Sunspot Series Research Articles

Parameter estimation based on stacked regression and evolutionary algorithms

A new parameter-estimation algorithm, which minimises the cross-validated prediction error for linear-in-the-parameter models, is proposed, based on stacked regression and an evolutionary algorithm. It is initially shown that cross-validation is very important for prediction in linear-in-the-parameter models using a criterion called the mean dispersion error (MDE). Stacked regression, which can be regarded as a sophisticated type of cross-validation, is then introduced based on an evolutionary algorithm, to produce a new parameter-estimation algorithm, which preserves the parsimony of a concise model structure that is determined using the forward orthogonal least-squares (OLS) algorithm. The PRESS prediction errors are used for cross-validation, and the sunspot and Canadian lynx time series are used to demonstrate the new algorithms.

A fuzzy identification procedure for nonlinear time series: With example on ARCH and bilinear models

In this paper we propose a detecting scheme for nonlinear time-series model classification by using a knowledge-based construction and fuzzy statistical decision making. ARCH and bilinear models are frequently applied in economic or financial time-series modeling, and both models exhibit certain kind of pattern similarity, such as unusual jumps and a diversity trend. So we take these two models as our illustration example for demonstration. Simulation results presented here demonstrate that our detecting procedure can classify ARCH and bilinear models effectively. The designed detection process also exhibits a significant rate of correct recognition for sunspot series and exchange rates.

Hierarchical autoregression using fuzzy measures

A hierarchical autoregressive model of time series using fuzzy measures and the Choquet integral is presented. The fitting and forecasting quality of the presented model is analysed by applying it to well-known time series data (the sunspot series) and compared with the forecasting qualities of other conventional models.

Optimization of bilinear time series models using fast evolutionary programming

This letter presents a new algorithm, fast evolutionary programming (FEP), for determining the model orders and parameters of reduced parameter bilinear (RPBL) models used for predicting nonlinear and chaotic time series. FEP is a variant of the conventional evolutionary programming (EP) algorithm with a new mutation operator. This new mutation operator enhances EP's ability to escape from local minima resulting in a significantly faster convergence to the optimal solution. Both the model order and the parameters are evolved simultaneously. Experimental results on the sunspot series and Mackey-Glass series show that FEP is capable of determining the optimal model order and, in comparison with conventional evolutionary programming, evolves models with lower normalized mean squared error.

Local S-spectrum analysis of 1-D and 2-D data

The local changes of the spectrum with time are often more interesting than the spectrum of the whole time series. For example, there is an apparent drift in the nominal 28 day fluctuations of sunspot numbers over the period of the sunspot cycle, averaging ∼ 11.1 years. This time-local change in spectrum is due to a combination of Spörer's Law and the differential rotation of the sun. Similarly, the space-local variations in the 2-D spectrum on an image conveys visual information on textures, boundaries and shapes. In this paper we use the recently developed S-transform to analyse two segments of the Wolf Sunspot series, a seismogram, and a synthetic 2-D image as examples of applications of the S-transform for time-local and space-local spectral analysis.

Forecasting chaotic time series with genetic algorithms

This paper proposes the use of genetic algorithms---search procedures, modeled on the Darwinian theories of natural selection and survival of the fittest---to find equations that describe the behavior of a time series. The method permits global forecasts of such series. Very little data are sufficient to utilize the method and, as a byproduct, these algorithms sometimes indicate the functional form of the dynamic that underlies the data. The algorithms are tested with clean as well as with noisy chaotic data, and with the sunspot series.

Combining Neural Network Forecasts on Wavelet-transformed Time Series

We discuss a simple strategy aimed at improving neural network prediction accuracy, based on the combination of predictions at varying resolution levels of the domain under investigation (here: time series). First, a wavelet transform is used to decompose the time series into varying scales of temporal resolution. The latter provides a sensible decomposition of the data so that the underlying temporal structures of the original time series become more tractable. Then, a dynamical recurrent neural netork is trained on each resolution scale with the temporal-recurrent backpropagation algorithm. By virtue of its internal dynamic, this general class of dynamic connections network approximates the underlying law governing each resolution level by a system of non-linear difference equations. The individual wavelet scale forecasts are afterwards recombined to form the current estimate. The predictive ability of this strategy is assessed with the sunspot series.

