Transcript Curve Fitting

Curve Fitting and Regression
EEE 244
Descriptive Statistics in MATLAB
• MATLAB has several built-in commands to compute
and display descriptive statistics. Assuming some
column vector s:
– mean(s), median(s), mode(s)
• Calculate the mean, median, and mode of s. mode is a part of the
statistics toolbox.
– min(s), max(s)
• Calculate the minimum and maximum value in s.
– var(s), std(s)
• Calculate the variance (square of standard deviation) and standard
deviation of s
• Note - if a matrix is given, the statistics will be
returned for each column.
Measurements of a Voltage
Drain Current in mA at
intervals t=10am-10pm
6.5
6.3
6.2
6.5
6.2
6.7
6.4
6.4
6.8
Find mean, median, mode , min, max,
variance and standard deviation using
appropriate Matlab functions.
Histograms in MATLAB
• [n, x] = hist(s, x)
– Determine the number of elements in each bin of data in s. x is
a vector containing the center values of the bins.
– Open Matlab help and run example with 1000 randomly
generated values for s.
• [n, x] = hist(s, m)
– Determine the number of elements in each bin of data in s using
m bins. x will contain the centers of the bins. The default case
is m=10
– Repeat the previous example with setting m=5
• Hist(s) ->histogram plot
EEE 244
REGRESSION
Linear Regression
• Fitting a straight line to a
set of paired observations:
(x1, y1), (x2, y2),…,(xn, yn).
y=a0+a1x+e
a1- slope
a0- intercept
e- error, or residual,
between the model and the
observations
6
Linear Least-Squares Regression
• Linear least-squares regression is a method to
determine the “best” coefficients in a linear model
for given data set.
• “Best” for least-squares regression means minimizing
the sum of the squares of the estimate residuals. For
a straight line model, this gives:
n
n
Sr   e   yi  a0  a1 xi 
2
i
i1

i1
2
Least-Squares Fit of a Straight Line
• Using the model:
y  a0  a1x
the slope and intercept producing the best fit
 using:
can be found
a1 
n xi yi   xi  yi
n x 
a0  y  a1 x
2
i
 x 
2
i
Example
V
(m/s)
F
(N)
a1 
i
xi
yi
(xi)2
x iy i
1
10
25
100
250
2
20
70
400
3
30
380
900
1400

11400
4
40
550
1600
22000
5
50
610
2500
30500
6
60
1220
3600
73200
7
70
830
4900
58100
8
80
1450
6400
116000

360
5135
20400
312850

n xi yi   xi  yi
n x 
2
i
 x 
2
i

8312850  3605135
820400  360
2
 19.47024
a0  y  a1 x  641.875 19.47024 45  234.2857
Fest  234.2857 19.47024v
Standard Error of the Estimate
• Regression data showing (a) the spread of data around the mean
of the dependent data and (b) the spread of the data around the
best fit line:
• The reduction in spread represents the improvement due to
linear regression.
MATLAB Functions
• MATLAB has a built-in function polyfit that fits a
least-squares nth order polynomial to data:
– p = polyfit(x, y, n)
•
•
•
•
x: independent data
y: dependent data
n: order of polynomial to fit
p: coefficients of polynomial
f(x)=p1xn+p2xn-1+…+pnx+pn+1
• MATLAB’s polyval command can be used to compute
a value using the coefficients.
– y = polyval(p, x)
Polyfit function
• Can be used to perform REGRESSION if the
number of data points is a lot larger than the
number of coefficients
– p = polyfit(x, y, n)
•
•
•
•
x: independent data (Vce, 10 data points)
y: dependent data (Ic)
n: order of polynomial to fit n=1 (linear fit)
p: coefficients of polynomial (two coefficients)
f(x)=p1xn+p2xn-1+…+pnx+pn+1
Polynomial Regression
•
•
The least-squares procedure
can be readily extended to fit
data to a higher-order
polynomial. Again, the idea
is to minimize the sum of the
squares of the estimate
residuals.
The figure shows the same
data fit with:
a) A first order polynomial
b) A second order polynomial
Process and Measures of Fit
• For a second order polynomial, the best fit would mean
minimizing:
n
n
Sr   e   yi  a0  a1 xi  a x
i1

