Letters

TOWARD RELIABLE BENCHMARKING OF SOLAR FLARE FORECASTING METHODS

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Published 2012 February 24 © 2012. The American Astronomical Society. All rights reserved.
, , Citation D. Shaun Bloomfield et al 2012 ApJL 747 L41 DOI 10.1088/2041-8205/747/2/L41

2041-8205/747/2/L41

ABSTRACT

Solar flares occur in complex sunspot groups, but it remains unclear how the probability of producing a flare of a given magnitude relates to the characteristics of the sunspot group. Here, we use Geostationary Operational Environmental Satellite X-ray flares and McIntosh group classifications from solar cycles 21 and 22 to calculate average flare rates for each McIntosh class and use these to determine Poisson probabilities for different flare magnitudes. Forecast verification measures are studied to find optimum thresholds to convert Poisson flare probabilities into yes/no predictions of cycle 23 flares. A case is presented to adopt the true skill statistic (TSS) as a standard for forecast comparison over the commonly used Heidke skill score (HSS). In predicting flares over 24 hr, the maximum values of TSS achieved are 0.44 (C-class), 0.53 (M-class), 0.74 (X-class), 0.54 (⩾M1.0), and 0.46 (⩾C1.0). The maximum values of HSS are 0.38 (C-class), 0.27 (M-class), 0.14 (X-class), 0.28 (⩾M1.0), and 0.41 (⩾C1.0). These show that Poisson probabilities perform comparably to some more complex prediction systems, but the overall inaccuracy highlights the problem with using average values to represent flaring rate distributions.

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1. INTRODUCTION

Solar flares result from the release of enormous quantities of energy (up to ∼1027 J; Kane et al. 2005) from twisted, non-potential magnetic fields. Along with coronal mass ejections (CMEs), flares are a major contributor to space weather that adversely affects the near-Earth environment (Hapgood & Thomson 2010). The magnetic energy to power solar flares is stored primarily in active regions (ARs) that are routinely classified in terms of complexity. The Mount Wilson scheme (Hale et al. 1919; Künzel 1960) describes magnetic polarity mixing, while the McIntosh (1990) scheme describes spatial structuring of the magnetic field "footprints" in sunspot groups. We concentrate on the McIntosh scheme that allows up to 60 classes, yielding reasonable resolution in terms of the observed structural complexity. In contrast, the Mount Wilson scheme allows up to eight classes, each with flare rate distributions broader than the McIntosh classes.

Recent years have seen a resurgence in the field of solar flare prediction. A sample of the techniques employed includes Poisson statistics (Gallagher et al. 2002), Bayesian statistics (Wheatland 2005), support vector machines (Li et al. 2007), discriminant analysis (Barnes et al. 2007), ordinal logistic regression (Song et al. 2009; Yuan et al. 2010), neural networks (Colak & Qahwaji 2009; Yu et al. 2009; Ahmed et al. 2012), wavelet predictors (Yu et al. 2010a), Bayesian networks (Yu et al. 2010b), predictor teams (Huang et al. 2010), superposed epoch analysis (Mason & Hoeksema 2010), and empirical projections (Falconer et al. 2011). It is worth noting that none of these techniques are based on physical models of the flare process. Most of the methods give a probability for an X-ray flare with peak flux above some magnitude in a time interval. If the aim of a prediction method is to provide a result that can be readily interpreted as "flare imminent" or "no flare expected," the predicted probabilities need to be converted into yes/no forecasts and the forecast success determined. However, it is extremely important that appropriate performance measures are used when comparing the success of different forecasts.

In this Letter, a case is presented for the adoption of an existing (but rarely utilized) performance measure for comparisons between different solar flare forecasts (Section 2). As an example, we investigate the performance of Poisson probabilities in predicting X-ray flares from ARs within 24 hr of a McIntosh classification being issued. The data and their sources are detailed in Section 3, while the method to determine forecast performance is described in Section 4. The effect of varying the threshold that is used in converting Poisson probabilities into yes/no predictions is studied in Section 5.1, while optimum performance measures are compared to the performance of other methods in Section 5.2. Finally, our conclusions and ideas for further work are given in Section 6.

