The North Atlantic Oscillation (NAO) is the dominant mode of climate variability over the North Atlantic basin and has a significant impact on seasonal climate and surface weather conditions. This is the result of complex and nonlinear interactions between many spatio-temporal scales. Here, the authors study a number of linear and nonlinear models for a station-based time series of the daily winter NAO index. It is found that nonlinear autoregressive models, including both short and long lags, perform excellently in reproducing the characteristic statistical properties of the NAO, such as skewness and fat tails of the distribution, and the different timescales of the two phases. As a spin-off of the modelling procedure, we can deduce that the interannual dependence of the NAO mostly affects the positive phase, and that timescales of 1 to 3 weeks are more dominant for the negative phase. Furthermore, the statistical properties of the model make it useful for the generation of realistic climate noise.

The large-scale atmospheric flow has attracted the attention of climate scientists since the days of Gilbert Walker almost a century ago (see, e.g., Walker and Bliss, 1932, Rossby, 1940, Horel and Wallace, 1981, and the review of Hannachi et al., 2017, and the references therein). An important driving force behind this interest is the prospect of obtaining a better understanding of the different mechanisms involved and using this knowledge to predict the system on long lead times. Of particular interest is the challenge posed by the extratropical atmospheric variability, which exhibits both nonlinearity and considerable complexity.

The extratropical large-scale circulation can be described by a number of interacting teleconnection patterns, including the North Atlantic Oscillation (NAO) and the Pacific North America (PNA) pattern; see e.g. Wallace and Gutzler (1981), Hannachi et al. (2017) and Feldstein and Franzke (2017). These teleconnections are characterized by typically large spatial scales and low-frequency variability (timescales of more than 10 d; Feldstein, 2000, 2002, 2003; Franzke and Feldstein, 2005), which provides the potential for longer term predictability compared, for example, to synoptic scales.

Being one of the main teleconnection patterns of the Northern Hemisphere (NH), the NAO controls much of the atmospheric variability, particularly over the North Atlantic region, the Mediterranean and the Eurasian continent. It also interacts with other teleconnections, such as the PNA and the El-Niño–Southern Oscillation (ENSO), to produce remote responses. The NAO is known to have a direct, strong impact on surface weather and climate through changes in the atmospheric mass and shifts of the jet stream. It varies on a wide range of timescales, ranging from days to decades and longer (Woollings et al., 2014, 2015). Descriptions of the physical characteristics of the NAO can be found in Benedict et al. (2004), Franzke et al. (2004), Stendel et al. (2016), Hannachi and Stendel (2016) and the references therein.

Several meteorological centres issue regular forecasts of the NAO for
various purposes (e.g. NOAA CPC,

The NAO has a number of characteristic features. Its probability density function (PDF) is non-Gaussian, both in the bulk and in both tails of the distribution. Furthermore, the NAO has non-zero autocorrelation for short and long lags (see, e.g., Önskog et al., 2018). A good forecasting model for the NAO should, in principle, be able to reproduce these main properties. The non-Gaussianity implies, in particular, that linear models such as autoregressive (AR) models cannot reproduce the main features. Note, however, that linear models with non-Gaussian noise can of course produce non-Gaussianity (Franzke 2017; Önskog et al., 2018). The NAO is a nonlinear phenomenon and is related to synoptic Rossby wave-breaking (Benedict et al., 2004; Franzke et al., 2004) and regime behaviour (Hannachi et al., 2017). Based on this observation, it is reasonable to expect that nonlinear probabilistic models can be better suited to fit the NAO time series and be able to reproduce the properties mentioned above.

In a previous article (Önskog et al., 2018), we studied the properties of the time series of the daily NAO index, in particular the station-based time series published by Cropper et al. (2015), and found that the distribution of the NAO index has clear non-Gaussian features and long-range dependence (Franzke et al., 2020). An autoregressive model with non-standardized t-distributed noise, taking the values of the index during the last three days into account (AR), provided a good model for the daily NAO index on timescales of up to 2 weeks. By investigating forecasts of the future distribution of the NAO (considering both the expected value and quantiles), we found that some, but not all, properties were well described by the model. Features that the model was unable to replicate included the long-range dependence on timescales of the order of 20 d or more, the different timescales of the positive and negative phases of the NAO and the fact that the negative tail of the NAO distribution is fatter than the positive tail. In this article, we study nonlinear time series models and derive models that reproduce all these features of the daily NAO index. In addition to considering other classes of models compared to our previous article (Önskog et al., 2018), we also develop the test statistics used to compare and evaluate the models further.