2 2
2 i
2
i
i1
• In general, this would mean minimizing:

n
n
Sr   e   yi  a0  a1 xi  a x 
2
i
i1
2
2 i
i1

m 2
m i
a x
EEE 244
INTERPOLATION
Polynomial Interpolation
• You will frequently have occasions to estimate
intermediate values between precise data points.
• The function you use to interpolate must pass
through the actual data points - this makes
interpolation more restrictive than fitting.
• The most common method for this purpose is
polynomial interpolation, where an (n-1)th order
polynomial is solved that passes
through
n data
2
n1
points: f (x)  a1  a2 x  a3 x   an x
MATLAB version :
f (x)  p1 x n1  p2 x n2 
 pn1 x  pn
Determining Coefficients using Polyfit
• MATLAB’s built in polyfit and polyval
commands can also be used - all that is
required is making sure the order of the fit for
n data points is n-1.
Newton Interpolating Polynomials
• Another way to express a polynomial
interpolation is to use Newton’s interpolating
polynomial.
• The differences between a simple polynomial
and Newton’s interpolating polynomial for
first and second order interpolations are:
Order
1st
2nd
Simple
f1 (x)  a1  a2 x
f2 (x)  a1  a2 x  a3 x 2
Newton
f1 (x)  b1  b2 (x  x1 )
f2 (x)  b1  b2 (x  x1 ) b3(x  x1 )(x  x2 )
Newton Interpolating Polynomials
(contd.)
• The first-order Newton
interpolating polynomial may
be obtained from linear
interpolation and similar
triangles, as shown.
• The resulting formula based
on known points x1 and x2 and
the values of the dependent
function at those points is:
f x2   f x1 
f1 x  f x1  
x  x1 
x2  x1
Newton Interpolating Polynomials
(contd.)
• The second-order Newton
interpolating polynomial
introduces some curvature to the
line connecting the points, but
still goes through the first two
points.
• The resulting formula based on
known points x1, x2, and x3 and
the values of the dependent
function at those points is:
f x3   f x2  f x2   f x1 

f x2   f x1 
x 3  x2
x2  x1
f2 x  f x1 
x

x

 1
x  x1 x  x2 
x2  x1
x3  x1
Lagrange Interpolating Polynomials
• Another method that uses shifted value to express an
interpolating polynomial is the Lagrange interpolating
polynomial.
• The differences between a simply polynomial and
Lagrange interpolating polynomials for first and second
order polynomials is:
Order
1st
2nd
Simple
f1 (x)  a1  a2 x
f2 (x)  a1  a2 x  a3 x 2
Lagrange
f1 (x)  L1 f x1  L2 f x2 
f2 (x)  L1 f x1  L2 f x2  L3 f x3 
where the Li are weighting coefficients that are
functions of x.
Lagrange Interpolating Polynomials
(contd.)
• The first-order Lagrange
interpolating polynomial may be
obtained from a weighted
combination of two linear
interpolations, as shown.
• The resulting formula based on
known points x1 and x2 and the
values of the dependent
function at those points is:
f1 (x)  L1 f x1  L2 f x2 
x  x2
x  x1
L1 
, L2 
x1  x2
x2  x1
x  x2
x  x1
f1 (x) 
f x1 
f x2 
x1  x2
x2  x1
x  x0
x  x1
f1 ( x) 
f ( x0 ) 
f ( x1 )
x0  x1
x1  x0


x  x0  x  x2 
x  x1 x  x2 
f 2 ( x) 
f ( x0 ) 
f ( x1 )
x0  x1 x0  x 2 
x1  x0 x1  x 2 

x  x0  x  x1 

f ( x2 )
x2  x0 x2  x1 
•As with Newton’s method, the Lagrange version has an
estimated error of:
n
Rn  f [ x, xn , xn 1 , , x0 ] ( x  xi )
i 0
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Figure 18.10
24
Lagrange Interpolating Polynomials
(contd.)
• In general, the Lagrange polynomial
interpolation for n points
is:
n
fn1 xi    Li x  f xi 
i1
where Li is given by:
x  xj
Li x  
x  xj
j1 i
n

ji
