Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a...
Transcript of Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a...
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Least square fit of a line/polynomial toinput/output data
Dr. Madhu N. Belur
Control & Computing groupDepartment of Electrical Engineering
Indian Institute of Technology BombayEmail: [email protected]
30th Nov, 2010
![Page 2: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/2.jpg)
Outline Scilab Least squares Scilab commands
Outline
1 Scilab
2 Least squares
3 Scilab commands
Madhu Belur, CC group, EE, IITB Least square fit
![Page 3: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/3.jpg)
Outline Scilab Least squares Scilab commands
Today’s focus
Scilab is free.
Matrix/loops syntax is same as for Matlab.
Scilab provides all basic and many advanced tools.
Today: best fit: line and polynomial : reglin command
Madhu Belur, CC group, EE, IITB Least square fit
![Page 4: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/4.jpg)
Outline Scilab Least squares Scilab commands
Today’s focus
Scilab is free.
Matrix/loops syntax is same as for Matlab.
Scilab provides all basic and many advanced tools.
Today: best fit: line and polynomial : reglin command
Madhu Belur, CC group, EE, IITB Least square fit
![Page 5: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/5.jpg)
Outline Scilab Least squares Scilab commands
Today’s focus
Scilab is free.
Matrix/loops syntax is same as for Matlab.
Scilab provides all basic and many advanced tools.
Today: best fit: line and polynomial : reglin command
Madhu Belur, CC group, EE, IITB Least square fit
![Page 6: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/6.jpg)
Outline Scilab Least squares Scilab commands
Today’s focus
Scilab is free.
Matrix/loops syntax is same as for Matlab.
Scilab provides all basic and many advanced tools.
Today: best fit: line and polynomial : reglin command
Madhu Belur, CC group, EE, IITB Least square fit
![Page 7: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/7.jpg)
Outline Scilab Least squares Scilab commands
Linear fit
Given n samples of (x , z) pairs:xi and zi for i = 1, . . . , n, we expect following equation is satisfied
zi = a1xi + a0 for i = 1, . . . , n (1)
for some constants a1 and a0.
xi and zi fall on some line with slope a1 and ‘z-intercept’=a0.The ‘line fit’ problem:Find these constants a1 and a0.‘Best’ fit?
Madhu Belur, CC group, EE, IITB Least square fit
![Page 8: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/8.jpg)
Outline Scilab Least squares Scilab commands
Linear fit
Given n samples of (x , z) pairs:xi and zi for i = 1, . . . , n, we expect following equation is satisfied
zi = a1xi + a0 for i = 1, . . . , n (1)
for some constants a1 and a0.xi and zi fall on some line with slope a1 and ‘z-intercept’=a0.The ‘line fit’ problem:Find these constants a1 and a0.‘Best’ fit?
Madhu Belur, CC group, EE, IITB Least square fit
![Page 9: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/9.jpg)
Outline Scilab Least squares Scilab commands
Best fit
Real situtation:Equation (1) will not be satisfied ‘exactly’.Least-square-fit problem:Given n samples of (xi , zi ) pairs,
find constants a1 and a0 such that the ‘total square error’
n∑i=1
(zi − a1xi − a0)2 (2)
is least.Least square error
Madhu Belur, CC group, EE, IITB Least square fit
![Page 10: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/10.jpg)
Outline Scilab Least squares Scilab commands
Best fit
Real situtation:Equation (1) will not be satisfied ‘exactly’.Least-square-fit problem:Given n samples of (xi , zi ) pairs,find constants a1 and a0 such that the ‘total square error’
n∑i=1
(zi − a1xi − a0)2 (2)
is least.
