Wolf Sunspot Research Articles

Weighted MLP-ARIMA series hybrid model for time series forecasting

With the increasing importance of forecasting with the utmost degree of accuracy, utilizing hybrid frameworks become a must for obtaining more accurate and more reliable forecasting results. Series hybrid methodology is one of the most widely-used hybrid approaches that has encountered a great amount of popularity in the literature of time series forecasting and has been applied successfully in a wide variety of domains. In such hybrid methods is assumed that there is an additive relationship among different components of time series. Thus, based on this assumption, various individual models can apply separately on decomposed components, and the final forecast can be obtained. However, developed series hybrid models in the literature are constructed based on the decomposing time series into linear and nonlinear parts and generating linear-nonlinear modeling order for decomposed parts. Another assumption considered in the traditional series model is assigning equal weights to each model used for modeling linear and nonlinear components. Thus, contrary to traditional series hybrid models, to improve the performance of series hybrid models, these two basic assumptions have been violated in this paper. This study aims to propose a novel weighted MLP-ARIMA model filling the gap of series hybrid models by changing the order of sequence modeling and assigning weight for each component. Firstly, the modeling order is changed to nonlinear-linear, and then Multi-Layer Perceptron Neural Network (MLPNN) -Auto-Regressive Integrated Moving Average(ARIMA) models are employed to model and process nonlinear and linear components respectively. Secondly, each model's weights are computed by the Ordinary Least Square (OLS) weighting algorithm. Thus, in this paper, a novel improved weighted MLP-ARIMA series hybrid model is proposed for time series forecasting. The real-world benchmark data sets, including Wolf's sunspot data, the Canadian lynx data, and the British pound/US dollar exchange rate data, are elected to verify the effectiveness of the proposed weighted MLP-ARIMA series hybrid model. The simulation results revealed that the weighted MLP-ARIMA model could obtain superior performance compared to ARIMA-MLP, MLP-ARIMA, as well as the ARIMA and MLPNN individual models. The proposed hybrid model can be an effective alternative to improve forecasting accuracy obtained by traditional series hybrid methods.

Productivity of dairy cows and quality of milk-raw material with different cosmophysic activity

The results of studies on the relationship between cosmophysical activity and animal productivity (the rhythm of milk yield of cows and the chemical composition of milk, the biochemical composition of raw milk) are presented. It was found that during the phase of growth of the activity of the Sun and the increase in the values of the Wolf index (sunspot) in the first half of the 24-year eleven-year cycle of solar activity, the average value of Wolf (sunspot) numbers varied from 0 to 100. For 2-3 years, at the lower peak of solar activity, and 2-3 years - at the top peak of activity, minimal and unreliable correlation relationships were observed. The most stable connections occur in the middle of an upswing or a decrease in solar activity. An increase in the relationship between the studied indicators was observed when the animals achieved maximum productivity. With an increase in milk yield from 7500 kg to 8500 kg, and further to 10,000 kg, the correlation coefficient was + 0.37, + 0.52 and + 0.64, respectively. The maximum influence of solar activity Wolf (sunspot) numbers) on the average productivity of one animal in the household is manifested at the peak of solar activity growth. It has been established that during the years of the rise of solar activity with low absolute values of Wolf (sunspot) numbers, low yields are more likely, and in high solar activity years, higher dairy values (record) are more likely. It was noted that all the components of milk decreased on days of increasing solar activity. The correlation coefficients for protein and milk fat content with Wolf (sunspot) numbers were r = -0.34 and r = -0.22, P 0.95. Correlation analysis revealed a reliable correlation of bacterial contamination and the number of somatic cells r = +0.28 and r = +0.31 (at P 0.95), respectively, with the Wolf (sunspot) numbers. In the collection milk, the negative correlation by fat content r = -0.12, density r = -0.18, acidity r = -0.17, protein content r = -0.19, sugar content r = -0.14, minerals r = -0.17 was observed with an increase in solar activity.