There Is Nothing New under the Sunspots

Life on earth is totally dependent on solar energy. Consequently, it is a straightforward assumption that any regularity in solar activity should be observed in at least some biotic production processes as well. Indeed, the number of sunspots, reflecting solar activity, is known to fluctuate with regular intervals, period length averaging 10-11 yr (Waldmeier 1961). Although it is controversial whether this regular fluctuation in solar activity causes any weather variable to fluctuate in phase with sunspot numbers (Molnar 1981) it is tempting to think so. Presumably this was the reasoning behind Charles Elton's classic paper (1924) in which he states that periodic fluctuations in the numbers of certain animals must be due to climatic factors. Elton (1924) suggested that the causal link between the fluctuation in sunspot numbers and animal populations are recurring favorable climatic conditions for vegetation due to solar activity. It is interesting that although Elton, as well as several others, has abandoned the above reasoning later on (MacLulich 1937, Elton and Nicholson 1942, Moran 1949, 1953a, b, Royama 1992), the idea of sunspot regulated population dynamics has survived over decades, as recently evidenced by Sinclair et al. (1993). They suggest that there is a solar cycle-climate-hare population link. As snowshoe hare, Lepus americanus, is the main prey of lynx, Lynx canadensis (e.g., Parker et al. 1983, Koehler 1990), it is a reasonable assumption that changes in the hare population numbers should affect the lynx fluctuation as well. Here, our aim is to explore the MacKenzie River series of lynx fur returns 1821-1934 (Elton and Nicholson 1942) and Wolfer sunspot series for the corresponding period (Waldmeier 1961) with the aid of time series analysis. We shall use an autoregressive (AR; Box and Jenkins 1970) time series model as an adequate enough characterization of the process:

Quasi-biennial oscillation in ionospheric parameters measured at Juliusruh (55°N, 13°E)

When seasonal variations were eliminated by evaluating 12-month running means, the ionospheric parameters foE, foF2 and hmF2 at Juliusruh (54.6°N, 13.4°E) showed large solar cycle variations. However, when further 3-yr running averages were evaluated and subtracted, QBO (Quasi-biennial oscillations) were noticed in all these parameters. Sunspot series did not reveal a QBO, but geomagnetic Ap did show a QBO. The peaks of the ionospheric QBO and QBO of Ap could be roughly compared, with lags or leads of a few months. Also, these compared roughly with the well-known QBO peaks of tropical stratospheric (50 mb) zonal winds. Similar analyses at other locations are warranted.

Solar Rotation, Irradiance Changes and Climate

Magnetic activity cycles for solar-type stars are believed to originate from non-uniform internal rotation. To determine this depthwise angular velocity distribution, helioseismology is a valuable source of information. Surface rotation, as traced by sunspot motion, is a well-observed parameter with data going back to the beginning of the telescopic era. This long sunspot series can be used in understanding the behaviour of the Sun’s surface rotation, the connection with its internal rotation, and thereby its magnetic activity. Apparent solar diameter is another important parameter. This is related to the structure of the convective envelope and how it reacts to the presence of magnetic fields. Both these parameters are related to the solar output, and can provide a surrogate for total solar irradiance, by way of a theoretical modeling of the response of the convective zone to the emergence of periodic magnetic fields. The impact of solar variability on the terrestrial climate is also addressed.