Least square error
Madhu Belur, CC group, EE, IITB Least square fit
![Page 11: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/11.jpg)
Outline Scilab Least squares Scilab commands
Best fit
Real situtation:Equation (1) will not be satisfied ‘exactly’.Least-square-fit problem:Given n samples of (xi , zi ) pairs,find constants a1 and a0 such that the ‘total square error’
n∑i=1
(zi − a1xi − a0)2 (2)
is least.Least square error
Madhu Belur, CC group, EE, IITB Least square fit
![Page 12: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/12.jpg)
Outline Scilab Least squares Scilab commands
Higher order polynomial best fit
Least-square-fit problem (of higher order):
Suppose we expect zi satisfies the following equation:
zi = a2x2i + a1xi + a0
Points (xi , zi ) are sitting on a parabola.Problem (more generally):Given n samples of (xi , zi ) pairs and some choice of degree d .Find constants ad , . . . a1 and a0 such that the ‘total square error’
n∑i=1
(adxdi + ad−1x
d−1i + · · · + a1xi + a0 − zi )
2 (3)
is least.
Madhu Belur, CC group, EE, IITB Least square fit
![Page 13: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/13.jpg)
Outline Scilab Least squares Scilab commands
Higher order polynomial best fit
Least-square-fit problem (of higher order):Suppose we expect zi satisfies the following equation:
zi = a2x2i + a1xi + a0
Points (xi , zi ) are sitting on a parabola.Problem (more generally):Given n samples of (xi , zi ) pairs and some choice of degree d .Find constants ad , . . . a1 and a0 such that the ‘total square error’
n∑i=1
(adxdi + ad−1x
d−1i + · · · + a1xi + a0 − zi )
2 (3)
is least.
Madhu Belur, CC group, EE, IITB Least square fit
![Page 14: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/14.jpg)
Outline Scilab Least squares Scilab commands
Higher order polynomial best fit
Least-square-fit problem (of higher order):Suppose we expect zi satisfies the following equation:
zi = a2x2i + a1xi + a0
Points (xi , zi ) are sitting on a parabola.
Problem (more generally):Given n samples of (xi , zi ) pairs and some choice of degree d .Find constants ad , . . . a1 and a0 such that the ‘total square error’
n∑i=1
(adxdi + ad−1x
d−1i + · · · + a1xi + a0 − zi )
2 (3)
is least.
Madhu Belur, CC group, EE, IITB Least square fit
![Page 15: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/15.jpg)
Outline Scilab Least squares Scilab commands
Higher order polynomial best fit
Least-square-fit problem (of higher order):Suppose we expect zi satisfies the following equation:
zi = a2x2i + a1xi + a0
Points (xi , zi ) are sitting on a parabola.Problem (more generally):
Given n samples of (xi , zi ) pairs and some choice of degree d .Find constants ad , . . . a1 and a0 such that the ‘total square error’
n∑i=1
(adxdi + ad−1x
d−1i + · · · + a1xi + a0 − zi )
2 (3)
is least.
Madhu Belur, CC group, EE, IITB Least square fit
![Page 16: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/16.jpg)
Outline Scilab Least squares Scilab commands
Higher order polynomial best fit
Least-square-fit problem (of higher order):Suppose we expect zi satisfies the following equation:
zi = a2x2i + a1xi + a0
Points (xi , zi ) are sitting on a parabola.Problem (more generally):Given n samples of (xi , zi ) pairs and some choice of degree d .
Find constants ad , . . . a1 and a0 such that the ‘total square error’
n∑i=1
(adxdi + ad−1x
d−1i + · · · + a1xi + a0 − zi )
2 (3)
is least.
Madhu Belur, CC group, EE, IITB Least square fit
![Page 17: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/17.jpg)
Outline Scilab Least squares Scilab commands
Higher order polynomial best fit
Least-square-fit problem (of higher order):Suppose we expect zi satisfies the following equation:
zi = a2x2i + a1xi + a0
Points (xi , zi ) are sitting on a parabola.Problem (more generally):Given n samples of (xi , zi ) pairs and some choice of degree d .Find constants ad , . . . a1 and a0 such that the ‘total square error’
n∑i=1
(adxdi + ad−1x
d−1i + · · · + a1xi + a0 − zi )
2 (3)
is least.
Madhu Belur, CC group, EE, IITB Least square fit
![Page 18: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/18.jpg)
Outline Scilab Least squares Scilab commands
Higher order polynomial best fit
Least-square-fit problem (of higher order):Suppose we expect zi satisfies the following equation:
zi = a2x2i + a1xi + a0
Points (xi , zi ) are sitting on a parabola.Problem (more generally):Given n samples of (xi , zi ) pairs and some choice of degree d .Find constants ad , . . . a1 and a0 such that the ‘total square error’
n∑i=1
(adxdi + ad−1x
d−1i + · · · + a1xi + a0 − zi )
2 (3)
is least.
