Parameters Derived from the SDO/HMI Vector Magnetic Field Data: Potential to Improve Machine-learning-based Solar Flare Prediction Models

We use the SDO/HMI vector magnetic field data (Hoeksema et al. 2014) to produce the PIL mask and then calculate the parameters for the ARs. The HMI instrument is a filtergraph with full disk coverage at 4096 × 4096 pixels. The spatial resolution is about 1'' with a 05 pixel size. The spectral line is the Fei 6173 Å absorption line formed in the photosphere (Norton et al. 2006). The Stokes parameters [I, Q, U, V], measured at six wavelength positions and sampled every 720 seconds using a 1350 second weighted average, are inverted to retrieve the vector magnetic field using a Milne–Eddington based inversion algorithm, the Very Fast Inversion of the Stokes Vector (VFISV; Borrero et al. 2011; Centeno et al. 2014). The 180° degree ambiguity of the azimuth is resolved using a "minimum energy" algorithm (Metcalf 1994; Metcalf et al. 2006; Leka et al. 2009). The location and extent of the ARs are automatically identified and bounded by a feature recognition model (Turmon et al. 2010), and the disambiguated vector magnetic field data are deprojected to heliographic coordinates (Bobra et al. 2014). Here we use the Lambert (cylindrical equal area) projection method centered on the region for the remapping. The remapped data of the vector magnetic field in ARs is then used to compute the parameters.

We collect SHARP vector magnetic field data from 2010 May 1 to 2018 November 30 to form two data samples. One includes the data of eruptive ARs one hour before the major flares (M-class and above); the other includes the data of noneruptive ARs (one data point per day). Eruptive ARs are defined here as ARs that produced at least one major flare (M-class and above) during their disk passage; noneruptive ARs are those that do not produce any major flares during disk passage. The data of flares are obtained from the NOAA Space Weather Prediction Center. Only the ARs within 60° from the central meridian are selected. Following the above procedures, we have sampled a total of 313 data points for the eruptive ARs and 6633 data points for the noneruptive ARs. We use these two data samples for our study.

For the vector magnetic field data, we first use radial field, B_r, to generate a PIL mask. We then multiply this mask with the SHARP parameter distribution maps derived from the vector magnetogram. Finally, we sum up all the pixels resulting in parameter maps to obtain the modified SHARP parameters.

The PIL mask is generated from B_r using the method described in Schrijver (2007). We first produce two bitmaps from the B_r map: one for positive field in which the pixels with B_r > 200 G are set to 1 and the rest are 0; the other for negative field in which the pixels with B_r < −200 G are −1 and the rest are 0. 200 G is about 2σ for HMI vector field data (Hoeksema et al. 2014). We then derive the positive and negative masks from the bitmaps using the Density-Based Spatial Clustering of Application with Noise (DBSCAN; Sander et al. 1998) method. We finally multiply the two masks after convolving with the masks a Gaussian kernel with a width of 10 pixels to generate the PIL mask. The SHARP parameter maps are calculated from the vector field data using the formulas listed in Table 1 that are modified from Table 3 in Bobra et al. (2014). The final modified SHARP parameters are then obtained after multiplying the parameter maps with the PIL mask. The modified SHARP parameters retain all the original ones except the parameters that are related with Lorentz force, because these parameters are global quantities (Fisher et al. 2012) which must be calculated using the data over all of the ARs. The parameters calculated from the area enclosed by the PIL mask only do not have physical meaning. As a result, the modified SHARP parameters include 16 physical quantities.

Table 1.  Some of the SHARP Parameters, Modified from Table 3 in Bobra et al. (2014)

