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Blog Author Note: This is a paper written with Dr. Scott McIntosh of Lynker Space. A PDF of this article is available by clicking here. Reproduction with attribution is permitted.

Sunspots to amateur radio operators are central to daily operations, especially on the HF bands. Ever since Schwabe began counting and plotting daily observed sunspots in 1826, the leading perspective has been that sunspots follow a given, time-ordered, sinusoidal pattern of rise and fall, largely around an eleven-year cycle. The pattern of differences among cycles has been the topic of much speculation, generally without any actual empirical observation of those factors. Until recently, the sunspot cycle is virtually just a given construct based largely on daily sunspot counts, summarized to each month.

There is almost no theoretical discussion of sunspot cycle antecedents, only their effects on propagation. For instance, Nichols (2015) exclaimed, the “top experts” are unable to identify the peak or trough of the solar cycle or the timing of the transition from one cycle to another. Previous work, especially in amateur radio, has focused almost wholly on atheoretical “curve-fitting” style models of the sunspot cycle with observed sunspots gathered for almost two centuries being just a given phenomenon (see Howell and McIntosh 2022a).

With the open acknowledgement that previous predictions have not been very accurate, it is puzzling as to why better theoretical explanations have not been sought by amateur radio experts on propagation patterns (e.g. Luetzelschwab n.d.). This devotion to the theoretically unexplained eleven-year sine wave function as a sterile paradigm in the face of the empirical anomalies reduces scientific progress in understanding the sunspot cycle. Here is why.

Philosophers of science have long debated the role of prediction and explanation in scientific progress. Douglass (2009) summarizes the debate as follows. “Prediction is important because we can be surer that the scientist generating the theory has not fudged or somehow subtly made his theory inconsistent or less clearly applicable to certain contexts by virtue of some torturous, ad hoc accommodation. Prediction also allows for the generation of new (hopefully supporting) evidence. Explanation is important because it helps us think our way through to new predictions.” To make progress in the scientific understanding of the sunspot cycle, we need both theoretical understanding coupled with better predictive capability.

This was the thrust of our 2022 article series in RadCom. We published papers in the July and August issues outlining the long prevailing scientific paradigm on the sunspot cycle, noting that it was largely devoid of a formal theory predicting its rise and fall. We outlined a major challenging theoretical paradigm, led in its creation by the second author, on not only predicting Cycle 25 but offering the beginnings of why the amplitude and modulation of such cycles behave the way they do. In essence, this marked a change from mere prediction toward explanation, a cardinal sign of growth in any area of science. As we have reached the midpoint of Cycle 25, it is time to see how this argument is faring.

Competing Sunspot Cycle Paradigms

The expert panels convened by the NASA/NOAA/ISED organizations (hereafter, NNI) over the past several sunspot cycles have published their own forecasts of the next one. They have done this without any disclosure of the specific model used or the specific substantive theoretical perspective driving them. They do not disclose their methodology but state that it is the consensus opinion of an expert panel reviewing more than fifty various models submitted to them for consideration. Unlike most all peer-reviewed scientific work, the official sunspot cycle forecasts are a theoretically unexplained given resulting from an expert opinion panel whose deliberations are not open to public inspection. Their forecasts have largely failed to be very accurate when later compared to the observed sunspot numbers in the predicted cycle (see Howell and McIntosh 2022a,b for a full discussion).

The second author’s team, hereafter called the McIntosh team, developed both a theoretical foundation and empirical forecast of Cycle 25, publishing the methods they utilized and what substantive concepts shaped them. Unlike the official NNI forecasts, the McIntosh team’s work is public for all to read. We strongly encourage the readers of this paper to review our 2022 articles for details as they are indeed nuanced arguments.

Suffice it to say that the competing McIntosh paradigm emphasizes not the mere curve-fitting exercise that so many amateur radio prognosticators subscribe to in their forecasts (e.g., Cohen 2020) but two new key conceptual elements of the Sun’s dynamo. This was new ground. As we illustrate below, much of the scientific community resisted an open consideration of these ideas at the beginning.

