CPE 332 Computer Engineering Mathematics II

1
CPE 332Computer Engineering Mathematics II

• Chapter 12
• Curve Fitting

2
Curve Fitting

• ?????????? ???????????????????????????
  ??????????????????? ????????????????? Function
  ??????????????????????????????????????
• Function ????????????????? Polynomial ??? Degree
  ?????
• Function ???????????????????????? Cosine Function
• ???????????????? ?????????????????? Function
  ?????????????????????????? ????????????
  Interpolation

3
Interpolation
y
x
4
Linear Interpolation
y
x
5
Second Degree Piecewise Interpolation
y
x
6
Third Degree Piecewise Interpolation
y
x
7
Higher Order Degree

• ?????????? ?????????????? Oscillate ???
• Function ???????????? Polynomial ??????????????
• N Degree ????? N1 ????????????????? Function

8
Spline Interpolation

• ?????????? Polynomial Degree ???? ??????????
  Third Degree ???? Cube
• ????? Cubic Spline
• Cubic Spline ???? Polynomial ??????? 2 ???(??????
  4) ?????????????????????????????? Polynomial
  ??????????????? ??????????? Polynomial
  ??????????? First Derivative ??? Second
  Derivative ???????
• 2 ?????? 2 ????? ???????????????????? First ???
  Second Derivative ???? 4 ????? 4 Unknown
• ????????????????????? First Derivative
  ??????????????? ????????? Second Derivative
  ???????????? ??????? ???????? Natural Spline
• ??????????????????????

9
Spline Interpolation
y
x
10
???????? Cubic Spline Interpolation (1)

• ????????????????????????
• ????????????? Curve ??????????????? ?????? Third
  Degree Polynomial
• ?? 4 Unknown ????????? ???????????????? 4 ?????
  ?????? 4 ?????? Data ????????
• ????? 3rd Degree Polynomial ????????????? 4 ???
  ????????????????????? 4 ??? ??????????????
  ?????????????? 4 ???????? Polynomial ??????????
• Polynomial ?????????????????????????????????

11
???????? Cubic Spline Interpolation (2)

• Example ????? Data 7 ?????????
• Yi 2 0 3 4 5 3 1 Xi 0 1 2 3 4 5 6
• ??????? Third Degree Polynomial ????? Fit Data
  ????????

12
???????? Cubic Spline Interpolation (3)
y1
y0
?????????? Cubic Polynomial 2 ?????? Fit 4 ??????
?????? 2 Fit 4 ???????? ????????? 4 ???????????
13
Cubic Spline Int.(4)

• Polynomial ?????? ??????????????
• ???????????

14
Cubic Spline Int.(5)

• Polynomial ????????? ??????????????
• ???????????

15
Cubic Spline Int.(6)
16
Cubic Spline Int.(7)
17
Cubic Spline Int.(8)
18
Cubic Spline Int.(9)

• ???????????????????????????????????????
• ????????????????? First Derivative ??????????
  Polynomial ?????? ?????????? First Derivative
  ???????????? Polynomial ?????? 2
• ??????????? ??????????????? First Derivative
  ???????????????? Polynomial ???????????
  ????????????
• ???????????????????? 4 ????? ??????? 4
  Coefficient ????????????????? Function ?????????
  ????????? First Derivative ????????????? (4 ???
  ???)
• ???????? ???????????? Data (?????????????)
  ?????????? Third Degree Polynomial ????????

19
Cubic Spline Int.(10)

• ?????????? Data n ?????? xii1,..,n ??? yi
  i1,..,n ???????????? Cubic Polynomial, Si(x)
  ??????? n-1 function ?????????? Interpolate
  ????????????? ?????????????????????????????

20
Cubic Spline Int.(11)

• ????? Polynomial ??????????????????????? Variable
  ????????? (Delay) ??????????? xi
  ??????????????????? ?????
• ??? First ??? Second Derivative ????????
  Polynomial ???????????????????

21
Cubic Spline Int.(12)

• ?????? xi ????????????????????? Polynomial
  ????????
• ????????? xi ?????????????? Function ???
• ????????? ??????(?????????????????????????????????
  ???????????????)

