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()

plot of chunk unnamed-chunk-2

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.
White noise.
  • White noise corresponds, typically, to a Gaussian process as above such that for any two time points, \( t_0, t_1 \):
    • \( \mathbb{E}\left[X_0\right]=\mathbb{E}\left[X_1\right]=\overline{x} \) and \( \mathrm{var}\left(X_0\right) = \mathrm{var}\left(X_1\right) = \sigma^2 \); and
    • \( X_0 \) and \( X_1 \) are independent.
  • A formal mathematical definition can actually be quite a bit more complicated, but this is a good intuitive way to understand the white noise Gaussian process.
  • We can think of white noise as adding random shocks to a system, with an average value of the shock being \( \overline{x} \).
  • When \( \overline{x}=0 \), this corresponds to a sequence of perturbations that “fuzz”, but do not bias, the underlying process.

The Weiner process

  • A related application of Gaussian processes is one that we will use frequently in stochastic, continuous-time models.
  • Wiener process
    A continuous-time stochastic process is denoted a Wiener process \( W_{t} \) if it has the following properties:
    • \( W_0:= 0 \),
    • \( W \) has independent increments, as defined earlier;
    • The increments \( W_{t+s} - W_{t} \sim N(0,s) \); and
    • \( W_t \) is continuous in \( t \).
  • The Wiener process is a common model for the interaction of a particle suspended in fluid as it bounces off other particles in its evolution.
  • In particular, this is also known as a Brownian motion, as pictured to the right.
Brownian motion.

Courtesy of Lookang et al., CC BY-SA 3.0, via Wikimedia Commons

  • The Wiener process is a very special example of a function that is everywhere continuous but is nowhere differentiable.
  • It is also used commonly to model the time evolution of a physical process, defined by differential equations, when this is perturbed by additive white noise.
  • In particular, the Wiener process can be formulated as an integral of a white noise process, using the proper mathematical formalism.