MATLAB QUESTIONS

Description

Lecture 11 – Interpolation, Curve Fitting, and Extrapolation
Step 1 – In your MATLAB Workspace, load Tesla stock price data using this command
load TSLA_stock_history_Mar_2022_2023.mat
NOTE 1: Variable “Stock_Date” is the number days since 7 March 2022
NOTE 2: Date numbers don’t increment uniformly – market isn’t open every day
NOTE 3: Variable “Stock_Price” is the rounded closing stock price
Step 2 – In the MATLAB “Curve Fitter” app, use the “Select Data” button and load in “Stock_Date” as the “X value” and “Stock_Price” as the “Y value”
Step 3 – Make “FIT TYPE” (see figure immediately below) “Interpolant”
Step 4 – “Export” by selecting “Generate Code” (as show in figure above on right side) to create a function called “createFit” that you paste into your script and call using this line:
[fitresult, gof] = createFit(Stock_Dates, Stock_Price)
a. Interpolation: Compare and show “fitresult” and “Stock_Price” values for days 1, 100, and 361 are identical – goal is to show interpolation result passes through the values being interpolated. Show that you can compute a price for day numbers 275.5 and 351.1 – the goal is to show you can now compute values at any date. Hint – use command “fitresult([1 100 361])” to get the fitresult values where the numerical values are the days, not indices
[deliverables: table of “fitresult” values, “Stock_Price” values, and 2 sentences explaining the different prices]
b. Curve Fit – Use “polyfit” command to generate polynomial fits to the data for N = 3, 5, & 7 (type “help polyfit” in the command window for information for what N means) for the same 5 dates provided in part a. Here is an example line of code where N=7:
p7 = polyfit(Stock_Dates, Stock_Price, 7)
price_p7 = polyval(p7,[1 100 361])
[deliverables: table of “Stock_Price” values versus the 3 polynomial fits (as shown below) and 2 sentences explaining the different prices].
c. Extrapolate – compare part a. interpolation vs part b. polyfit solutions for these dates: 380, 400, & 410.
[deliverables: plot all Stock_Price, interpolation, and 3 polynomial curve fits (days 1 through 361) plus the 12 total extrapolation prices as markers, ensure plot has a legend]
NOTE: your actual extrapolation values are not graded, you are only graded on your ability to extrapolate nor will these values be taken seriously to buy or sell stock ?

2. Lecture 12 – Singular Value Decomposition (SVD)
GOAL: Your client wants you to create an algorithm to compress image data to reduce mission data transmission time for a new satellite design. They want you to only keep 90% of the cumulative sum of singular values – meaning keep largest singular values starting with the largest and only keep those that add up to 90% of the total (see plot below).
Use this MATLAB stock image to test your approach and these commands to read image:
X = imread(‘ngc6543a.jpg’);
X = double(rgb2gray(X));
If you want more info on the image, see: Cat’s Eye Nebula – Wikipedia
a. Calculate the SVD of the image using the ‘econ’ option.
Plot the singular values (similar to how we did in the
“Lecture_12_SVD_face_demo.mlx” code.
[deliverables: plot of singular values]
b. Calculate the cumulative sum of the singular values
(your plot should look similar to example plot on right which came from
the clown image we discussed in class).
Determine the index (# of singular values) of the cumulative sum when
the percentage of cumulative sum of singular values is 90%.
[deliverables: plot of percentage of cumulative sum of
singular values & report the index]
c. Using the “whos” command, calculate the compression
ratio of original image to compressed image sizes in bytes (original divided by compressed)
[deliverables: show calculations & report your final compression ratio (bytes:bytes) should be greater than 3]

