Lecture 6 - Approximation

Question 1

Regression

Answer 1

Another name of approximation

Question 2

Using of approximation

Answer 2

We use it when the data have some error and we want to represent for example the general trend of data. There is no sens to make an effort to intersect every point because they may be incorrect.

Question 3

Measurements

Answer 3

Lagrange interpolation is ok, but when we take into consideration that some points was measured with some accuracy then the STRAIGHT LINE would be more proper.

Question 4

Trends

Answer 4

They can be shown by polynomials with different degrees (very often low-degree ones) instead large tables

Question 5

Linear Regression start

Answer 5

We create a straight line “y=a0+a1x”, which pass directly through a single point and we can shift this line up or down using error “e”

Question 6

Linear Regression with N points

Answer 6

When we consider N points and one straight line, each point has its own error ei

Question 7

Linear Regression strategy

Answer 7

The most popular strategy for fitting the best line is to minimize the SUM OF THE SQUARES of errors

Question 8

Least squares criterion

Answer 8

The approach with minimizing sum if the squares of errors

Question 9

Error in the linear regression

Answer 9

Residual between the measured y and the y calculated with the linear model; Value that was given and value of our line function in the same point

Question 10

Linear regression error equation

Answer 10

ei=(yi-a0-a1xi)^2

Question 11

Determining coefficients on line/polynomial step 1

Answer 11

Calculate the DERIVATIVE of “sum of the squares” with respect to each coefficient and compare them to zero

Question 12

Determining coefficients on line/polynomial step 2

Answer 12

Divide the sum of whole bracket to sums of particular elements. Change the equation in such way that elements with Y are on the right side and rest on the left.

Question 13

Note for sums

Answer 13

When you calculate SUM form i=1 to n of free coefficients a0 it just n*a0

Question 14

Polynomial Regression

Answer 14

Works exactly the same as linear, just more coefficients

Question 15

Drawback of polynomial regression

Answer 15

RUNGE’s effect - big oscillations at the edges

Question 16

Trigonometric functions

Answer 16

We can use trig functions instead of monomials with w(omega) as a pulsation

Question 17

Determining coefficients on line/polynomial step 3

Answer 17

Change obtained set of equations to matrices and calculate coefficients.
Ma=y => a = y*inv(M)

Question 18

Trigonometric functions -basic function

Answer 18

y=a0+a1sin(wx)+ a2cos(wx)+ a3sin(2wx)+ a4cos(2wx)+…

We can divide it into separate functions: fi0(x)=1, fi1(x)=sin(wx), fi2=cos(wx)…

Question 19

Trigonometric function - coefficients

Answer 19

We can obtain coefficients by solving Sa=y