2. FORECAST PERFORMANCE MEASURES

The success of a forecast method that provides yes/no forecasts should be studied using a forecast contingency table and calculating verification measures (an excellent comparison of different evaluation measures is given in Woodcock 1976). Quantitative measures are essential to compare the relative performance of different prediction methods. The flare forecast contingency table format is presented in Table 1, containing the elements TP (true positives, "flare" predicted and observed), FN (false negatives, "no flare" predicted and flare observed), FP (false positives, "flare" predicted and none observed), and TN (true negatives, "no flare" predicted and none observed). Numerous skill scores exist to quantify the performance of forecasts, but the Heidke (1926) skill score (HSS),

Equation (1)

is most frequently used in flare forecasting (e.g., Barnes & Leka 2008). The strength of the HSS lies in its use of the whole contingency table to quantify the accuracy of achieving correct predictions relative to random chance. The Hanssen & Kuipers (1965) discriminant, known as the true skill statistic (TSS), also uses all of the elements,

Equation (2)

However, only TSS is unbiased when confronted with varying event/no-event sample ratios (Woodcock 1976). This is demonstrated by considering a new forecast that achieves the same prediction success with two times the number of flare ARs (i.e., TPnew = 2TP; FNnew = 2FN; TPnew/FNnew = TP/FN). Equation (1) becomes

Equation (3)

while Equation (2) becomes

Equation (4)

This simple example shows that HSS changes despite the prediction success being held constant, highlighting the problem with using HSS to compare between different methods (or different trials of the same method). Note that we do not dismiss the usefulness of HSS as a measure within a particular forecast method trial. However, we propose TSS to be the standard measure for comparing between flare forecasts, given that different studies use differing flare/no-flare sample ratios.

Table 1. Flare Forecast Contingency Table

Flare Forecast
Observed "Flare" "No flare"
Yes TP FN
No FP TN

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3. DATA SOURCES

3.1. Training Set

In order to facilitate the calculation of flare probabilities, we obtained historical flare rates for each McIntosh class from two locations that share the same data source. The National Oceanic and Atmospheric Administration (NOAA) Space Weather Prediction Center (SWPC) provided total numbers of Geostationary Operational Environmental Satellite (GOES) C-, M-, and X-class flares and the originating ARs for each McIntosh classification over 1988 December 1 to 1996 June 30 (C. C. Balch 2011, private communication). Additional M- and X-class flares and McIntosh class numbers were taken from Kildahl (1980) over 1969–1976, but relate to the same data source (i.e., NOAA-collated ground-based AR observations and GOES flare events). These were included to increase the rare M- and X-class samples so that the rates were more statistically significant. Table 2 presents the recorded McIntosh classes with the numbers of observed regions and flares produced.

