A simple, practical way to Random Processes
For you if...
This page contains some materials I've prepared in a recent past, re-organized for clarity and ease of use. If you have some practical interest in the field of "random processes", these little reporta may be useful to you.
Reports are listed here, more or less in meanings order. They are in PDF form: to access you may just click on the link.
What's inside?
When I prepared this material I was thinking to applications of random processes in the specific field of environmental fluid mechanics and dispersion moeling. The contents reflects this point of view, with a strong emphasis on "generating realizations" and a lesser (but present) focus on classification. I gave no attention to "prediction" instead, and this is, I'm afraid, quite evident in the contents. The interest span of random processes is broad however, and chances are good other application profiles may find the material useful.
Instead of adopting an in-depth, rigorous style (this is used in many excellent textbooks, easily reperible and, more important, already written) I had chosen to use a "simple and intuitive" level, by showing and evoking rather than explaining all. You may think, if you like, this work as part, from my side, of a co-creation process. The other will be provided by you, and if these reports are successful you will be curious enough to fill the gaps relevant to you, search and gather from the relevant literature, and make experiments on yourself. I did not imagined my mini-reports as a programming or usage manual on R, but I used R and RStudio to realize them: I imagine you too may consider this wondrous environment useful and inspiring. Other computational means exist however (I also use Mathematica, Julia and Python in my life as data analyst, and C/C++, D and Fortran as a developer - if you look around you can find some contributions by me to open source projects, mostly in the area of atmospheric dispersion modeling). (By the way, I'm still one of the extremely few female "coders": I'm not that proud of that, and be more content if some others - maybe you if you identify like that and wish to - to join the open-source community: I'd egoistically feel less alone, but, more importantly, "we" may contribute a voice which, I see, is somewhat different, and no less human, than the more usual way.)
List of reports
Practical introduction to "random processes". What they are, the concept of realizations, and ways I use in my professional life to show them.
The concept of "ensemble mean" as a tool for gathering some insight on an underlying random process when all we have in our hands is a set of its realizations, like for example "replicas" of a laboratory experiment. Basic properties of the ensemble mean are also introduced.
Here we look at the (multi-variate) distributions associated to random processes, and take contact with the correlation structure underlying and characterizing random processes.
An interesting, cozy case: the speciation-extinction process. Here you can see an example of random process which, as simple as it is, can only be generated using an algorithm, and not a more usual recurrence formula, or something like that.
Back to work! Here you will find an important property: stationarity
Stationary random processes stay behind their realizations, which often are their only observable manifestation. In some fields, unique. Is it then possible to infer something about the underlying random process by analysing one realization of it? Here we deal with ergodicity, which provides a link.
First contact with natural time-series, and the need for developing a new language (and tools) to deal with them. Focus shifts here from random processes, seen as "generating tools", to time series, or one-shot realizations, with the need to re-define "ergodicity" and other familiar terms.
The analysis of natural time series demands techniques which, although derived from the theory of random processes, is not the same thing. Here we'll meet with the generalized mean, in a sense the basis of them all, and explore its meaning, mainly from a mathematical standpoint (with a bit of physics).
Once the generalized means is introduced, moments can be defined based on it. They are important tools, in the analysis of data series, and here you may find them.
About use, licensing and distribution
All my mini-reports are distributed under the permissive license Creative Commons 4.0 BY, allowing you to distribute, remix, adapt, and build upon the material in any medium and format, under the sole condition attribution is given to me (by the way, commercial use is admitted). Any changes you make do not, of course, imply or require my endorsement. I would be very interested in your feedback, and will be grateful if you send me your comments at my e-mail address, patti.favaron@gmail.com.