Curve Fitting and Interpolation2

1
Curve Fitting and Interpolation-2
2
APPROXIMATION
OF DATA
Least Squares Method of Curve Fitting
3
Data Approximation

• Data points (xi, yi) can be approximated by a
  function
• y f(x) such that the function passes
  close to the data
• points but does not necessarily pass through
  them.

• Data fitting is necessary to model data with
  fluctuations such
• as experimental measurements.

• The most common form of data fitting is the
• least squares method.

4
Multi-Valued Data Points
y
x
5
Fitting Data With a Linear Function
y

x
Using the least squares algorithm, ensure that
all the data points fall close to the straight
line/function.
6
Linear Least Squares Algorithm

• Data points

• Choice of fitting function (linear)

• Errors between function and data points

• Sum of the squares of the errors

• In compact notation

7
Linear Least Squares Algorithm-cont.

• Our goal is to determine the values of a and b
  that will minimize z, the sum of the squares of
  the errors.

To find the minimum value for z, Matlab uses the
same technique that we would use analytically
(i.e., setting the derivative of z to zero and
solving for a and b)
8
Plot of Linear Fit using Matlab
70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
9
Plot of Linear Fit using Matlab

• What about
• this point?
• Is it really a
• good data point?
• What do you know about the data?

70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
10
A Linear Fit Ignoring One Data Point
70
60
50
40
30
20

• But what if we
• "convicted" the
• wrong data point?

10
0
0
1
2
3
4
5
6
7
8
9
10
11
Second Degree Polynomial Fit, Ignoring a
Different Data Point

• Ignoring a different data point allows us to
  approximate the data pretty well with a second
  degree polynomial.

70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
12
Choosing the Right Polynomial

• The degree of the correct approximating
  function depends on the type of data being
  analyzed.

When a certain behavior is expected, we know what
type of function to use, and simply have to solve
for its coefficients.
When we dont know what sort of response to
expect, ensure your data sample size is large
enough to clearly distinguish which degree is the
best fit.
13
Data Approximation with Matlabs polyfit

• The steps used in Matlab for approximating
  data are very similar to those used for
  interpolation

The degree of the approximating function depends
on the type of fit desired, rather than on the
number of data points.
14
Example Ohms Law

• This graph shows the amount of current through an
  unknown resistor, plotted against the voltage
  applied to the resistor.
• Knowing Ohms Law (I V/R), find the resistance,
  and plot an approximating function.

15
Example-cont.

• Ohms Law shows a linear dependence between
  current and voltage, so we will use a first
  degree fit.
• Once again, there is what appears to be an
  "outsider" in the data set (0.08 A at 20 V) that
  is most likely a data collection error, and will
  be ignored.

30
16
IN MATLAB

• Rewrite I V/R as I (1/R)V, which is a
  linear relationship between V and I, with
  coefficient 1/R.

Do not include data point V20 xdata 0 5 10
15 25 ydata 0.001 0.881 2.1637 3.1827
4.961 degree 1 Linear
relationship coef polyfit(xdata, ydata,
degree) xx -5 0.5 30 Range for
plotting yy polyval(coef, xx) plot(xdata,
ydata, 'o', xx, yy) resistance 1 / coef(1)
Output the answer
17
IN MATLAB-The Plot
plot(xdata, ydata, 'o', xx, yy) resistance 1 /
coef(1) Output the answer

• The output from Matlab is
• resistance
• 4.9515
• Thus, the resistance is 4.95 O.

18
Exponential Fit

• Not all experimental data can be approximated
  with polynomial functions.
• Exponential data can be fit using the least
  squares method by first converting the data to a
  linear form.

19
Relationship between an Exponential Linear
Form
This graph shows a population that doubles
every year. Notice that for the first fifteen
years, growth is almost imperceptible at this
scale. It would be difficult to approximate this
raw data.
20
Population Example-cont.

• What if we plot the same data with the y-axis
  against a logarithmic scale?
• Notice that the change in population is not
  simply perceptible, but is clearly linear.
• If we can transform the numeric exponential data,
  it would be simple to approximate with the
• least-squares method.

21
Population Example-cont.

• This graph shows what happens if we take the
  natural logarithm of each y value.
• Notice that the shape of the graph did not change
  from the last example, only the scaling of the
  y-axis.

22
Relationship between an Exponential Linear
Form

• An exponential function,
• y aebx
• can be rewritten as a linear polynomial by
  taking the natural logarithm of each side
• ln y ln a bx
• By finding ln yi for each point in a data set, we
  can solve for a and b using the least squares
  method.

23
Using Matlab to Approximate Exponential Data
An Example

• A radiation detector is used to record counts of
  the decay of an unknown material according to
• A(t)A0 e-?t
• where A0 is the count rate at t 0 and ? is the
  decay constant.
• Using data taken at one-minute intervals, find A0
  and ? and plot a function in Matlab that
  approximates the decay of the material.

24
Example-Cont.

• A(t)A0 e-?t
• ln(A(t)) ln(A0) -?t

• t 010
• cr 2095 1252 766 452 266 162 98 70 39 21 12
• crlog log(cr)
• coefs polyfit(t, crlog, 1) find ln(a) and
  b
• A0 exp(coefs(2)) coefs(2)ln(a) so
  aexp(coefs(2))
• lambda -coefs(1) b -lambda so lambda -b
• tt 00.110 the range over which to
  approximate
• approxcr A0exp(-lambdatt)
• plot(t,cr,'o', tt,approxcr)

25
Example-Cont.

• The plot of the exponential approximation

26
Interpolation vs. Approximation

• Splines are used
• To produce smooth surfaces, shapes, and motions.
• When interpolation best used
• When you wish to find an unknown intermediate
  point within a dataset with well-defined
  behavior.
• When a relatively small data set exists.
• When is approximation best used
• When evaluating fluctuating data.
• To create a mathematical model of a process.

27
Fundamental to Engineering Practice and Research

• Analysis of data in engineering research involves
  relating data to theory or a model.
• This starts with fitting the data to
  mathematical functions so that relationships can
  be explored.
• The next step is to compare the approximating
  (data) function with the theoretical function.
• The theoretical function is often the solution to
  a set of differential equations.
• For a given theory, when the data function is
  similar to the theoretical function, we say that
  we have a good theory or model of the phenomenon.
  
• If the functions are not similar, the theoretical
  function may need to be modified, or completely
  new equations may be needed.