Talk:Stochastic matrix

Right vs. left convention

Currently, this page says in the introduction that

It is a convention in English language mathematic literature to use the right or row sum of 1 version.

However, then in the sections below, I think right (row sum) matrices are used in the section "Definition and properties", and then left (column sum) matrices in the section "Example computation: the cat and the mouse". The "Definition and properties" doesn't even say that it is using right stochastic matrices, it talks of "**the** stochastic matrix". So, I have two suggestions. First, if the intro says that right stochastic matrices are the default convention, use right stochastic matrices everywhere on this page. Second, state explicitly that this page is using that convention. Bayle Shanks 21:56, 14 May 2007 (UTC)

A common convention in English language mathematics literature is to use row vectors of probabilities and right stochastic matrices rather than column vectors of probabilities and left stochastic matrices; this article follows that convention. As a post-graduate Physics/Math student, I have never seen this convention and think it can be confusing to readers. I'm not saying it is not true - I don't know the field well enough; I just hope it's not just the opinion of one single person here. At least, the use of this (at least in other areas of Math) unusual convention should be mentioned again in the definition. (I never read the introduction blah blah of Math articles carefully and usually jump right to the definitions, and I guess I am not the only one.) 84.248.100.68 (talk) 22:49, 19 July 2017 (UTC)

I also find this a bit confusing, although the "row sum = 1" convention is confirmed by another source [Asmussen's "Applied Probability and Queues"]. It would be great if someone with good subject knowledge further clarified the issue. One way the article could be expanded is by discussing that the matrix can be viewed as an abstract linear operator, as highlighted in Asmussen on page 7:

"...Accordingly, we can think of the transition matrix P as an operator acting on measures to the left and on functions to the right, and we sometimes write νP as νP and Pf as Pf. A particularly important function is the constant 1 which we write as 1 in vector notation." [Asmussen]

AVM2019 (talk) 16:11, 28 August 2020 (UTC)

Odd redirect?

Transition Matrix redirects here, whilst the page refers exclusively, barring the first paragraph, to stochastic matrices. Transition matrices in linear algebra, (at least can) refer to the a matrix which transforms any matrix from one basis-representation into another. I think this article should either include reference to this fact, or the redirect should be removed. Fish-Face (talk) 00:37, 22 January 2009 (UTC)

The redirect is now a disambiguation page.—Anita5192 (talk) 06:39, 29 December 2017 (UTC)

Definition

Shouldn't the notion for the probability of moving from i to j in one time step be $Pr(j|i)$ -- instead of $Pr(i|j)$ that is currently in the article. — Preceding unsigned comment added by 106.51.78.194 (talk) 03:52, 18 February 2014 (UTC)

See "Right vs Left" issue discussed above. AVM2019 (talk) 16:12, 28 August 2020 (UTC)

Adding section on history of the stochastic matrix

Hi! As a class project, I was assigned to improve this page, and am planning to add a section on the history of the Markov matrix in various fields. Fairly short. Here's my initial proposed language (I have citations for all this lot), let me know if you have any issues!

″The stochastic (or Markov) matrix was developed alongside the Markov chain by Andrey Markov, a Russian mathematician who first published on the topic in 1906. (2) His initial intended uses were for linguistic analysis and other mathematical subjects like card shuffling, but both Markov chains and matrices rapidly found use in other fields. (1, 2) Stochastic matrices were further developed by scholars like Andrei Kolmogorov, who expanded their possibilities by allowing for continuous-time Markov processes (4). By the 1950s, Markov matrices had begun to be used outside of their original mathematical fields, and articles appeared in the fields of econometrics (JSTOR 1), circuit theory (JSTOR 2) In the 1960s, stochastic matrices appeared in an even wider variety of scientific works, from behavioral science (JSTOR 3) to geology (JSTOR 4 and 5) to residential planning (JSTOR 6). In addition, much mathematical work was also done through these decades to imporove the range of uses and functionality of the stochastic/Markov matrix and Markov processes more generally. From the 1970s to present, stochastic matrices have found use in almost every field that requires formal analysis, from structural science (JSTOR 7) to medical diagnosis (JSTOR 8) to personnel management (JSTOR 9). In addition, stochastic matrices have found wide use in land change modeling (link to land change modeling page), usually under the term Markov matrix. "

- Ganesha811 (talk) 14:10, 29 April 2017 (UTC)

Use of stochastic matrix in land change modeling

Hello, This is a group project member (saronow). I would like to add a section to the stochastic matrix page on the use of the stochastic matrix in land change modeling. My section would be something like this:

Basic Use "Transition matrices are commonly used in Geographic Information Systems and remote sensing applications to quantitatively estimate the rate of change in a land type category. In land change modeling, the stochastic matrix can be used to show the transition potential of one land category to another and create quantitative comparisons between land types, locations of study, and time intervals. The output shows the conditional probability that one land type will change into another land type within the specified time interval. The diagonal values in the transition probabilities file represent persistence of land type change. In land change modeling, one can multiply the transition potential values by the number of pixels at the first time point in a specified time interval to derive the absolute pixel count for that land cover type at the end of the specified time interval.[1] Another useful application of the stochastic matrix in land change modeling is predicting future land change, which helps to envision future land change, plan future land development, and notice warning signs in land change. It is helpful to have an understanding of the current background of land changes in the area of study in predicting future land change. Variations in transition probabilities of one land type to another are often caused by historical, political, economic, or biological changes in the research site of interest. [2]

Evaluation and Calibration In land change modeling, one can use Correct Rejections, False Alarms, Misses, and Hits evaluate model output. The Correct Rejection explains no reference land type change and no diagnostic change in land type. The False Alarm explains no reference land type change, but diagnostic change in land type. The Miss explains reference land type change, but no diagnostic change in land type. The Hit explains reference land type change, and diagnostic change in land type.

Jump up^ Takada, Takenori; et al. (2009). "Derivation of a yearly transition probability matrix". Springer Science+Business Media B.V (25): 561. doi:10.1007/s10980-009-9433-x. More than one of |pages= and |page= specified (help); Check date values in: |access-date= (help); Jump up^ Takada, Takenori; et al. (2009). "Derivation of a yearly transition probability matrix". Springer Science+Business Media B.V (25): 562. doi:10.1007/s10980-009-9433-x. More than one of |pages= and |page= specified (help); Check date values in: |access-date= (help); — Preceding unsigned comment added by Saronow (talk • contribs) 23:09, 6 May 2017 (UTC)