Bayesian Spectral Analysis With Student-t Noise

We introduce a Bayesian spectral analysis model for one-dimensional signals where the observation noise is assumed to be Student-t distributed, for robustness to outliers, and we estimate the posterior distributions of the Student-t hyperparameters, as well as the amplitudes and phases of the component sinusoids. The integrals required for exact Bayesian inference are intractable, so we use variational approximation. We show that the approximate phase posteriors are Generalised von Mises distributions of order 2 and that their spread increases as the signal to noise ratio decreases. The model is demonstrated against synthetic data, and real GPS and Wolf's sunspot data.

Likelihood-Based Inference in Autoregressive Models with Scaled t-Distributed Innovations by Means of EM-Based Algorithms

This article applies the EM-based (ECM and ECME) algorithms to find the maximum likelihood estimates of model parameters in general AR models with independent scaled t-distributed innovations whenever the degrees of freedom are unknown. The ECME, sharing advantages with both EM and Newton–Raphson algorithms, is an extension of ECM, which itself is an extension of the EM algorithm. The ECM and ECME algorithms, which are analytically quite simple to use, are then compared based on the computational running time and the accuracy of estimation via a simulation study. The results demonstrate that the ECME is efficient and usable in practice. We also show how our method can be applied to the Wolfer's sunspot data.

An ARIMA‐ANN Hybrid Model for Time Series Forecasting

Autoregressive integrated moving average (ARIMA) model has been successfully applied as a popular linear model for economic time series forecasting. In addition, during the recent years, artificial neural networks (ANNs) have been used to capture the complex economic relationships with a variety of patterns as they serve as a powerful and flexible computational tool. However, most of these studies have been characterized by mixed results in terms of the effectiveness of the ANNs model compared with the ARIMA model. In this paper, we propose a hybrid model, which is distinctive in integrating the advantages of ARIMA and ANNs in modeling the linear and nonlinear behaviors in the data set. The hybrid model was tested on three sets of actual data, namely, the Wolf's sunspot data, the Canadian lynx data and the IBM stock price data. Our computational experience indicates the effectiveness of the new combinatorial model in obtaining more accurate forecasting as compared to existing models. Copyright © 2013 John Wiley & Sons, Ltd.

Indicative proxy evidences of the changing monsoonal precipitation during the last millennium over central west coast of India: A possible link to solar variability

Proxy records of paleomonsoonal variations were obtained, utilizing foraminiferal tracers in two shallow water sediment cores collected from the inner shelf region off Karwar, central west coast of India. The preliminary results record the average signature of relatively better monsoonal precipitation at around 1400–1410, 1575–1590 and 1750 AD with intermittent dry phases from around (1290–1355 AD), (1425–1535 AD) and (1600–1665 AD) due to general reduction in the intensity of the rainfall. A comparison with the solar variability (Wolf, Sporer and Maunder sun spot minima) during the last thousand years suggests a possible linkage between the rainfall pattern and solar variability. The presented results though preliminary in nature due to limited dating of the sediments put forth only a possibility of inter-correspondence between solar variability and rainfall intensity. To further augment the reasoning, a number of sediment cores with close spaced dating control need to be studied from geographically distinct regions.

Generalized and subset integrated autoregressive bilinear time series models with applications

Generalized integrated autoregressive bilinear model which is capable of achieving stationary for all non linear series is proposed and compared with subset generalized integrated autoregressive bilinear model using the residual variance to see which perform better. The parameters of the proposed models are estimated using Newton-Raphson iterative method and Marquardt algorithm and the statistical properties of the derived estimates were investigated. An algorithm was proposed to eliminate redundant parameters from the full order generalized integrated autoregressive bilinear models. To determine the order of the models, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) were adopted. Generalized integrated autoregressive bilinear models are fitted to Wolfer sunspot numbers and stationary conditions are satisfied. Generalized integrated autoregressive bilinear model performed better than subset generalized integrated autoregressive bilinear model.

