Students study problems arising from the
physical, biological, and/or social sciences and the algorithms and theory
used to solve them computationally. Included among the problems are
numerical methods for maximizing a function and solving a differential
equation. Prerequisite: MATH 130 and CSCI 150.
Learning Goals
By the end of this course, among other things you should be able to
visualize data from a wide variety of sources,
analyze data using exploratory data analysis and clustering techniques
build and solve system dynamics problems,
construct a Monte-Carlo simulation model,
understand and perform basic techniques in signal processing,
develop agent-based models for complex simulations,
approximate the roots of a continuous function by several techniques and understand the strengths and limitations of these techniques.
Software
Anaconda
Anaconda is the leading open data science platform powered by Python.
Jupyter
The Jupyter Notebook is a web application that allows you to create and share documents that contain live code, equations, visualizations and explanatory text.
SciPy
SciPy (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering.
Learn to project 3D map data from geographical information systems
into a 2D representation. Incorporate API data to visualize
patterns across state and county maps.
Learn to project 3D map data from geographical information systems
into a 2D representation. Incorporate API data to visualize
patterns across state and county maps.
The purpose of this project, worth 25% of your final grade, is to improve your research, writing and communication skills as well as give you an opportunity to explore in-depth a particular domain with your scientific computing skills.
Analysis of patterns, thresholds, and filters for
processing time-series signals.
Dec 5 Last Day
Dec 9 Final
Details
Disabilities
It is the policy of Hendrix College to accommodate students with disabilities, pursuant
to federal and state law. Students should contact Julie Brown in the Office of Academic
Success (505.2954; brownj@hendrix.edu) to begin the accommodation process. Any student
seeking accommodation in relation to a recognized disability should inform the instructor
at the beginning of the course.
After assignments are returned, you are welcome to revise and resubmit your work. Each submitted revision will be graded anew, the original and revised grades will be averaged to produce a new grade for that assignment. Revisions may be submitted anytime until the start of the final exam period.
Extensions
No late work will be accepted. Any work not submitted on time is a zero. However, you may submit a solution after the deadline to qualify under the revision policy. In effect, this means that late work can earn up to half credit.
Absences
You may miss three class days with no penalty. These can be for sports travel, school sanctioned
activities, sick, etc. Every subsequent absence will result in a 4% penalty on your final grade.
To refresh some Python skills, learn new Python modules, and practice using IPython,
our first assignment is to attempt
The Python Challenge!
Work you way through and complete levels 0-5. As you do, record all of your progress in
an IPython notebook, both successes and failures. Record your solutions and attempts in
Code blocks, and annotate what you did with MarkDown blocks.
You may work with one partner on this assignment.
We will complete level 0 together in class, to start you off with a 50%. All subsequent
solved levels with complete annotations will earn an additional 10%.
FEV1 is the amount of air that a subject can expel in one second.
This is an indicator used to diagnose Asthma and other pulmonary diseases. Linked above is
a dataset recording information about individuals and their FEV1 score.
Analyze this file using Jupyter and matplotlib, attempting to discover if there are any
relationships between the data fields. Specifically, you should
Draw histograms
Draw pie charts
Draw scatter plots
Divide the data into subcategories
Use Linear Regression to look for patterns with high correlation
Be sure to label your axes, title your figures, and write descriptions of your
findings in Markdown blocks. What can you conclude from exploring this data?
For this lab, you will be creating a Choropleth, incorporating the tools we learned in
class about data analysis and matplotlib.
I have linked in a number of possible sources for datasets above, but this is by no
means exhaustive, please feel free to use data from a topic you find relevant and
interesting.
There are many examples of Choropleths in the news, also linked above. You should similarly
strive to a visualization not seen before and that helps
explain a hypothesis and visualizes the supporting evidence. Make something awesome
that you are proud of and is facebook/reddit/hackernews worthy.
You will be working in pairs on this assignment.
Specifically, you should
Find a geospatial dataset and note its provenance (collection time, accuracy, method, etc).
Draw a map using this data to shade the regions with colors. This can be on a state, national, or county level.
Denote the steps you used so that your analysis is repeatable by others.
Write descriptions in your IPython document of your project for a general audience. This should be in the
style of a blog post,
intermixed with the relevant python code and map figures.
Give a 3 minute presentation to the class on Friday about your findings.
Doing well on the above will earn you a B on this lab.
For a 100, do something extra. For example, add in specific points as symbols on the map
with Circles or Text, draw a path across the map
connecting major important pieces, or animate the map over time by
assembling an animated gif. Be creative!
Specify in your writeup what takes your
lab beyond the required elements above.
This assignment is due at 6:00 pm on Thursday, September 8th in Moodle. This is to give
me time to assemble your documents for easy presentation in class on Friday, and to
allow you to practice your presentation.
For this lab, you will be analyzing text using the unsupervised learning techniques
of K-means clustering and Principal Component Analysis.
You will be working individually on this assignment.
Specifically, you should create an annotated IPython notebook that performs the following steps.