One concept is the Terminator, a landmark event in the sunspot cycle delineating the start, end, and overlap of sunspot and magnetic activity cycles. This event does not correspond to the statistical minimum or maximum in the number of sunspots but to an underlying shift in part of the sun’s dynamo that shapes the entire cycle’s behavior. It arises from the famous Hale Magneto Cycle (Howell and McIntosh 2022a).

A second concept is the timing of the Terminator within the approximate eleven-year period. Taken from our earlier paper:

“This variability, when viewed through the lens of an insular sunspot cycle, lends itself to the anomalies noted by prominent amateur radio propagation enthusiasts. The delay in Termination frequently leads to the forecast of a poor cycle approaching, even another Maunder Minimum, by hams. More critically, the longer the time between terminators, the weaker the next cycle would be. Conversely, the shorter the time between terminator events, the stronger the next Solar Cycle would be. This is the cornerstone thesis in the new competing paradigm which successfully addresses several anomalies observed by Nichols (2015), Nichols (2016) and Luetzelschwab et al. (2021).” (Howell and McIntosh 2022a: 40).

We suggested in 2022 that the key question is whether we are indeed in a crisis stage of a paradigm shift, using the perspective of the well-established Kuhnian model of scientific revolutions (Kuhn 1962). The evidence of such a crisis state would include the following two elements. Firstly, if the competing McIntosh team’s model produces a better empirical forecast than the official NNI’s forecast does, then the theoretically-based paradigm that is a better forecast pushes toward a crisis state. Secondly, what further shapes a crisis state is if other scientists flock to the empirically-superior, theoretically-explicated one. This pattern of behavior, measured largely through citations of the competing paradigm’s exemplars, propagates the new paradigm to the field. If scientists use the competing paradigm’s exemplary papers to shape their own work, then the revolution is taking shape through the collective behavior of other scientists (Kuhn 1962).

We offer a narrated illustration in Figure 1 of the stages and processes of Kuhn’s classic explanatory model applied to these two competing paradigms. We begin on the right side of the wheel of paradigm change. Effectively, the initial “boundary maintenance,” or resistance by adherents to shift from the traditional paradigm embedded in the NASA/NOAA/ISED predictions, eventually gave way to peer reviewers’ objectivity. This occurred through reviewers and editors evaluating the increasingly massive empirical evidence based upon all sunspot cycles for which there were data constructed by the McIntosh team as they revised their initial 2012 work. This was not accomplished very quickly or very easily. As Kuhn (1962) stipulated, this is not at all unusual for a competing set of ideas which threaten the “normal science” embedded in a reigning paradigm.

Nevertheless, the existing normal science “puzzle solving” produced many anomalies in the prediction of both the amplitude and the timing of adjacent sunspot cycles. This acknowledgement that we just do not have a sufficient understanding to produce very accurate forecasts created increasing doubt in adherents to the current paradigm after facing the massive amount of evidence from the original McIntosh team’s paper.

The “boundary maintenance” from 2012 when Science Magazine rejected the initial paper on the new theory began to give way some years later. This occurred through the McIntosh team’s surprising Cycle 25 prediction of a far higher peak in sunspots than the NASA/NOAA/ISED (NNI) official predictions and why they made this forecast. Remember, the official sunspot forecast for Cycle 25 contained no explanations of how they were derived, only that a panel of experts came up with them. This “exemplar” article was published in 2020, some eight years after the initial “boundary maintenance” rejection in 2012. As these results are compared to the errors in previous expert panel forecasts by more and more scientists, this set in motion increasing collective doubt being attached to NNI’s undisclosed methods. This behavior is shown as “model drift” in Figure 1.

