Programming Homework Help

Lab 2 MSc: Time Series Prediction with GP

You need to implement one program that solves Exercises 1-3 using any programming language. In Exercise 5, you will run a set of experiments and describe the result using plots and a short discussion.

(In the following, replace abc123 with your username.) You need to submit one zip file with the name niso3-abc123.zip. The zip file should contain one directory named niso3-abc123 containing the following files:

the source code for your program
a Dockerfile (see the appendix for instructions) a PDF file for Exercises 4 and 5

1

In this lab, we will do a simple form of time series prediction. We assume that we are given some historical data, (e.g. bitcoin prices for each day over a year), and need to predict the next value in the time series (e.g., tomorrow’s bitcoin value).

We formulate the problem as a regression problem. The training data consists of a set of m input vectors X = (x(0), . . . , x(m−1)) representing historical data, and a set of m output values Y=(x(0),…,x(m−1)),whereforeach0≤j≤m−1,x(j) ∈Rn andy(j) ∈R. Wewillusegenetic programming to evolve a prediction model f : Rn → R, such that f(x(j)) ≈ y(j).

Candidate solutions, i.e. programs, will be represented as expressions, where each expression eval- uates to a value, which is considered the output of the program. When evaluating an expression, we assume that we are given a current input vector x = (x0 , . . . , xn−1 ) ∈ Rn . Expressions and eval- uations are defined recursively. Any floating number is an expression which evaluates to the value of the number. If e1, e2, e3, and e4 are expressions which evaluate to v1, v2, v3 and v4 respectively, then the following are also expressions

• (add e1 e2) is addition which evaluates to v1 + v2, e.g. (add 1 2)≡ 3
• (sub e1 e2) is subtraction which evaluates to v1 − v2, e.g. (sub 2 1)≡ 1
• (mul e1 e2) is multiplication which evaluates to v1v2, e.g. (mul 2 1)≡ 2
• (div e1 e2) is division which evaluates to v1/v2 if v2 ̸= 0 and 0 otherwise, e.g., (div 4 2)≡ 2, and (div 4 0)≡ 0,
• (pow e e ) is power which evaluates to vv2 , e.g., (pow 2 3)≡ 8 121
• (sqrt e1) is the square root which evaluates to √v1, e.g.(sqrt 4)≡ 2
• (log e1) is the logarithm base 2 which evaluates to log(v1), e.g. (log 8)≡ 3
• (exp e1) is the exponential function which evaluates to ev1 , e.g. (exp 2)≡ e2 ≈ 7.39
• (max e1 e2 ) is the maximum which evaluates to max(v1 , v2 ), e.g., (max 1 2)≡ 2
• (ifleq e1 e2 e3 e4) is a branching statement which evaluates to v3 if v1 ≤ v2, otherwise the expression evaluates to v4 e.g. (ifleq 1 2 3 4)≡ 3 and (ifleq 2 1 3 4)≡ 4
• (data e1) is the j-th element xj of the input, where j ≡ |⌊v1⌋| mod n.
• (diff e1 e2) is the difference xk −xl where k ≡ |⌊v1⌋| mod n and l ≡ |⌊v2⌋| mod n
• (avg e1 e2) is the average 1 max(k,l)−1 xt where k ≡ |⌊v1⌋| mod n and l ≡ |⌊v2⌋| |k−l| t=min(k,l)

mod n
In all cases where the mathematical value of an expression is undefined or not a real number (e.g.,

√
−1, 1/0 or (avg 1 1)), the expression should evaluate to 0.

We can build large expressions from the recursive definitions. For example, the expression

                             (add (mul 2 3) (log 4))

2

evaluates to

2·3+log(4) = 6+2 = 8.
To evaluate the fitness of an expression e on a training data (X,Y) of size m, we use the mean

square error

f(e)=m

j=0

m−1
1 (j) (j) 2

y −e(x ) ,
where e(x(j)) is the value of the expression e when evaluated on the input vector x(j).

3

Exercise 1.

Implement a routine to parse and evaluate expressions. You can assume that the input describes a syntactically correct expression. Hint: Make use of a library for parsing s-expressions1, and ensure that you evaluate expressions exactly as specified on page 2.

