As a matter of fact, my application scenario is a recommender system in which the interests/preferences of the users change. I have such a global user-interest matrix: the rows are the records of many users (one row for each user); the columns are some interests or properties of the users. Some of the properties or interests are possibly related to each other. Through the time dimension, this matrix is dynamically changing. But because these are the properties or interests of users, the matrix changes in a way that the element values of the last matrix in the time series might be predictable. My objective is to develop a way to describe the future state of this system, especially to predict the future state of the element values (one user's one interest) in this matrix.

Some known works on this use statistics or Numerical Analysis approaches to fit a line or curve to represent a user's interest's change progress, so as to predict the user's preferred items (which correspond the user's interests) in very near future.

Are there any mathematical/physical concepts or theories for dealing with a matrix in which the values are changing in a certain way?

Are there anything I can get from Physics about this? Is this tensor-related? Should I read something about "Multilinear algebra"? If so, what part of this subject should I take notice of especially?

I would be very appreciated if anybody can point out an direction for me to follow.

EDIT: It was when I submit this question, I realized such a concept in mathmatics called "dynamical system theory" and also "chaotic theory" that might be related to my work (application). I don't know much about this field "dynamical system theory", what do they use this theory to do? Is it possible for me to use this theory to computationally increase prediction accuracy (calculation) OR just only can help me to explain some dynamic phenomenon? Or maybe I can not use it in my recommender system scenario at all?

  • $\begingroup$ Do you mean you need a dynamical system whose time varying variables formed a matrix $\endgroup$
    – Shuchang
    Sep 1 '13 at 11:36
  • $\begingroup$ If the elements of the matrix are just functions, you most likely have a matrix over a ring of [insert type] functions which is a common object of study in commutative algebra. $\endgroup$
    – Dan Rust
    Sep 1 '13 at 12:01

Let $t$ denote time, and take $f(t) = A_t$, where $A_t$ is the matrix at time $t$. Now try to answer the following questions :

  1. What is the range of this function : For instance, if each entry of $A_t$ is a complex number, and all matrices are $n\times n$, then you will get $f:(0,+\infty) \to M_n(\mathbb{C})$

  2. How does this function change with time : For instance, is it true that $f(t+s) = f(t)f(s)$, where the product on the right hand side is matrix multiplication.

Once you know this, perhaps something like http://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups might be of use.

Your eventual goal, of course, is to find $f(t_0)$ for some fixed time $t_0$.


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