Regressions without calculator

[jmsdg7]

im a senior in high school taking ap calculus, I've done regressions here and there with the calculator but i was wondering how do do it without the calculator, i obviously know how to get a linear equation out of 2 points, but how is it done with more points and higher degrees?

i understand that you need 2 points for a line, 3 for a quadradic, 4 for a cubic and 5 for a quartic but i was hoping someone could show me how it is done.

 

[Gunthi]

The simplest method i know is the Least Squares, there are many others.

I can show you the general form for any number of points and any type of equation, but it takes a little hard work and algebra.

In short therms, you have to find the parameters that minimize the difference $$\sum [f(x)-q(x)]^2$$, where $$q(x)$$ is the function you want to minimize, and $$f(x)$$ are the points you are given.

I'll try to post something more elaborate this week.

 

[EnumaElish]

Let the model be y = a + b x + u. Parameters of the model are a and b, u is the error term.

Variables y, x (and u) are each N-by-1 vectors.

Let X = [1 x] be the N-by-2 matrix. The first column of X is a vector of 1's. The second column of X is identical to vector x.

Let Z be the inverse of X'X. Z is a 2-by-2 matrix.

Then we can write [a b]' = ZX'y, which is 2-by-1. Parameter a is the first (top) element of ZX'y. Parameter b is the second element of ZX'y.

For higher order polynomials, substitute [x x^2 x^3 ...] for x, and [b1 b2 b3 ...] for b.

 

[statdad]

Of course, for simple regression, the matrix approach mentioned above is equivalent to the following equations:
The slope is given by

$$b = \frac{\sum{(x-\bar x)(y - \bar y)}}{\sum (x-\bar x)^2}$$

and the y-intercept is given by

$$a = \bar y - b \bar x$$

 

[CellCoree]

As a scientist, it is important to understand the fundamentals of mathematical concepts. While using a calculator can be helpful in performing regressions, it is also important to understand the underlying principles and methods behind them. In order to perform regressions without a calculator, you can use the method of least squares. This involves finding the line or curve of best fit that minimizes the sum of the squared distances between the data points and the line or curve.

For linear regression with more than two points, you can use the formula y = mx + b, where m is the slope and b is the y-intercept. To find the values of m and b, you can use the least squares method by setting up a system of equations using the given data points. This can be done by finding the sum of the x-values, the sum of the y-values, the sum of the x-values squared, and the sum of the x-values multiplied by the y-values. These values can then be used to solve for m and b, which will give you the equation for the line of best fit.

For higher degree regressions, such as quadratic, cubic, or quartic, the process is similar. You will need to set up a system of equations using the given data points and the coefficients of the polynomial. These equations can then be solved using techniques such as substitution or elimination to find the coefficients and ultimately the equation for the curve of best fit.

It is important to note that performing regressions without a calculator may be more time-consuming and prone to human error. However, it is a valuable skill to have and can help deepen your understanding of the mathematical concepts involved. I suggest practicing with different sets of data and using online resources or textbooks for further guidance and examples. Good luck with your studies!