Table 2. McIntosh Classification Flare Statistics

McIntosh SWPC (1988–1996) Kildahl (1969–1976)b Combined Flare Rate (24 hr−1) Poisson Flare Probability (%)
Region Region Total Flares Region Total Flares In GOES Class In GOES Class Above GOESd
Classesa Count C M X Count Cc M X C M X ±σ C M X M1.0 C1.0
AXX 2748 82 10 0 2517 75.1 31 3 0.03 0.01 0.00 0.01 3 1 0 1 4
BXO 3342 217 18 1 1906 123.8 41 2 0.06 0.01 0.00 0.01 6 1 0 1 7
BXI 0 0 0 0 334 0.0 20 0 0.00 0.06 0.00 0.05 0 6 0 6 6
HRX 336 21 1 0 211 13.2 7 1 0.06 0.01 0.00 0.04 6 1 0 2 8
HSX 1968 94 21 0 1963 93.8 99 6 0.05 0.03 0.00 0.02 5 3 0 3 8
HAX 598 49 13 0 222 18.2 14 0 0.08 0.03 0.00 0.03 8 3 0 3 11
HHX 53 3 1 0 150 8.5 16 2 0.06 0.08 0.01 0.07 6 8 1 9 14
HKX 49 11 2 0 38 8.5 7 0 0.22 0.10 0.00 0.11 20 10 0 10 28
CRO 745 102 3 0 368 50.4 20 2 0.14 0.02 0.00 0.03 13 2 0 2 15
CRI 6 2 0 0 152 50.7 7 0 0.33 0.04 0.00 0.08 28 4 0 4 31
CSO 1504 284 27 0 1020 192.6 40 1 0.19 0.03 0.00 0.02 17 3 0 3 19
CSI 14 8 2 0 211 120.6 16 2 0.57 0.08 0.01 0.07 44 8 1 9 48
CAO 1455 361 38 2 232 57.6 18 1 0.25 0.03 0.00 0.02 22 3 0 3 25
CAI 27 14 6 0 166 86.1 19 0 0.52 0.13 0.00 0.07 40 12 0 12 48
CHO 88 21 2 1 112 26.7 8 1 0.24 0.05 0.01 0.07 21 5 1 6 26
CHI 2 1 0 0 29 14.5 6 0 0.50 0.19 0.00 0.18 39 18 0 18 50
CKO 135 59 11 0 52 22.7 13 2 0.44 0.13 0.01 0.07 35 12 1 13 44
CKI 17 14 6 0 28 23.1 6 2 0.82 0.27 0.04 0.15 56 23 4 27 68
DRO 63 12 3 0 75 14.3 6 0 0.19 0.07 0.00 0.09 17 6 0 6 23
DRI 2 7 0 0 54 189.0 7 1 3.50 0.12 0.02 0.13 97 12 2 13 97
DSO 546 198 26 1 553 200.5 51 6 0.36 0.07 0.01 0.03 30 7 1 7 36
DSI 39 34 6 0 246 214.5 31 1 0.87 0.13 0.00 0.06 58 12 0 12 63
DSC 0 0 0 0 20 0.0 5 2 0.00 0.25 0.10 0.22 0 22 10 30 30
DAO 1775 784 124 4 288 127.2 28 2 0.44 0.07 0.00 0.02 36 7 0 7 40
DAI 391 419 70 6 324 347.2 58 7 1.07 0.18 0.02 0.04 66 16 2 18 72
DAC 8 5 3 0 46 28.8 12 1 0.62 0.28 0.02 0.14 46 24 2 26 60
DHO 46 26 1 1 43 24.3 11 0 0.57 0.13 0.01 0.11 43 13 1 14 51
DHI 11 14 1 0 41 52.2 3 0 1.27 0.08 0.00 0.14 72 7 0 7 74
DHC 0 0 0 0 6 0.0 2 0 0.00 0.33 0.00 0.41 0 28 0 28 28
DKO 217 178 55 5 43 35.3 14 2 0.82 0.27 0.03 0.06 56 23 3 25 67
DKI 223 288 69 6 88 113.7 42 6 1.29 0.36 0.04 0.06 73 30 4 33 81
DKC 57 93 35 5 100 163.2 72 10 1.63 0.68 0.10 0.08 80 49 9 54 91
ESO 95 37 6 0 82 31.9 14 0 0.39 0.11 0.00 0.08 32 11 0 11 39
ESI 18 33 1 0 78 143.0 22 2 1.83 0.24 0.02 0.10 84 21 2 23 88
EAO 459 267 61 0 47 27.3 10 4 0.58 0.14 0.01 0.04 44 13 1 14 52
EAI 295 370 83 2 82 102.8 48 1 1.25 0.35 0.01 0.05 71 29 1 30 80
EAC 3 5 1 0 17 28.3 6 3 1.67 0.35 0.15 0.22 81 30 14 39 89
EHO 42 31 6 0 39 28.8 6 0 0.74 0.15 0.00 0.11 52 14 0 14 59
EHI 15 24 6 0 45 72.0 28 4 1.60 0.57 0.07 0.13 80 43 6 47 89
EHC 2 9 0 0 4 18.0 8 0 4.50 1.33 0.00 0.41 99 74 0 74 100
EKO 185 173 35 3 52 48.6 20 1 0.94 0.23 0.02 0.06 61 21 2 22 69
EKI 423 703 173 23 81 134.6 103 11 1.66 0.55 0.07 0.04 81 42 7 46 90
EKC 103 278 132 17 63 170.0 149 21 2.70 1.69 0.23 0.08 93 82 20 85 99
FRI 0 0 0 0 2 0.0 1 0 0.00 0.50 0.00 0.71 0 39 0 39 39
FSO 14 9 3 0 13 8.4 6 1 0.64 0.33 0.04 0.19 47 28 4 31 64
FSI 6 12 0 0 8 16.0 15 0 2.00 1.07 0.00 0.27 86 66 0 66 95
FAO 73 63 16 0 3 2.6 0 0 0.86 0.21 0.00 0.11 58 19 0 19 66
FAI 91 106 35 3 12 14.0 8 0 1.16 0.42 0.03 0.10 69 34 3 36 80
FHO 9 5 1 0 10 5.6 0 0 0.56 0.05 0.00 0.23 43 5 0 5 46
FHI 10 17 9 0 18 30.6 15 0 1.70 0.86 0.00 0.19 82 58 0 58 92
FHC 0 0 0 0 5 0.0 4 0 0.00 0.80 0.00 0.45 0 55 0 55 55
FKO 97 165 29 1 19 32.3 6 0 1.70 0.30 0.01 0.09 82 26 1 27 87
FKI 235 517 161 17 47 103.4 106 17 2.20 0.95 0.12 0.06 89 61 11 66 96
FKC 93 233 146 24 27 67.6 39 13 2.51 1.54 0.31 0.09 92 79 27 84 99