22
Cubic Spline Int.(13)

• ???????????????????????? First Derivative
  ???????????????????
• ??????? ??????
• ????????????? ??? Second Derivative
  ??????????????????????????

23
Cubic Spline Int.(14)

• ??????
• ???????
• ???
• ???
• ??????

24
Cubic Spline Int.(15)

• ??????????????????? ????????
• ???????
• ????????????????????
• ???? ai ??? ci ??????????????????????

25
Cubic Spline Int.(16)

• ???????????????????????????
• ?????????????????? Matrix ??????

26
Cubic Spline Int.(17)

• ???
• ???????????????????? Matrix ?????????

27
Cubic Spline Int.(18)

• ???????????????? n-2 ???????? n unknown
  ???????????????????? ??? Condition
  ???????????????????????????
• ??? Condition ????????????????????? ?????????
  Variation ??? Cubic Spline Interpolation ????????

28
Cubic Spline Int.(19)

• ?????????????????? Spline ?????? ??? ???
  Condition ?????????????????
• Natural Spline
• ??????????? M1Mn0
• Parabolic Runout Spline
• ??????????? M1M2 ??? MnMn-1
• Cubic Runout Spline
• ????????? M12M2-M3 ??? Mn2Mn-1-Mn-2
• ?????????????????????????????? Natural Spline

29
Cubic Spline Int.(20)

• Natural Spline
• ???????? M1Mn0 Matrix ???????????? n-2 ?????
  ??? n-2 Unknown ??????
• ?????????????????????? Mi ?????????????????
  Coefficient ???????? Polynomial ???

30
Cubic Spline Int.(21)

• ?????????? Coefficient Matrix ??????????????????
  Diagonal ??????????????? ???? Matrix ???????
  Matrix ????? ????? Toeplitz Matrix
• Toeplitz Matrix ????????????????? ?????? O(n2)
  Computation Time ???????????? O(n3)
• Algorithm ????????? Levinson Algorithm
• Toeplitz Matrix ???????? LU Decomposition ??????
  O(n2) ????????????
• ?????????????????????????????? Matrix ??? ???
  Levinson Algorithm ???????????????????????
  ????????????????????????????????????????? Linear
  Algebra ??? Numerical Method

31
Example Natural Spline

• ??????????????? ???????? Spline ??????? Fit Data
  ????????? n7, h 1, M0M70
• Yi 2 0 3 4 5 3 1 Xi 0 1 2 3 4 5 6
• ??????????? Matrix ??????
• ??????????????

32

• ??????? Coefficient ??? Cubic ???? 6 ??????

33
Polynomial Coefficient Matrix

• gtgt pa b c d
• p
• 1.4987 0 -3.4987 2.0000
• -2.4936 4.4962 0.9974 0
• 1.4756 -2.9846 2.5090 3.0000
• -1.4090 1.4423 0.9667 4.0000
• 1.1603 -2.7846 -0.3756 5.0000
• -0.2321 0.6962 -2.4641 3.0000
• ????????????? Plot ????? Polynomial ???????????
  ??????? 6 Polynomial ??? 6 ????

34
Natural Spline Interpolation
35
Natural Spline Interpolation RFirst Der G
Second Der B
36
Natural Spline Interpolation RFirst D G
Second D B vs 6th D Poly K
?????????????? Polynomial ????? Degree ?????????
Overshoot ??? Spline ????? Curve ???????????????
????????????????????
37
Regression

• ???????? Data ????????????? ?????? ????????
  Interpolation ???? Polynomial ????????????
• ???????????? Data ????????????????????????????????
  ? ?????????????? Random
• ???????????? Noise ????????
• ??? Fit ???? Polynomial ?????????? Spline
  ??????????????????????????????