Our aim here is to develop a statistical NAO model with which we will be able to identify important dynamical processes controlling the NAO. In order to make progress towards this aim, we here address the following specific research questions:

What are the nonlinearities and state-dependencies of the NAO distribution?

Is there a nonlinear time series model which reproduces the properties of the NAO distribution?

Does the inclusion of long lags on seasonal and interannual timescale improve a nonlinear model for the NAO?

Our article is organized as follows. First, in Sect.

In this study, we are using the daily time series of the NAO index published
by Cropper et al. (2015). The index is calculated from actual sea level pressure (SLP) observations on Iceland and
the Azores, but reanalysis data have been used to fill in the gaps (1888–1905, 1940–1941 and on 145 other occasional days) in the observations. These station-based data are freely available (

We have restricted the present study to the winter season, which we
here define as starting on 1 December and ending on 28 February
(excluding 29 February for all leap years). The study is based on data
for 142 consecutive winters from 1872/1873 to 2013/2014. We have
chosen the 71 winters for which the month of December falls on an odd
year (1873/1874, 1875/1876 and so on) as our training data for
fitting of the models. The remaining 71 winters (1872/1873, 1874/1875
and so on) constitute our testing data used for validation of the
models. Both training and testing data consist of 6390 data points
each. We have chosen to spread the training and testing data evenly
during the time period to reduce the effect of any non-stationarity in
the data. Note that some of the nonlinear models studied in Sect.

The analyses of the models that we propose for the NAO will focus on
the extent to which the models capture the distribution,
autocorrelation and timescale of the NAO. The ability of the models
to reproduce the distribution of the NAO will be visualized by means
of density plots and Q–Q plots but also quantified in terms of the
Kullback–Leibler divergence (Kullback, 1959; Kullback and Leibler, 1951; Cover and Thomas, 2006; Lacasa et al., 2012; Kowalski et al., 2011). The symmetrized Kullback–Leibler divergence (KLD) between two probability densities

It should be noted that the comparisons based on KLD that we carry out are
somewhat flawed by the fact that the simulated distributions are not fixed
but influenced by statistical simulation error. However, in this study each
simulated distribution is based on almost

It is well known that the NAO distribution is non-Gaussian in the
sense that both the skewness and excess kurtosis of its distribution
are non-zero. Moreover, the values of higher moments and, hence, the
size of the departure from normality, tend to depend on the current
state of the NAO. In Fig.

Relation between higher moments of the daily North Atlantic Oscillation (NAO) winter index and the sample mean for 71 winters. The plots show the sample standard deviation

A simple measure for the persistence of NAO events is provided by the
sample autocorrelation function (ACF). Figure

Overview of the relations between higher moments and sample autocorrelation function (ACF) of the daily NAO winter index on one hand and the sample mean for 71 winters on the other hand. As described in the captions of Figs.

Relation between sample ACF of the daily NAO winter index and the sample mean for 71 winters. The plots show the sample ACF for lags 1, 8, 15, 22, 29, and 36, respectively. The red lines are fitted regression lines. Point estimates and 95 % confidence intervals for the intercepts and slopes of these lines are given above each plot.

In a previous study (Önskog et al., 2018), we found that an
autoregressive (AR) model of order 3 with non-standardized

Figure

Plot of the ability of the AR

We have also investigated the timescales of the two phases of the NAO. Following the procedure in Woollings et al. (2010), we first introduce two thresholds at

Durations of the positive and negative phase of the NAO index as compared to durations derived from the AR

It is unlikely that the skewness and different timescales of the two phases of the NAO can be reproduced by a linear autoregressive model, as such models stipulate the same probabilistic behaviour regardless of the current state of the NAO. To validate this conjecture, we conduct a statistical test on the hypothesis that all 142 winters in the data set can be successfully modelled by the same AR model. To this end, we have fitted AR models for all 142 winters separately using the Yule–Walker equations, first removing the mean of the particular winter. Let

Under the null hypothesis that all winters are described by the same
AR model with fixed, unknown parameters

An analysis of the parameters of the 142 AR models for the various years shows that there is no significant autocorrelation in the parameters. As these parameters are related to the autocorrelation of the daily winter NAO index for the different years, the absence of significant autocorrelation in the parameters should be interpreted as a lack of interannual dependence between the perturbations of the sample ACF from its mean. Conclusively, any interannual dependence in the winter NAO is due to the dependence of the actual value on past states of the index and not due to a dependence in its autocorrelation function.

