to produce conditional estimates when there is new information arriving sequentially in time.

• Firstly, however, we will extend this discrete Gauss-Markov model to systems that are generated by stochastic differential equations.

• In doing so, we will also seek to explain how one simulates such systems numerically, and how some of these simulation techniques are connected to our ultimate goal, i.e.,

• nonlinear estimation with the Gauss-Markov approximation.