Almost all climate time series have some degree of nonstationarity due to external driving forces perturbing the observed system. Therefore, these external driving forces should be taken into account when constructing the climate dynamics. This paper presents a new technique of obtaining the driving forces of a time series from the slow feature analysis (SFA) approach, and then introduces them into a predictive model to predict nonstationary time series. The basic theory of the technique is to consider the driving forces as state variables and to incorporate them into the predictive model. Experiments using a modified logistic time series and winter ozone data in Arosa, Switzerland, were conducted to test the model. The results showed improved prediction skills.

Many previous studies have concluded that the climate system is essentially nonstationary (Trenberth, 1990; Tsonis, 1996; Yang and Zhou, 2005; Boucharel et al., 2009). However, the lack of any general theory for predicting nonstationary processes has become one of the main barriers in the field of climate prediction. To unravel this issue, in recent years, increasing effort has been devoted to devising methods to analyze and predict nonstationary time series (e.g., Hegger et al., 2000; Verdes et al., 2000; Wan et al., 2005; Wang and Yang, 2005; Yang et al., 2010). The most used method in such studies was to remove or reduce the nonstationarity of the predicted system using some mathematical techniques, thereby improving the prediction skills.

The nonstationarity exists due to the fact that the driving forces that perturb the observed system change with time (Manuca and Savit, 1996). Consequently, the most effective way to remove the nonstationarity may be to incorporate all the driving forces into the constructed dynamical system, and to consider them as the state variables of that system when establishing a prediction equation within a general circulation model (GCM). Based on this principle, lately a data-driven modeling path that is compatible with a GCM was proposed to predict several artificial nonstationary time series with known external forces. It has achieved success in improving predictions when driving forces were included in some ideal or climate systems, such as the Lorenz system, a logistic model, or global temperature over seasonal timescales including the North Atlantic Oscillation (NAO), the Pacific Decadal Oscillation (PDO), the El Niño–Southern Oscillation (ENSO), and the North Pacific Index (NPI) variability (Wang et al., 2012, 2013). However, a disadvantage of this technique is that it can not differentiate between the assumed driving forces from the predictive model. Therefore, in the present study, we considered the extraction of driving forces from the time series itself and established a predictive model by incorporating the constructed driving forces. As a result, the extraction of driving forces became the focus of this study.

Wiscott (2003) developed a technique called slow feature analysis (SFA) to extract driving forces from time series. This technique has been applied to nonstationary time series with some success (Wiskott, 2003; Berkes and Wiskott, 2005; Gunturkun, 2010; Konen and Koch, 2011). In this paper, we used SFA to construct the driving forces from a testing time series, and then established a predictive model that incorporated the driving forces. The paper is organized as follows: a brief description of the predictive technique is presented in Sect. 2. In Sect. 3, results are reported from applying the approach to a modified logistic time series and the total ozone data of Arosa, Switzerland. A summary is provided in Sect. 4.

SFA is a method that extracts slowly varying driving forces from a quickly
varying nonstationary time series. In this section, we provide a brief
overview of SFA and its application for the extraction and construction of
the driving forces from the time series. The details of SFA are presented in
Wiscott (2003), but the basic steps of the technique are provided here for
convenience and completeness. Let us assume that we have a single variable
time series

Embed the above time series into an m-dimensional space (also named the
length of the m window), a phase trajectory in the m-dimensional space
denoted as

Generate an expanded signal

Normalize the expanded signal H(t) by an affine transformation to
generate

By means of the Schmidt algorithm, the function space Eq. (5) is
orthogonalized as

Establish the covariance matrix of

In this study, the SFA was tested on a logistic map

The true and constructed driving forces.

So far we have two time series: one is the original time series

The comparison of prediction skills between models combined with or without driving force.

Based on this trajectory, a predictive model to predict the future state of
the system can be established as

The slowest driving force with different embedding dimensions for total ozone data.

We applied the prediction technique described above to perform some prediction experiments using several given nonstationary time series. The experiment presented in Sect. 3.1 was performed with data from the modified logistic model given above.

RMSE comparison of the prediction experiments (unit: Dobson units).