Keyword	Description	Sharp Formula
TOTUSJH	Total unsigned current helicity	${H}_{{c}_{\mathrm{total}}}\propto \sum \left|{B}_{z}\cdot {J}_{Z}\right|$
TOTPOT	Total photospheric magnetic free energy density	${\rho }_{\mathrm{tot}}\propto \sum {\left({{\boldsymbol{B}}}^{{obs}}-{{\boldsymbol{B}}}^{\mathrm{Pot}}\right)}^{2}{dA}$
TOTUSJZ	Total unsigned vertical current	${J}_{{z}_{\mathrm{total}}}\propto \sum \left|{J}_{z}\right|{dA}$
ABSNJZH	Absolute value of the net current helicity	${H}_{{c}_{{abs}}}\propto \left|\sum {B}_{z}\cdot {J}_{Z}\right|$
SAVNCPP	Sum of the modulus of the net current per polarity	${J}_{{z}_{\mathrm{sum}}}\propto \left|{\displaystyle \sum }_{{B}_{z}^{+}}{J}_{z}{dA}\right|+\left|{\sum }_{{B}_{z}^{-}}{J}_{z}{dA}\right|$
USFLUX	Total unsigned flux	${\rm{\Phi }}\propto \sum \left|{B}_{z}\right|{dA}|$
MEANPOT	Mean photospheric magnetic free energy	$\overline{\rho }\propto \tfrac{1}{N}\sum {\left({{\boldsymbol{B}}}^{\mathrm{Obs}}-{{\boldsymbol{B}}}^{\mathrm{Pot}}\right)}^{2}$
R_VALUE	Sum of flux near polarity inversion line	${\rm{\Phi }}=\sum \left|{B}_{z}\right|{dA}$ within R mask
MEANSHR	Mean shear angle	$\overline{{\rm{\Gamma }}}=\tfrac{1}{N}\sum {\cos }^{-1}\tfrac{{{\boldsymbol{B}}}^{\mathrm{Obs}}\cdot {{\boldsymbol{B}}}^{\mathrm{Pot}}}{\left|{{\boldsymbol{B}}}^{{obs}}\right|\cdot \left|{{\boldsymbol{B}}}^{\mathrm{Pot}}\right|}$
MEANGAM	Mean angle of field from radial	$\overline{\gamma }=\tfrac{1}{N}\sum {\tan }^{-1}\tfrac{{B}_{h}}{{B}_{z}}$
MEANGBT	Mean gradient of total field	$\overline{\left|{\rm{\nabla }}{B}_{\mathrm{tot}}\right|}=\tfrac{1}{N}\sum \sqrt{{\left(\tfrac{\partial {B}_{\mathrm{tot}}}{\partial x}\right)}^{2}+{\left(\tfrac{\partial {B}_{\mathrm{tot}}}{\partial y}\right)}^{2}}$
MEANGBZ	Mean gradient of vertical field	$\overline{\left|{\rm{\nabla }}{B}_{z}\right|}=\tfrac{1}{N}\sum \sqrt{{\left(\tfrac{\partial {B}_{z}}{\partial x}\right)}^{2}+{\left(\tfrac{\partial {B}_{z}}{\partial y}\right)}^{2}}$
MEANGBH	Mean gradient of horizontal field	$\overline{\left|{\rm{\nabla }}{B}_{h}\right|}=\tfrac{1}{N}\sum \sqrt{{\left(\tfrac{\partial {B}_{h}}{\partial x}\right)}^{2}+{\left(\tfrac{\partial {B}_{h}}{\partial y}\right)}^{2}}$
MEANJZH	Mean current helicity (Bz contribution)	$\overline{{H}_{c}}\propto \tfrac{1}{N}\sum {B}_{z}\cdot {J}_{z}$
MEANJZD	Mean vertical current density	${F}_{y}\propto \sum {B}_{y}\cdot {B}_{z}\cdot {dA}$
MEANALP	Mean characteristic twist parameter, α	$\overline{\alpha }\propto \tfrac{\sum {B}_{Z}\cdot {J}_{Z}}{\sum {B}_{Z}^{2}}$

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As an example, Figure 1 demonstrates the procedures to generate the PIL mask for AR 11158. The top left panel is B_r taken at 01:12 UT, 2011 February 15. The top right and bottom left are bitmap masks for positive and negative fields, respectively. The PIL mask is shown in the bottom right panel. It can be clearly seen that the PIL is well characterized by the mask.

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Figure 2 shows the SHARP parameter maps for AR 11158. The panels in the first and third columns are the original parameter maps, while the second and fourth columns denote the modified maps multiplied with the PIL mask. In the modified maps, the parameters are excluded from most pixels in the images, except for the area enclosed by the PIL mask. The pixels finally chosen to calculate the modified SHARP parameters are indeed enclosed by the PIL mask, and also satisfy a high-confidence disambiguation threshold where the vector field has been disambiguated with high confidence (Bobra & Couvidat 2015). Thus, the modified SHARP parameters represent the physical quantities in the PIL areas.

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Figures 3 and 4 show distributions of the original and modified SHARP parameters, respectively. The black (red) curves refer to distributions of the parameters for the noneruptive (eruptive) AR sample. The two groups of data are described in Section 2.1. X-axis refers to the value of the parameters. The parameters are normalized. We use distributions of the total unsigned current helicity (TOTUSJH, top left panels) as an example to describe the plots. For both the original and modified TOTUSJH, most noneruptive ARs have low values and the peak of the distribution is thus on the low-value end (black); the value becomes high for most eruptive ARs, and thus the peak for eruptive ARs is on the high-value end (red). The peak separation between the noneruptive and eruptive ARs represents a discrimination distance between the two groups. Discriminating the two groups becomes easy when the distance is large. TOTUSJH is therefore an effective parameter to separate the noneruptive and eruptive ARs.

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Parameters TOTPOT, TOTUSJZ, ABSNJZH, SAVNCPP, USFLUX, MEANPOT, and R_VALUE in both original and modified SHARP data are also effective for discrimination. The rest of the parameters (the third and fourth rows) that perform poorly in discriminating the noneruptive and eruptive ARs have been improved substantially after modification. Thus this modification increases the number of effective SHARP parameters by a factor of two that are able to discriminate the two groups of ARs.

