Climate Variability and Forecasting: Statistical, Stochastic and Machine Learning Approaches

Definition. This entry provides an overview of climate variability and forecasting using statistical, stochastic, and machine learning approaches. It focuses on time-series analysis, Markov chain models, long-memory processes, ARIMA/ARFIMA modelling, and the analysis of large-scale climate phenomena such as El Niño–Southern Oscillation (ENSO). It also explores solar activity and potential solar–terrestrial relationships with climate variability, highlighting modelling techniques, forecasting applications, methodological challenges, and future research directions. The entry aims to provide an accessible and scientifically grounded resource for researchers, students, and professionals in climate science, environmental statistics, and space science.

1. Introduction

Climate variability is the natural fluctuation of climate variables, such as temperature, precipitation, pressure and sea-surface temperature, around their long-term mean state. These fluctuations occur on timescales from months to decades. They arise from internal ocean–atmosphere interactions, external forcings (volcanic eruptions, solar output, greenhouse gases) and the interaction of the two. Anticipating them matters for agriculture, water management, energy planning and disaster preparedness.

Two broad strategies are used for prediction. Dynamical models solve the physical equations of the climate system numerically. Statistical, stochastic and machine learning models instead learn patterns directly from observed records. The second group is cheaper to run, often easier to interpret, and frequently competitive at seasonal-to-interannual horizons. This entry concentrates on that group and on its application to ENSO and to solar activity, a possible external driver of climate variability.

2. Climate Data as Time Series

A climate record is a time series: a set of observations ordered in time, usually at regular intervals. Such series are commonly described as a combination of trend, seasonal cycle, quasi-periodic oscillations and irregular noise. Climate series often violate the assumptions of classical statistics. They are autocorrelated, may be non-stationary, may be non-Gaussian (precipitation is the typical example), and may contain several interacting timescales.

Standard preprocessing includes removing the seasonal cycle to form anomalies, detrending, testing stationarity, and examining the autocorrelation (ACF) and partial autocorrelation (PACF) functions. These diagnostics guide model choice. The same workflow applies to climate indices such as ENSO and to solar-activity indices such as sunspot number, the coronal index and the K-index.

3. Statistical and Stochastic Models

3.1 ARIMA models

The autoregressive integrated moving average (ARIMA) family models a series using its own past values and past forecast errors. An ARIMA(p, d, q) model combines p autoregressive terms, d differencing steps to remove non-stationarity, and q moving-average terms. Model orders are chosen from the ACF and PACF and from information criteria such as AIC or BIC, and adequacy is checked by testing whether residuals resemble white noise.

ARIMA models have been fitted to solar-activity cycles. Akhter et al. [1] developed predictive ARIMA models for cycles of the coronal index, a measure of solar activity in the Sun’s corona. Related autoregressive modelling, again guided by ACF and PACF, has been applied to coronal mass ejection cycles [2]. Such models give short-range forecasts of solar variability with prediction intervals, which is useful because solar activity is a candidate input for climate forecasting.

3.2 Markov chain models

A Markov chain describes a system that moves between discrete states, where the next state depends only on the current one (first order). It is specified by a transition probability matrix whose entries give the chance of moving from one state to another. Markov chains provide the long-run state frequencies, the expected duration of events and the probability of transitions. They suit ENSO naturally, because it can be classified into three states: El Niño, La Niña and non-active (neutral).

Markov methods have been used to examine the relationship between solar activity and ENSO. Hassan et al. [3] applied a Markov method to sunspots and ENSO. Akhter et al. [4] studied the periodic behaviour of coronal index cycles and the associated ENSO using a Markov process, and a related analysis considered El Niño, La Niña and non-active ENSO events in relation to solar activity [5]. The limitations of this approach are the memoryless assumption, the loss of information caused by discretizing continuous variables, and the difficulty of choosing the chain order.

3.3 Long-memory processes and ARFIMA/FARIMA

Many hydro-climatic and solar series show long memory (long-range dependence): their autocorrelations decay slowly, as a power law, rather than exponentially. The strength of this persistence is summarized by the Hurst exponent H. The autoregressive fractionally integrated moving average model, called ARFIMA or FARIMA, extends ARIMA by allowing the differencing parameter d to be fractional. For 0 < d < 0.5 the process is stationary with long memory, and H = d + 0.5.

FARIMA has been applied to solar data. Hassan et al. [6] modelled the full sunspot series (1932–2014) and the four Hale cycles with FARIMA models, and a further study examined heavy-tail behaviour of sunspot cycles within a stochastic-modelling framework [7]. Estimating d reliably requires long records, and apparent long memory can be produced by trends or regime changes, so results need careful interpretation.

3.4 Fractal dimension, persistence and probability distributions

Fractal dimension offers another way to quantify persistence. It is linked to the Hurst exponent, and values that indicate persistent behaviour imply that increases tend to be followed by increases. The approach has been used for the solar–terrestrial relationship [8] and for solar flare hemispheric data compared with K-index geomagnetic activity [9]. The same technique has also been applied to non-climatic series, such as stock-exchange index data, to detect persistence epoch by epoch [10].

Fitting probability distributions to whole cycles is a complementary descriptive tool. It has been applied to solar flare data from the northern and southern hemispheres [11], to coronal index cycles [12], to sunspots and K-index data [13], and to the 24 sunspot cycles together with geomagnetic K-index activity [14]. These fits describe the length, duration and peakedness of cycles and support probabilistic statements about future activity.