Notes. aOnly includes classifications producing ⩾1 C-, M-, or X-class flare in either time range. bFrom Kildahl (1980). cNon-integer flare numbers result from use of observed C-class rates from SWPC (1988–1996). d"Above GOES X1.0" is equivalent to "In GOES Class X."

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3.2. Testing Set

The AR and flare data that are used for testing were gathered from the online archives of NOAA/SWPC.3 McIntosh classes of regions that have predictions issued and tested were taken from the daily NOAA Solar Region Summary files over 1996 August 1 to 2010 December 31. In this work, each daily record of an NOAA region was treated as an individual measurement, yielding 22276 AR samples. GOES flares with originating NOAA numbers assigned to their entry were extracted from the edited daily NOAA Solar Event Reports over the same date range as the McIntosh classes. NOAA region numbers attributed to any associated Hα flares were used for those GOES flares with no NOAA region directly assigned.

4. ANALYSIS METHOD

4.1. Historical Poisson Probabilities

Following Bornmann & Shaw (1994), GOES-class flare rates in 24 hr intervals were calculated for each McIntosh class by combining the number of flares that classification produced over 1969–1976 and 1988–1996 and dividing by the number of times the McIntosh class was observed in both periods, Ntot. It should be noted that C-class flares were not provided in Kildahl (1980). In order to provide C-class related forecasts comparable to those for M- and X-classes, rates measured over 1988–1996 were taken to hold for 1969–1976. The relative numbers of McIntosh observations in the time periods were then used to determine the expected number of C-class flares for 1969–1976 (Table 2, Column 7). The C-, M-, and X-class flare rates combined over 1969–1976 and 1988–1996 are presented in Columns 10–12 of Table 2, with the error on the average rate (σ = N−1/2tot) given in Column 13.

To achieve a probability of flaring we follow the Poisson statistics technique of Gallagher et al. (2002). Under the assumption of flares being a Poisson-distributed process,4 the probability of observing N flares in a time interval is related to the average flare rate, μ, over that interval by

Equation (5)

When μ is calculated over 24 hr intervals, the probability of observing one or more flares in any 24 hr interval is

Equation (6)

Poisson probabilities for a McIntosh class to produce at least one flare within a 24 hr interval are displayed in Columns 14–16 of Table 2 for the C-, M-, and X-classes, with those for flaring ⩾M1.0 (M- and X-classes) and ⩾C1.0 (C-, M-, and X-classes) in Columns 17 and 18.