Madhu Belur, CC group, EE, IITB Least square fit
![Page 19: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/19.jpg)
Outline Scilab Least squares Scilab commands
reglin
Linear regression problem.The unknowns ai enter the problem linearly.
(i.e. ai ’s are not getting squared, or multiplied to each other.)[a1, a0] = reglin(x,z)where x and z are vectors with same number of columns.∑
i
(zi − a1xi − a0)2
is minimized.[a1, a0, sig] = reglin(x,z)sig : standard deviation of the residual.(Smaller sig means better fit.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 20: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/20.jpg)
Outline Scilab Least squares Scilab commands
reglin
Linear regression problem.The unknowns ai enter the problem linearly.(i.e. ai ’s are not getting squared, or multiplied to each other.)[a1, a0] = reglin(x,z)where x and z are vectors with same number of columns.∑
i
(zi − a1xi − a0)2
is minimized.
[a1, a0, sig] = reglin(x,z)sig : standard deviation of the residual.(Smaller sig means better fit.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 21: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/21.jpg)
Outline Scilab Least squares Scilab commands
reglin
Linear regression problem.The unknowns ai enter the problem linearly.(i.e. ai ’s are not getting squared, or multiplied to each other.)[a1, a0] = reglin(x,z)where x and z are vectors with same number of columns.∑
i
(zi − a1xi − a0)2
is minimized.[a1, a0, sig] = reglin(x,z)sig : standard deviation of the residual.
(Smaller sig means better fit.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 22: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/22.jpg)
Outline Scilab Least squares Scilab commands
reglin
Linear regression problem.The unknowns ai enter the problem linearly.(i.e. ai ’s are not getting squared, or multiplied to each other.)[a1, a0] = reglin(x,z)where x and z are vectors with same number of columns.∑
i
(zi − a1xi − a0)2
is minimized.[a1, a0, sig] = reglin(x,z)sig : standard deviation of the residual.(Smaller sig means better fit.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 23: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/23.jpg)
Outline Scilab Least squares Scilab commands
More than one independent variables
Suppose z depends on independent variables x1, x2, etc.[a1,a0]=reglin(x,z)where x and z are matrix/vector with same number of columns.
(but x has many rows.)If x has more than one rows:components in a1 = number of rows of x(number of independent variables.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 24: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/24.jpg)
Outline Scilab Least squares Scilab commands
More than one independent variables
Suppose z depends on independent variables x1, x2, etc.[a1,a0]=reglin(x,z)where x and z are matrix/vector with same number of columns.(but x has many rows.)
If x has more than one rows:components in a1 = number of rows of x(number of independent variables.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 25: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/25.jpg)
Outline Scilab Least squares Scilab commands
More than one independent variables
Suppose z depends on independent variables x1, x2, etc.[a1,a0]=reglin(x,z)where x and z are matrix/vector with same number of columns.(but x has many rows.)If x has more than one rows:components in a1 = number of rows of x(number of independent variables.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 26: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/26.jpg)
Outline Scilab Least squares Scilab commands
Exercise plan
Generate data using known (actual) values of a0 and a1.
Add noise.Use noisy data to estimate a0 and a1:
Madhu Belur, CC group, EE, IITB Least square fit
![Page 27: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/27.jpg)
Outline Scilab Least squares Scilab commands
Exercise plan
Generate data using known (actual) values of a0 and a1.Add noise.
Use noisy data to estimate a0 and a1:
Madhu Belur, CC group, EE, IITB Least square fit
![Page 28: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/28.jpg)
Outline Scilab Least squares Scilab commands
Exercise plan
Generate data using known (actual) values of a0 and a1.Add noise.Use noisy data to estimate a0 and a1:
Madhu Belur, CC group, EE, IITB Least square fit
![Page 29: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/29.jpg)
Outline Scilab Least squares Scilab commands
Random numbers
rand(3) generates a 1 × 1 random number (uniformly distributedbetween 0 and 1).