3.5 Trend and harmonic models of solar cycles

Where cycles are regular, deterministic components can be combined with stochastic ones. Sinusoidal models have been fitted to sunspot cycles [15], and linear and non-linear techniques have been compared for modelling sunspot maxima and minima [16]. Such models are simple and interpretable, but they assume a stable period and amplitude, which real cycles do not strictly have.

4. Multiscale and Spectral Methods

Because climate and solar signals contain oscillations at several scales, spectral and multiscale tools are important. Fourier analysis identifies dominant periodicities but assumes stationarity. Wavelet analysis localizes variability in both time and frequency, showing how the strength of an oscillation, such as the two-to-seven-year ENSO band, changes over time. The discrete wavelet transform splits a signal into approximation (low-frequency) and detail (high-frequency) components, which can be examined separately. Akhter et al. [17] used the discrete wavelet transform to decompose coronal index and ENSO signals over six solar cycles and compared the detail and approximation components of the two series. Decompositions of this kind can also be combined with forecasting models to improve accuracy.

5. Machine Learning Approaches

Machine learning methods learn nonlinear relationships without a specified functional form. Commonly used methods include support vector regression, random forests, gradient boosting and artificial neural networks. Recurrent networks, especially long short-term memory (LSTM) networks, are designed for sequences and can retain information over many time steps. Convolutional networks process gridded fields such as sea-surface temperature maps.

Hybrid approaches are common: wavelet or decomposition steps combined with neural networks, or ARIMA combined with neural networks so that the linear part and the nonlinear residual are modelled separately. The stochastic and multiscale models in Sections 3 and 4 [1,6,17] are natural baselines and building blocks for such hybrids. Machine learning is limited by small sample sizes (a few decades of reliable observations), overfitting, sensitivity to non-stationarity and limited interpretability. Physics-informed designs and explainable-AI techniques are being developed to address the last point.

6. El Niño–Southern Oscillation

ENSO is the dominant mode of interannual climate variability. It is a coupled ocean–atmosphere phenomenon in the tropical Pacific, with warm (El Niño) and cold (La Niña) phases, and it influences weather worldwide through teleconnections. It is monitored through indices such as Niño 3.4 sea-surface temperature anomalies, the Oceanic Niño Index and the Southern Oscillation Index.

ENSO is a standard test case for the approaches above. Markov chains describe its phase transitions [3–5], and wavelets reveal its changing periodicity [17]. Forecast skill drops sharply for predictions made across the boreal spring, a limitation known as the spring predictability barrier. Decadal modulation and changes in ENSO behaviour under warming further complicate forecasting.

7. Solar Activity and Climate Variability

The Sun’s activity varies over the average 11-year sunspot cycle, and one solar magnetic cycle spans about 22 years, because the magnetic poles reverse polarity after each sunspot cycle [13]. Solar activity is described by indices such as sunspot number, solar flare index, the K-index of geomagnetic activity and the coronal index [9,11,12]. Proposed physical pathways to climate include changes in the ultraviolet part of the solar spectrum acting through the stratosphere, and changes in irradiance acting through the ocean surface.

Statistical studies search for solar signals in climate records using correlation, wavelet analysis, Markov-chain linkage, fractal dimension and regression [3–5,8,17]. These studies face serious difficulties. Climate records are short compared with the number of solar cycles, other forcings and internal variability mask any signal, and apparent correlations can arise by chance between two oscillating series. Solar–climate relationships nevertheless remain relevant for understanding regional and decadal variability and for improving forecasting inputs.

8. Forecast Evaluation and Methodological Challenges

Forecasts should be assessed on out-of-sample data, using measures such as root mean square error, mean absolute error, anomaly correlation and skill scores relative to a baseline like climatology or persistence. Cross-validation must respect time order (for example, rolling-origin evaluation), because random shuffling leaks future information. Key challenges are:

• non-stationarity caused by climate change;
• short and noisy records and measurement inhomogeneities;
• the risk of spurious correlations when many predictors are tested;
• the trade-off between accuracy and interpretability;
• quantifying uncertainty with probabilistic rather than single-value forecasts.

9. Future Directions

Promising directions include hybrid models that combine physical knowledge with statistical learning, probabilistic and ensemble machine learning forecasts, and multiscale methods that handle non-stationarity. Combining wavelet decomposition [17] with ARIMA/FARIMA [1,6] and neural network forecasters is a natural extension of the work cited above. Improved treatment of long memory, regime changes and extreme events will also be important. For solar–terrestrial studies, longer and better-homogenized records and rigorous significance testing against realistic noise models are needed.

10. Conclusions

Statistical, stochastic and machine learning methods form a complementary toolkit for understanding and predicting climate variability. ARIMA and FARIMA models describe linear and persistent behaviour, Markov chains describe state transitions, fractal and distributional measures characterize persistence and cycle shape, multiscale methods separate overlapping oscillations, and machine learning captures nonlinearities. ENSO shows both the value and the limits of these tools, and the possible influence of solar activity illustrates the need for cautious, well-tested statistical inference. Combining these approaches with physical understanding is the most promising route to reliable climate forecasts.

This entry is adapted from: https://www.researchgate.net/profile/Danish-Hassan-2?ev=hdr_xprf