Least Squares.

1. Programming
Meeting #8
EEC: T.Hideyuki

2. Programming
Meeting #8
Least Squares
Practice
What's next?
+

3. Programming
Meeting #8
Least Squares
x
y
O
1
2
3
4
1 2 3 4 65 7
Error

4. Programming
Meeting #8
Least Squares
x
y
1
2
3
4
1 2 3 4 65 7
y- intercept
Slope

5. Programming
Meeting #8
Least
Squares
Data
Slope
y-intercept
Least Squares

6. Programming
Meeting #8
Data
Least Squares

7. Programming
Meeting #8
DataLeast Squares
Coordinates
(X,Y)

8. Programming
Meeting #8
DataLeast Squares
Variable Array
Coordinates
(X,Y)

9. Programming
Meeting #8
DataLeast Squares
Array
x = [1.0, 2.0, 3.0, 4.0, 5.0]
y = [1.3, 1.8, 2.8, 2.5, 3.9]

10. Programming
Meeting #8
x
y
O
1
2
3
4
1 2 3 4 65 7
x = [1.0, 2.0, 3.0, 4.0, 5.0]
y = [1.3, 1.8, 2.8, 2.5, 3.9]
Least Squares
0
2 3
4
Data
1

11. Programming
Meeting #8
n xy x y
n x x2
2
α
( )
=
Σ −Σ Σ
Σ − Σ
The slope
The number of points
Arrays
Least Squares

12. Programming
Meeting #8
n xy x y
n x x2
2
α
( )
=
Σ −Σ Σ
Σ − Σ
The slopeLeast Squares
∑xy ∑x ∑y
∑x^2 ∑x
Function

13. Programming
Meeting #8
The slopeLeast Squares
xΣ
SummationArray Float

14. Programming
Meeting #8
The slopeLeast Squares
xΣ for ine x
end
sum = 0.0
sum = sum + e

15. Programming
Meeting #8
The slopeLeast Squares
xΣ for ine array
end
sum = 0.0
sum = sum + e
def
end
return sum
Summation ( )array

16. Programming
Meeting #8
The slopeLeast Squares
Summation( )
x = [1.0, 2.0, 3.0, 4.0, 5.0]
y = [1.3, 1.8, 2.8, 2.5, 3.9]
x 15.0
xΣ

17. Summation(
Programming
Meeting #8
The slopeLeast Squares
x = [1.0, 2.0, 3.0, 4.0, 5.0]
y = [1.3, 1.8, 2.8, 2.5, 3.9]
y 12.3)
Σy

18. Programming
Meeting #8
The slopeLeast Squares
x = [1.0, 2.0, 3.0, 4.0, 5.0]
y = [1.3, 1.8, 2.8, 2.5, 3.9]
def Summation(array)
sum = 0.0
for e in array
sum = sum + e
end
return sum
end

19. Programming
Meeting #8
n xy x y
n x x2
2
α
( )
=
Σ −Σ Σ
Σ − Σ
The slopeLeast Squares
Σx2
∑x ∑y
∑x

20. Programming
Meeting #8
The slopeLeast Squares
Σx2 for ine array
end
sum = 0.0
sum = sum + e
def
end
return sum
Summation2 ( )array
**2

21. Programming
Meeting #8
The slopeLeast Squares
xyΣn xy x y
n x x2
2
α
( )
=
Σ −Σ Σ
Σ − Σ
∑x ∑y
∑x∑x^2

22. Programming
Meeting #8
The slopeLeast Squares
xyΣ
x = [1.0, 2.0, 3.0, 4.0, 5.0]
y = [1.3, 1.8, 2.8, 2.5, 3.9]
* * * * *
1.3 3.6 8.4 10.0 19.5
42.8

23. Programming
Meeting #8
The slopeLeast Squares
xyΣ for in

24. Programming
Meeting #8
The slopeLeast Squares
xyΣ while
sum = 0.0
i = 0
i < x.length
sum = sum + x[i] * y[i]
i = i + 1
end
The number
of elements in x

25. Programming
Meeting #8
The slopeLeast Squares
xyΣ while
sum = 0.0
i = 0
i < x.length
sum = sum + x[i] * y[i]
i = i + 1
end
in other languages
for (i = 0; i < x.length; i++) {
sum = sum + x[i] * y[i];
}
keep in
mind…

26. Programming
Meeting #8
The slopeLeast Squares
xyΣ while
sum = 0.0
i = 0
i < x.length
sum = sum + x[i] * y[i]
i = i + 1
end
def SumXY(x, y)
end
return sum

27. n xy x y
n x x2
2
α
( )
=
Σ −Σ Σ
Σ − Σ
∑x ∑y
∑x^2 ∑x
Programming
Meeting #8
The slopeLeast Squares
∑xy

28. Programming
Meeting #8
The slopeLeast Squares
(n*SumXY(x,y) - Summation(x) * Summation(y))
/
(n*Summation2(x) - Summation(x)**2)
n xy x y
n x x2
2
α
( )
=
Σ −Σ Σ
Σ − Σ
∑xy ∑x ∑y
∑x^2 ∑x

29. Programming
Meeting #8
Least Squares
n
x.length
y.length
or
The slope
Array.length
The number
of elements
in Array

30. Programming
Meeting #8
y x
n
β
α
=
Σ − Σ
The slope
Least Squares The y-intercept
(Summation(y) - a*Summation(x)) / n

31. Programming
Meeting #8
What's next?

32. Programming
Meeting #8
What's next?
Practice

33. Programming
Meeting #8
Questions

34. Programming
Meeting #8
Thank you for Meeting!