Advanced Aspects of Nature Inspired Search and Optimisation 2019/2020

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. (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 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. (10 % of the marks) **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. (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:

- -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. (10 % of the marks)** - 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. (20 % of the marks)**- 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.

**a portion of the work is done and attached below , but there are mistakes in it that needs to be corrected . please do not bid if you don’t already know the material because that has already happened before and it is highly time sensitive . Thank you. **