3. Lecture 13 – Fast Fourier Transform (FFT)
Load “chime2.mat” (it’s on CANVAS)
“N” is the number of samples
“audio_recording_data” is the not so good audio recording of the chime
“dt” is the time between samples
“t” is the time array
For example, you can plot or hear the recording:
plot(t,audio_recording_data)
sound(audio_recording_data)
a.
Compute the FFT of “audio_recording_data” using the “fft” command, compute the PSD, and plot the Power Spectral Density (PSD) (like how it was done in class)
[deliverables: plot with labels and legend. X axis must in units of Hz. Like how it was done in class]
b. 
Report the first 6 frequencies in Hz that have a PSD value above 1E6
[deliverables: Table of frequencies in Hz]
c. 
Denoise the data by the following (see class notes and code):
(1) Magnitude filter – only keeping PSD values above 1E6
(2) Frequency filter – only keep PSD values below 2000 Hz
[deliverables: plot PSD of denoised data – plot should look similar to the one below]
d. 
Compute inverse FFT using the “ifft” command and plot the time domain signal from 0 to just under 2 seconds
[deliverables: plot time domain denoised data]Eigenanalysis and
Singular Value Decomposition (SVD)
1
Overview
• If we think of matrices a set of column vectors, can we figure out
the most important vectors (directions & magnitudes) and
disregard the rest?
• We will study 2 different approaches to computing the MOST
important characteristics (vectors) from our data:
(1) Eigenanalysis (only applies to square matrices)
(2) Singular Value Decomposition (SVD) (applies to any matrix)
Remember to take the daily quiz
Intro
We learned to think about matrices as a collections of vectors,
Now – “which vectors provide us the most important information about a matrix?”
• For example, what is the most important vector in this rank 1 matrix?
1 1 1
= 1 1 1
1 1 1
• Eigenanalysis (use “eig” command in MATLAB)
, = eig
0.41
= 0.41
−0.81
0.71
0.71
0
0.5774
0.5774
0.5774
0
& = 0
0
1
Ԧ1 = 1
1
0.5774
ො3 = 0.5774
0.5774
0 0
0 0
0 3
• Singular Value Decomposition (SVD) (use “svd” command in MATLAB)
, , = svd
−0.5774
= −0.5774
−0.5774
0.82
−0.41
−0.41
3 0
0.0
−0.71 & = 0 0
0 0
0.71
0
−0.5774
0 = −0.5774
0
−0.5774
0.82
−0.41
−0.41
0.0
−0.71
0.71
Intro
• Next example, what is the most important vector in this rank 2 matrix?
1
= 1
0
0.1 1.1
−0.1 0.9
0
0
• Eigenanalysis (use “eig” command in MATLAB)
, = eig
0.7641
= 0.6451
0
−0.0841
0.9965
0
0.1
Ԧ 2 = −0.1
0
Ԧ2 =
−0.0841
0.9965
0
1.0844
& =
0
0
−0.5774
−0.5774
0.5774
Think “direction” when thinking about eigenvectors
1
Ԧ1 = 1
0
0.7641
Ԧ1 = 0.6451
0
0
−0.1084
0
0
0
0
Think “magnitude” when thinking about eigenvalues
• Singular Value Decomposition (SVD) (use “svd” command in MATLAB)
, , = svd
−0.7418 −0.6706
0
2.0025
= −0.6706 0.7418
& =
0
0
0
0
1.00
0
0
0.173
0
0
−0.7053
0 = −0.0036
0
−0.7089
Note – sign flips are expected and not an issue
0.4113
−0.8165
−0.4052
−0.5774
−0.5774
0.5774
Eigenanalysis
• The word “eigen” is adopted from the German word that means
“characteristic” or “proper” [1]
• Originally eigenanalysis was used to study principle axes of rotation matrices
and inertia matrices
• Now, eigenanalysis is also used for stability, vibration, atomic orbitals, facial
recognition, matrix diagonalization, and other analyses/functions [1]
• Applies only to square matrices
• Equations:
Ԧ = Ԧ
where is a × matrix, Ԧ is a vector, and is a scalar
Also stated equivalently as
− Ԧ = 0ത
where is the × identity matrix
Eigenanalysis
• The “characteristic” polynomial has degree and can be factored in terms of
− = 1 − 2 − … −
• Where the “eigenvalues” 1 , 2 , … , are the “roots” of the polynomial
• The eigenvectors Ԧ corresponding to each eigenvalue can be computed from the
components of
− Ԧ = 0ത
• Using the “eig” command, MATLAB provides unit normalized eigenvectors in matrix
with corresponding eigenvalues in a diagonal matrix
, = eig
Where the first column of is ො1 which corresponds with the first eigenvalue 1 in on its
diagonal – see example below from before
0.7641
= 0.6451
0
−0.0841
0.9965
0
, = eig
1.0844
−0.5774
& =
0
−0.5774
0
0.5774
0
−0.1084
0
0
0
0
Eigenanalysis
• The matrix A can now be
represented as = −1
• One powerful application for
eigenanalysis is data reduction or
reduced order modeling
• We can use the most important
eigenpairs (values and vectors) to
represent the matrix
• In this case we take a rank 3 matrix
and represent it with a rank 1 matrix