A SOLAR CYCLE LOST IN 1793-1800: EARLY SUNSPOT OBSERVATIONS RESOLVE THE OLD MYSTERY

Because of the lack of reliable sunspot observation, the quality of sunspot number series is poor in the late 18th century, leading to the abnormally long solar cycle (1784--1799) before the Dalton minimum. Using the newly recovered solar drawings by the 18--19th century observers Staudacher and Hamilton, we construct the solar butterfly diagram, i.e. the latitudinal distribution of sunspots in the 1790's. The sudden, systematic occurrence of sunspots at high solar latitudes in 1793--1796 unambiguously shows that a new cycle started in 1793, which was lost in traditional Wolf's sunspot series. This finally confirms the existence of the lost cycle that has been proposed earlier, thus resolving an old mystery. This letter brings the attention of the scientific community to the need of revising the sunspot series in the 18th century. The presence of a new short, asymmetric cycle implies changes and constraints to sunspot cycle statistics, solar activity predictions, solar dynamo theories as well as for solar-terrestrial relations.

CYCLES AND UNIVERSALITY IN SUNSPOT NUMBER FLUCTUATIONS

We analyze the famous Wolf's sunspot numbers. Surprisingly, we discovered that the distribution of the sunspot number fluctuations for both the ascending and descending phases is close to the universal nonparametric Bramwell-Holdsworth-Pinton (BHP) distribution. Since the BHP probability density function appears in several other physical phenomena, our result reveals a universal feature of the Wolf's sunspot numbers.

Estimation of semi-parametric additive coefficient model

In the multivariate regression setting, we propose a flexible varying coefficient model in which the regression coefficients of some predictors are additive functions of other predictors. Marginal integration estimators of the coefficients are developed and their asymptotic properties investigated. Under β -mixing, it is found that the estimators of the parameters in the regression coefficients have rate of convergence 1 / n , and the nonparametric additive components are estimated at the same rate of convergence as in univariate smoothing. A data-driven bandwidth selection method is developed based on asymptotic considerations. Its effectiveness is confirmed in a Monte-Carlo study. The procedure is applied to the real German GNP and Wolf's Sunspot data, where the semi-parametric additive coefficient model demonstrates superior performance in terms of out-of-sample forecasts.

Correlations of life-span variation parameters in 128 successive generations of Drosophila melanogaster with changes in atmospheric pressure and geomagnetic activity

Correlations between the parameters of life-span (LS) distribution of Drosophila melanogaster, including mean LS (MLS) and the time of 10 and 90% population mortality, and some geophysical parameters that are usually beyond the control of researchers dealing with laboratory cultures, including atmospheric pressure, solar activity indices (Wolf's sunspot numbers and 2,800-MHz radio flux), and geomagnetic activity (planetary index, K(p)), were studied. Geophysical data were obtained from free-access official web sites of the National Oceanic & Atmospheric Administration of the US Department of Commerce and the Institute of Terrestrial Magnetism and Radiowave Propagation of the Russian Academy of Sciences. The geophysical parameters were calculated only for the period corresponding to 10 days of preimaginal development of the flies from egg to imago. Canonical correlation analysis, calculation of the non-parametric Spearman rank-order correlation coefficients, and graphical data analysis were used. Highly significant correlations between parameters of LS distribution in males and females and environmental factors, such as the atmospheric pressure on the 4th and 5th day of development and geomagnetic activity indices (K(p)) on the 6th and 10th day of development were found, with correlation coefficients varying from 0.31 to 0.37 (P<0.02). Assuming a causal relationship between geophysical factors and LS, it may be hypothesized that energetically weak environmental factors determine the formation of LS oscillatory dynamics in laboratory populations. The possible mechanisms underlying the contribution of these environmental factors to the LS variation in successive generations are discussed.

