Data Science

A dynamical system is a system in which a function describes the time dependence of a point in a geometrical space
Said another way, a dynamical system is a collection of random variables that is indexed by time
A stochastic dynamical system is a dynamic system subjected to the effects of noise
- In other words, a dynamic system is considered to be stochastic if the system requires mapping its inputs to outputs using probabilities
A deterministic dynamical system is a dynamic system that isn't subjected to the effect of noise
We will be interested in stochastic dynamical systems in the majority of our research

In our case, a system refers to a collection of states indexed by time, which represents some broader notion
- Some examples of this include the stock market, weather, etc.
A state refers to a distinct observation of some random variable
- Some examples of this include if the stock market goes up, the weather being sunny, etc.

State Spaces are used in Dynamics
- State spaces imply that there is a time progression and that our system will assume different states as time progresses
- For example, the state space of the largest number up to $n^{th}$ roll is a number from the following state space: ${1, 2, 3, 4, 5, 6}$
Sample Spaces are used more often in Statistics
- A sample space represents a set of distinct observations of some random variable
- There is typically a probability distribution associated with each state
- For example, the sample space of a single die roll is the following
$\Omega = {1, 2, 3, 4, 5, 6}$
In summary, we expect to be thinking about probabilities when we hear sample spaces, unlike state spaces, which do not carry the same connotation
State spaces are typically indexed by time, unlike sample spaces, which do not carry the same connotation

Credible Intervals

Stochastic Processes

Dynamic Systems