≈ ො1 1 ො1−1

Very
similar
8
Exercises
See file “Lecture12_examples.mlx” on CANVAS
(1) Compute the eigenvalues and corresponding eigenvectors for matrix , , and using
the eig command. For example,
, = eig
(2) Create a rank 1 approximation for matrices A and B
(3) Create a rank 2 approximation for matrix C
(4) What is the norm of the difference between the original and low-rank approximations?
Singular Value Decomposition (SVD)
• You can think of SVD as a generalized version of the
eigendecompsition because it doesn’t require rectangular
matrices
• SVD is useful for:
– Low-rank approximations to matrices
– Pseudo-inverses of non-square matrices — least-squares solutions
– Principal components analysis (PCA)
• Reduced order data sets or models allow us to handle large
complex systems & data sets such as audio, image, or video
• We want to extract the dominant patterns or behaviors
• “Provides a systematic way to determine the dominant patterns
underlying a high-dimensional system” → provide a “low rank”
approximation to high-dimensional data [1]
Singular Value Decomposition (SVD)
• Say we want to analyze a large data matrix that has millions of rows and
thousands of columns
• For example, the columns of could be measurements collected from multiple
experiments or complete audio files or images
• Using SVD, this large rectangular matrix can expressed as = Σ ∗
– is any × matrix (typically ≫ meaning a tall skinny matrix)
– ∈ ℂ × and ∈ ℂ × are unitary matrices. “*” is the complex-conjugate
transpose. Columns of are called the “left singular” vectors and columns of are
the “right singular” vectors
– Σ is a diagonal matrix with non-negative singular values on the diagonal in rank order
• SVD example from earlier (use “svd” command in MATLAB): , , = svd
3 0 0
1
0.1 1.1
−0.5774 0.82
0.0
−0.5774 0.82
0.0
= −0.5774 −0.41 −0.71 & = 0 0 0 = −0.5774 −0.41 −0.71 where = 1 −0.1 0.9
0 0 0
0
0
0
−0.5774 −0.41 0.71
−0.5774 −0.41 0.71
Singular Value Decomposition (SVD)
The matrix X can be represented as the full or reduced SVD without loss
Fig from [2]
Matrix X can be compressed further by truncating and only keeping the largest
singular values
Fig from [2]
Example — Singular Value Decomposition (SVD)
• Think of a digital photo as collection of
pixels and their respective values
• From a SVD of this photo, we can rebuild
it starting from the highest going to
lowest singular values
• Original image 200 x 300 pixels & has a
rank of 200
• Low-rank approximations is one way to
compress the image data
• The magnitude of each singular value is
shown at in bottom 2 plots – we can
determine the best place to truncate to
get a good low-rank approximation
Example
– keep
first 50
Example
– keep
first 50
14
Exercises
(1)
Compute the singular values and unitary matrices for
matrix , , and using the svd command For example,
, , = svd
(2)
Create a rank 1 approximation for matrices A and B
(3)
Create a rank 2 approximation for matrix C
(4)
What is the norm of the difference between the original
and low-rank approximations?
Figure from [2]
Conclusions
• As engineers, we analyze large matrices that can have thousands
of columns and many hundreds of thousands of rows
• In this lecture, we studied 2 different approaches to computing
the MOST important characteristics from our data:
– Eigenanalysis
• PROS: produces eigenvectors and eigenvalues so we can focus on
the most important vectors
• CONS: applies to square matrices only
– Singular Value Decomposition (SVD)
• PROS: Applies to any matrix, we can focus on the most important
vectors, we can easily create low-rank approximations, typically
cutting out the noise
Interpolation, numerical
derivatives, and other topics
Don’t forget to open CANVAS during class and take the daily quiz
1
Overview
• Typical challenges or issues with measured data: noise,
discretization, incomplete, incorrect, inconsistent
• Interpolation – we will discuss 2 types
(1) Single polynomial through entire dataset
(2) Piecewise – linear and cubic splines
• Curve Fitting (a.k.a. linear regression or least-squares)
• Extrapolation (make predictions outside of measured range)
• How do we typically compute numerical derivatives?
Noisy Data
• Almost all
measurements have
noise
• Most of the time we
need to filter the data,
especially before trying
to compute its
derivative
• You can see the
amplitude of the
derivative of the noisy
data (blue) quickly
exceeds the derivative
of the cosine function
(both were computed
numerically)
Cosine function with random
noise and without
Numerical derivatives of
cosine function with random
noise and without (red line)
Discrete Data
Continuous data can be divided into smaller and smaller
units (i.e. non-divisible)
Examples: instantaneous time, temperature, velocity
Typically plot functions as lines
• Discrete data is counted and can
not be divided into smaller units
(example – # of students in
classroom, continuous data that
we sample, like 1x per second)
– Typically plot discrete data as
markers
– People tend to connect markers,
but this may not represent
interpolated data
