1.10 Linear models in buisiness, science, and engineering

本文为《Linear algebra and its applications》的读书笔记

目录

• Difference Equations 差分方程

Linear models are important because natural phenomena are often linear or nearly linear when the variables involved are held within reasonable bounds. Also, linear models are more easily adapted for computer calculation than are complex nonlinear models.

Difference Equations 差分方程

In many fields such as ecology, economics, and engineering, a need arises to model mathematically a dynamic system that changes over time. Several features of the system are each measured at discrete time intervals, producing a sequence of vectors $\boldsymbol x_0, \boldsymbol x_1, \boldsymbol x_2,...$x0​,x1​,x2​,... The entries in $\boldsymbol x_k$xk​ provide information about the state of the system at the time of the $k$kth measurement.

If there is a matrix $A$A such that $\boldsymbol x_1 = A\boldsymbol x_0$x1​=Ax0​, $\boldsymbol x_2 = A\boldsymbol x_1$x2​=Ax1​, and, in general,
$\boldsymbol x_{k+1} = A\boldsymbol x_k\ \ \ \ \ for\ k=0,1,2,...\ \ \ \ \ \ \ (5)$xk+1​=Axk​     for k=0,1,2,...       (5)then (5) is called a linear difference equation(线性差分方程) (or recurrence relation(递归方程)). Given such an equation, one can compute $\boldsymbol x_1$x1​, $\boldsymbol x_2$x2​, and so on, provided $\boldsymbol x_0$x0​ is known. Sections 4.8 and 4.9, and several sections in Chapter 5, will develop formulas for $\boldsymbol x_k$xk​ and describe what can happen to $\boldsymbol x_k$xk​ as $k$k increases indefinitely. The discussion below illustrates how a difference equation might arise.

A subject of interest to demographers(人口统计学家) is the movement of populations or groups of people from one region to another(人口迁移). The simple model here considers the changes in the population of a certain city and its surrounding suburbs(郊区) over a period of years.

Fix an initial year—say, 2014—and denote the populations of the city and suburbs that year by $r_0$r0​ and $s_0$s0​, respectively. Let $\boldsymbol x_0$x0​ be the population vector

For 2015 and subsequent years, denote the populations of the city and suburbs by the vectors

Our goal is to describe mathematically how these vectors might be related.

Suppose demographic studies show that each year about 5% of the city’s population moves to the suburbs (and 95% remains in the city), while 3% of the suburban population moves to the city (and 97% remains in the suburbs). See Figure 2.

After 1 year, the original $r_0$r0​ persons in the city are now distributed between city and suburbs as

The $s_0$s0​ persons in the suburbs in 2014 are distributed 1 year later as

The vectors in (6) and (7) account for all of the population in 2015. Thus

That is,
$\boldsymbol x_1=M\boldsymbol x_0$x1​=Mx0​where $M$M is the migration matrix(移民矩阵) determined by the following table:

In general,
$\boldsymbol x_{k+1} = M\boldsymbol x_k\ \ \ \ \ for\ k=0,1,2,...\ \ \ \ \ \ \ (9)$xk+1​=Mxk​     for k=0,1,2,...       (9)