Transcript Lecture29.pptx

Recap
 Summary of Chapter 6
 Interpolation
 Linear Interpolation
Cubic Spline Interpolation
 Connecting data points with straight lines probably isn’t the best way to
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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 third-order
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 = 3.5:
ans = 3.9417
The cubic spline technique can be used to create an array of new estimates
for y for every member of an array of x -values:
 new_x = 0:0.2:5;
 new_y_spline = interp1(x,y,new_x,'spline');
 A plot of these data on the same graph as the measured data using the
command
plot(x,y,new_x,new_y_spline,'-o')
results in two different lines
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
403 807 1210 1614];
 Now use interp1 to find the values of z at y = 3 for all the x -values:
 new_z = interp1(y,z,3) returns
 new_z =
30.50 62.00 93.00 124.00
 Finally, since we have z -values at y = 3, we can use interp1 again to find z at y = 3
and x = 1.5:
 new_z2 = interp1(x,new_z,1.5)
 new_z2 =
46.25
Continued….
 Although the previous approach works, performing the calculations
in two steps is awkward
 MATLAB includes a two-dimensional 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 the values associated with each row
(in this case, y )
 The array z must have the same number of columns as the number of
elements in x and must have the same number of rows as the number
of elements in y
 The fourth and fifth fields correspond to the values of x and of y for
which you would like to determine new z -values
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
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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 built-in curve-fitting functions that allow us to
model data empirically
These models are good only in the region where we’ve
collected data
If we don’t understand why a parameter such as y changes as it
does with x , we can’t predict whether our data-fitting equation
will still work outside the range 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
points to get a rough
model of the data’s
behavior
 This process is
sometimes called
“eyeballing it”—
meaning that no
calculations were done,
but it looks like a good
fit
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
difference between the
actual y -value and the
value calculated from
the estimate. This
difference is called the
residual
Continued….
 The equation of line in graph at x=0, y=0 and x=5, y=0
can be found
 The slope of line is
𝑟𝑖𝑠𝑒 ∆𝑦 𝑦2 − 𝑦1 0 − 15
=
=
=
= −3
𝑟𝑢𝑛 ∆𝑥 𝑥2 − 𝑥1
5−0
 The line crosses the y -axis at 15, so the equation of the
line is
𝑦 = −3𝑥 + 15
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((y-y_calc).^2)
 which gives us
sum_of_the_squares =5
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 first-order polynomial, enter the number 1 into the polyfit
function:
 polyfit(x,y,1)
 ans = -2.9143 14.2857
 The results are the coefficients corresponding to the best-fit first-order polynomial
equation:
 y = -2.9143x + 14.2857
 Calculate the sum of the squares to find out that if this is a better fit than “eyeballed”
model or not:
 best_y = -2.9143*x+14.2857;
 new_sum = sum((y-best_y).^2)
 new_sum = 3.3714
 Since the result of the sum-of-the-squares calculation is indeed less than the value
found for the “eyeballed” line, we can conclude that MATLAB found a better fit to
the data
 We can plot the data and the best-fit line determined by linear regression to try to get
a visual sense of whether the line fits the data well:
 plot(x,y,'o',x,best_y)
Polynomial Regression
 Straight lines are not the only equations that could be analyzed with the
regression technique. For example, a common approach is to fit the data
with a higher-order polynomial of the form
 𝑦 = 𝑎1 𝑥 𝑛 + 𝑎2 𝑥 𝑛−1 + 𝑎3 𝑥 𝑛−2 + ⋯ … … … … + 𝑎𝑛 𝑥 + 𝑎𝑛+1
 Polynomial regression is used to get the best fit by minimizing the sum of
the squares of the deviations of the calculated values from the data
 The polyfitfunction allows us to do this easily in MATLAB
 We can fit our sample data to second and third-order equations with the
commands
 a=polyfit(x,y,2)
 a = 0.0536 -3.1821 14.4643
 and
 a=polyfit(x,y,3)
 a = -0.0648 0.5397 -4.0701 14.6587
 which correspond to the following equations
 𝑦2 = 0.0536𝑥 2 − 3.1821𝑥 + 14.4643
 𝑦3 = −0.0648𝑥 3 + 0.5397𝑥 2 − 4.0701𝑥 + 14.6587
Continued….
 We can find the sum of the squares to determine whether
these models fit the data better:
 y2 = 0.0536*x.^2-3.182*x + 14.4643;
 sum((y2-y).^2)
 ans = 3.2643
 y3 = -0.0648*x.^3+0.5398*x.^2-4.0701*x + 14.6587
 sum((y3-y).^2)
 ans = 2.9921
 The more terms we add to our equation, the “better” is the
fit, at least in the sense that the distance between the
measured and predicted data points decreases
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.^2-3.182*smooth_x + 14.4643;
 subplot(1,2,1)
 plot(x,y,'o',smooth_x,smooth_y2)
 smooth_y3 = -0.0648*smooth_x.^3+0.5398*smooth_x.^2-4.0701*
 smooth_x + 14.6587;
 subplot(1,2,2)
 plot(x,y,'o',smooth_x,smooth_y3)
The Polyval Function
 The polyfit function returns the coefficients of a polynomial that best fits the
data, at least on the basis of a regression criterion
 In the previous section, we entered those coefficients into a MATLAB
expression for the corresponding polynomial and used it to calculate new
values of y
 The polyval function can perform the same job without having to reenter the
coefficients
 The polyval function requires two inputs
 The first is a coefficient array, such as that created by polyfit
 The second is an array of x -values for which we would like to calculate
new y –values
 For example:
 coef = polyfit(x,y,1)
 y_first_order_fit = polyval(coef,x)
 These two lines of code could be shortened to one line by nesting functions:
 y_first_order_fit = polyval(polyfit(x,y,1),x)
Continued….
