NB! This coursework is only compulsory for MSc students taking the 20cr module. We released a different Lab 2 with an earlier deadline for UG students taking the 20cr module.
You need to implement one program that solves Exercises 13 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 niso3abc123.zip. The zip file should contain one directory named niso3abc123 containing the following files:
ˆ the source code for your program
ˆ a Dockerfile (see the appendix for instructions)
ˆ a PDF file for Exercises 4 and 5
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)}), where for each 0 ≤ j ≤ m − 1, x^{(}^{j}^{)} ∈ Rn and y^{(}^{j}^{)} ∈ R. We will use genetic 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 = (x_{0}, . . . , x_{n−}_{1}) Rn. Expressions and eval uations are defined recursively. Any floating number is an expression which evaluates to the value of the number. If e_{1}, e_{2}, e_{3}, and e_{4} are expressions which evaluate to v_{1}, v_{2}, v_{3} and v_{4} respectively, then the following are also expressions
ˆ (add e_{1} e_{2}) is addition which evaluates to v_{1} + v_{2}, e.g. (add 1 2)≡ 3
ˆ (sub e_{1} e_{2}) is subtraction which evaluates to v_{1} − v_{2}, e.g. (sub 2 1)≡ 1
ˆ (mul e_{1} e_{2}) is multiplication which evaluates to v_{1}v_{2}, e.g. (mul 2 1)≡ 2
ˆ (div e_{1} e_{2}) is division which evaluates to v_{1}/v_{2} if v_{2} ƒ= 0 and 0 otherwise, e.g., (div 4 2)≡ 2, and (div 4 0)≡ 0,

ˆ (pow e_{1} e_{2}) is power which evaluates to v1 , e.g., (pow 2 3)≡ 8
ˆ (sqrt e_{1}) is the square root which evaluates to √v_{1}, e.g.(sqrt 4)≡ 2
ˆ (log e_{1}) is the logarithm base 2 which evaluates to log(v_{1}), e.g. (log 8)≡ 3
ˆ (exp e_{1}) is the exponential function which evaluates to e^{v}1 , e.g. (exp 2)≡ e^{2} ≈ 7.39
ˆ (max e_{1} e_{2}) is the maximum which evaluates to max(v_{1}, v_{2}), e.g., (max 1 2)≡ 2
ˆ (ifleq e_{1} e_{2} e_{3} e_{4}) is a branching statement which evaluates to v_{3} if v_{1} ≤ v_{2}, otherwise the expression evaluates to v_{4} e.g. (ifleq 1 2 3 4)≡ 3 and (ifleq 2 1 3 4)≡ 4
ˆ (data e_{1}) is the jth element x_{j} of the input, where j ≡ v_{1}∫ mod n.
ˆ (diff e_{1} e_{2}) is the difference x_{k} − x_{A} where k ≡ v_{1}∫ mod n and A ≡ v_{2}∫ mod n


ˆ (avg e_{1} e_{2}) is the average ^{1} Σmax(k,A)−1 x_{t} where k ≡ v_{1}∫ mod n and A ≡ v_{2}∫

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))
evaluates to
2 · 3 + log(4) = 6 + 2 = 8.

To evaluate the fitness of an expression e on a training data ( , ) of size m, we use the mean square error
f (e) =
1 mΣ−1 .
y^{(}^{j}^{)} − e(x^{(}^{j}^{)})Σ2 ,
j=0
where e(x^{(}^{j}^{)}) is the value of the expression e when evaluated on the input vector x^{(}^{j}^{)}.
Exercise 1. (30 % of the marks)
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 sexpressions^{1}, and ensure that you evaluate expressions exactly as specified on page 2.
Input arguments:
ˆ expr an expression
ˆ n the dimension of the input vector n
ˆ x the input vector Output:
ˆ 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. (10 % of the marks) Implement a routine which computes the fitness of an expression given a training data set.
Input arguments:
ˆ expr an expression
ˆ n the dimension of the input vector
ˆ m the size of the training data (X , Y)
ˆ 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.
ˆ The fitness of the expression, given the data.
1See e.g. implementations here http://rosettacode.org/wiki/SExpressions
Exercise 3. (30 % of the marks)
Design a genetic programming algorithm to do time series forecasting. You can use any genetic operators and selection mechanism you find suitable.
Input arguments:
ˆ lambda population size
ˆ n the dimension of the input vector
ˆ m the size of the training data (X , Y)
ˆ 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.
ˆ time budget the number of seconds to run the algorithm Output:
ˆ The fittest expression found within the time budget.
Exercise 4. (10 % of the marks)
Describe your algorithm from Exercise 3 in the form of pseudocode. The pseudocode should be sufficiently detailed to allow an exact reimplementation
Exercise 5. (20 % of the marks)
In this final task, you should try to determine parameter settings for your algorithm which lead to as 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.
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