Stairstep graph – think
“sample and hold”
Incomplete, incorrect, & inconsistent Data
• Incomplete data typically has
gaps, such there is a period of
time where the data was
collected or data is missing
between samples
• Incorrect data commonly shows
up when the data is being
transmitted
• Inconsistent data have variable
sampling periods and you may
need to resample the data to
ensure a fixed sampling period
Resample uniform or nonuniform data to
new fixed rate – MATLAB resample
(mathworks.com)
5
Interpolating Data
Interpolation produces estimates between
known observations by developing functions
that cross through a discrete set of known
data points and give us values in between the
known data
– Typical choices for spline interpolation:
We know the red dots, but we want a
function that gives us values in between
the known values
– Linear – straight lines between data points
– PROS: easy, continuous
– CONS: discontinuous derivatives, unrealistic
– Quadratic – 2nd order polynomials between data points
– PROS: continuous, smooth
– CONS: path doesn’t account for final velocity
– Cubic – 3rd order polynomials between data points
– PROS: continuous, most realistic
– CONS: continuous derivatives
6
MATLAB Example Linear Interpolation
– Using the interp1
command, we can
create 10 linear
functions that pass
through the 11
known values (see
circles) but provide
values at a 100x
smaller increment
size — see green
line
MATLAB Example Cubic Interpolation
– Using the interp1
command, we can
create 10 cubic
functions that pass
through the 11
known values (see
circles) but provide
values at a 100x
smaller increment
size — see green
line
Curve Fitting
– Two lectures ago, we looked at
the polyfit command — here
we will use MATLAB’s fit
command
– “Coefficient of determination,
in statistics, R2 (or r2), a
measure that assesses the
ability of a model to predict or
explain an outcome in the
linear regression setting.”
– Normalized between 0 to 1
where 1 means the function(s)
pass through all the measured
data
[F_poly1,gof]= fit(x,y,’poly1′)
[F_poly2,gof] = fit(x,y,’poly2′)
plot(F_poly1,’r-‘);
plot(F_poly2,’b-‘);
legend(‘data’,’1st order’,’2nd order’);
Curve Fitting App
(1) Get the Curve Fitter App
(2) Code in Command window or mfile:
x = linspace(0.1,10,100)’;
noise = (2*randn(size(x)));
y = exp((x – 0.5)/2)/10 + noise;
(3) “Select Data” x and y
(4) Select Fit type – try various types
and study R-square and RMSE
(5) Try to set exclusion rules
(6) Export Fit
Extrapolation
– Estimate of the value of
some function outside
the range of known
values
– “Extrapolation is a type of
estimation, beyond the
original observation
range” [1]
– Extrapolation is subject to
greater uncertainty and a
higher risk of producing
meaningless results [1]
See “Extrapolation” in AEE_3150_Lecture_11_examples.mlx
Methods to Differentiate Noisy Data
• If measured data is believed to be low noise, one can compute
interpolation functions and then compute derivatives – pick
interpolation functions that have continuous derivatives
• If data is noisy, then a curve fit may be the best initial step
followed by differentiating the curve fit
• We will study a different method in the upcoming Fast Fourier
Transform (FFT) lecture data that can be represented with sines
and cosines
Exercises
– There are 4 data sets in the file
called “Lecture11_data.mat” with
the following names:
– “x1” “y1_validation” &
“y1_noisy”
– Each data set is a matched set –
meaning x2 goes with y2
– Use the Curve Fitting App to curve
fit, interpolate, and extrapolate
– Share your R-square and Root
Mean Square Error (RMSE) for
each data set
For each plot, the red dots are
the “noisy” data, and the blue
lines is the “validation” data
13
Exercise — for each data set,
perform these actions
Data Set 1: curve fit the noisy data to estimate the amplitude and frequency
1 = cos 2 ∙ 1
Data Set 2: What is the best curve fit function you can find – how does it compare to
the validation data and function:
1
2 =
1 + 22
Data Set 3: Compare your best curve fit to extrapolate values for x values between
10 and 15. Compare your values to the function:
3 −0.5
3 = 0.1 ∙ 2
Data Set 4: Compare your slope and intercept estimate to the original function:
4 = 0.867 4 − 2.022
14
Conclusions
• There are many typical challenges or issues with measured data, such as
noise, discretization effects, incomplete, incorrect, and/or inconsistent data
• We studied 3 approaches to model data sets:
(1) Interpolation – we are passing functions through known data points using piecewise
linear and cubic splines
(2) Curve Fitting (a.k.a. linear regression or least-squares) – is a great approach for
approximating noisy data with 1 smooth function
(3) Extrapolation (make predictions outside of measured range)
• We used the “Curve Fitting App” – this app is very useful because you can
quickly produce interpolate data, create curve fits, and extrapolate data
• How do we typically compute numerical derivatives?
Fourier Series and the
Fast Fourier Transform (FFT)
1
Overview
• Taylor Series approximates use polynomials [1]

