# How can these matrices solve a first order differential equation?

The below PDF is the one I am working through from a conceptual perspective, but I don't understand enough about matrix notation (or advanced calculus in general) to follow the logic the authors use to find their values to solve the equations on pages 7 and 8. I will attempt to provide context below; my main question is if the matrices on page 8 follow a certain notation that I can learn/research more.

Background:

Both n, the day of the week, and t, the hour of the day, are defining characteristics of each value x.

X0 = The increasing set of absolute value deviations from the linear regression for all values x

X1 = The set containing the cumulative sum of the deviations in X0

Z = The set containing the average between each value in set X1

Sets Y and Z are intended to help predict the next value x. In the below first order differential equation, "a" is a function or value describing the increasing speed of numbers in set X, and "u" is an endogenous control coefficient in system.

(dY/dt) + aY = u

The problem:

I do not know/fully understand the below matrices and notation provided to help determine "a" and "u".

U = [a, u] ^T

U = (B^T B)^-1 B^T Y

Y is a (n-1)x1 matrix and B is a (n-1)x2 matrix.

The values in Y are X(2), X(3)...X(n) in a column.

The values in B are -Z(2), -Z(3)...-Z(n) in the left column and all 1s in the right column.

In addition, there is an unknown value k that appears in the predictive equations; is this a standard constant/coefficient I'm not familiar with?

Thank you very much for any advice/input.

• $U = (B^T B)^{-1} B^T Y$ is the so-called "least square solution" to linear system $BU=Y$ when $B$ is non-square (also called "normal equations"). Notation [a,u] is nothing more than $a$ placed with $u$ to make a new vector. See "Linear algebra viewpoint" here – Jean Marie Mar 26 at 2:33
• Thank you for the link - that was very helpful to work through. My only other question is about Equations 17/18 on Page 8 - variable k appears in the exponent of e, but it didn't appear anywhere prior. Is this a standard variable that I can derive from somewhere/is there a similarly helpful link you know of that would explain as such? Thanks again! – Eddie Mar 30 at 1:14
• In fact, $k=n$. It's a typo in a paper that I find rather ill-written IMHO. Here is an interesting reference : (see in particular formula (7)) here "Forecasting and analyzing the competitive diffusion of mobile cellular broadband and fixed broadband in Taiwan with limited historical data" by Chiun-SinLin ("Economic modelling" Volume 35, September 2013, Pages 207-213). You will see a large similarity in this article compared with yours. I obtained it by googling with keyword IAGO, a method I didn't know. – Jean Marie Mar 30 at 7:10
• Ah! Interesting. Thank you for your help with this. I haven’t fully gone through it yet, but this looks very interesting. Thank you very much! – Eddie Apr 1 at 20:38