Secular variation of the aurora for the past 500 years

Knowledge of solar variability and its effects on the Earth is essential since the Sun affects almost every aspect of our lives. Direct observations of the Sun, usually of sunspots, with some continuity, exist only since about 1700. Understanding of long‐term solar variability must then depend on proxy data, such as visual auroral observations, measurements of magnetic activity, and the radiocarbon record. These also give us information on the interaction between the solar wind, the interplanetary field, and the terrestrial magnetosphere, as well as, for the radiocarbon record, heliospheric conditions. This paper uses a data base of visual auroral observations for a period of about 500 years, from 1450 to 1948, comprising about 45,000 observations, in addition to the well‐known sunspot series and the magnetic activity index aa, from 1868 to 1990. The secular variation of the aurora is examined and compared, where possible, to sunspot data and magnetic activity data. Blackman‐Tukey power spectra are used to determine periodicities. The study confirms the variability of the periodicities in both frequency and amplitude. In particular, the well‐known 11.1‐year cycle disappears during the Maunder minimum and at the end of the eighteenth and beginning of the nineteenth century. While the 11.1‐year period is normally strongly dominant for sunspots, other shorter periods become important, and even dominant, for auroras and magnetic activity. Consequently, the temporal behavior of these three variables differs. Prolonged solar activity minima are clearly evident. In addition to the known Spörer, Maunder, Dalton, and 1901–1913 minima, a previously unrecognized minimum about 1765 is clearly evident in the data. Comparison of the depth of these minima shows that the Dalton minimum may be the deepest, or at least rivals the Maunder minimum in importance. This minimum clearly deserves further study. Combining the polar data base with that of mid‐latitudes provides for the first time a globally comprehensive historical record of auroral occurrence. The data provide confirmation of the anticorrelation of auroral occurrence in the polar regions with sunspot activity, as a result of displacement of the auroral oval with changes in solar and magnetic activity. Assuming the validity of some current models of the solar origin of geomagnetic activity, the data provide a basis for understanding the variation over time of the general magnetic field of the Sun, in particular the polar field.

Spectral analysis of annual sunspot series?An update

Maximum Entropy Spectral Analysis (MESA) of annual sunspot series for 1700–1987 (288 points) indicated the most prominent peak atT=10.91 years, followed byT=10.62, 10.00 and 103 years of comparable amplitudes and twelve others of smaller amplitudes, all significant at a 3σa priori level.MacDonald (1989) has suggested that the method of spectral analysis should depend upon the choice of the chosen model. However, our results compare well with his results from a periodogram study, indicating that the choice of the model, generally difficult in geophysical problems, is not vital. Since amplitude estimates in MESA are unreliable, our prescription is to use MESA only for detecting possible periodicities and then, estimate amplitudes and phases by a separate Multiple Regression Analysis.

Evolutive spectral analysis of sunspot data over the past 300 years

We analyse the series of the Wolf sunspot number in the frequency domain to determine the dimension of the solar cycle system by using the properties of its strange attractor and to study the stability in time of this dimensionality and of the main quasiperiodicities. The two classical methods of time series analysis, Fourier harmonic and Blackman-Tukey spectral analysis, have been applied first to the series of the annual Wolf sunspot numbers to determine its overall character. To detect stationarity, periodic regression based upon the three most statistically significant quasi-periods and especially a moving form of the maximum entropy spectrum analysis (mesa) have been used. Both analyses show a splitting of the 11-year cycle before 1800, when a ± 55-year cycle is dominant, and a single 11-year and + 100-year peak after 1800. Moreover, these quasi-periods are very sensitive to the time interval over which the analysis is carried out. The reason is that the sunspot numbers constitute a widely non-stationary process, which therefore implies that Fourier techniques are not useful to predict solar activity and must be used as fitting procedure only. The minimum cross-entropy method serves to improve the maximum entropy spectrum. With a good a priori estimate and data containing a low noise level, this method allows the detection of very close peaks and the refinement of the main frequencies; it does not split nor introduce artificial peaks. The Thomson model was also applied for its superior bias control, its excellent leakage resistance and a better statistical information. The same methods were then used to study the 22-year magnetic cycle, which is formed by taking into account the change in polarity of the succeeding 11 -year cycle. The moving form of mesa confirms the 22-year cycle to be highly stable in contrast to the instability in the period of the 11-year sunspot series. This suggests the importance of working with the more invariant 22-year magnetic series to explain the complex, non-stationary behaviour of the sunspot series and of the solar—terrestrial interactions. Finally, we tried to see if the system generated by the sunspot data was allowing the existence of an attractor and tried to determine the minimum number of variables necessary to describe this system. It is shown that the dimension of the attractor is highly unstable varying from 2.21 to 4.95 in a quasi-cyclic way.

Precambrian cyclic rhythmites: solar-climatic or tidal signatures ?