Once this model drift occurred after the McIntosh team’s Cycle 25 forecasts were published (2020), the empirical race was on to see which forecast would be more accurate. Modern website technology made this a monthly comparison with the release of each new count of sunspots (shown in Figure 2 below). When the second “exemplar” paper on the timing of the Terminator event appeared in 2023, this undoubtedly significantly enhanced the motivation of other scientists to read and consider the competing paradigm’s exemplars to use as a basis of their own work.

Modern technology speeds up scientific awareness of new works as compared to periodic print journal publication. So, this social network technology makes the process identified by Kuhn back in the 1960s as a “revolution” in paradigm-change an even more valid metaphor today. Should a growing number of other scientists base their published work on the exemplars of the McIntosh team, then direct “model competition” sets in. These collective acts by others in the scientific community are behaviorally manifested through increasing citations of the exemplars in the competing paradigm. If the competing paradigm’s empirical superiority continues, it is only a matter of time before the full model revolution occurs, quickly resulting in a rapid change to a new accepted paradigm. It is our assessment, as illustrated in Figure 1, that we are clearly in the model competition stage as we stand today.

Where Is Paradigm Competition at the Peak of Cycle 25?

In this paper, we evaluate the status of this potential revolution in our shared understanding of the important sunspot cycle. This is based on the two elements described above:

1. Empirical superiority of the McIntosh team exemplars which introduced their paradigm to this field of science. Is the McIntosh team forecast for Cycle 25 demonstrably more accurate than those offered by the NASA/NOAA/ISED Panel of experts?
2. Does the pattern of citations of the two exemplar articles published by the McIntosh team show that other scientists are adopting them? If this adoption is considerable, then the evidence compounds in favor of their new paradigm.

We now provide evidence on both elements of the issue at the approximate middle of Cycle 25. It will show that the results underscore our assessment of where things are in Figure 1.

Statistical Comparisons of the Two Cycle 25 Forecasts

Using the Austrian Space Weather Office website, we produce in Figure 2 the smoothed monthly sunspot numbers for Cycle 25 and for the two competing forecasts. We use the approximate peak time in Cycle 25 to delineate our comparisons. In other words, if this were an athletic competition, what is the score at the end of the first half of play?

In Figure 2, the vertical line is this demarcation as of August 2024 in the time series. Note that the NASA/NOAA/ISED (hereafter NNI) forecast had somewhat of a “false start,” to borrow a track-and-field metaphor, in that after their first set of numbers went public (light blue line), they released another revised forecast. This one shifted their forecast start back some six months (dark blue line). No public explanation was given by the NNI group. We use this revised NNI forecast in our analysis. The McIntosh team forecast is in the red line.

For comparative illustration, there are four other data series. The average monthly sunspot cycle number since 1750 is in green. The three observed sunspot numbers include the daily sunspots (light green line) and the key smoothed monthly sunspots in black. (There is a short series of estimated daily numbers in orange, shown after the final monthly figure.) This represents a visualization of the forecast and observed monthly sunspot numbers. The series includes 32 months of data, our approximation of the first half of Cycle 25.

In this graph, the NNI forecast does appear consistently lower than the observed monthly sunspot data after the summer months of 2022 while the McIntosh team numbers appear generally higher. The exception is near mid-cycle where the observed sunspots spike above both projections. Neither set anticipated this sharp rise in monthly sunspots. But are these two forecasts just a random walk around the observed monthly sunspots? Statisticians have addressed questions like this for some time because time series graphs are somewhat subject to various interpretations. We make statistical comparisons using standard methods for this in Figures 2 and 3.

In our 2022 RadCom paper series, we presented the McIntosh team forecast for a complementary index of solar propagation influence, the Solar Flux Index (SFI, abbreviated as f10.7). We use the NNI forecast for SFI to further compare the statistical accuracy of an atheoretical expert opinion forecast versus the theoretically-driven McIntosh team model.

We use the standard text by Theil (1966) for the analysis of forecast comparisons. One measure of the statistical accuracy of two time series is the mean absolute error (MAE) represented by the formula of

where yⁱ and xⁱ are the respective data values for each time series compared at the i^th time interval. That sum is divided by the number of points in the time series (or n) to yield this average absolute error in numbers of monthly sunspots.