+

1!
− +
′′
2!

2
+
′′′
3!

3
+⋯
• Taylor Series does not work well on sinusoids
because you need infinite terms, approximating
sinusoids with sine and cosine functions works
really well → Fourier Series [2]


=0

2
2
~ 0 + ෍ cos
+ ෍ sin

=0
( )

!
Taylor Series Approximation of sine
wave. Fig from [3]
=0
• Fast Fourier Transform (FFT) computes coefficients
for large data sets very quickly and the inverse FFT
(iFFT) allows you to transform back
• We only look at discrete data and 2 different forms
of Fourier Coefficients: real- and complex-valued
Fourier Series
Approximation
of square
wave. Fig
from [4]
FFT History
• The Fourier Series is named in
honor of Jean-Baptiste Joeseph
Fourier (1768–1830) [2]
• Gauss originally invented FTT in
1805 but it wasn’t published or
recognized until later [2]
• Cooley and Tukey created 1965
and get the credit for
implementation [2]
Example
seismometer -mass moves up and
down with the
motion of Earth’s
crust
Fig from [2]

Example seismometer data
Fig from [1]
Watch Intro: 0-33 sec
Watch description: 7:26 through 10:20
Fourier Series
• Imagine we have collected discrete samples
−1
1
0
0
0 1 2

−1

0
(could be spatial, temporal, or both)
• Approximate signal with a truncated series of sines and cosines

2
2
≅ 0 + ෍ cos
+ ෍ sin

=1

1
0 = ෍ −1

=1
“The average”
ℎ ℎ #
=1

2

= ෍ −1 cos

=1

2

= ෍ −1 sin

=1
“Magnitude of each cosine term” “Magnitude of each sine term”
• Example file: “AEE_3150_Lecture_13_Fourier_Series_Example.mlx”
Fourier Series – MATLAB Example
(we are only discussing key lines of code, not plotting and tables)
Step 1) Create delta X – what size do we want to cut L up into?
dx = 0.005;
Step 2) What is the length of one period (time or space)?
L = 1; % could be time or a spatial dimension that defines the units
Step 3) How do we create a “x” array over the length L? What is N?
x = dx:dx:L;
Step 4) Create the function – what are the magnitudes and frequencies?
f = 1*sin(2*pi*x) + 0.5*sin(4*2*pi*x)+ 0.25*sin(8*2*pi*x);
Step 5) Create variable for the Fourier Series (FS)
f_FS = zeros(size(x));
Step 6) Calculate the A0 “average” term for the FS
A0 = (1/L)*sum(f.*ones(size(x)))*dx;