In this section, we consider a couple of classes of nonlinear time series models and investigate whether they provide a better fit of the daily winter NAO index than standard linear AR models.

We conclude from Figs.

As the optimal order of AR models for the daily winter NAO index was found to be three in Önskog et al. (2018), we also use

Parameter estimates for the SETAR model for the daily winter NAO index. Here, and throughout the article,

The distributional properties of the SETAR model are shown in the second row of Fig.

Instead of assuming distinct regimes for the different phases of the
NAO, we may assume that the parameters of an AR model depend on the
present state of the NAO. If we assume parameters of an AR(

As for the AR and SETAR models, we use

Parameter estimates for an SDNAR model for the daily winter NAO index.

The distributional properties of the SDNAR model are shown in the lower row of Fig.

From Figs.

The final model, which we refer to as the extended SDNAR (ESDNAR) model,
is very similar to the SDNAR model. The variables included in the
model are the same, except that the ESDNAR model includes an additional

Plots of the ability of the extended SDNAR (ESDNAR)

In a similar fashion, the SETAR model can be supplemented by additional terms with longer lags. Using the thresholds

Parameter estimates for an ESETAR model for the daily winter NAO index.

Figure

Properties of the ESETAR

Up to now we have assumed the noise in all models to be independent and identically normally distributed. A necessary condition for the independence of the elements of a time series is that the sample ACF of the squared entries in the series is close to zero, and this fact can be used as a test of independence. If a nonlinear model captures all the dependencies in the data, we expect the sample ACF of squared residuals of the model to be close to zero for all lags. But the lag 1 sample ACF of the squared residuals of all models investigated up to now is between 0.035 and 0.045, which is significantly larger than zero (at 5 % significance level, the values differing more than 0.025 from zero are significantly non-zero for time series of the length used here), whereas for larger lags the sample ACF is not significantly non-zero. This implies that large residuals in terms of absolute value are likely to be followed by residuals with large absolute value and vice versa, and that the error variance can be well modelled by an autoregressive model.

The noise clustering described above is often successfully modelled by
a generalized autoregressive conditional heteroskedasticity (GARCH)
process (see e.g. De Gooijer, 2017). A GARCH(

Considering now, instead of the residuals from the ESDNAR and ESETAR models, the standardized residuals obtained by dividing the residuals by

GARCH parameter estimates for the noise in the extended models. Note: “ESDNAR and GARCH” denotes the ESDNAR model with GARCH noise, and “ESETAR and GARCH” denotes the ESETAR model with GARCH noise.

Applying the BDS test to the standardized residuals, we obtain the values shown in Table 5 below. Under the null hypothesis of independence of the standardized
residuals, 95 % of the entries in Table 5 should be within

BDS test results for the standardized residuals of the GARCH noise models. The test has been performed with parameter values

Although the application of GARCH noise in the model explains most of the dependence in the noise, it does not improve the fit of the distributional properties of the daily winter NAO index. The last two rows in Fig.

Plots of the ability of the ESDNAR model with GARCH noise, denoted as ESDNAR and GARCH, (first row); the ESETAR model with GARCH noise, denoted as ESETAR and GARCH, (second row); ESDNAR and correlated additive and multiplicative
(CAM) noise, denoted as EDSNAR and CAM, (third row); and the ESETAR model with CAM noise, denoted as ESETAR and CAM, (fourth row) models to reproduce the higher moments and sample ACF of the daily NAO winter index. The plots are generated in the same way as the plots in Fig.

The application of GARCH noise in the models explains most dependence
in the noise, but it does not improve the fit of the NAO distribution
and does not give a clear improvement of the fit of the sample ACF. We
therefore implement another type of noise which has previously been
suggested in the literature (Sardeshmukh and Sura, 2009; Majda et al.,
2009; Franzke, 2017), namely correlated additive and multiplicative
(CAM) noise. We define a time series model

Properties of the ESDNAR model with CAM noise

These sample means are easily calculated given the choice

CAM parameter estimates for the noise in the extended models. Note: “ESDNAR and CAM model” denotes the ESDNAR model with CAM noise, and “ESETAR and CAM model” denotes the ESETAR model with CAM noise.

Comparing Figs.