Another test that is metric-free is the mean absolute percentage error (MAPE) which is the percent version of MAE. It is the sum of the actual minus forecast divided by the actual which is averaged over the number of temporal observations:

The third consideration we make is to test for the equivalence of the two forecasts (i.e., are the different forecast series just random walks?). In this part, we use the Diebold-Mariano test or D-M (Diebold and Mariano 1995) which compares the mean difference in the squared-error or absolute error of each forecast to the observed data. This S₁ test is applied to both the MAE and the MAPE (Theil 1966). The D-M S₁ test can use alternate kernel densities in this computation. To safeguard our comparisons, we compute tests using both a uniform and a Bartlett kernel for the standard error estimation. Each produced similar results so the uniform kernel is presented in our results. See Diebold and Mariano (1995) for details. We used Stata 17 software for our computations using the dmariano script (StataCorp 2017).

As shown in Figure 3, the McIntosh team series results in an average monthly forecast error of 26.8 sunspots. For the NNI forecast, the average monthly error is 45.3 sunspots. The McIntosh team forecast is 18.73 sunspots more accurate on average each month. The D-M test shows that this is statistically significant: the McIntosh forecast is significantly more accurate than the NNI prediction for the first half of Cycle 25. The most recent surge in monthly sunspots during 2024 was predicted by both forecasts but the McIntosh predictions were more on track in the graph with those observations.

The second panel of Figure 3 contains the percent form (MAPE) of the forecast errors for each group’s projections. The McIntosh forecast averaged a 25.4% monthly error, lower than the NNI expert panel’s 38.3% error each month. This is a 12.9% difference between the two, reflecting a statistically more accurate forecast (p=.0000).

The Solar Flux Index (SFI) is also a critical index for propagation. It rivals the SSN in importance for daily HF operations. We use this forecast to complement the ones for monthly sunspots. The SFI graph is in Figure 4. For NNI, all monthly errors are on the high side whereas the McIntosh team’s hover on the low side of zero (i.e., matching the observed SFI). Both anticipated the rise upward during the summer months of 2024 but were off in their respective predictions. Over the first-half of Cycle 25, the McIntosh forecast averaged 25.65 Index points closer to the observed SFI (17.2 vs. 42.9). The D-M test suggests that this is a significant edge in favor of the McIntosh prediction (p= .0000).

Putting the SFI forecast errors in percent form, the lower panel illustrates a consistent over-prediction of the monthly Solar Flux Index by both. The NNI’s numbers are visibly off-base by 20 percent or greater in the graph. Some segments of the McIntosh predictions are also off by 10 to 20 percent. Overall, however, the average percent error is 27.8% for the NNI forecast and some 16.6% less for McIntosh at 11.2%. As with the MAE metric, this difference in percent form is statistically in favor of the McIntosh series (p = .0000).

In short, the McIntosh team has empirically superior forecasts for both monthly sunspots and the Solar Flux Index, two leading indices for propagation used by amateur radio and many other spheres of radio transmission practice. They are uniformly statistically significant in favor of the McIntosh theory-driven approach as compared to the expert panel forecasts from NNI.

Evidence of Paradigm Change Through Bibliometric Analysis

To examine evidence on other scientists adopting the new McIntosh team paradigm, we used methods of bibliometric citation analysis (De Bellis 2009: Chapter 8; Prabhakaran et al. 2018). This is a set of methods used to measure the impact and influence of scholarly works through the patterns and frequency of citations in various contexts (Andres 2009; Alphasoft.com n.d.). This set of metrics measures the behavior of the scientific community toward the competing paradigm which Kuhn (1962) shows is the key to paradigm change.