The prediction experiments were based on 5000 data points from the
above-verified logistic map (Eq. 8) with the assumed driving force (Eq. 9).
The first 4800 data points were applied to establish the predictive model,
and the remaining 200 data points were used to test the prediction and
estimate the correlation coefficient between the actual and predicted values
as a function of the prediction time step. The embedding dimension of the
verified logistic time series, namely

Many studies have sought to explain the variables involved in ozone dynamics,
such as the Quasi-Biennial Oscillation (QBO), the 11-year solar cycle,
volcanic eruptions, the El Niño–Southern Oscillation (ENSO), and the
North Atlantic Oscillation (NAO) (e.g., Brasseur and Granier, 1992; Hood,
1997; Schmidt et al., 2010; Rieder et al., 2010). In this paper, we focused
on prediction experiments with total ozone data. The total ozone data were
from Arosa, Switzerland, and were the world's longest total ozone record.
Homogenized total ozone data from 1927 to 2007 were obtained from the World
Ozone and Ultraviolet Radiation Data Centre (WOUDC;

The comparison of prediction skills between models combined with or without driving force.

Errors (Dobson units) at prediction steps with or without forcing input.

By using the SFA technique on Arosa's daily total ozone data in winter (from
January to March) for the period 1927 to 2007, we obtained the first output
of the driving force

We established a prediction model for winter ozone data by incorporating the
driving force constructed by SFA. The prediction was based on 7305 data
points. Out of the 7305 data points, the first 7125 data points were used to
build the predictive model, and the remaining 180 data points were used to
test the prediction using root-mean-square error (RMSE) and the correlation
coefficient between observed and predicted values. The time lag

The experimental results for this case are listed in Table 1, and are also shown in Figs. 4 and 5. From Table 1, it can be seen that all RMSE values given by the forcing model were much lower than those given by the stationary model. Figure 4 presents the correlation coefficients between the observed and predicted values. The forcing model outperformed the stationary model, especially at the first two steps. At the first prediction step, the correlation coefficients reached 0.61 for the stationary model but 0.91 for the forcing model. At the eighth prediction step, the correlation coefficients were reduced to 0.39 for the stationary model, but still maintained at 0.45 for the forcing model. At the twelfth prediction step, the correlation coefficients were 0.22 and 0.33 for the stationary model and the forcing model, respectively. This has clearly shown that, when the constructed driving force is introduced, the accuracy of prediction is dramatically improved. The average correlation over the prediction time steps is improved by 50 % when the driving force extracted through the SFA technique is included. Figure 5 illustrates the error between the prediction and the observation. The prediction error for every time step is lower for the forcing model than for the stationary model. All these results indicate that the inclusion of the driving force constructed by the SFA approach in the prediction model largely improves the predictive skill of winter total ozone in Arosa. Some sensitivity analysis with different training/verifying lengths does not alter this conclusion.

In this study, we first constructed the driving forces of a time series based on the SFA approach, and then the driving forces were introduced into a predictive model. By doing so, we extend the study by Wang et al. (2012, 2013) and present a novel technique to predict nonstationary time series. Unlike the former works by Wang et al. (2012, 2013) with assumed driving forces, in this study, the driving force was extracted from original time series. The experimental results obtained from a modified logistic time series and winter ozone data in Arosa confirmed the effectiveness of the model.

The driving force construction technique based on SFA represents a progress for climate causal relations. Such an approach may provide a compatible and direct window for studying causality using external driving forces. We constructed the driving forces with SFA and then combined these driving forces to establish the predictive model. Although we found this approach was able to effectively improve the predictive ability, the constructed driving force time series still lacks physical explanation. In order to understand the real background of these, one has to further explore the physical processes behind it. One recommended method, provided by Verdes (2005), suggests using a measure called “transfer entropy” to analyze the causality; another recommended method is named “convergent cross mapping”, provided by Sugihara et al. (2012), which measures causality in nonlinear dynamic systems. Work in this area is in progress and will be reported in future publications.

This research was supported by the National Natural Science Foundation of China under grant 41275087. The authors are grateful to the editor and three anonymous referees for useful comments during the discussion phase in Nonlin. Processes Geophys. Discuss.Edited by: V. Lucarini