A dynamic architecture for artificial neural networks

Artificial neural networks (ANN), have shown to be an effective, general-purpose approach for pattern recognition, classification, clustering, and prediction. Traditional research in this area uses a network with a sequential iterative learning process based on the feed-forward, back-propagation algorithm. In this paper, we introduce a model that uses a different architecture compared to the traditional neural network, to capture and forecast nonlinear processes. This approach utilizes the entire observed data set simultaneously and collectively to estimate the parameters of the model. To assess the effectiveness of this method, we have applied it to a marketing data set and a standard benchmark from ANN literature (Wolf's sunspot activity data set). The results show that this approach performs well when compared with traditional models and established research.

SOME IMPLICATIONS USING THE GROUP SUNSPOT NUMBER RECONSTRUCTION

The Wolf or Zurich sunspot number Rz, (values available from 1700 onwards) is considered the primary descriptor of solar activity for the past 100 years or more. Recently, another, more accurate time series, called the Group sunspot number, R G, has become available. Comparison of the indices suggests that Rzmay be grossly overestimated in the early sunspot records. In this paper, some results based previously on Rzare reexamined using R G. Conclusions based on data from about 1850 to the present remain the same. For the 18th century, however, small differences are seen. For estimates of solar irradiance in the Maunder Minimum (1645–1715), because both Rzand R Gwere near zero then and are almost comparable today, irradiance estimates remain about the same, whether Rzor R Gis used to calculate them. However, the evolutions of Rzand R Gfrom their near zero value in the Maunder Minimum to their present similar value, are different, with Rzincreasing more rapidly than R Gfrom 1700 to 1790. While the temperature of the Earth has increased almost monotonically for the past few centuries, the relationship of both Rzand R Gwith the temperature trend is not especially great.

ARMA MODEL SELECTION USING PARTICLE SWARM OPTIMIZATION AND AIC CRITERIA

This paper presents a new method for determining ARMA model parameters using Particle Swarm Optimization (PSO). PSO is a new optimization method that is based on a social-psychological metaphor. Each ARMA model is represented as a particle in the particle swarm. Particles in a swarm move in discrete steps based on their current velocity, memory of where they found their personal best fitness value, and a desire to move toward where the best fitness value that was found so far by all of the particles during a previous iteration. PSO is applied for determining the ARMA parameters for the Wolfer Sunspot Data. The method is extended using Akaike's Information Criterion (AIC). PSO is used to simultaneously optimize and select an estimated “best approximating ARMA model” based on AIC. Several plots are included to illustrate how the method converges for various PSO parameter settings.

Relative sunspot numbers in the first half of eighteenth century

We revised relative sunspot numbers in the time interval 1700 –1748 for which Wolf derived their annual means. The frequency of daily observations, counting simultaneously the number of sunspots and the number of sunspot groups necessary for determinating Wolf's relative sunspot numbers, is in this time interval very low and covers, on average, 4.8% of the number of all days only. There also exist incomplete observations not convenient to determine relative sunspot numbers. To enlarge the number of daily relative sunspot numbers we used the nonlinear, two-step interpolation method derived earlier by Letfus (1996, 1999). After interpolation, the mean value increased to 13.8%. Waldmeier (1968) found that the scaling factor k can be derived directly from the observed number of spots f and from the number of sunspot groups g. From the observations made at Zurich (Wolf and his assistants, Wolfer), at Peckeloh, and at Moncalieri during the years 1861–1928, we derived a new, more correct empirical relation. The resulting annual relative sunspot numbers are given in Table II. However, only for 26 years (53.0%) from the total number of 49 years was it possible to derive annual relative sunspot numbers. The observations were missing for the other years. This corresponds with results of Wolf, which gives the annual relative sunspot numbers for all 49 years. For the years when the data were missing, he marked these values as interpolated or very uncertain ones. Most of the observations originate from two data series (Kirch, Plantade), for which Wolf derived a higher scaling factor (k=2.0) than followed from the newly derived relation (k=1.40). The investigated time interval covers four solar cycles. After our results, the height of the first cycle (No. −4), given by Wolf, should be lowered by about two-thirds, the following two cycles (Nos. −3 and −2) lowered by one-third, as given by Wolf, and only the height of the fourth one (No. −1) should be unchanged. The activity levels of the cycles, as represented by group sunspot numbers, are lower by about one-fourth and, in the case of the first one (No. −4) even by two-thirds of the levels derived by us. The group sunspot numbers, derived from a much greater number of observations, have also greater credibility than other estimates. The shapes of the cycles, as given by Wolf, can be considered only as their more or less idealized form.

