# Stochastic Processes and Gauss-Markov Models Part I

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## Outline

• The following topics will be covered in this lecture:
• Extending the concept of random vectors – stochastic processes
• Continiuity in probability
• Separability
• Gaussian stochastic processes
• Wiener Processes

## Stochastic processes

• We are now familiar with the Gaussian model for a random state, and its centrality due to the central limit theorem.

• Likewise, we have developed a variety of tools to extend the Gaussian model, and to efficiently simulate it numerically.
• We will now make a fundamental leap that will allow us to model time-varying random processes – i.e., we will introduce stochastic processes.

• A stochastic process is an infinite collection of random variables.

• We will ultimately use this object to model uncertain signals based on mechanistic processes;

• however, we want to emphasize from the beginning that a stochastic process is just a generalization of a random variable / vector.
• However, unlike a random vector $$\pmb{x}\in\mathbb{R}^{N_x}$$, a stochastic process requires considerably more care.

• This is due to the fact that a stochastic process $$\pmb{x}_t$$ is the generalization of a random vector into infinite dimensions.
Stochastic Processes
A stochastic process is a family of random variables, $$X(\omega, t)$$, indexed by a real parameter $$t \in T$$ and defined on a common probability space $$(\Omega, A, \mathcal{P} )$$.
• The “real parameter $$t$$” in the above is almost always taken a time variable.

• Thus, a random process can be thought of as a function that takes both the sample point and time as arguments.

• Hence, we will often use the notation $$X(\omega, t)$$, or $$X_t$$ for short, to denote a random variable depending on time.

### Stochastic processes

• Note, in many cases we will actually consider time to be discrete in nature;

• this is a consequence, in part, of the fact that we often have measurements discretely and not continuously in time.
• If the parameter set $$T = \mathbb{N}$$, the natural numbers, then $$X_t$$ is often called a random sequence to reflect the discrete nature of the time parameter.

• In this case, we will often use a notation such as $$X_k$$ corresponding to a sequence of times $$\{t_0, t_1, \cdots, t_k, \cdots\}$$ to denote the discrete nature of the time variable.
• On the other hand, if $$T = \mathbb{R}$$, then $$X_t$$ is called either a random function or process.

• We should also note that the values of $$X_t$$, for a fixed value of $$t$$, can also be a discrete or continuous random variable.

• If we look at random processes in a certain way, we can see that they are a natural extension of the idea of random vectors to infinite collections:

\begin{align} &\begin{matrix}\text{Random variable} & X \end{matrix} & \begin{matrix}\text{Random vector} & & \pmb{x}^\top:= \begin{pmatrix} X_1 & \cdots & X_{N_x}\end{pmatrix}\end{matrix}\\\\ &\begin{matrix}\text{Random sequence} & \begin{pmatrix} \cdots & X_k & \cdots \end{pmatrix}\end{matrix} & \begin{matrix} \text{Random process} & \begin{pmatrix} \cdots & X_t & \cdots \end{pmatrix}\end{matrix} \end{align}

• The key notion to take away is that a random process is a set of time functions, each of which is one possible outcome, $$\omega$$, out of the set of all possible outcomes, $$\Omega$$.
• A random variable returns a real number, and a random process returns a continuously indexed collection of real numbers or a real-valued function.

### Stochastic processes

• As a quick example, consider a sine wave with an amplitude given as a random variable taking on values from −1 to 1 with a uniform distribution:

\begin{align} X(\omega, t) := A(\omega) \sin(t): \Omega \times \mathbb{R} \rightarrow \mathbb{R} & & A(\omega) \sim U[-1, 1] \end{align}

• Again, we emphasize that the random variable $$X$$ takes $$\omega$$ and returns a function.

• In particular, if we know which sample point, $$\omega$$, we have selected, we know the plot of the function $$X_t$$ for all $$t\in \mathbb{R}$$.
• We can consider this in Python as follows, where A below is a uniform random variable on $$[-1,1]$$

import matplotlib.pyplot as plt
import numpy as np
np.random.seed(123)
A = np.random.uniform(low=-1, high=1)
A

0.3929383711957233

• Having determined the particular outcome for $$\omega$$, we now define a function in time given by

\begin{align} 0.3929383711957233 * \sin(t). \end{align}

### Stochastic processes

• We plot the function defined on the last slide for 5 cycles of sine:
time_points = np.linspace(start=0, stop=10*np.pi, num=10000)
plt.subplots(figsize=(24, 6))

(<Figure size 2400x600 with 1 Axes>, <matplotlib.axes._subplots.AxesSubplot object at 0x7fec2c398940>)

plt.plot(time_points, A*np.sin(time_points))

[<matplotlib.lines.Line2D object at 0x7fec2bdd3ba8>]

plt.show() ### Stochastic processes

• We just showed how fixing $$\omega$$ yields an outcome for a random function defined for all time.

• This particular function, defined over the time variable $$t$$, is known as a sample path or a realization of the random process.
• On the other hand, if we fix $$t$$ and let $$X$$ vary over $$\omega$$, i.e.,

\begin{align} X_{t_0}(\omega) := X(\omega,t)|_{t=t_0}, \end{align}

• we get a random variable $$X_{t_0}$$ defined on $$(\Omega, \mathcal{A}, \mathcal{P})$$ with

\begin{align} \{\omega : X_{t_0} (\omega) \leq x \} \in \mathcal{A}, \end{align} i.e., these form the generating sets of the observable events for the probability space.

• This is what is meant that the parameters $$t$$ indexes a family of random variables.

### Stochastic processes

• We can consider random processes as infinite-dimensional extensions of the idea of random vectors.

