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The use of a BDF method as a tool to correct the direction of predictions made using curve fitting techniques is investigated. Random data is generated in such a fashion that it has the same properties as the data we are modelling. The data is assumed to have “memory” such that certain information imbedded in the data will remain within a certain range of points. Data within this period where “memory” exists—say at time steps

In this brief note we show how a BDF method can be used as a corrector for predictions. BDF methods are backward differentiation formulae which are a family of multistep implicit methods. They are designed to solve initial value ordinary differential equations. The derivative of a function is approximated using information computed from earlier time steps, thereby increasing the accuracy of the approximation. This characteristic makes BDF methods ideal for our purposes where we seek to improve upon already existing data in the form of predictions. The predictions aim at accurately reflecting the direction of random data and are made using curve fitting techniques. This research comes out of a project undertaken to predict the direction of a subset of the South African market. The approach taken in analysing the data assumes that the data has a “memory.” More precisely this presupposes that a time series will have certain periods when the data has the same inherent information and dynamics. This allows us to conclude that the same information embedded in a previous selection of data points is still contained within the data point to be predicted from that set. While the data we generate in this paper is random with zero mean we are still able to show how an application of a BDF method improves the degree of accuracy in predicting the direction the data takes. The BDF formulae are constructed by satisfying the differential equation exactly at one point

The random walk hypothesis has had its fair share of attention as a means of explaining stock price movements. This financial theory states that stock market prices evolve according to a random walk and thus the prices of the stock market cannot be predicted. While the work undertaken in this paper concurs with this theory with regard to the random walk followed by the relevant data, we still maintain an assumption that there exists embedded information within the relevant data which can be modelled. Thus we assume a consistent underlying dynamic which can be identified and used as a means of extrapolating beyond known data points, that is, predict future movements in market prices. Theoretical developments in mathematics of finance have centred around the random walk hypothesis [

While the mathematical theory used to develop the mathematical aspects of finance has not really focused on predicting returns there has been a strong interest in developing tools which can give a sense of the direction the market is going to take or when possible turning points will occur. The inclusion of ideas from the social sciences in financial mathematics has heralded the potential development of tools that can be used to aid the prediction of market trends. Among these ideas are aspects of behavioral science [

The paper is set out as follows. In Section

The first part of this analysis is to generate random data that may be used to simulate an actual financial time series. Here we use the MATLAB function

repeat

for

for

end

Plot of three different simulated data series.

We then choose an appropriate length of data to indicate what we term “memory” in the data. For the purposes of this paper we assume that using four initial data points is sufficient to account for the memory. Given the fact that the behaviour observed in Figure

Our aim is to predict—and improve upon—the direction of the data, that is, whether the quantitative value is positive or negative, which is indicative of whether market prices are moving up or down. Since most forecasting systems are far from accurate when predicting the quantitative value, predicting direction is far easier and can be equally profitable. For instance, for a system that draws its conclusion of how to trade tomorrow from the closing price of today’s action, getting the direction is vitally important. We are not looking at this from a multisignal/asset point of view which would require the determination of how much to invest in each asset. This would be along the lines of an efficient frontier in Modern Portfolio Theory which would take factors like standard deviation and error in the forecast into account. Rather, we are simply wishing to trade on the back of successfully predicting direction. Thus while considering the distribution of the time series itself and the relevant mean and standard deviation may be more accurate quantitatively, trading on the predictions of the direction the prices are moving in can in itself be very profitable.

As a consequence the success of our methodology is calculated by considering the percentage of times the sign of the predicted data matches the sign of the actual originally generated data. To improve the accuracy of predicting direction we make use of the fact that the direction is just the gradient of the data. By creating a vector of gradients we have the numerical representation of an ordinary differential equation. We then use the structure of a BDF method to numerically solve the ordinary differential equation. BDF methods are appropriate because they depend on previous values. Some examples of BDF methods for solving the first order ordinary differential equation

Using a forward difference approximation to the derivative we find that

As stated in their names, BDF methods are backward approximations of the first order derivative in a first order ordinary differential equation. In this instance, however, we are not applying the BDF method to an ODE but rather to actual discrete data points. Equations (

Thus (

It is important to note that we are not implementing the BDF method in a direct fashion. We are incorporating a lower order approximation into the method in order to obtain a means of improving on already generated data. In the computational implementation of this work, the BDF method given by (

In this section we investigate the convergence properties of the difference equations (

When either (

The results and concluding remarks are presented in the next section.

As a means of evaluating the effectiveness of the BDF method as implemented above to improve upon the prediction of the direction of data we compare the accuracy of the originally fitted data and the corrected data. Direction success rate can be seen as a simple bimodal result. If we let

In Table

Table comparing percentage accuracy of predicting direction.

Linear | Spline | ||
---|---|---|---|

Fitted | Corrected | Fitted | Corrected |

51.0417% | 80.0000% | 56.2500% | 88.4211% |

52.0833% | 92.6316% | 50.0000% | 86.3158% |

52.0833% | 90.5263% | 52.0833% | 86.3158% |

45.8333% | 87.3684% | 47.9167% | 88.4211% |

55.2083% | 92.6316% | 41.6667% | 85.2632% |

46.8750% | 85.2632% | 53.1250% | 91.5789% |

48.9583% | 84.2105% | 48.9583% | 85.2632% |

42.7083% | 85.2632% | 59.3750% | 86.3158% |

35.4167% | 87.3684% | 44.7917% | 76.8421% |

48.9583% | 88.4211% | 52.0833% | 85.2632% |

Table

In this paper we have shown how the implementation of the BDF method with a lower order approximation of the gradient of discrete data can improve the accuracy of predicting the direction of random data. The motivation for using a numerical ordinary differential equation solver comes from the fact that direction is just a gradient. We form a discrete ordinary differential equation from our vector of predictions. This discrete ordinary differential equation is solved with the implementation of a lower order approximation of the gradient with a BDF method. We find that the accuracy of our predictions improves from an accuracy of approximately 50% to an accuracy of approximately 87%.

An advantage of the approach taken in this paper is that the BDF scheme is a marching scheme. Irrespective of how big the data set is, the scheme will march through the data accordingly. The “speed” of the algorithm on very large data sets can be improved upon by using a computer with a faster CPU.

E. Momoniat and C. Harley acknowledge support from the National Research Foundation of South Africa.