Traditional citation counts from the print medium tend to be much slower than scientific discovery is actually produced because of the circulation of print media (De Bellis 2009: Chapter 8). Because of this, alternative metrics were developed to measure how Internet-based tools enhance the sharing of scholarship. These tools include paper pre-print servers, online exchange of papers, and other discussion networks that are in daily use to stay abreast of the latest emerging knowledge. These “alternative metrics” including social media and online publishing are used in this part of our analysis through the Altmetric system (see Astrophysics Data System).

Following Kuhn’s approach to paradigm-change, we studied the two papers that the second author identified as the exemplars (Kuhn 1962) introducing and illustrating his team’s competing paradigm. The Astrophysics Data System (or ADS) maintained by Harvard University on behalf of the Smithsonian Astrophysical Observatory (SAO) under a NASA grant was the source of our citation analysis. The ADS system (available at (https://ui.adsabs.harvard.edu/) facilitates meta-analysis of papers in astrophysics with both bibliometric computation and visualization of results. We utilize this system to analyze citation and discussion metrics for both exemplar papers as well as all publications for the leader of this scientific team (the second author here). While we do not report a full bibliometric analysis (for example, see Prabhakaran et al. 2018), the compilation of citation metrics does suffice to gauge the initial attention and influence that this competing paradigm is having on the field of solar physics and amateur radio itself.

Figure 5 summarizes these metrics for the two exemplars. The first paper, introducing the overlapping Hale magnetic activity cycles and the relationship they have to sunspot amplitude, has 73 total citations. The bar chart on the upper right shows the trends in citations for this paper (note that 2025 is not yet fully realized). The scientific output analysis by Altmetric gives it an “attention score” of 858. This ranks number one of almost two thousand articles in the journal. Among over one-half million articles published during the same period, it ranks 813. It is in the top 5% of all research ever tracked by Altmetric.

The second exemplar introduced the Terminator timing construct, illustrating it using all the data on sunspot cycles in existence, by associating it to patterns within the 22-year Hale Cycle. This paper extended the first exemplar’s idea of potential causes of the amplitude by adding a conceptual basis for what “kick starts” the next cycle. This paper has 18 citations, moving up quickly in the year after publication as shown in the bar chart. It has an attention score thus far of 654. This article is ranked number one of almost 1,600 articles in this leading journal, Solar Physics. By comparison to about one-half million articles published in the field at the same time, it ranks 999. More importantly, it too ranks in the top 5% of all research articles scored by Altmetric.

Turning to all papers published by this scientific team’s leader, Figure 6 summarizes the same type of citation analysis. Emphasizing the period of 2020-2025 for when the competing paradigm was introduced, there are 457 papers considered in this figure. There are 5,901 citations of these papers, only a thousand of which are self-citations, necessary to build and continue a line of scholarship. The citations by other scholars are the key element for exemplar adoption. There are about two thousand citations, “normalized” to the volume of other articles published around the same time. This puts the citation patterns into the context of the scientific problem as a comparison to numbers of citations per se (see the ADS website for details). The bar chart extends the scope prior to the year 2020 to check the scholarly output by this team’s intellectual leader. The result is a steady increase in citations by other scientists in peer-reviewed papers, an indication of a very productive scholar growing in a career that is being recognized by other scientists in their own work.

The H-index is the most popular one in use for scientific comparisons. A value for H means that the author has that number of papers that are each cited by a minimum of the same number. The H-index number in Figure 6 increases prominently in 2020 to about 45, continuing through 2025. Note that an H-index value of 40 is outstanding and over 60 is exceptional (Hirsch 2005).

The read10-index reflects a decade swath of readership citations of the author’s publications. It shows the works published in 2010 (pre-paradigm introduction) and 2020 (paradigm introduction) as having the highest values, well over 100. This trend line shows the immediate interest in the author’s works over a long period of time (a decade), a sign of prominence in science.