If P is an n × p matrix, thenB=rand(P)defines a random matrix B of the size of P.(P is not overwritten. A new matrix B is defined.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 30: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/30.jpg)
Outline Scilab Least squares Scilab commands
Random numbers
rand(3) generates a 1 × 1 random number (uniformly distributedbetween 0 and 1).If P is an n × p matrix, then
B=rand(P)defines a random matrix B of the size of P.(P is not overwritten. A new matrix B is defined.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 31: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/31.jpg)
Outline Scilab Least squares Scilab commands
Random numbers
rand(3) generates a 1 × 1 random number (uniformly distributedbetween 0 and 1).If P is an n × p matrix, thenB=rand(P)defines a random matrix B of the size of P.
(P is not overwritten. A new matrix B is defined.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 32: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/32.jpg)
Outline Scilab Least squares Scilab commands
Random numbers
rand(3) generates a 1 × 1 random number (uniformly distributedbetween 0 and 1).If P is an n × p matrix, thenB=rand(P)defines a random matrix B of the size of P.(P is not overwritten. A new matrix B is defined.)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 33: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/33.jpg)
Outline Scilab Least squares Scilab commands
Exercises:
Generate data, add noise, and estimate the parameters back:One independent variable (first).a0a = 3;a1a = 6;(actual)x = 1:10;
dev = 0.5;y = a1a*x + a0a + dev*(rand(x)-0.5)Use this x and y to estimate a0 and a1Compare a0 & a1 with actual a0a & a1a.For sig to compare with dev, note that sig is standard deviation(most relevant for normal distribution, and we used uniformdistribution).
Madhu Belur, CC group, EE, IITB Least square fit
![Page 34: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/34.jpg)
Outline Scilab Least squares Scilab commands
Exercises:
Generate data, add noise, and estimate the parameters back:One independent variable (first).a0a = 3;a1a = 6;(actual)x = 1:10;dev = 0.5;
y = a1a*x + a0a + dev*(rand(x)-0.5)Use this x and y to estimate a0 and a1Compare a0 & a1 with actual a0a & a1a.For sig to compare with dev, note that sig is standard deviation(most relevant for normal distribution, and we used uniformdistribution).
Madhu Belur, CC group, EE, IITB Least square fit
![Page 35: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/35.jpg)
Outline Scilab Least squares Scilab commands
Exercises:
Generate data, add noise, and estimate the parameters back:One independent variable (first).a0a = 3;a1a = 6;(actual)x = 1:10;dev = 0.5;y = a1a*x + a0a + dev*(rand(x)-0.5)Use this x and y to estimate a0 and a1Compare a0 & a1 with actual a0a & a1a.
For sig to compare with dev, note that sig is standard deviation(most relevant for normal distribution, and we used uniformdistribution).
Madhu Belur, CC group, EE, IITB Least square fit
![Page 36: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/36.jpg)
Outline Scilab Least squares Scilab commands
Exercises:
Generate data, add noise, and estimate the parameters back:One independent variable (first).a0a = 3;a1a = 6;(actual)x = 1:10;dev = 0.5;y = a1a*x + a0a + dev*(rand(x)-0.5)Use this x and y to estimate a0 and a1Compare a0 & a1 with actual a0a & a1a.For sig to compare with dev, note that sig is standard deviation(most relevant for normal distribution, and we used uniformdistribution).