For more than 60 years geologists have sought evidence of solar-climatic cyclicity in rhythmically laminated sedimentary rocks, but claims in general have not been persuasive. Three Precambrian rhythmite sequences in Australia that comprise varve-like laminae recently have received attention, however, as their conspicuous cycles ofca. 10-14 and/or 20-25 laminae have been ascribed a sunspot-cycle origin. They are the 2500 Ma old Weeli Wolli Formation, the 1750 Ma old Wollogorang Formation, and the 650 Ma old Elatina Formation. New observations for the Weeli Wolli Formation, a siliceous banded iron-formation, suggest a cycle period exceeding the 23 microband couplets proposed by Trendall, casting doubt on the solar interpretation. The Weeli Wolli cyclicity may record Earth-tidal rhythms that modulated the discharge and composition of silica- and iron-bearing fumarolic waters. The structure of the cycles of silty dolomite and mudstone in the Wollogorang Formation does not support a sunspot-cycle origin, and a tidal control on sedimentation should be considered. The Elatina sequence of cyclic sandstone and silts tone laminae displays several empirical similarities to the sunspot series. The discovery of thicker, more complex lamina-cycles in the correlative Reynella Siltstone has, however, caused reappraisal of the solar interpretation of the Elatina rhythmites. The Reynella cycles consistently contain 14 or 15 laminae, with many laminae comprising pairs of‘semi-laminae’; the overall structure of the cycles is similar to the mixed (semidiurnal and diurnal) fortnightly tidal growth patterns of modern bivalves. Also, recent observations that fine material can be transported in suspension to deeper waters offshore by ebb-tidal currents provides a tidal mechanism that can explain the deposition of thin, laterally extensive, graded laminae whose thickness would be a measure of tidal range and current speed. By this tidal model, the Elatina laminae are regarded as diurnal increments, and the lamina-cycles as commonly abbreviated fortnightly groupings. The Elatina series so interpreted may encode unique information on lunar orbital periods and the Earth’s palaeorotation: the data indicateca. 30.5 days per lunar month, 13.1 lunar months andca. 400 days per year, and lunar apsides and lunar nodal cycles of 9.7 andca. 19.5 years respectively some 650 Ma ago. Evidence for significant modulation of terrestrial climate by the solar activity cycle in the geological past may prove as elusive as have convincing indications of solar signals in modern patterns of weather and climate.

PREDICTING THE FUTURE: A CONNECTIONIST APPROACH

We investigate the effectiveness of connectionist architectures for predicting the future behavior of nonlinear dynamical systems. We focus on real-world time series of limited record length. Two examples are analyzed: the benchmark sunspot series and chaotic data from a computational ecosystem. The problem of overfitting, particularly serious for short records of noisy data, is addressed both by using the statistical method of validation and by adding a complexity term to the cost function ("back-propagation with weight-elimination"). The dimension of the dynamics underlying the time series, its Liapunov coefficient, and its nonlinearity can be determined via the network. We also show why sigmoid units are superior in performance to radial basis functions for high-dimensional input spaces. Furthermore, since the ultimate goal is accuracy in the prediction, we find that sigmoid networks trained with the weight-elimination algorithm outperform traditional nonlinear statistical approaches.

On the persistence of the 22 y solar cycle

We briefly discuss the existence of precise periodicities of the Hale cycle (∼-22 y), the Gleissberg cycle (∼-88 y), and ∼-132 y cycle, in various direct and indirect indicators (as aurorae, 14C from tree rings and 10Be from polar ice) of solar activity. We consider also the behaviour of the 11y cycle and its first harmonic in modern sunspot series. It appears that the frequencies of ∼-1 c/88 y and ∼-1 c/132 y might be two subharmonics of the Hale cycle. The results support the hypothesis that the Sun behaves as a nonlinear system forced by an oscillator having the Hale frequency. The forcing element inside the Sun can be identified with a magnetofluid torsional oscillator.

Cyclicity in the late Precambrian Elatina Formation, South Australia: Solar or tidal signature?