New Method of Time Series Analysis and Its Application to Wolf's Sunspot Number Data. II: Periodicities of The Daily Sunspot Numbers

The daily sunspot numbers data from 1979 October 1 to 1993 December 31 were analyzed using a new method of time series analysis proposed by the authors, and the long- and short-term periodicities were investigated. A three-dimensional (3D) spectral array of the normalized power spectral densities (PSD's) obtained by the segment time series analysis revealed the temporal behavior of the 27- and 155-d periodicities in the solar activity minimum. In the normalized 3D spectral array, the solar rotation cycle of 27-d accompanied by its multiplets was confirmed to migrate gradually from the low frequency region toward the high frequency region in the solar activity minimum as the solar activity increases, while the 155-d periodicity exists in the solar activity maximum only. Moreover, the behavior of the daily sunspot numbers variation was conjectured to be deterministic chaos, based on the exponential characteristics of the PSD's and the subharmonic relations of the spectral lines.

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:

Generalized autoregression and the analysis of dynamical processes

The dynamics of nonlinear systems can be characterized in terms of response functions. Mathematical representations of data that contain mixed stochastic-deterministic components are likely to have coefficients that are not well behaved. In these circumstances, statistical averaging can be used to obtain a moment hierarchy that has well behaved coefficients and that can be inverted for the response function values. The moment hierarchy is generated by operating on a convolution series expansion that is truncated so that a tractable set of equations with well behaved coefficients can be solved. The moment heirarchy is used in two contexts. First the analysis of noiseless deterministic mappings is considered as the precursor to the analysis of experimental data. A chaotic numerical example is used to demonstrate the accuracy of the moment heirarchy method to identify the order and from of the mapping and to predict the future behavior of the chaotic sequence. Second, Wolf's annual sunspot number, which is theoretically predicted to be a delayed logistic map, is analyzed and discussed. Finally, experimental data from a driven electronic anharmonic oscillator that exhibits period doubling and chaotic behavior is analyzed and discussed.

New Method of Time Series Analysis and Its Application to Wolf's Sunspot Number Data

A newly devised procedure of time series analysis, which is a linearized version of the nonlinear least squares method combined with the maximum entropy spectral analysis method, was proposed and applied to the annual sunspot number data from 1700 to 1991. Multiple periodicities of the temporal variation were elucidated in detail. The solar cycle of a 11.04-year period accompanied with the solar cycle multiplets, the periods of 50.41 years, the so-called “Yoshimura cycle”, and 107.11 years corresponding to the cycle of century-scale minima, for example, were clearly observed. The optimum least squares fitting curve for the data was extended over the past two millennia and the next millennium. The past grand minima such as the Maunder minimum were confirmed in the past extrapolation curve, and the next centennial minimum was predicted to occur between 2000 and 2030.

Predicting Wolf's sunspot numbers with and without the assumption of periodicity

This paper provides an alternate model to the Box Jenkins time series for forecasting sunspot numbers. Both periodic and nonperiodic models are considered. The purpose is to find a model which will give both direction and magnitude of the large changes in sunspot numbers but which also gives error terms comparable to those of the Box-Jenkins techniques. We will use the statistical technique of nonparametric density estimation to develop these models