High School
SAT
ACT
University
MCAT
LSAT
DAT

Example: Least Squares for non-Polynomials

Related Topics

Wize University Linear Algebra Textbook > Least Squares Approximation

Least Squares Approximation

4 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Example: Least Squares for non-Polynomials

Find the function of form c1sin⁡(x)+c2cos⁡(x)c_1\sin(x)+c_2\cos(x) which best fits the data points (−π,−3),(0,−4),(π/2,−5)(-\pi,-3),(0,-4),(\pi/2,-5) in the least squares sense.

Solution:

M=[sin⁡(−π)cos⁡(−π)sin⁡(0)cos⁡(0)sin⁡(π/2)cos⁡(π/2)]=[0−10110],y⃗=[−3−4−5],MTM=[1002],MTy⃗=[−5−1]M=\left[\begin{array}{ll}\sin(-\pi)&\cos(-\pi)\\\sin(0)&\cos(0)\\\sin(\pi/2)&\cos(\pi/2)\end{array}\right]=\left[\begin{array}{rr}0&-1\\0&1\\1&0\end{array}\right],\vec y=\left[\begin{array}{r}-3\\-4\\-5\end{array}\right],M^TM=\left[\begin{array}{rr}1&0\\0&2\end{array}\right],M^T\vec y=\left[\begin{array}{r}-5\\-1\end{array}\right]

The normal equations row reduce as: [10−502−1]→[10−501−1/2]\left[\begin{array}{rr|r}1&0&-5\\0&2&-1\end{array}\right]\rightarrow \left[\begin{array}{rr|r}1&0&-5\\0&1&-1/2\end{array}\right] , so the desired approximating function is f(x)=−5sin⁡(x)−(1/2)cos⁡(x)f(x)=-5\sin(x)-(1/2)\cos(x) .

More Least Squares Approximation Questions:
Practice: Least Squares Approximation
Find the cubic polynomial which best fits the data points (−2,−1),(−1,1),(0,0),(1,−2),(2,3)(-2,-1),(-1,1),(0,0),(1,-2),(2,3).
19.4F_133_8.2_Mock_F1_$\tkco{eg2}$_$\key{Final}$_Builder_$\tkcth{8.2.}\tkcf{2}$_
Show that, in the case that the matrix A‾\A is invertible, the least squares solution corresponding to the system:
A‾ x⃗  =  b⃗, \A \, \vx \; = \; \vb,
is the same as the solution to the system itself.
Least Squares for Mix of Quadratic and Linear Functions
Find a least squares approximation of form a+bx+cx2+dya+bx+cx^2+dy for the following data:
x‾y‾result‾11112−121−3222011\begin{array}{r|r|r}\underline{x}&\underline{y}&\underline{result}\\1&1&1\\1&2&-1\\2&1&-3\\2&2&2\\0&1&1\end{array}
Least Squares
Find a function of the form a0+a1x+a3x3a_0+a_1x+a_3x^3 (note there is no squared term) which is a best approximation in the sense of least squares to the following data points: (0,1),(−1,1),(1,2),(−2,1)(0,1),(-1,1),(1,2),(-2,1) .
Practice Question: Least Squares for Polynomials
Find the cubic polynomial which best fits the data points (−2,−1),(−1,1),(0,0),(1,−2),(2,3)(-2,-1),(-1,1),(0,0),(1,-2),(2,3) in the least squares sense.
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7).
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7).
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible, in the least squares sense, to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7) .
Solution:
Example: Least Squares for Polynomials
Example: Least Squares for Polynomials
Find the equation of a quadratic polynomial that comes as close as possible to passing through the points: (1,2),(1,3),(2,3)(1,2),(1,3),(2,3).