4.2. Contingency Table Construction

Two sets of binary (yes/no) information are required to build the forecast contingency tables—flare truth and flare prediction. The first is achieved by cross-referencing the SWPC-extracted AR and GOES event lists over the testing period (1996–2010). For each AR observed each day, the list of AR-associated flares within 24 hr of the McIntosh class being issued is searched for the NOAA number of that AR (i.e., the same UT day; McIntosh classes are published at 00:30 UT based on data before 00:00 UT). Flare truth is set to "no" for ARs when no flares occurred with peak magnitude at the appropriate level or "yes" when ⩾1 flare occurred. This results in the number of flare ARs, Nfl, being 3667, 810, and 92 for C-, M-, and X-class events, respectively. Similarly, Nfl is 858 and 3912 for ARs with flares ⩾M1.0 and ⩾C1.0, respectively.

The second set of information is achieved by applying a flare/no-flare discriminating threshold to the Poisson probabilities achieved in Section 4.1. All ARs in the test period had the corresponding McIntosh class flare probabilities (Table 2) assigned to the 24 hr interval after observation. Probabilities were converted into predictions by choosing a threshold (varying in 1% increments from 0% to 100%) and predicting "no flare" for values below the threshold and "flare" for those at or above the threshold.

The contingency table elements (Section 1 and Table 1) are the number of each pair combination of flare truth and prediction. The variation of the HSS and TSS measures is shown in Figure 1 and Table 3 for separate forecasts of C-, M-, and X-class events and forecasts ⩾M1.0 and ⩾C1.0. It is worth noting that the approach applied here changes occurrences of TP to FN and FP to TN as the threshold probability rises ("flare" predictions become "no-flare" predictions, but flare truth is unchanged).

Figure 1.

Figure 1. Threshold probability variation of HSS (a)–(b), TSS (c)–(d), and FN/FP (e)–(f). Curves in panels (a), (c), and (e) are forecasts over 24  hr of at least one C-class (solid), M-class (dashed), or X-class (dotted) flare, while those in panels (b), (d), and (f) are forecasts of at least one flare ⩾C1.0 (solid), ⩾M1.0 (dashed), and ⩾X1.0 (dotted). Arrows in panels (a) and (b) mark thresholds where FN/FP ≈ 1, while those in panels (c) and (d) mark thresholds where FN/FP ≈ Nfl/Nnf. Only FN/FP ≈ Nfl/Nnf is marked for X-class (≪1), as 0.35 is the largest finite value.

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Table 3. Flare Forecast Contingency Table and Skill Score Dependence on Threshold Poisson Probability

Prob. Flaring In GOES M-class Within 24 hr
  Contingency Table Elements Skill Scores FN/FP
% TP FN FP TN HSS TSS  
0 810 0 21466 0 0.000 0.000 0.00
10 568 242 3832 17634 0.167 0.523 0.06
20 452 358 2163 19303 0.221 0.457 0.17
30 330 480 1129 20337 0.256 0.355 0.43
40 288 522 850 20616 0.264 0.316 0.61
50 209 601 471 20995 0.256 0.236 1.28
60 202 608 458 21008 0.250 0.228 1.33
70 149 661 308 21158 0.215 0.170 2.15
80 59 751 173 21293 0.099 0.065 4.34
90 0 810 0 21466 0.000 0.000
100 0 810 0 21466 0.000 0.000

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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5. RESULTS AND DISCUSSION

5.1. Skill Score Variation with Prediction Threshold

Figure 1 shows HSS peaking at FN/FP ≈ 1 (panels (1a) and (1b)). This indicates that the HSS measure of forecast accuracy is maximized5 when the absolute frequency of incorrect predictions are equal, FN ≈ FP. Sensitivity to FN/FP confirms the HSS dependence on sample ratio (Equation (3) here; Woodcock 1976). Table 1 shows that TP and FN increase if additional flaring ARs are included (FN/FP increases and unity occurs at higher thresholds). Conversely, FP and TN increase if additional no-flare ARs are included (FN/FP decreases and unity occurs at lower thresholds). Note that varying the number of ARs included in the verification test does not have the same effect as varying the threshold used to construct the contingency tables: adding ARs alters the sample ratio, but maintains the forecast success ratio (if the added sample is random); varying the threshold maintains the sample ratio, but alters the forecast success ratio.