Load in the 50 books provided above from Project Gutenberg.
Calculate the 20 most frequent words in the whole corpus. Be sure to clean up the
data (standardize capitalization, remove punctuation, etc.). Regular
expressions might help.
Create a 20-dimensional vector for each book based on the normalized frequency
of the top 20 words for each book.
Cluster these books using K-means clustering, determining the best K by creating
an elbow chart for K from 2 to 10.
Make sure to note which books go into which clusters. You can find the title for each
book near the beginning of each text file. Do these clusters make sense
based on what you can find out about these books?
Perform Principal Component Analysis on the data vectors.
Plot the data using the first two principal components, and color the data according
to the clusters found above.
Examine the first two principal components, and determine which words were the most
important factors in these new transformed dimensions.
Successfully completing the above steps will earn you an 85 on this assignment.
For a 100%, complete the following
Download a book not listed in the top 50 from Project Gutenberg and determine its
cluster membership. Project this book into the new axes orientation produced by PCA and
plot it, noting its location.
Remove the Stop
Words from the corpus first, and compare your analysis.
This assignment is due at 4 pm on Wednesday, September 21st in Moodle.
Use Moodle to turn in your IPython notebook.
For this lab, we will investigate and model the accumulation of caffeine in the human body. From
Wikipedia:
Caffeine is the world's most widely consumed psychoactive substance, but unlike many other
psychoactive substances it is legal and unregulated in nearly all jurisdictions. In North
America, 90% of adults consume caffeine daily. The U.S. Food and Drug Administration
lists caffeine as a "Multiple Purpose Generally Recognized as Safe Food Substance."
We are interested in the differences between moderate and excessive amounts of caffeine usage in
the human body.
Research
To create your model, you need to research some facts about caffeine. At a minimum, you should
find
Average number of cups of coffee per day.
Amount of caffine in cup of coffee.
Absorption rate of caffeine into bloodstream.
Half-life of caffeine in system, breakdown due to metabolism.
Document each reference found with a proper citation. Go beyond wikipedia to find your sources.
Models
You will be creating four models for this lab.
moderate intake, one compartment
excessive intake, one compartment
moderate intake, two compartment
excessive intake, two compartment
The one compartment model assumes instantaneous absorption of caffeine, while the two compartment
model should use the absorption rate of caffeine into the bloodstream from the stomach that you found through your
research.
Run each of your models to simulate three days of caffeine intake.
Diagrams
Include diagrams to explain the relationships present within the one
compartment model and the two
compartment model. Be sure to include which differential equations affect the simulation
in various ways, so someone can follow the flow of caffeine into and out of the system
Analysis
Describe the caffeine levels present over time in each of the two types of coffee drinkers.
In your graphs, describe each axis and the information conveyed in the graph.
Is an average equilibrium reached in either situation? If so, what is this equilibrium?
What are the major differences (if any) between the results of the one vs. two compartment models?
Sensitivity
Discuss the sensitivity of your models to the parameters you found through research. Back up your
statements with numerical results based on altering the parameters of the original models.
Some researchers
are recommending 7 grams of dark chocolate per day to prevent heart disease.
Decide how to incorporate this into your models, and rerun. How does this
change the results of your models?
What else did we ignore in our models? Suggest one way we could enrich the model to improve
the accuracy of our results.
One way to generate fractal images
is to use a version of a Monte Carlo simulation
known as the Chaos Game.
Starting with a particular point, (x0,y0),
one randomly chooses between a number of functions that will transform
(xi,yi) into (xi+1,yi+1). These
iterated points quickly converge to an attractor that forms a fractal.
Game 1
Plot 40,000 iterations of the Chaos Game with a scatter plot using the following set of equations. Begin
with x0 = 0 and y0 = 0. Set the aspect ratio of the final figure to be equal.
Each function
should be chosen with equal probability.
Game 2
Plot 40,000 iterations of the Chaos Game with a scatter plot using the following set of equations. Begin
with x0 = 0 and y0 = 0. Set the aspect ratio of the final figure to be equal.
Each
function is chosen according to the probability distribution below.
Infinity Game
A casino offers a game of chance for a single player in which a fair coin is tossed at
each stage. The pot starts at 2 dollars and is doubled every time a head appears. The
first time a tail appears, the game ends and the player wins whatever is in the pot. Thus
the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears
on the first toss and a tail on the second, and so on. In short, the player wins 2k
dollars, where k equals number of tosses.
If we assume the casino can pay out any winning
amount, the expected value for this game is infinity. This is weird.
Explore different values for the
maximum payout from the casino using a Monte Carlo simulation. Plot the maximum payout
versus your calculated expected values, for payouts from 1 to 1,000,000.
What do you think would be a fair price to pay the casino for entering the game?
Integration
Write a general method mc_integrate(func, a, b)to integrate
a given function over a specified interval from a to b using
Monte Carlo integration.
Demonstrate your method by estimating the integral for the following three functions.
We will be recording accelerometer data in the x, y, and z direction from four activities.
