Several researchers emphasise the potential of market predictions to improve important financial decisions29, from helping businesses make sounder investment decisions to helping governments make more efficient fiscal and monetary policy decisions8. These time series are amongst the most complex time series because of the number of parameters involved. Our results are compared with respect to long-term predictions with ARIMA, GARCH and VAR30, which are the most widely used and most efficient methods for making long-term time series predictions. We also compared our results for short-term predictions with those obtained by two existing methods: the Learning Financial Agent Based Simulator (L-FABS)21 and the MLP model22.

For the first period, we considered the DJIA (Dow Jones Industrial Average) time series between 1993 and 2001, when markets were stable with no major changes and no financial crisis. In the second period considered, the US stock market peaked in October 2007, but by March 2009, the Dow Jones average had reached its minimum, which reflects the most serious effects of a financial crisis. In the third period (August 2004–August 2012), the recession was in the middle of the considered range.

Another important application of time series predictions is in medical science. Approximately 1% of the world population suffers from epilepsy31. Epileptic seizures are the result of unusual and irregular neuronal activity in the brain32,33. Many recent methods have been proposed for predicting epileptic seizure26,34,35 but none of them as shown their ability to perform accurate predictions more than 10 minutes in advance on a large number of patients. To evaluate the performance of our new method for predicting epileptic seizures, we examined the EEG time series measured by five electrodes, generating five different time series, for 21 patients. For each EEG time series, the exact time of the seizure is known.

Predicting the monthly records of global temperature anomalies is currently one of the most pressing and controversial environmental concerns36. As a third experiment, we used the global temperature anomaly data from 1880 to 1983 to train for the prediction of global temperatures during 1983–2013. Global temperature anomaly data come from the Global Historical Climatology Network-Monthly (GHCN-M) data set and International Comprehensive Ocean-Atmosphere Data Set (ICOADS), which have data from 1880 to the present. These two datasets are blended into a single product to produce the combined global land and ocean temperature anomalies.

Our new method for complex time series prediction is based on the concepts of chaos theory and an optimisation process. The general idea is to extract a unique characteristic from an existing time series that somehow represents the behaviour of the time series and to subsequently generate successive new values that continue the time series, each value minimising the difference between the characteristic of the new time series and the initial one. The details of the GenericPred method for long-term time series prediction are as follows. We consider a time series S N :

A nonlinear measure V() is computed on S N The fractal dimension37 and the Lyapunov exponent38 are examples of such nonlinear measures that return a single value for a time series. A possible mapping may be required, forming a new time series , for different applications as follows:

otherwise, , where 0 < L < N is the size of a sliding window used to compute the local level of chaos measured by V(). Therefore, when the mapping is applied, the new considered time series corresponds to the variation in time of the local non-linear measure in the initial time series S N .

We consider as a reference value that will be used for predicting the next k values of the time series:

The parameter σ of a normal distribution N(y i , σ2) is estimated by computing the variation between every two consecutive values (y i to y i +1 ) of the time series . This distribution represents the probability distribution P(y i |y i −1 ) (see Fig. 3). Several data sets have been considered to determine that a normal distribution is a good approximation of the real distribution. However, the same method has been applied using other distributions without significant degradation is the prediction.

Figure 3 Successive steps of the GenericPred method for time series prediction. Full size image

For predicting y N +i , a set Pos(y N +i ) of N rand random values are generated following the distribution N(y N +i−1 , σ2) (Fig. 3):

N rand is a parameter that can impact the quality of the prediction because having more values will increase the chance of finding an optimal value. However, no significant improvement was observed for the data considered when N rand was greater than 10. For this reason, we chose 10 as the value of N rand for each experiment. y N +i is then computed by selecting the that makes the new nonlinear measure the closest to :

The value is chosen to make as close as possible to .

The important point is that the reference value is always , which is the calculated nonlinear measure from the original time series. Therefore, the GenericPred method uses two basic rules:

R1: Always endeavour to keep the value of a nonlinear measure as steady as possible during prediction (Fig. 3).

R2: The new value must be chosen from a set of potential values generated from a probability distribution.

The prediction has to be pursued one step at a time because the predicted value in the current step is needed for determining the valid range of change for the next step. For those problems for which a binary prediction (‘yes’ or ‘no’) is required, (e.g., the epileptic seizure prediction), a threshold t is computed from the learning data. Whenever the value y N +i is greater than the threshold t, the prediction is positive. For example, yes there is an epileptic seizure at time N + i if y N +i > t; otherwise, there will be no seizure at time N + i.

Classical model-based prediction approaches consider a unique value for the next step, whereas in the GenericPred method, several points are considered simultaneously. Our method is also able to constantly adjust the information regarding the current time series, whereas classical predictive methods apply the model without taking into account the concordance between the original time series and the predicted ones. Technically, any nonlinear measure could be used for the time series characterisation. However, here, we used the P&H method25 because it has been shown that this method can efficiently discriminate between different types of nonlinear behaviour39.