GP Approach and Multi-objective Characterization

4.2.1 GP Approach

The overall approach we used in this chapter is in many details the same as the overall approach used for single-objective configurations in the previous chapter, but this time we explore the use of GP with a multi-objective methodology. Nevertheless we present a reminder of the basic GP approach with a brief description in this section. The GP tree (see Figure 3-1) comprises two types of binary operators at internal nodes and various kinds of indicators at leaf nodes, and the same function and terminal set as in the single-objective approach are used to construct the GP tree; they are Boolean operators and relational operators for the function set, and six groups of technical indicators for terminal set: Price, Volumes, Moving Averages (MA), Rate of Change (ROC), Price Resistance and Trend Line. When evaluating the GP tree, it returns the result as a Boolean value which is interpreted as a trading signal. If the value is True, then this is a buy signal; otherwise it is a sell signal. Please refer to sections 3.2.1, 3.2.2 and 3.2.4 for more details of the GP approach used, which applies equally to all the experiments in this chapter.

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In the next section we give some details of multi-objective configurations used in the experiments in this chapter. Essentially, what we mean by a ‘configuration’ is a subset of either two or three specific objectives. Each individual objective is essentially the same as the corresponding single-objective method described in the previous chapter.

For this reason, details of how to calculate each objective are not given here, since they have been given already in section 3.2.3.

4.2.2 Single and Multi-objective Approaches

The experiments in this chapter are mainly classified into two groups: (a) profit driven approaches that reward trading rules on the basis of their returns, and (b) risk-adjusted approaches that incorporate penalties based on the chance of loss.

4.2.2.1 Profit driven approaches

The experiments reported based on profit-driven approaches involve three main separate types of objective, specified in Table 4-1.

# Objective Description

1 CMR Market Return (essentially equivalent to excess return) 2 CPC_LK12 Performance Consistency with 12-unit periods

3 CPC_LK24 Performance Consistency with 24-unit periods Table 4-1: Three objectives of single-objective approach.

In each case, the initial “C” indicates that fitness is modified by the complexity-penalizing factor as indicated in eq. 3-9. In CPC_LK12, for example, and an experiment involving monthly trading on which the unseen test data cover a 60-month period, fitness (before the complexity modification) is either 0, 1, 2, 3, 4 or 5, according to in how many of the separate 12-month periods the rule was able to outperform both buy and hold and risk-free return.

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In the multi-objective approaches tested, each used a combination of either two or three objectives, chosen from the Table 4-1 as well as the additional list of objectives in Table 4-2.

# Objective Description

1 MR Market Return

2 2MR

2 Separated Market Return (Divide the period into 2 sub-periods and use MR as fitness value for each one) – where used, obviously this counts as two objectives

3 PC_LK12 Performance Consistency with 12-unit periods 4 PC_LK24 Performance Consistency with 24-unit periods

5 CXP Complexity Penalizing Factor – standalone measure of the tree complexity – simply the depth of the tree

6 2CMR 2MR weighted by complexity penalizing factor Table 4-2: Six objectives of multi-objective approach.

In total, to test profit-driven approaches, we test 9 distinct multi-objective configurations, as specified in Table 4-3.

# Configuration No. of

Objective Description

1 MR-CXP 2 MR and CXP

2 PC_LK12-CXP 2 PC with 12-unit periods, and CXP 3 PC_LK24-CXP 2 PC with 24-unit periods, and CXP

4 2MR-CXP 3 MR for two sub-periods, and CXP

5 MR-PC_LK12-CXP 3 MR, PC with 12-unit periods, and CXP 6 MR-PC_LK24-CXP 3 MR, PC with 24-unit periods, and CXP

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7 2CMR 2 as 2MR, but both objectives complexity

penalized

8 CMR-CPC_LK12 2 MR and PC with 12-unit periods, but both objectives complexity penalized

9 CMR-CPC_LK24 2 MR and PC with 24-unit periods, but both objectives complexity penalized

Table 4-3: Nine multi-objective configurations of profit driven approach.

4.2.2.2 Risk-adjusted approaches

Two basic risk-adjusted fitness functions are involved in our risk-adjusted approaches, as specified in Table 4-4.

# Objective Description

1 SHARO Sharpe Ratio

2 MSTLRO Modified Stirling Ratio

Table 4-4: Three risk-adjusted objectives of single-objective approach.

In the risk-adjusted multi-objective approaches, each configuration comprised either two, three or four objectives, picked up from Table 4-4 and Table 4-5.

# Objective Description

1 MR Market Return

2 PC_LK12 Performance Consistency with 12-unit periods 3 PC_LK24 Performance Consistency with 24-unit periods

4 MMDD Modified Drawdown – max(drawdown, threshold% of current position); drawdown are measured as a percentage of traded

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5 CXP Complexity Penalizing Factor – standalone measure of the tree complexity – simply the depth of the tree

Table 4-5: Five objectives used in risk-adjusted multi-objective approaches.

In total we have 20 distinct multi-objective configurations to test risk-adjusted approaches. These are detailed in Table 4-6.

# Configuration No. of

Objective Description

1 SHARO-CXP 2 Sharpe Ratio and CXP

2 MSTLRO-CXP 2 Modified Stirling Ratio and CXP

3 MR-MMDD-CXP 3 MR, Modified Drawdown and CXP

4

SHARO-PC_LK12-CXP 3 Sharpe Ratio, PC with 12-unit periods and CXP

5

SHARO-PC_LK24-CXP 3 Sharpe Ratio, PC with 24-unit periods and CXP

6

MSTLRO-PC_LK12-CXP 3 Modified Stirling Ratio, PC with 12-unit periods and CXP

7

MSTLRO-PC_LK24-CXP 3 Modified Stirling Ratio, PC with 24-unit periods and CXP

8

MR-PC_LK12-SHARO-CXP 4 MR, PC with 12-unit periods, Sharpe Ratio and CXP

9

MR-PC_LK24-SHARO-CXP 4 MR, PC with 24-unit periods, Sharpe Ratio and CXP

10

MR-PC_LK12-MSTLRO-CXP 4 MR, PC with 12-unit periods, Modified Stirling Ratio and CXP

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11

MR-PC_LK24-MSTLRO-CXP 4 MR, PC with 24-unit periods, Modified Stirling Ratio and CXP

12 CMR-MMDD 2 MR complexity penalized and Modified Drawdown

13 SHARO-CPC_LK12 2 Sharpe Ratio and PC with 12-unit periods complexity penalized

14 SHARO-CPC_LK24 2 Sharpe Ratio and PC with 24-unit periods complexity penalized

15 MSTLRO-CPC_LK12 2 Modified Stirling Ratio and PC with 12-unit periods complexity penalized

16 MSTLRO-CPC_LK24 2 Modified Stirling Ratio and PC with 24-unit periods complexity penalized