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+--- Thread: (11C) Quadratic Regression (/thread-19180.html)
(11C) Quadratic Regression - Thomas Klemm - 11-20-2022 10:28 PM
Formula
Given a set of points \(\{(x_1, y_1), (x_2, y_2), \cdots, (x_n, y_n)\}\) where \(n \geqslant 3\) we want to find the best fit for a quadratic polynomial:
\(
\begin{align}
y = a \cdot x^2 + b \cdot x + c
\end{align}
\)
This leads to the following linear system of equations:
\(
\begin{bmatrix}
n & \Sigma x & \Sigma x^2 \\
\Sigma x & \Sigma x^2 & \Sigma x^3 \\
\Sigma x^2 & \Sigma x^3 & \Sigma x^4 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
c \\
b \\
a \\
\end{bmatrix}
=
\begin{bmatrix}
\Sigma y \\
\Sigma xy \\
\Sigma x^2y \\
\end{bmatrix}
\)
We can perform Gauss elimination on the first row to get:
\(
\begin{bmatrix}
n & \Sigma x & \Sigma x^2 \\
0 & n \cdot \Sigma x^2 - (\Sigma x)^2 & n \cdot \Sigma x^3 - \Sigma x \cdot \Sigma x^2 \\
0 & n \cdot \Sigma x^3 - \Sigma x \cdot \Sigma x^2 & n \cdot \Sigma x^4 - (\Sigma x^2)^2 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
c \\
b \\
a \\
\end{bmatrix}
=
\begin{bmatrix}
\Sigma y \\
n \cdot \Sigma xy - \Sigma x \cdot \Sigma y \\
n \cdot \Sigma x^2y - \Sigma x^2 \cdot \Sigma y \\
\end{bmatrix}
\)
We define the following:
\(
\begin{align}
S_{x^2} &= n \cdot \Sigma x^2 - (\Sigma x)^2 \\
S_{xy} &= n \cdot \Sigma xy - \Sigma x \cdot \Sigma y \\
S_{xx^2} &= n \cdot \Sigma x^3 - \Sigma x \cdot \Sigma x^2 \\
S_{x^2x^2} &= n \cdot \Sigma x^4 - (\Sigma x^2)^2 \\
S_{x^2y} &= n \cdot \Sigma x^2y - \Sigma x^2 \cdot \Sigma y \\
\end{align}
\)
This allows us to rewrite the linear system for \(b\) and \(a\) as:
\(
\begin{bmatrix}
S_{x^2} & S_{xx^2} \\
S_{xx^2} & S_{x^2x^2} \\
\end{bmatrix}
\cdot
\begin{bmatrix}
b \\
a \\
\end{bmatrix}
=
\begin{bmatrix}
S_{xy} \\
S_{x^2y} \\
\end{bmatrix}
\)
Since the matrix is symmetric we can use the built-in function L.R. to solve it.
In case of linear regression we have the following linear system:
\(
\begin{bmatrix}
n & \Sigma x \\
\Sigma x & \Sigma x^2 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
b \\
a \\
\end{bmatrix}
=
\begin{bmatrix}
\Sigma y \\
\Sigma xy \\
\end{bmatrix}
\)
Registers
We use the following registers:
\(
\begin{align}
R_{0} &: n \\
R_{1} &: \Sigma x \\
R_{2} &: \Sigma x^2 \\
R_{3} &: \Sigma y \\
R_{4} &: \Sigma y^2 \\
R_{5} &: \Sigma xy \\
R_{6} &: \Sigma x^3 \\
R_{7} &: \Sigma x^4 \\
R_{8} &: \Sigma x^2y \\
R_{9} &: c \\
R_{.0} &: b \\
R_{.1} &: a \\
I &: \text{scratch} \\
\end{align}
\)
Thus the function L.R. allows to solve the following linear system:
\(
\begin{bmatrix}
R_{0} & R_{1} \\
R_{1} & R_{2} \\
\end{bmatrix}
\cdot
\begin{bmatrix}
b \\
a \\
\end{bmatrix}
=
\begin{bmatrix}
R_{3} \\
R_{5} \\
\end{bmatrix}
\)
This means we have to do the following assignments:
\(
\begin{align}
R_{0} &\leftarrow R_{0} \cdot R_{2} - R_{1}^2 \\
R_{1} &\leftarrow R_{0} \cdot R_{6} - R_{1} \cdot R_{2} \\
R_{2} &\leftarrow R_{0} \cdot R_{7} - R_{2}^2 \\
R_{3} &\leftarrow R_{0} \cdot R_{5} - R_{1} \cdot R_{3} \\
R_{5} &\leftarrow R_{0} \cdot R_{8} - R_{2} \cdot R_{3} \\
\end{align}
\)
We temporarily store the previous values in the following registers:
\(
\begin{align}
R_{9} &\leftarrow R_{0} \\
R_{.0} &\leftarrow R_{1} \\
R_{.1} &\leftarrow R_{2} \\
I &\leftarrow R_{3} \\
\end{align}
\)
Forecast
The linear estimate function ŷ allows to calculate:
\(
\begin{align}
\hat{y} &= a \cdot x + b
\end{align}
\)
This can be used to calculate:
\(
\begin{align}
\hat{y} \cdot x + c
&= (a \cdot x + b) \cdot x + c \\
&= a \cdot x^2 + b \cdot x + c \\
\end{align}
\)
Program
Code:
001 - 42,21,11 LBL A
002 - 36 ENTER
003 - 36 ENTER
004 - 43 11 x↑2
005 - 20 ×
006 - 44,40, 6 STO+ 6
007 - 33 R↓
008 - 43 36 LST×
009 - 43 33 R↑
010 - 34 x<>y
011 - 20 ×
012 - 44,40, 8 STO+ 8
013 - 33 R↓
014 - 43 36 LST×
015 - 43 11 x↑2
016 - 44,40, 7 STO+ 7
017 - 33 R↓
018 - 49 Σ+
019 - 43 32 RTN
020 - 42,21,12 LBL B
021 - 36 ENTER
022 - 36 ENTER
023 - 43 11 x↑2
024 - 20 ×
025 - 44,30, 6 STO- 6
026 - 33 R↓
027 - 43 36 LST×
028 - 43 33 R↑
029 - 34 x<>y
030 - 20 ×
031 - 44,30, 8 STO- 8
032 - 33 R↓
033 - 43 36 LST×
034 - 43 11 x↑2
035 - 44,30, 7 STO- 7
036 - 33 R↓
037 - 43 49 Σ-
038 - 43 32 RTN
039 - 42,21,13 LBL C
040 - 45 0 RCL 0
041 - 44 9 STO 9
042 - 45 2 RCL 2
043 - 44 .1 STO .1
044 - 20 ×
045 - 45 1 RCL 1
046 - 44 .0 STO .0
047 - 43 11 x↑2
048 - 30 -
049 - 44 0 STO 0
050 - 45 9 RCL 9
051 - 45 6 RCL 6
052 - 20 ×
053 - 45 .0 RCL .0
054 - 45 .1 RCL .1
055 - 20 ×
056 - 30 -
057 - 44 1 STO 1
058 - 45 9 RCL 9
059 - 45 7 RCL 7
060 - 20 ×
061 - 45 .1 RCL .1
062 - 43 11 x↑2
063 - 30 -
064 - 44 2 STO 2
065 - 45 9 RCL 9
066 - 45 5 RCL 5
067 - 20 ×
068 - 45 .0 RCL .0
069 - 45 3 RCL 3
070 - 44 25 STO I
071 - 20 ×
072 - 30 -
073 - 44 3 STO 3
074 - 45 9 RCL 9
075 - 45 8 RCL 8
076 - 20 ×
077 - 45 .1 RCL .1
078 - 45 25 RCL I
079 - 20 ×
080 - 30 -
081 - 44 5 STO 5
082 - 42 49 L.R.
083 - 45 .0 RCL .0
084 - 34 x<>y
085 - 44 .0 STO .0
086 - 20 ×
087 - 34 x<>y
088 - 45 .1 RCL .1
089 - 34 x<>y
090 - 44 .1 STO .1
091 - 20 ×
092 - 40 +
093 - 45 25 RCL I
094 - 34 x<>y
095 - 30 -
096 - 45 9 RCL 9
097 - 10 ÷
098 - 44 9 STO 9
099 - 45 .0 RCL .0
100 - 45 .1 RCL .1
101 - 43 32 RTN
102 - 42,21,14 LBL D
103 - 42 48 ŷ,r
104 - 43 36 LST×
105 - 20 ×
106 - 45 9 RCL 9
107 - 40 +
108 - 43 32 RTNUsage
- A: add data point (similar to Σ+)
- B: remove data point (similar to Σ-)
- C: calculate best fit (similar to L.R.)
- D: forecast (similar to ŷ)
Example
Q H
(flow) (head)
[m3/h] [m]
----------------
0 90.0
30 90.2
60 86.4
90 79.0
120 67.7
150 52.4
90.0 ENTER 0 A
1
90.2 ENTER 30 A
2
86.4 ENTER 60 A
3
79.0 ENTER 90 A
4
67.7 ENTER 120 A
5
52.4 ENTER 150 A
6
C
-0.002132936508 (a)
R↓
0.06955952381 (b)
R↓
89.99642857 (c)
100 D
75.62301587 (y)
Hint
A linear regression can be performed before calculating the best fit.
This result can then later be compared with the quadratic regression.
References
- Quadratic fit without linear algebra (32SII)
- Small Solver Program
- Polynomial regression
- Least Squares Fitting--Polynomial
- Quadratic regression Calculator
RE: (11C) Quadratic Regression - Csaba Tizedes - 11-24-2022 01:24 PM
Thanks for the credit at the end!
In the first matrix-coefficient equation row 3-coloumn 3 the sum(x^3) is wrong. Pls update.
Cs.
RE: (11C) Quadratic Regression - Thomas Klemm - 11-24-2022 06:01 PM
Thank you for pointing out the typo.