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- Lecture 29
- Recap
Summary of Chapter 6
Interpolation
Linear Interpolation
- Cubic Spline Interpolation
Connecting data points with straight lines probably isn’t
the best way to estimate intermediate values, although it is
surely the simplest
A smoother curve can be created by using the cubic
spline interpolation technique, included in the interp1
function. This approach uses a thirdorder polynomial to
model the behavior of the data
To call the cubic spline, we need to add a fourth field to
interp1 :
interp1(x,y,3.5,'spline')
This command returns an improved estimate of y at x =
- Multidimensional Interpolation
Suppose there is a set of data z that depends on two
variables, x and y . For example
- Continued….
In order to determine the value of z at y = 3 and x = 1.5,
two interpolations have to performed
One approach would be to find the values of z at y = 3
and all the given x values by using interp1 and then do a
second interpolation in new chart
First let’s define x , y , and z in MATLAB :
y = 2:2:6;
x = 1:4;
z = [ 7 15 22 30
54 109 164 218
- Continued….
Although the previous approach works, performing the
calculations in two steps is awkward
MATLAB includes a twodimensional linear interpolation
function, interp2 , that can solve the problem in a single
step:
interp2(x,y,z,1.5,3)
ans =
46.2500
The first field in the interp2 function must be a vector
defining the value associated with each column (in this
case, x ), and the second field must be a vector defining
- Continued….
MATLAB also includes a function, interp3 , for three
dimensional interpolation
Consult the help feature for the details on how to use this
function and interpn , which allows you to perform n
dimensional interpolation
All these functions default to the linear interpolation
technique but will accept any of the other techniques
- Curve Fitting
Although interpolation techniques can be used to find
values of y between measured x values, it would be more
convenient if we could model experimental data as y=f(x)
Then we could just calculate any value of y we wanted
If we know something about the underlying relationship
between x and y , we may be able to determine an
equation on the basis of those principles
MATLAB has builtin curvefitting functions that allow
us to model data empirically
These models are good only in the region where we’ve
collected data
- Linear Regression
The simplest way
to model a set of
data is as a straight
line
Suppose a data set
x = 0:5;
y = [15, 10, 9, 6,
2, 0];
If we plot the data,
we can try to draw
a straight line
through the data
- Continued….
Looking at the plot,
there are several of
the points appear to
fall exactly on the
line, but others are
off by varying
amounts
In order to compare
the quality of the fit
of this line to other
possible estimates,
we find the
- Continued….
- Continued….
The linear regression technique uses an approach called
least squares fit to compare how well different equations
model the behavior of the data
In this technique, the differences between the actual and
calculated values are squared and added together. This has
the advantage that positive and negative deviations don’t
cancel each other out
MATLAB could be used to calculate this parameter for
data. We have
sum_of_the_squares = sum((yy_calc).^2)
- Continued….
Linear regression is accomplished in MATLAB with the
polyfit function
Three fields are required by polyfit :
a vector of x –values
a vector of y –values
an integer indicating what order polynomial should be used
to fit the data
Since a straight line is a firstorder polynomial, enter the
number 1 into the polyfit function:
polyfit(x,y,1)
ans = 2.9143 14.2857
- Polynomial Regression
- Continued….
We can find the sum of the squares to determine whether
these models fit the data better:
y2 = 0.0536*x.^23.182*x + 14.4643;
sum((y2y).^2)
ans = 3.2643
y3 = 0.0648*x.^3+0.5398*x.^24.0701*x + 14.6587
sum((y3y).^2)
ans = 2.9921
The more terms we add to our equation, the “better” is the
- Continued….
In order to plot the curves defined by these new
equations, more than the six data points are used in the
linear model
MATLAB creates plots by connecting calculated points
with straight lines, so if a smooth curve is needed, more
points are required
We can get more points and plot the curves with the
following code:
smooth_x = 0:0.2:5;
smooth_y2 = 0.0536*smooth_x.^23.182*smooth_x +
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