Input arguments:

-expr an expression
-n the dimension of the input vector n -x the input vector

Output:
the value of the expression

Example:

   [pkl@phi ocamlec]$ niso_lab3 -question 1 -n 1 -x "1.0" 
           -expr "(mul (add 1 2) (log 8))"

   9.0
   [pkl@phi ocamlec]$ niso_lab3 -question 1 -n 2 -x "1.0 2.0" 

           -expr "(max (data 0) (data 1))"

2.0

Exercise 2. Implement a routine which computes the fitness of an expression given a training data set.

Input arguments:

• -expr an expression
• -n the dimension of the input vector
• -m the size of the training data (X,Y)
• -data the name of a file containing the training data in the form of m lines, where each line contains n + 1 values separated by tab characters. The first n elements in a line represents an input vector x, and the last element in a line represents the output value y.Output:
  The fitness of the expression, given the data.1See e.g. implementations here http://rosettacode.org/wiki/S-Expressions4

Exercise 3.
Design a genetic programming algorithm to do time series forecasting. You can use any genetic

operators and selection mechanism you find suitable. Input arguments:

• -lambda population size
• -n the dimension of the input vector
• -m the size of the training data (X,Y)
• -data the name of a file containing training data in the form of m lines, where each line contains n + 1 values separated by tab characters. The first n elements in a line represents an input vector x, and the last element in a line represents the output value y.
• -time budget the number of seconds to run the algorithm Output: The fittest expression found within the time budget.
• 
  
• Exercise 4.
• Describe your algorithm from Exercise 3 in the form of pseudo-code. The pseudo-code should besufficiently detailed to allow an exact re-implementation
• .Exercise 5.
  In this final task, you should try to determine parameter settings for your algorithm which lead toas fit expressions as possible.Your algorithm is likely to have several parameters, such as the population size, mutation rates, selection mechanism, and other mechanisms components, such as diversity mechanisms.Choose parameters which you think are essential for the behaviour of your algorithm. Run a set of experiments to determine the impact of these parameters on the solution quality. For each parameter setting, run 100 repetitions, and plot box plots of the fittest solution found within the time budget.5

A. Docker Howto

Follow these steps exactly to build, test, save, and submit your Docker image. Please replace abc123 in the text below with your username.

1. Install Docker CE on your machine from the following website:

https://www.docker.com/community-edition

2. Copy the PDF file from Exercises 4 and 5 all required source files, and/or bytecode to an empty directory named niso2-abc123 (where you replace abc123 with your username).

3. Create a text file Dockerfile file in the same directory, following the instructions below.

mkdir niso2-abc123 cd niso2-abc123/
cp ../exercise.pdf . cp ../abc123.py .

# Do not change the following line. It specifies the base image which # will be downloaded when you build your image.

FROM pklehre/niso2020-lab2-msc

• # Add all the files you need for your submission into the Docker image,
• # e.g. source code , Java bytecode , etc. In this example , we assume your
• # program is the Python code in the file abc123.py. For simplicity, we
• # copy the file to the /bin directory in the Docker image. You can add
• # multiple files if needed.ADD abc123.py /bin# Install all the software required to run your code. The Docker image # is derived from the Debian Linux distribution. You therefore need to # use the apt-get package manager to install software. You can install # e.g. java, python, ghc or whatever you need. You can also# compile your code if needed.
  # Note that Java and Python are already installed in the base image.# RUN apt-get update
  # RUN apt-get -y install python-numpy# The final line specifies your username and how to start your program. # Replace abc123 with your real username and python /bin/abc123.py
  # with what is required to start your program.CMD [“-username”, “abc123”, “-submission”, “python /bin/abc123.py”]

6

1. Build the Docker image as shown below. The base image pklehre/niso2019-lab3 will be downloaded from Docker Hubdocker build . -t niso2-abc123
2. Run the docker image to test that your program starts. A battery of test cases will be executed to check your solution.docker run niso2-abc123
3. Once you are happy with your solution, compress the directory containing the Dockerfile as a zip-file. The directory should contain the source code, the Dockerfile, and the PDF file for Exercise 4 and 5. The name of the zip-file should be niso2-abc123.zip (again, replace the abc123 with your username).Following the example above, the directory structure contained in the zip file should be as follows:

         niso2-abc123/exercise.pdf
         niso2-abc123/abc123.py
         niso2-abc123/Dockerfile
   

   Submissions which do not adhere to this directory structure will be rejected!

4. Submit the zip file niso2-abc123.zip on Canvas.