Running
Walking
Jumping
Falling
Each team member should record each activity for 30 seconds, with 5 seconds of no movement before, and 5 seconds of no movement after. Since falling down will not take 30 seconds, only about 5 seconds, instead record 11 falls per team member.
Load up your saved csv files into Jupyter. (Yes, Ryan, you can use Pandas.) For each axis from each recording, create a Wave object. Then, make a segment to keep only the relevant 30 seconds from the data, so that you chop out the 5 seconds of still before and after the activity. Normalize your waves, and plot one for each activity.
Next, divide these segments up further into 5 second segments, overlapping by 2.5 seconds. This will make 11 segments for the Running, Walking, and Jumping activities. This is why each team member needed to fall down 11 times. You should now have 6 * 4 * 11 * 3 = 792 Wave segments.
For each segment, calculate a Spectrum. Use the peaks() method and save the top 30 frequencies, exempting the first frequency, which will be 0. Then, concatenate the x, y, and z found frequencies to make one 90 element vector for each 5 second instance of the above activities. This should result in a 264 x 90 matrix of data.
Now, perform PCA analysis and k-means clustering on this dataset. Do you see an elbow at 4 clusters? Plot the data colored by cluster on first two dimensions from PCA.
Label each cluster based on the majority class found in that cluster. Replot your data, and indicate the minority elements in each cluster with a different symbol. How well do the clusters of your data match the four original categories of activities?
This is an individual lab, you may discuss the high-level concepts with other students, but your code
should be your own.
Forest Fires
Complete the TODO items to answer the following questions for the Forest Fire Ipython document.
Explore forest density values between 35% and 65%, and plot this versus the average percent of initial trees burned over 20 iterations.
Questions to answer:
For the VonNeuman neighborhood, what density of forest does it take for a fire to spread?
Does this change for the Moore neighborhood?
Does treating the world as a bounded square versus a torus make a difference?
Add in decay of burnt trees, new growth, and lightning, and plot the distribution of the states over time.
Diffusion
Alter the Diffusion Ipython document,
such that the pollutant
from the factory can be affected by a breeze from West to East. This breeze is
controlled by a parameter between 0 and 1 called WINDSTRENGTH.
Termites
Implement an Agent-Based Model to model the collective behavior of
termites. In particular,
you will be modeling how they organize to build large structures without a leader.
The world of the termites will have two states, DIRT and EMPTY. Make the DIRT state
appear yellow, and the EMPTY state be black. Look to the Forest Fire model for
how to represent this.
Initialize your CA with a size of 40x40 and 200 pieces of dirt. In this model, the
CA will not need to update its state, only hold information for the termites about
the environment.
We will use three states to model our termite behavior, FORAGE, LOOK, and DROP.
When a termite is in the FORAGE state, they wander randomly around the world. If they
step on top of a cell that contains DIRT, they pick it up, making their current cell EMTPY,
and change to the LOOK state.
In the LOOK state, the termite is looking for another piece of DIRT. They will again wander
around the world randomly until they are standing on a cell with DIRT. When this happens,
they change their state to DROP.
Finally, a termite in the DROP state is now looking for an EMPTY cell in the world. They
wander randomly until they find this EMPTY cell, then put their DIRT into the cell.
They then take 5 random steps, and finally change to the FORAGE state. This last piece
helps them not to immediately pick up the piece of DIRT they just dropped.
Create 50 termites, and place them randomly in the world to start.
Run your model for 1000 iterations. Describe what you see in terms of
the formation of piles of dirt.
An oil spill has fouled 200 miles of Pacific shoreline. The oil company responsible has
been given 13 days to clean up the shoreline, after which a fine will be levied in the
amount of $8,000/day. The local cleanup crew can scrub five miles of beach per week at
a cost of $600/day. Additional crews can be brought in at a fixed cost of $14,000 plus an extra $800/day
for each crew.
How many additional crews should be brought in to minimize cost? What is total cost?
How long will the cleanup take?
TSP Extensions
Create a Tour with 80 random points.
Draw a plot of the cost of a tour over time using gradient descent.
Draw a plot of the cost of a tour over time using simulated annealing.
Draw a scatter plot of the probability of accepting a bad move over time using simulated annealing.
Extend the Tour object to include an "interchange" function. Similar to swap,
it will bring in an ith and jth city, and reverse the order of all cities
in between them, including the ith and jth city.
Rerun your simulated annealing algorithm and gradient descent algorithm above, using
the interchange neighborhood function. How do the results compare?
GA Extensions
Implement one of the crossover techniques below and compare your results to the
method presented in class.
You are planning a journey for a robotic probe to survey all of the
closest stars.
Your probe will start at Earth, visit each star once, and return home.
Convert the distance, right ascension and declination for each
star to cartesian coordinates, and plan
out a close-to-optimal journey using the techniques we have learned in class.
Assume the probe can travel at half the speed of light.
State any other assumptions you need to make to solve this problem. Draw a
3D plot of your resulting journey, and show how long this journey will take.