38
Regression Data with Noise
Y
X
39
Polynomial Fit Noise Data
Y
X
40
Ordinary Simple Linear Least-Squares Regression

• ???????? ?????????????????? Function ?????
  ??????????? ???? Exponential ?????????????
• ????????????????????????????????????????
  ????????????? Data ??????????????????????????
• ??????????????? Fit ?????? ???????? Error ??????
  ????????????????? Regression
• ??????????????????? (?????????????????)????????
  Linear Regression
• ?????????????? Simple Linear Regression
  ???????????????????? ???????????????????????????
• ??????????? Multiple Linear Regression
  ???????????????????????? ???????????? Linear
  Equation ??????????
• ??????????????? yaxb (?????????? y a bx
  ???????????????????????? a ??? b
  ?????????????????????????)
• Parameter 2 ??????????????? a slope ??? b
  y-intercept

41
Ordinary Simple Linear Least Square Regression

• ???????? Fit ???????????? Error ?????????
  ?????????????????? Least-Square ????? Ordinary
  Least Square Regression (OLS) (???? Linear Least
  Square)
• ???????? Error ???????????????????????? Data
  ????????????????????? ???????????
• Linear Regression ???????????????????
  ???????????????????????????????? Random Variable
  ??????
• Linear Regression ????????????????????????? ???
  OLS ????????????

42
Linear Least-Squares Regression
Y
Error
X
43
Ordinary Linear Least Square Regression (OLS)

• ??? a ??? b ????????????? Summation ??? Error
  ?????????? ?????????
• ???????????????????????????? Derivative
  ????????????????????????? 0 (??? Minima)
• ??????? Unknown ??? a ??? b ?????????????
  Derivative ????????????????????????? 0 ??????
• df(a,b)/da 0
• df(a,b)/db 0
• f(a,b) ??? Sum of Square Error Function
• ???????????????????????????? r correlation
  coefficient ????????????????????????????? Fit
  ????????????, r ?????????????? -1,1

44
Linear Least-Squares Regression
Regression line
(xi,yi)
yi
ei
axib
xi
???????????? Data Pair (xi,yi) i 1,2,,n
??????? n ???
Variable X ??? Y ????????? ??????? Continuous
??? Discrete
45
(No Transcript)
46
R, r Correlation Coefficient
???? Correlation Coefficient ??? Pearson
Product-Moment Correlation Coefficient
?????????????????????????????? Covariance
?????????????? Standard Deviation ??????????
Variable (???????????) ??? ?????????????????????
r 1 ???? -1 ??????????????????????????????? ????
???????? (????????? ????????????) ??? r
???????????????????????????????????????
47
Nonlinear Regression

• ??????????????????????????????????????
  ?????????????? Function ???????? Regression ????
• Exponential Model
• Power Equation
• Saturation-Growth-Rate Equation
• Polynomial Regression
• ?????????????????????????????

48
Example

• 1. ???????????????????????? ???? CPE 332 ???????
  ????? ?????????????? ?????????????????????????????
  ???????????? ????????? ?????????????????????????
• ????? OLS Regression ?????????????????????????????
  ??? ????? Plot Scatter Diagram ????????????????
  ??????????????? Correlation Coefficient

Data Hour Grade
1 5 72
2 8 51
3 2 86
4 6 75
5 4 88
Data Hour Grade
6 10 40
7 7 68
8 6 63
9 12 46
10 8 65
49
Example
50
Example
51
y-5.0159x 99.5079
r -0.9069
52
End of Chapter 12

• Download
• Homework 12 ????????????????? 12.00 ?.(??????????
  ???????? ???? 5-310)

53
Course Ends

• Prepare For Exam
• ???????? 6 ??? ???????? (?????????????????????)
• ?????????? Midterm ?????????
• 1. Function Approximation (1 ???)
• Taylor Series/McLauren Series
• 2. Roots of Function (1 ???)
• Bisection
• Newton-Ralphson
• 3. Linear Equations (1 ???) ???????????????????
• Gauss Elimination
• Gauss Jordan (Including Matrix Inverse)
• Gauss Seidel
• LU Decomposition (Crout Decomposition)

54
Course Ends

• Prepare For Exam
• 4. Numerical Integration (1 ???) ?????? Finite
  Difference
• Trapezoidal Rule
• Simpson 1/3 Rule
• Richardson Extrapolation
• 5. ODE (1 ???)
• Classical Forth Order RK Method ???????????
• 6. Curve Fitting (1 ???)
• Natural Spline
• OLS Regression

55
Formulas
56
END OF CPE 332 T1-56

• Minimum 40 ?????????????????????
• A ????????? 80 ??????