Figure 1 also shows TSS peaking at FN/FP ≈ Nfl/Nnf (panels (1c) and (1d)), where Nnf is the number of no-flare ARs (Nnf = 22276 − Nfl). This indicates that the TSS measure of accuracy is maximized when the fractional frequency of incorrect predictions for flare ARs equals the fractional frequency of incorrect predictions for no-flare ARs, FN/Nfl = FP/Nnf. This dependence on the fractional form of incorrect frequencies again illustrates that forecasts with differing sample ratios will keep the same TSS value: changes in FN or FP are absorbed by corresponding changes in Nfl or Nnf (Equation (4)). Note that HSS = TSS when Nfl = Nnf, but this is seldom the case in flare forecasting as flares are rare events.

5.2. Inter-forecast Skill Score Comparison

Flare forecasting studies do not usually quote values of TSS and rarely use equal flare/no-flare sample sizes that make HSS equal TSS.6 Unfortunately, most do not show contingency tables that would enable TSS or other unpublished measures to be calculated. Optimum values of TSS and HSS achieved by Poisson probabilities in Section 5.1 are compared to other methods in Table 4, restricted to those with a contingency table (or values one can be inferred from) and those quoting HSS. Other measures used in flare forecasting include the probability of detection: POD = TP/[TP + FN]; the false alarm ratio: FAR = FP/[TP + FP]; and the odds ratio or accuracy: ACC = [TP + TN]/[TP + FN + FP + TN]. Table 4 includes these to allow broad assessment of each method.

Table 4. Inter-forecast Skill Score Comparison

Forecast Verification Measure Reference
Flare Level Interval (hr) TSS FN/FP HSS POD FAR ACC  
C-class... 24 ... ... 0.493 0.772 0.319 0.811 Colak & Qahwaji (2009)
............ 24 0.650 0.429 0.623 0.850 0.292 0.818 Song et al. (2009)a
............ 24 0.090 7.000 0.116 0.138 0.471 0.722 Yuan et al. (2010)
............ 24 0.443 0.176 0.296 0.737 0.670 0.711 This work: optimum TSS
............ 24 0.399 0.836 0.384 0.513 0.531 0.824 This work: optimum HSS
M-class... 24 ... ... 0.470 0.865 0.688 0.944 Colak & Qahwaji (2009)
............ 24 0.621 6.000 0.676 0.647 0.083 0.873 Song et al. (2009)a
............ 24 0.054 1.963 0.061 0.221 0.643 0.652 Yuan et al. (2010)
............ 24 0.526 0.070 0.177 0.693 0.864 0.829 This work: optimum TSS
............ 24 0.272 1.002 0.273 0.299 0.701 0.949 This work: optimum HSS
X-class... 24 ... ... 0.169 0.917 0.967 0.981 Colak & Qahwaji (2009)
............ 24 0.693 2.000 0.739 0.714 0.167 0.945 Song et al. (2009)a
............ 24 0.160 3.000 0.205 0.206 0.562 0.843 Yuan et al. (2010)
............ 6 0.312 0.005 0.008 0.617 0.992 0.694 Mason & Hoeksema (2010)b
............ 24 0.740 0.005 0.049 0.859 0.971 0.881 This work: optimum TSS
............ 24 0.241 0.348 0.142 0.250 0.896 0.988 This work: optimum HSS
⩾M1.0... 24 ... ... 0.153 ... ... 0.922 Barnes & Leka (2008)c
............ 48 0.650 1.105 0.650 0.817 0.169 0.825 Yu et al. (2009)d
............ 48 ∼0.66 ... ∼0.66 ∼0.90 ... ... Huang et al. (2010)
............ 24 0.539 0.072 0.190 0.704 0.854 0.830 This work: optimum TSS
............ 24 0.273 1.089 0.280 0.298 0.684 0.948 This work: optimum HSS
⩾C1.0... 24 ... ... 0.512 0.814 0.301 0.805 Colak & Qahwaji (2009)
............ 24 0.641 0.952 0.636 0.662 0.349 0.961 Ahmed et al. (2012)e,f
............ 24 0.456 0.178 0.315 0.753 0.649 0.712 This work: optimum TSS
............ 24 0.412 0.942 0.407 0.520 0.495 0.826 This work: optimum HSS