 So according to new understanding of the polyfit and polyval
functions to write a program to calculate and plot the fourthand fifth-order fits for the data
 y4 = polyval(polyfit(x,y,4),smooth_x);
 y5 = polyval(polyfit(x,y,5),smooth_x);
 subplot(1,2,1)
 plot(x,y,'o',smooth_x,y4)
 axis([0,6,-5,15])
 subplot(1,2,2)
 plot(x,y,'o',smooth_x,y5)
 axis([0,6,-5,15])
The Interactive Fitting Tools
 MATLAB includes new interactive plotting tools that
allow to annotate plots without using the comman window
 MATLAB also included
 Basic curve fitting
 More complicated curve fitting
 Statistical tools
Basic Curve Fitting
 To access the
basic fitting
tools, first create
a figure:
 x = 0:5;
 y=
[0,20,60,68,77
,110]
 plot(x,y,'o')
 axis([-1,7,20,120])
 These
commands
produce a graph
with some
sample data
Continued….
 To activate the
curve-fitting
tools, select
Tools -> Basic
Fitting from the
menu bar in the
figure
 The basic
fitting window
opens on top of
the plot
 By checking
linear , cubic ,
and show
equation, plot is
generated
Continued….
 Checking the
plot residuals
box generates a
second plot,
showing how
far each data
point is from
the calculated
line
Continued….
 In the lower
right-hand
corner of the
basic fitting
window is an
arrow button.
 Selecting that
button twice
opens the rest
of the
window
Continued….
 The center panel
of the window
shows the results
of the curve fit
and offers the
option of saving
those results into
the workspace
 The right-hand
panel allows to
select x -values
and calculate y values based on
the equation
displayed in the
center panel
Continued….
 In addition to the basic
fitting window, the data
statistics window can be
accessed from the figure
menu bar
 Select Tools -> Data
Statistics from the
figure window
 The data statistics
window allows to
calculate statistical
functions such as the
mean and standard
deviation interactively,
based on the data in the
figure, and allows to
save the results to the
workspace
Curve-Fitting Toolbox
 In addition to the basic fitting utility, MATLAB contains
toolboxes to help to perform specialized statistical and
data-fitting operations
 In particular, the curve-fitting toolbox contains a graphical
user interface (GUI) that allows to fit curves with more
than just polynomials
 The curve-fitting toolbox must be installed in copy of
MATLAB before execution the examples that follow
Example
 The data we’ve
 x = 0:5;
 y = [0,20,60,68,77,110];
 To open the curve-fitting
toolbox, type
 cftool
 This launches the curve-
fitting tool window
 Now tell the curve-fitting
tool what data to use
 Select the data button, which
will open a data window
 The data window has access
to the workspace and will let
you select an independent
(x) and dependent (y)
variable from a drop-down
list
Example Continued….
 Choose x and y, respectively, from the drop-down lists
 A data-set name can be assigned, or MATLAB will assign one
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automatically
Once chosen variables have been chosen, MATLAB plots the
data
At this point, the data window can be closed
Going back to the curve-fitting tool window, now select the
Fitting button that offers you choices of fitting algorithms
Select New fit , and select a fit type from the Type of fit list
Chose an interpolated scheme that forces the plot through all
the points, and a third-order polynomial
Numerical Integration
 An integral is often thought of as the area under a curve
 The area under the curve can be found by dividing the area into
rectangles and then summing the contributions from all the
rectangles:
 𝐴=
𝑛−1
𝑖=1
𝑥𝑖+1 − 𝑥𝑖 𝑦𝑖+1 − 𝑦𝑖 /2
 The MATLAB commands to calculate this area are
 avg_y = y(1:5)+diff(y)/2;
 sum(diff(x).*avg_y)
 This is called the trapezoid rule, since the rectangles have the same
area as a trapezoid drawn between adjacent elements
 MATLAB includes a built-in function, trapz , which gives the same
result, and which uses the syntax
 trapz(x,y)
Continued….
 We can approximate the area under a curve defined by a function instead of
data by creating a set of ordered x–y pairs
 Better approximations are found as we increase the number of elements in
our x and y vectors
 For example, to find the area under the function
 y = f(x) = 𝑥 2
 from 0 to 1, we would define a vector of 11 x -values and calculate the
corresponding y -values:
 x = 0:0.1:1;
 y = x.^2;
 The calculated values are used to find the area under the curve:
 trapz(x,y)
 This result gives us an approximation of the area under the function:
 ans= 0.3350
 The preceding answer corresponds to an approximation of the integral from
x = 0 to x = 1, or

1 2
𝑥 𝑑𝑥
0
Continued….
 MATLAB includes two built-in functions, quad and quadl ,
which will calculate the integral of a function without requiring
the user to specify how the rectangles (shown in previous slide)
are defined
 The two functions differ in the numerical technique used
 Functions with singularities may be solved with one approach
or the other, depending on the situation
 The quad function uses adaptive Simpson quadrature:
 quad('x.^2',0,1)
 ans = 0.3333
 The quadl function uses adaptive Lobatto quadrature:
 quadl('x.^2',0,1)
 ans =0.3333
Continued….
 Both functions require the user to enter a function in the
first field. This function can be called out explicitly as a
character string or can be defined in an M-file or as an
anonymous function
 The last two fields in the function definethe limits of
integration, in this case from 0 to 1
 Both techniques aim at returning results within an error of
1 × 10−6