Least Squares Fitting
Wanho Choi
(wanochoi.com)
Goals
키 - 체중 상관관계
강수량 - 생산량 상관관계
흡연기간 - 폐암발생률 상관관계
What We Want
Prediction
Problem Description
x
y
Given data that have some noise
Problem Description
x
y
y = ax + b
What is the best fitting line?
Problem Description
x
y
x
i
y
i
P(x
i
, y
i
)
y = ax + b
ax
i+ b
y
i− ax
i− b
Mathematical Formulation
y
i= ax
i+ b (i = 1,2,3,⋯, N)
Mathematical Formulation
y
i= ax
i+ b (i = 1,2,3,⋯, N)
Model:
Total Error:
E = 1 N N ∑ i=1 yi− axi− bMathematical Formulation
y
i= ax
i+ b (i = 1,2,3,⋯, N)
Model:
Total Error:
E = 1 N N ∑ i=1 yi− axi− b E = ∑N i=1 yi − axi− b : const. NMathematical Formulation
y
i= ax
i+ b (i = 1,2,3,⋯, N)
Model:
Total Error:
E = 1 N N ∑ i=1 yi− axi− b E = ∑N i=1 yi − axi− b : const. N E = N ∑ i=1 (yi − axi− b)2Mathematical Formulation
y
i= ax
i+ b (i = 1,2,3,⋯, N)
Model:
Total Error:
E = 1 N N ∑ i=1 yi− axi− b E = ∑N i=1 yi − axi− b : const. N argmin a,b EWhat we want:
We have to find & that make
a b
∂E
.
∂a
= ∂E
∂b
= 0
E = ∑N
i=1 (yi
