Basic Polynomial Regression?

Question

For a Polynomial Regression ^2, Microsoft Excel returns the following formula for the data listed below: y=0.0841x^2-0.8481x+143.09

As for the data below, the numbers 10 to 1 represent my independent variable (x) and the corresponding values represent my data. My most recent data value is 143.24. Could you please explain how the equation was derived. What does .0841, .8481 and 143.09 represent?

10 143.24
9 142.16
8 141.54
7 141.07
6 141.19
5 140.54
4 141.67
3 141.37
2 141.62
1 142.21

Answer 1

For a linear regression,
y = a + bx
you have the matrix
n. ∑x. . a ∑F
∑x ∑x^2 b ∑xF

It can be shown that, for
y = a + bx + cx^2
you have the matrix
n . . ∑x. . ∑x^2 a ∑F
∑x. . ∑x^2 ∑x^3 b ∑xF
∑x^2 ∑x^3 ∑x^4 c ∑xF

This pattern extends to polynomials of any degree.

y = 0.0841x^2 - 0.8481x + 143.09 can be thought of as a line
y = - 0.8481x + 143.09
adjusted by a factor + 0.0841x^2

Putting your result into vertex form we have
y - 140.95 = 0.0841(x - 5.0422)^2

One more way of looking at it is
y = 143.09 - x(0.8481 + 0.0841x)

Answer 2

You need to study a numerical methods text to fully understand the procedure.
For a quick reference, the logic behind your regression is as follows:
# You want a second degree fit, that means you want an equation of the
form y=Ax^2+Bx+C for your dependent variable y in terms of x. We
need to find the coefficients A, B and C.
# Ideally 3 equations from 3 (x,y) pairs will solve A,B,C, but we have more
than 3 data sets (i.e. equations). That brings statistics into picture.
# Now we need A,B,C which will provide good estimate of observed y. The
linear regression method is used here. We will find those A,B,C for
which the difference between the actual y and predicted (from equation)
y is minimized.
# So we actually calculate a quantity called Chi-square by adding the
squares of the difference between actual y and predicted y for each x.
# Next to get minimum, we differentiate Chi-square partially with respect
to A, B and C and set each derivative to 0. That gives 3 equations and 3 unknowns.
# What you see in the excel equation is the solution of these equations.

I personally feel that this answer will be too difficult to comprehend for most people, but you asked for the trouble right!!!

Answer 3

For a polynominal Regression you need two sets of data...a regression measures the length to which one variable (or more)influences another (correlation), the value of the result (R) varies between 1 and -1, the closer it comes to 1, the stronger the influence is.
So to answer your question it would be helpful to know exactly what you are regressing and why the values 1 thru 10 are a set of data if they are a set at all or if they are just a number list.
The equation Y= is the expression of a typical regression formula, which is really a linear equation representing a line...as for the constant 143.09...that is the value of the commonly known Y Intercept.
hope that helps

Answer 4

This is much easier than you think. You are basically plotting y vs x and finding a solution that yields y in terms of x:
y(x) = ax^2 + bx + c
Now there are two ways that I know of to find the values of a,b,c:
way 1.
use a multiple regression formula as given in the reference section below; a common one is called REGRES in which you input the values x,y, sigma y, etc. That is what was likely used in your case.

way 2.
consider the problem to an overdetermined one; using a least squares approach where the a, b, c are determined by the so-called "normal equations"; these equations are derived in the reference below.

Answer 5

In physics or engineering (the fields i be responsive to) the habit of issues is modeled with equations. each thing. How a radio wave scatters off a development or an merchandise. How sound propagates in a room. How the stresses on a bridge are disbursed. What the actual gravitational field of the earth is (the easy version you learn in first year physics with a superbly uniform sphere isn't sufficient for many functional purposes), and how satellites circulate over that floor. maximum of those issues at the instant are not purely equations yet differential equations (equations regarding calculus derivatives or integrals). you oftentimes learn 2 years of calculus and then a year of differential equations to benefit the thank you to sparkling up those. and then possibly a year of examining computer arithmetic to benefit the thank you to write down computer courses to sparkling up them. i don't be responsive to chemistry, even even though it in all danger has to do lots of quantum mechanics to type molecules. QM includes fixing very complicated differential equations. You have been asking enormously approximately algebra. that's a relentless, in spite of uncomplicated equations. you have equations bearing on quite a few parts, and in diverse situations quite a few issues are unknown. enable's say I even have the equation for a fashion far the seen horizon is, given the top of an merchandise (some thing I merely calculated for the umpteenth time some weeks in the past). That equation is D = approximately sqrt(2Rh) the place R = radius of earth and h = top. (by how, that's calculus that tells me that that's an approximation, how sturdy an approximation that's, and what approximation to make to the greater precise equation). yet think i needed to be responsive to the top to realize a primary horizon distance of, say, 20 km? Algebra.