Least squares; Projections of a vector on a subspace (11/25/2008)

1. Main Points

In todays reading I learned how to make a projection of vector on a line through the origin. For thus vector there is a multiple ˆmx of x that reaches closest to y. That multiple is called the projection of y on the line spanned by x. The step from the tip of the projection ˆmx to y is the residual vector r. I will never know the value of m; but I can find ˆm which is my best estimate of m. I can evaluate the quality of our estimate by the length of the residual vector r, because IIkII is the distance between ˆmx and y, the two sides of what we had hoped would be an equality between mx and y.
One can also do curve-fitting with two parameters leads to projections onto 2-dimensional subspaces. Allowing one to find the linear combination ˆmu +ˆbv that comes closest to s. Lastly, least-squares curve-fitting with J parameters leads to projection onto subspaces spanned by J
vectors.

2. Confusing

The fundamental problem of linear modeling was hard to grasp.

3. Intruiging

The applications of such linear modeling must be almost limitless. The example of the business who wanted to investigate the return on the money that they put on advertisement and used curve-fitting with two parameters to do so is a good example of this.

Vector projection (11/20/2008)

1. Main points

In today’s reading I learned that for equations that have any solution one can make approximate solutions. The main tool for doing this is using the dot product of vectors, obtained by multiplying corresponding coordinates of the vectors and adding. The most important fact about the dot product is that the dot product of two vectors is zero if the vectors are perpendicular and thus nonzero if the vectors are not perpendicular.

2. Challenging

Why is the dot product zero if the vectors are orthogonal?

3.Interesting

I found it interesting learning about methods to get approximate answers to equations and questions that do not have any solutions, I can see how this will can be very useful in statistics etc.

11/18: Linear combinations, linear independence, span

1. Main Points
One can combine the two operations, adding multiples of two vectors. A sum of multiples is a
linear combination.
A system of equations for unknowns x and y
ax + by = e
cx + dy = f
Has two fundamental but different geometric interpretations: the points interpretation and the vector interpretation.
Linear combinations of vectors occur so frequently that a special notation has been developed
using matrices. This is defined as an m × n matrix which is a rectangular array of numbers, arranged in m rows and n columns.

2. Challenging
Linear combinations in higher dimensions was a little challenging to think about.

3. Intruiging
I can see how linear combinations can be applied in many areas of science!

SIR and the spread of disease (11/11/2008)

1. Main Points

In todays reading I learned how to apply differential equations to predict epidemic outbreaks just like the CDC. The model used for an epidemic is the S-I-R model, where S is the number susceptible, I the number of infected and R the number of recovered.

dS/dt =-(Rate susceptible get sick)= -aSI

dI/dt=(Rate susceptible get sick)-(Rate of infected get removed=aSI-bI

The constant a measures how infectious the disease is, meaning how quickly it is transmitted from the infected to the susceptible.

a=-(dS/dt)/SI

The constant b represents the rate at which infected people are removed from the infected population.

Threshold population = b/a

If the initial number of susceptible is above b/a, then there is an epidemic, if the initial number of susceptible is below b/a, then there is no epidemic.


2. Challenging about the material

Looking at an epidemic in terms of a phase plane was a little confusing I thought, maybe we can go over this example in class?

3. Intriguing

This was one of the more interesting readings for me yet in this book. I am currently taking medical anthropology where we are talking a lot of epidemic outbreaks and containment of such. Therefore I found it very interesting how to interpret epidemics mathematically!

Interacting systems: modeling, phase plane, and trajectories (11/6/2008)

1. Main Points
In this reading I learned how to use differential equations to model the growth of two interacting populations, a system requiring two differential equations. One can get an idea of what the solutions for these equations look like from a slope field.

2. Challenging
I had a hard time wrapping my head around the idea of a phase plane, maybe you can explain these in detail in class?

3. Intriguing
Being a biology major and having the opportunity to learn how to model different interactions between populations is really cool! The population size of the two species worms and robins modeled as a function of time I thought was a really interesting example of this!

Exponential growth and decay / Applications of ODE, equilibria, and stability (11/4/2008)

Main Points
In today’s reading I learned about how to compute and model exponential growth and decay, equilibrium and Newton’s law of heating and cooling with ODEs. The most important concept in doing this is that the general solution to dy/dt=ky is y=Ce^kt and is considered exponential growth for k>0 and exponential decay for k<0.

Challenging
I did not quite understand what the difference between a stable/unstable equilibrium was.

Intriguing
A lot of things in our world exhibit exponential growth or decay, therefore it is very interesting to learn about this because of its wide applications to the natural sciences!