Madhu Belur, CC group, EE, IITB Least square fit
![Page 37: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/37.jpg)
Outline Scilab Least squares Scilab commands
Fit 2nd order polynomial
Find a2, a1 and a0 such that
y = a1x + a2x2 + a0
Suppose a0a = 3; a1a = 6; a2a = 2; (actual)
x = 1:10; dev = 0.5;y = a1a∗x + a2a∗x•ˆ2 + a0a + dev∗(rand(x)-0.5)x = 1:10; x2 = x.ˆ2;X=[x;x2];reglin(X,y)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 38: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/38.jpg)
Outline Scilab Least squares Scilab commands
Fit 2nd order polynomial
Find a2, a1 and a0 such that
y = a1x + a2x2 + a0
Suppose a0a = 3; a1a = 6; a2a = 2; (actual)x = 1:10; dev = 0.5;
y = a1a∗x + a2a∗x•ˆ2 + a0a + dev∗(rand(x)-0.5)x = 1:10; x2 = x.ˆ2;X=[x;x2];reglin(X,y)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 39: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/39.jpg)
Outline Scilab Least squares Scilab commands
Fit 2nd order polynomial
Find a2, a1 and a0 such that
y = a1x + a2x2 + a0
Suppose a0a = 3; a1a = 6; a2a = 2; (actual)x = 1:10; dev = 0.5;y = a1a∗x + a2a∗x•ˆ2 + a0a + dev∗(rand(x)-0.5)x = 1:10; x2 = x.ˆ2;X=[x;x2];reglin(X,y)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 40: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/40.jpg)
Outline Scilab Least squares Scilab commands
Read/write csv files
csv ≡ comma separated value
r = read csv(’lsquare data.csv’,ascii(9)); // read csv data into rr = strsubst(r,’,’,’.’); // string substituter = evstr(r); // convert string to numerical values
xval = r(1,:); //first row (time)yval = r(2,:); //second row (displacement)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 41: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/41.jpg)
Outline Scilab Least squares Scilab commands
Read/write csv files
csv ≡ comma separated value
r = read csv(’lsquare data.csv’,ascii(9)); // read csv data into rr = strsubst(r,’,’,’.’); // string substituter = evstr(r); // convert string to numerical values
xval = r(1,:); //first row (time)yval = r(2,:); //second row (displacement)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 42: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/42.jpg)
Outline Scilab Least squares Scilab commands
Read/write csv files
csv ≡ comma separated value
r = read csv(’lsquare data.csv’,ascii(9)); // read csv data into rr = strsubst(r,’,’,’.’); // string substituter = evstr(r); // convert string to numerical values
xval = r(1,:); //first row (time)yval = r(2,:); //second row (displacement)
Madhu Belur, CC group, EE, IITB Least square fit
![Page 43: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/43.jpg)
Outline Scilab Least squares Scilab commands
Displacement under gravity
Constant gravity: g = 9.8 m/s2 downwards.Initial velocity: v0Initial displacement: x0
x(t) = x0 + v0t −1
2gt2
Madhu Belur, CC group, EE, IITB Least square fit
![Page 44: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/44.jpg)
Outline Scilab Least squares Scilab commands
Spoken tutorial?
More spoken tutorials?
For least square fit: and remaining topics:Enthusiasts/volunteers required for spoken tutorials creation.Be in touch
Madhu Belur, CC group, EE, IITB Least square fit
![Page 45: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/45.jpg)
Outline Scilab Least squares Scilab commands
Spoken tutorial?
More spoken tutorials?For least square fit: and remaining topics:
Enthusiasts/volunteers required for spoken tutorials creation.Be in touch
Madhu Belur, CC group, EE, IITB Least square fit
![Page 46: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/46.jpg)
Outline Scilab Least squares Scilab commands
Spoken tutorial?
More spoken tutorials?For least square fit: and remaining topics:Enthusiasts/volunteers required for spoken tutorials creation.
Be in touch
Madhu Belur, CC group, EE, IITB Least square fit
![Page 47: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/47.jpg)
Outline Scilab Least squares Scilab commands
Spoken tutorial?
More spoken tutorials?For least square fit: and remaining topics:Enthusiasts/volunteers required for spoken tutorials creation.Be in touch
Madhu Belur, CC group, EE, IITB Least square fit
![Page 48: Least square fit of a line/polynomial to input/output …...2010/11/29 · Least square t of a line/polynomial to input/output data Dr. Madhu N. Belur Control & Computing group Department](https://reader034.fdocuments.net/reader034/viewer/2022042420/5f37bff0bda2ae008324946a/html5/thumbnails/48.jpg)
Outline Scilab Least squares Scilab commands
Spoken tutorial?
More spoken tutorials?For least square fit: and remaining topics:Enthusiasts/volunteers required for spoken tutorials creation.Be in touch
Madhu Belur, CC group, EE, IITB Least square fit