Distribution forecasting of high frequency time series

Abstract

The availability of high frequency data sets in finance has allowed the use of very data intensive techniques using large data sets in forecasting. An algorithm requiring fast k-NN type search has been implemented using AURA, a binary neural network based upon Correlation Matrix Memories. This work has also constructed probability distribution forecasts, the volume of data allowing this to be done in a nonparametric manner. In assistance to standard statistical error measures the implementation of simulations has allowed actual measures of profit to be calculated.

Introduction

Many techniques for forecasting nonlinear time series exist. Traditionally in finance forecasters have looked at daily or monthly prices. Recently the availability of large high frequency data sets has encouraged much more research into intra-day and even tick price forecasting. It has also made possible the use of very data-intensive forecasting techniques and the adjustment of traditional techniques. An introduction to this new field is Ref. [5].

The Farmer–Sidorowich [9] forecasting algorithm is well understood and many forecasting methods implement a variant of it. Delay coordinates are used to construct representations of the current and all previous states. The state at time t of a variable x, denoted S(x_t) is constructed from the observed values:

$$\text{x}{}_{\mathrm{t}}\text{,x}{}_{\mathrm{t-1}}\text{,x}{}_{\mathrm{t-2}}\text{,…,x}{}_{\mathrm{t-d+1}}$$

The parameter d is considered the window size or embedding dimension, the number of most recent historical values that will be used to construct a state. It is assumed that a functional relationship between the current state and the future state exists.

$$\text{x}{}_{\mathrm{t+1}}\text{=f(S(x}{}_{\mathrm{t}}\text{))}$$

For this function to exist the attractor must be static or evolving only slowly. At this stage, it is assumed that the time series shows this property (evidence of this relationship has been documented [19]) but it is tested for as part of this work.

The aim is to construct a predictor approximating the function f. A forecast of the next price is constructed from one or more ‘local’ states considered similar by some distance metric to the current state. For each local state, the next value is its evolution. The evolutions of the local states are used to forecast the evolution for the current state, often by fitting a linear polynomial. Delay coordinates require the window size to be determined but there are no clear rules on how this should be done. Several researchers have produced guidelines in line with the study of nonlinear dynamics [9], [14], [17], but embedding is still considered an art as much as a science. In the field of neural network design for forecasting, Genetic Algorithms have often been considered [16], [18], [24]. Other issues the technique introduces are which distance metric to use, how to implement it and how many neighbour states should be used to construct a forecast?

Traditional forecasting methods produce point forecasts, each prediction is a single value. Alternatively probability and interval forecasting has been reported in the literature, for example [1], [4]. Full probability distribution forecasts contain extra information, which can be utilised by modern risk management systems or allow improved trading models. This work produces as forecasts, discrete probability distributions and is part of a new but expanding area of research.