New observations from two late Precambrian (~800-650 Ma) sedimentary rhythmites in South Australia (Reynella Siltstone and Chambers Bluff Tillite) reveal cycles structurally similar to the ~12-laminae cycles of the ~650-Ma Elatina Formation but which comprise from 15 to 26 or more laminae. The new data are difficult to accomodate in depositional models for the Elatina Formation whereby its laminae are regarded as annual and the cyclicity ascribed to solar variability (Williams and Sonett, 1985) or to the combined influences of the sunspot cycle and the lunar nodal tide (Zahnle and Walker, 1987). If, however, the long-term ‘Elatina Cycle’ and not the individual lamina is taken as a yearly climatic signal, the Elatina and other rhythmites studied may be interpreted as the deposits of marine ebb-tidal deltas that record variability in the velocity and range of paleo-ebb tides. The basic laminae-cycles would represent lunar fortnightly cycles of diurnal and/or semidiurnal laminae, commonly truncated through non-deposition at neap tides. Despite numerous empirical similarities between the Elatina and sunspot series, the ebb-tidal model for deposition is preferred because it accounts plausibly for observations from all three rhythmites studied.

The thermoluminescence profile of a recent sea sediment core and the solar variability

The analysis of the thermoluminescence (TL) profile of the GT14 recent sea sedimentary core shows the existence of four main periodicities of 137.7, 59,12.06, and 10.8 years. Here we discuss the affinity of these waves to the known cycles of solar variability. The beats of the two high frequency components produce a modulated wavetrain with a carrier wave of 11.4 years and an amplitude modulation with period 206 years. The minima of this squared amplitude modulation fall in 1810 and 1913 A.D. and closely correspond to the periods of lowest solar activity as indicated by the sunspot series. The sum of the two low frequency waves can in turn be rewritten as a component with period 82.6 years which is amplitude modulated by a second component with period of 206 years. The 82.6-yr wave has the period commonly attributed to the Gleissberg cycle of solar activity. The maxima of the 82.6-yr wave occur in agreement with the dates of maximum solar radius as suggested by Gilliland (1981).

Spectral analysis of the Elatina series

The Elatina formation in South Australia, which provides a rich fossil record of presumptive solar activity in the late Precambrian, is of great potential significance for the physics of the Sun because it contains laminae grouped in cycles of about 12, an appearance suggestive of the solar cycle. The actual spectrum of the lamina-thickness series is rather complex, 20 or more spectral lines having been recognized by Fourier analysis. It is shown how these numerous lines arise as combination frequencies, from a much simpler intrinsic spectrum, by rectification. Optical studies of the Sun have shown that there is a magnetic polarity reversal on the Sun every 11 years approximately, but terrestrial consequences of solar activity, for example in the ozonosphere or ionosphere, do not respond to solar magnetism; thus the negative-going semi-cycles of the full magnetic cycle are in effect rectified according to a linear law. Application of this knowledge to the Elatina formation shows that derectification simplifies the spectrum of the laminathickness series in exactly the way that one would expect if the solar cycle were at work here also. Zig-zag effect, an alternation of cycle thickness, is taken to be due, not to a beat phenomenon, but to rectification in the presence of a weak 345-year oscillation; subtraction of this oscillation after derectification is essential to the simplifying procedure. The fundamental period is established at a new sharper value of 23.7 ± 0.2 years as compared with the looser 22.2 ± 1.8 years for the modern sunspot series. This paper treats the laminae as varves laid down yearly and modulated in thickness in accordance with the late Precambrian sunspot activity for the year of deposition. Since the difference between 23.7 and 22.2 is less than a standard deviation it is premature to speculate that the sunspot cycle period has undergone secular change; indeed the possibility that the solar oscillator has been secularly stable is not ruled out. The high Q now demonstrated for the varve oscillator (around 120 compared with a previous value of 12) weakens support for that part of solar dynamo theory that ascribes the solar cycle to a self-sustaining relaxation osculation; conversely, the evidence for an internal solar clock mechanism is strenghtened. A wave propagation zone intervening between the clock and the solar surface could produce the intrinsic spectrum.

Forecasting by volterra type models

An inverse Volterra type model for fitting time series data is investigated here. Based on polyspectral analysis of the process an identification and estimation technique is developed, the algorithm for which is presented here. The methodology has been applied to two well-known nonlinear processes viz., the Canadian Lynx Series and the Wolfer's Sunspot Series. A one step ahead predictor is constructed, forecast results show considerable improvement over linear fitting and in some situations may be better than those obtained from bilinear modelling.