Notes. aModel (4). bReported HSS contains miscalculation of expected correct random forecasts (J. P. Mason 2011, private communication). cTotal unsigned magnetic flux. dContingency table provided by X. Huang (2011, private communication). eTemporally segmented training and operational testing (test still spatially segmented to ARs ⩽ 60° from disk center). fContingency table calculated from reported forecast measures.

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5.2.1. Performance for Separate Flare-magnitude Classes

In forecasting flares in the separate GOES flare classes over 24 hr intervals, the ordinal logistic regression model (4) of Song et al. (2009) yields the highest TSS values for C- and M-classes, while the optimum TSS for Poisson probabilities is highest for X-class. Song et al. (2009) convert flare probabilities into predictions using static thresholds of 50% for C- and M-class events and 25% for X-class events. Improved performance might be achieved by the Song et al. (2009) technique by investigating its dependence on the prediction threshold, as studied here. Unfortunately, the Song et al. (2009) results are the most susceptible to noise (given a small sample of 55 ARs7) and weighted toward successful prediction of flaring ARs, since their samples of each flare-magnitude class have higher proportions of flaring ARs (36%, 31%, and 13% for C-, M-, and X-classes) than typically observed (16%, 4%, and 0.4% in cycle 23). It is unclear how this method would perform operationally when non-flaring ARs outnumber flaring ARs and successfully predicting no-flare periods has increased importance. The significantly lower performance of Yuan et al. (2010) in TSS and HSS is surprising when adding support vector machine classification to the Song et al. (2009) technique. It is worth noting that neural network operational forecasting of McIntosh classes by Colak & Qahwaji (2009) yields an HSS between that found here and Song et al. (2009) for all flare classes, but published values do not permit TSS calculation.

For X-class flares, the optimal TSS value for Poisson probabilities over 24 hr intervals is higher than that from the superposed-epoch analysis of Mason & Hoeksema (2010) over 6 hr intervals. The Mason & Hoeksema (2010) technique is segmented by predicting "no flare" for ARs with a magnetic quantity change over the previous 40 hr below one threshold and "flare" for ARs with changes above a second higher threshold. The forecast success would likely decrease if the unpredicted mid-range AR population were included. Note the optimum TSS found here has large FAR because it results from a yes/no prediction threshold of 1%, meaning that X-class flares are always predicted for all McIntosh classifications that historically produced any X-class activity.

5.2.2. Performance above the M1.0 Level

In forecasting flares ⩾M1.0, sequential supervised learning by Yu et al. (2009) and the predictor team work of Huang et al. (2010) yield the highest HSS values that equate to TSS from equal flare and no-flare sample sizes. However, they predict cumulative flare importance equivalent to at least one M1.0 event in a 48 hr interval (e.g., 10 C1.0, 5 C2.0, 2 C5.0). This raises uncertainty about these good skill scores representing the successful forecasting of events ⩾M1.0, as forecasting multiple C-class events from an AR may be easier than single M-class events. More importantly, both works only consider ARs that produce at least one flare ⩾C1.0 in their life. This segmentation weakens their interpretation for operational purposes (similar to the case of Song et al. 2009 in Section 5.2.1), as the number of AR no-flare periods considered in Yu et al. (2009) and Huang et al. (2010) are severely reduced by excluding all completely non-flaring NOAA numbers. It is worth noting that the optimum TSS achieved here equals that for the application of 1 decision tree in Huang et al. (2010) (with HSS, hence TSS, of ∼0.54).

The highest HSS achieved in the discriminant analysis study of Barnes & Leka (2008) was found using total unsigned magnetic flux. However, the value is low (notably also lower than the optimum HSS found here) and likely due to the overlap between flaring and non-flaring AR-parameter distributions. However, proper comparison to the performance of Poisson probabilities is not possible as TSS values from Barnes & Leka (2008) are not available.