1
0 = ෍ −1

Step 7) Create new figure and plot A0 term (average)
plot(x,A0*ones(size(x)),’k-‘,’Color’,[0.94 0.86 0.86]); hold on;
=1
Fourier Series (FS) – MATLAB Example
(we are only discussing key lines of code, not plotting and tables)
Step 8) How many FS coefficient terms do we want?
T = 10;
Step 9) Loop over FS coefficient terms – show how approximation improves
for term_ctr = 1 : T
Step 10) add in the first term – “the average”
f_FS = A0;
Step 11) Loop over K to calculate
for K = 1 : term_ctr

=
2

෍ −1 cos

=1
Step 12) calculate Ak coefficient
Ak(K) = (2/L)*sum(f.*cos(2*pi*K*x/L))*dx;

Step 13) calculate Bk coefficient
2

= ෍ −1 sin

Bk(K) = (2/L)*sum(f.*sin(2*pi*K*x/L))*dx;

=1
Step 14) calculate the FS time/spatial domain approximation
f_FS = f_FS + Ak(K)*cos(2*K*pi*x/L) + Bk(K)*sin(2*K*pi*x/L);

=1
=1
2
2
≅ 0 + ෍ cos
+ ෍ sin

Fourier Series – MATLAB Example
2 term FS
approximation
Original signal is
shown in black
1 term
approximation
is the average
3 term FS
approximation
4 term FS
approximation
Tabular Form
A0 term
B1 term
B4 term
B8 term
Exercise
Fast Fourier Transform (FFT)
• Imagine we have collected discrete samples
0

−1
−1
1
0
0
0 1 2

DFT
Data points in
temporal and/or
spatial domain

−1
መ0

መ −1
The “hat”
indicates
Fourier
coefficients in
the complex
frequency
domain
• Write Series using Euler’s Formula: = cos + sin
−1
• Discrete Fourier Transform (DFT) መ = ෍ −2 Τ
=0
There are 2 values
changing here: &
= −2 Τ where & will vary from 0 to − 1 in the
summation above
FFT
−1
መ = ෍ −2 Τ
= −2 Τ
=0

=0

መ0
1
=

መ −1
= −1
Fourier
coefficients
All complex
valued (magnitude
and phase)
1
1



1
1
1
1 1
1
2
3
−1

0
2 4





−1



2
−1 ⋯ ⋯ ⋯ −1
Column
all 1s
because
n=0
Row all 1s
because K=0
Data points in
temporal and/or
spatial domain
DFT matrix – it is a very large N by N matrix.
You never compute Fourier coefficients with
this matrix in this form – reorganized version
of this matrix is the FFT
Step 1) create dt
Step 2) create a
time vector
Step 3) Create the
clean discrete
signal
Step 4) Add noise
to the clean signal
Step 5) plot
Questions:
Is there a signal
hidden in the
noise?
Can we extract the
signal?
11
Step 6) N is number of samples
Step 7) Compute FFT of clean signal
Step 8) Compute Power Spectral
Density (PSD) of clean signal
Step 10) Create the x-axis
Step 11) Compute FFT of noisy
signal
Step 12) Compute PSD of noisy
signal
Step 13) Create the x-axis
Step 14) Create an index to only plot
½ of the FFT
QUESTIONS:
(1) What is the difference between
the clean and noisy PSDs?
(2) What are the 2 spikes in the
clean PSD?
(3) Can you filter the noise from the
noisy PSD?
Magnitude Filter
12
Step 15) find indices where
PSD is above the cutoff
Step 16) Create new PSD for
signal above cutoff
Step 17) Apply cutoff to the
FFT coefficients
Step 18) Inverse FFT to
create a denoised signal
Step 19) plot the denoised
signal
Step 20) plot the PSD of the
denoised signal
13
Exercise
See file: ‘Lec13_Noisy_Signal_Student.mat’
QUESTIONS:
(1) Is there a
hidden signal?
(2) What are its
frequencies and
magnitudes?
Conclusions
• We learned that approximating signals with sine and cosine
functions works really well with the Fourier Series
• We learned how the Fast Fourier Transform (FFT) computes
coefficients for large data sets very quickly and the inverse FFT
(iFFT) allows you to transform back
• We looked two different forms of the Fourier Series: real- and
complex-valued (both work well)

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