BDS test results for the standardized residuals of the CAM noise models. The test has been performed with parameter values

The models investigated in this article only use present and past values of the NAO index to compute the future NAO index and not any other covariates. For this reason, the predictive skill of these models will be inferior to that of dynamical models taking many other variables than the NAO index
itself into account. To give an example, comparing the forecast skill
of one of the nonlinear statistical models proposed in this article
to that of the dynamical forecast model issued by NOAA (

Simulation of weekly means of the NAO. Panel

We next investigate if simulations of the models proposed in this article
generate surrogate time series which have the same properties as the daily
winter NAO index on timescales other than days. To this end, we have
simulated

Figures

Simulations of monthly means of the NAO. The plots are identical to Fig.

Simulations of winter means of the NAO. The plots are identical to
Fig.

In this study we have shown that the NAO can be well described by nonlinear autoregressive models. Different models have different benefits and drawbacks, but in general the performances of the various nonlinear models are fairly similar. Putting all the analyses together, the time series model that provides the best description of the station-based daily winter NAO index is of polynomial form with GARCH-type noise. This model is able to reproduce the skewness and fat tails of the observed NAO index and the autocorrelation and timescales of the positive and negative phases of the NAO. The nonlinear and state-dependent NAO model gives some indication of dominant timescales for the NAO. Our model elucidated that the interannual variability mainly affects the positive phase of the NAO, while the negative phase is more affected by processes acting on timescales of 1–3 weeks. These findings are in agreement with the recent results of Caian et al. (2018), who analysed the interannual link between Arctic sea ice and the NAO using reanalysis data and climate model simulations. It was found, in particular, that anomalous Arctic sea ice, associated with a quasi-steady positive gradient of sea ice anomalies about coastal line, act to force precisely the positive NAO phase on interannual timescales.

Interestingly, the ESETAR model for the NAO index reveals significant association with slow timescales of the order of 7 years. A similar long-range behaviour for the NAO has previously been observed in several studies on the spectrum of the NAO, which found peaks of the NAO spectrum in the broad range of 7–10 years (e.g. Gámiz-Fortis et al., 2002; Wunsch 1999). This extended range dependence could be an imprint of the Hurst phenomenon (e.g. Franzke et al., 2020) and has been observed in various branches of natural sciences, such as hydrology (Mandelbrot and Wallis, 1968) and the atmospheric circulation (Franzke et al., 2015b). However, this long timescale did not show up in the sample ACF of the NAO time series, and it is unclear if this is due to the limited length of the data or that the dependence is not manifested in the sample ACF.

The better performance of the SDNAR model compared to the SETAR model might suggest that it is not possible to straightforwardly represent the nonlinearity by conditional linear models as was proposed by Horenko (2010), O'Kane et al. (2013), Franzke et al. (2015a) and Risbey et al. (2015). The idea, in these studies, is that the nonlinear behaviour of the atmospheric circulation can be captured by piecewise linear models. While models in these studies are not exactly similar to the SETAR class of models, they share the underlying idea.

The NAO is not an isolated and self-sustained phenomenon. It interacts with many other processes, including mainly the bottom boundary forcing. The sea surface temperature, for example, varies on a wide range of timescales, with the particular predominance of low frequencies, and can affect the NAO on those scales. Land surface and, in particular, sea ice interactions, can also affect NAO variability on those same scales. A comprehensive probabilistic model for the NAO that is able to provide a better long-term prediction should possibly include those processes as predictors, in addition to the nonlinear terms considered here. This is beyond the scope of this paper and is left for future research.

The station-based data for the daily NAO index are freely available online
at

TÖ constructed the models, carried out the simulations and wrote Sects. 2–5 of the paper. CLEF and AH provided ideas, suggested improvements during the entire process of conducting the research and contributed to Sects. 2–5. All authors contributed to the writing of Sects. 1 and 6.

The authors declare that they have no conflict of interest.

We would like to thank Stephen Baxter and Jon Gottschalck at NOAA's Climate Prediction Center for kindly providing us with data about the NOAA forecasts. CLEF was partially supported by the Collaborative Research Centre (grant no. TRR 181) “Energy Transfer in Atmosphere and Ocean”, funded by the Deutsche Forschungsgemeinschaft (DFG – German Research Foundation) under project 274762653 (grant no. FR3515/3-1). Four anonymous reviewers provided constructive comments that helped improve the paper.

This research has been supported by the DFG, German Research Foundation (grant no. 274762653).

This paper was edited by Chris Forest and reviewed by four anonymous referees.