The i100-index, however, might be the most illustrative for our purpose to ascertain how the paradigm is being adopted by others in this field of science. It illustrates the number of publications with at least 100 citations, a challenging hill to climb in science. The growth in the (purple) i100 line shows that a minimum of 100 citations for papers by McIntosh steadily increases after 2015 but especially after 2020, the year of publication for the first exemplar paper. This is also indicative of movement toward paradigm-adoption by others. The tori-index corroborates this trend as his papers become central citations by other scholars. The i10-index began to also spike when the 2020 paper came out and continued after the second exemplar appeared. This index surpasses 130, suggesting that many papers by the author have been each cited by a minimum of 10 other authors.

The bibliometric portion of our analysis shows strong evidence that the peer scientific community is heavily engaged in the competing paradigm. The two exemplars have been substantially growing in peer citations to a level of prominence. They have garnered the top attention in the respective scientific journals where they appeared, no small feat for any scientist. The H-index score shows that the lead scholar producing this new paradigm has reached an outstanding region, further evidence of movement toward direct model competition in Kuhn’s model of paradigm change.

Is There Demonstrable Progress in the Revolution?

Our goal has been to determine if there is Kuhnian movement (Kuhn 1962) toward a revolution in the long-standing paradigm for the sunspot cycle at the midpoint of Cycle 25. We identified two elements of evidence: the empirical superiority of the NNI versus the McIntosh team forecasts and the degree of scientific adoption of the competing paradigm’s exemplary papers.

Using long-established methods in forecast comparisons, our results leave little objective doubt that the theory-driven forecast by the McIntosh team is superior. For smoothed monthly sunspot counts covering the first 32 months of the Cycle, the McIntosh team forecast is 19 spots more accurate, a 13 percent and statistically significant improvement over the NNI numbers. (Note that we used the NNI’s revised forecast after they adjusted to some six months behind their original Panel’s predictions.) We included forecasts for the Solar Flux Index (f10.7) over the same time horizon. The McIntosh team’s SFI forecast is 26 index points or 17 percent more accurate. The empirical superiority, at least at mid-cycle, clearly favors the McIntosh paradigm.

The bibliometric analysis we presented on how the two exemplary papers have been received by the scientific field showed strong evidence of engagement and adoption with the competing paradigm. The overall standing of scholarship by the lead scientist was a second element surrounding this new paradigm. It too demonstrated a clear upturn in citation metrics after the publication of the two exemplar papers.

The citation numbers have been continually increasing since the first (2020) and second papers (2023) appeared in peer-reviewed journals. We noted in Figure 1 the boundary maintenance by keepers of the long-standing paradigm who rejected the original paper in 2012. It took nearly a decade (from 2012 to 2020) of continually increasing the amount of scientific evidence involving the linkages between the Hale Cycle to the sunspot cycle’s behavior to reach a successful peer-reviewed publication. With the observed rapid increase in citations of the two exemplar papers, Kuhn’s concept of a non-linear, revolutionary adoption of a competing paradigm appears indeed to fit the bibliometric results.

To underscore Kuhn’s notion, the more contemporary “attention” metrics for the two exemplars show that each is the number one ranked article in the respective publishing journal. Each is also in the top five percent of all research articles ever tracked by Altmetric. The two exemplars have clearly captured the attention of the field and the associated reporting on it challenging the status quo paradigm.

We find the bibliometric citation results to also be strong evidence that the competing paradigm is indeed now within direct model competition as illustrated in Figure 1. It may well take until the end of Cycle 25 to determine the extent that a paradigm revolution has occurred. It will depend on the continued reception of the McIntosh team’s published results as they continue their research program. This adoption would be spurred along by a continuing forecast superiority during the second half of Cycle 25. Those monthly comparative results are available on the Austrian Space Weather website for all to see.

We likened this study to that of checking the score at half-time of an athletic event. However, we must wait until this cycle is compete to render a full assessment of who wins the scientific competition. We plan to revisit this analysis at the appropriate time. The available evidence at half-time, nonetheless, clearly favors progress in the revolution involving our understanding of the critically important sunspot cycle.

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