• However, when we extend our ideas to infinite dimensions, new complications arise.

• With random vectors, we can completely define the probability of the vector as a collection of random variables by the joint probability,

\begin{align} P(x_1, \cdots, x_{N_x}) := \mathcal{P}\left( X_1(\omega) \leq x_1 , \cdots, X_{N_x}(\omega) \leq x_{N_x}\right) \end{align}

• However, when $$T$$ is an infinite set, we have to be more careful.

• If $$T$$ is a countably infinite set, then we can characterize the process with sets of the form

\begin{align} \{\omega: X(\omega,t_k) \leq a, \quad k= 1,\cdots\} = \cap_{k=1}^\infty \{\omega : X_{t_k} \leq a\}; \end{align}

• in the measure-theoretic approach, this type of countably-infinite intersection still generates the observable events of the process.
• However, trouble arises with the uncountable infinity of the continuum for such constructions, where

\begin{align} \{\omega: X(\omega,t_k) \leq a, \quad k= 1,\cdots\} = \cap_{t\in\mathbb{R}} \{\omega : X_{t_k} \leq a\} \end{align} an intersection over a continuum can lead to problems with our construction of observable events.

• In particular, if we follow the above alone, the probability of $$\mathcal{P}\left(X_t \geq 0 : \forall t\in[0,1]\right)$$ may not actually be defined.

## Continuity in probability

• To get around the problems which might arise from continuous-time stochastic processes, we need to add two more conditions to our process.
Continuous in probability
A stochastic process $$X(\omega, t)$$ is said to be continuous in probability at $$t$$ if \begin{align} \lim_{s\rightarrow t} \mathcal{P}\left(\vert X(\omega, s) - X(\omega, t) \vert \geq \epsilon\right) =0 \end{align} for all $$\epsilon > 0$$.
• Note that in the above, the probability we are referring to is over all possible outcomes $$\omega\in \Omega$$.

• This means there is such a small set of possible $$\omega$$ for which we would see a discontinuity, that the probability of observing such an event is zero.

• Being continuous in probability means then that the possibility of seeing a nonzero jump in zero time has probability zero.

## Separability

• Our second condition is known as separability, which is described as follows.
Separability
A stochastic process $$X(\omega, t)$$ is said to be separable if there exists a countable, dense set $$S \subset T$$ such that for any closed set $$K \subset [-\infty, \infty]$$ the two sets \begin{align} A_1 = \{\omega : X(\omega, t) \in K, \forall t \in T \} & & A_2 =\{\omega: X(\omega, t)\in K, \forall s \in S\} \end{align} differ by a set $$A_0$$ such that $$\mathcal{P}\left(A_0\right)=0$$.
• Separability means that for a continuous-parameter random process, it is possible to analyze it like a discrete-parameter one.

• When we say that $$S$$ is dense in $$T$$, we mean that $$S$$ must contain enough points of $$T$$ so as to provide essentially a complete representation of $$T$$.

• That is, if we plotted out the set $$\{X(\omega, t) : t \in S\}$$ versus $$t$$, it would look indistinguishable from $$\{X(\omega, t), t \in T \}$$.

• Lets suppose that $$T = \mathbb{R}$$, the real line, or perhaps the positive real line $$\mathbb{R}^+$$.

• The simplest way to get a separating set of the continuous-time interval $$T$$ is to simply choose $$S$$ to be the set of rational numbers $$\mathbb{Q} \cap T$$.

• Mathematically, what this means is that for any point $$t \in T$$, there exists a discrete sequence $$t_k \in S$$ such that $$t_k \rightarrow t$$.

• Thus, if $$X(t, \omega)$$ is a separable process, it is such that $$X(t_k , \omega) \rightarrow X(t, \omega)$$ except on the zero probability set mentioned in the definition.

• We, therefore, have well-defined discretized limits in time with probability one.

## Gaussian processes

• Just like we extended the Gaussian to multiple variables, we can now extend the (multivariate) Gaussian to a family of Gaussian random (vectors) variables parameterized by time.

• We will need to start with identifying the notion of independence properly for a stochastic process.

Independent increments
Let $$X$$ be a random process defined on the time interval, $$T$$. Let $$t_0 < t_1 < \cdots < t_n$$ be a partition of the time interval, $$T$$. If the increments, $$X(t_k) − X(t_{k−1})$$, are mutually independent random variables for any partition of $$T$$, then $$X$$ is said to be a process with independent increments.
• In the above, this is to say that if we take arbitrary, discrete time points $$t_{k-1} < t_{k} < t_{k+1} \in T$$, the following holds

\begin{align} \mathcal{P}\left( X_{k+1} - X_{k} \leq a | X_{k} - X_{k-1} = b\right) = \mathcal{P}\left( X_{k+1} - X_{k} \leq a\right), \end{align} and vice-versa in the time indices.

• In particular, the size of any past difference in realizations doesn't affect the probability of any future difference in realizations.

## Gaussian processes

• We now introduce the notion of a Gaussian process, and two of its quintessential applications.
• Gaussian processes
We say that a random process, $$X_t$$, is a Gaussian process if for every finite collection, $$X_{1}, X_{2}, \cdots , X_{N_x}$$, the corresponding density function, \begin{align} p(x_1 , . . . , x_n ), \end{align} is a Gaussian density function, for the joint, multivariate Gaussian in $$\pmb{x}^\top:= \begin{pmatrix}X_1, & \cdots, & X_{N_x} \end{pmatrix}$$.
• A commonly understood Gaussian process is actually “white noise”, visualized to the right.