5.2.3. Performance above the C1.0 Level

Finally, in forecasting flares ⩾C1.0 in 24 hr intervals, the application of neural networks by Ahmed et al. (2012) to magnetic properties with semi-operational testing yields the highest TSS. Semi-operational refers to no segmentation being applied based on flare history, while spatial segmentation was applied (only ARs within 60° of disk center). Optimum TSS values show that Poisson probabilities do not perform as well as the machine learning of Ahmed et al. (2012), possibly from truly operational application (e.g., ARs near the limb may be misclassified by foreshortening effects and inappropriately predicted). It is interesting that the neural network system of Colak & Qahwaji (2009) does not perform significantly better than the application of Poisson probabilities, but this is based on HSS as TSS is unavailable for their work.

6. CONCLUSIONS

To be operationally practical, flare forecasts should provide predictions for all ARs irrespective of properties or flare history (i.e., no minimum criteria in selecting ARs for flare prediction). We have presented the variation of forecast verification measures with the threshold Poisson probability used to define "flare" and "no-flare" predictions. Forecasts for different X-ray flare levels from all NOAA ARs over 1996 August 1 to 2010 December 31 were tested against observed flares.

Optimized forecasts from Poisson flare probabilities are found to perform to similar standards as some more sophisticated methods (e.g., in forecasting events ⩾M1.0). However, the relatively low levels of optimum skill score (HSS ≲ 0.4 and TSS8 ≲ 0.5) lend further support to the need to use flaring rate distributions (in, e.g., a Bayesian methodology like Wheatland 2005) rather than averages over an AR class. This will be a focus of future work in the construction of Bayesian prior distributions of AR-property-dependent flare rates.

Providing forecasts and quantifying their performance will be acutely necessary as we approach the activity maximum of cycle 24. It is foreseen that specific forecast requirements may be targeted by careful consideration of skill scores and particular contingency table elements, e.g., the threshold for interpreting flare probabilities as yes/no forecasts could be tailored to achieve relative failure ratios (FN/FP) within the tolerance of various groups in the scientific and space weather communities. However, complete flare forecasts will require a deeper physical understanding of magnetic energy release and partitioning of energy between flare emission at different temperatures, acceleration of CMEs, and acceleration of high-energy particles (Emslie et al. 2005).

In closing, it is imperative that the performance of flare forecasting methods with differing flare/no-flare sample ratios is compared in a suitable manner. This requires the use of a verification measure that is not sensitive to the flare/no-flare sample ratio. We have highlighted an issue with the commonly adopted HSS and instead propose the sample ratio invariant TSS for the reliable comparison of flare forecasts.

The authors thank the referee for comments that improved the paper, Christopher Balch (NOAA/SWPC) for providing flare-AR-association data over 1988–1996, the NASA All-clear Workshop 2008, and Graham Barnes for useful discussions. This work was supported by a Marie Curie Intra-European Fellowship (D.S.B.) and the HELIO e-Infrastructure Grant (P.A.H.) under the European Community's 7th Framework Programme.

Facilities: GOES (XRS) - Geostationary Operational Environmetal Satellite

Footnotes

  • Aschwanden & McTiernan (2010) show that flare waiting times are consistent with a nonstationary Poisson process. Application of Poisson probability here averages the time-dependent rates in 24 hr intervals and over the solar cycle.

  • The concept of a peak value of skill score is only possible here because forecast performance is altered by varying the threshold. Methods without a variable threshold can only achieve one value.

  • This behavior is good practice given the rarity of flare events. Forcing a balance between Nnf and Nfl results in discarding ∼80%, ∼96%, and >99% of the available Nnf sample when considering events ⩾C1.0, ⩾M1.0, and ⩾X1.0, respectively.

  • Changing 1 TP into FN (and vice versa) yields ±0.050, ±0.059, and ±0.143 in TSS for C-, M-, and X-class forecasts, respectively.

  • Optimum TSS of 0.74 is found here for X-class at a threshold of 1%, but this results in severe overprediction and large FAR.

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10.1088/2041-8205/747/2/L41