To further evaluate the effectiveness of the modified SHARP parameters for discrimination, we calculate the Fisher ranking score (F-score) for each parameter. The F-score is a useful statistics algorithm that measures both the distance between the data points in different groups and the distance between the data points in the same group for a selected feature (Lomax & Hahs-Vaughn 2013, p.10). For a good predictor, the former should be as large as possible, the latter should be as small as possible. For a binary classification (yes or no outcomes), F-score is defined as,

$$\begin{eqnarray}&&F(x)=\displaystyle \frac{{N}_{+}{\left(E[{x}_{+}]-E[x]\right)}^{2}+{N}_{-}{\left(E[{x}_{-}]-E[x]\right)}^{2}}{{N}_{+}\mathrm{Var}[{x}_{+}]+{N}_{-}\mathrm{Var}[{x}_{-}]},\end{eqnarray} \tag{ 1 }$$

where x₊ and x₋ are the subsamples of the feature belonging to two groups, respectively. E(x) and Var(x) are the expect value and variance of the samples. A low F-score indicates that the distance between two groups is blunt and there are many nearby values of parameter X related to both groups. Conversely, a high F-score indicates that the distance between two groups is sharp and clear. Similar to Bobra & Couvidat (2015), we calculate the F-score using the SelectKBest class of binary classifier in the Scikit-learn package (Pedregosa et al. 2011).

Figure 5 displays the F-scores for the modified (red dots) and original (black) SHARP parameters. All the modified SHARP parameters have a high F-score, while the F-score is fairly low for more than half of the original SHARP parameters. This once again demonstrates that the modified parameters have been improved significantly in discriminating the two groups of noneruptive and eruptive ARs, which shows potential for them to be used for predicting solar flares.

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In the aforementioned tests, the original SHARP parameters that perform poorly in discrimination are physical quantities that average over the entire images, i.e., the mean of the quantities. Modification that weighs the PIL areas to the parameters improves their performance significantly. This strongly suggests that the magnetic field property in PILs is tightly related to flare eruptions. This localized property has been significantly diluted when averaging over all of the ARs for the original SHARP parameters.

To assess the potential of predicting flares with the modified SHARP parameters, we use a binary classification method to classify the ARs based on each parameter. We adopt the random forest algorithm using the RandomForestClassifier in the Scikit-learn package (Breiman 2001; Pedregosa et al. 2011). For each AR, the method classifies whether the AR is in the eruptive group or the noneruptive group. To achieve unbiased results, for each parameter we perform a Monte Carlo analysis 500 times on the training and testing processes. For each time, we first shuffle the data points; then we split them into 60% training data set and 40% testing data set; we generate a random forest model based on the training data set to classify ARs; and we run a test on the independent testing data set; finally, we verify the model performance statistically.

As listed in Table 2, compared with the ground truth, this prediction for the AR can be true positive if the prediction agrees with the true and the AR is an eruptive AR, or true negative if the prediction agrees and the AR is a noneruptive AR. The prediction can also be false negative if the prediction disagrees and the AR is eruptive, or false positive if the prediction disagrees and the AR is noneruptive.

Following Zhang & Casey (2000) and Powers (2011), we use five metrics, recall, precision, F1 Score, True Skill Statistic (TSS), and Heidke Skill Scores (HSS), to evaluate the performance of the prediction. They are defined as,

$$\begin{eqnarray}&&\mathrm{recall}=\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\mathrm{precision}=\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{F}_{1}=\displaystyle \frac{2\mathrm{TP}}{2\mathrm{TP}+\mathrm{FP}+\mathrm{FN}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\mathrm{TSS}=\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}-\displaystyle \frac{\mathrm{FP}}{\mathrm{FP}+\mathrm{TN}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{HSS}\\ =\,\displaystyle \frac{2[(\mathrm{TP}\times \mathrm{TN})-(\mathrm{FN}\times \mathrm{FP})]}{(\mathrm{TP}+\mathrm{FP})\times (\mathrm{FP}+\mathrm{TN})+(\mathrm{TP}+\mathrm{FN})\times (\mathrm{FN}+\mathrm{TN})}.\end{array}\end{eqnarray} \tag{ 6 }$$

This binary classifier is deemed to be effective if the metrics are high.

Figure 6 shows metrics for both original (left panel) and modified SHARP parameters (right panel). The black, blue, red, orange, and purple circles represent recall, precision, TSS, HSS, and F1 scores for each parameter, respectively. The horizontal bars in each circle refer to the error bar (standard deviation) for the metrics. The TSS, HSS, and F1 becomes much smaller for half of the original SHARP parameters. Apparently these parameters are not effective classifiers for eruptive and noneruptive ARs. Again these parameters are means of the physical quantities, agreeing with the result in Section 3.1. Performance of these parameters becomes much better for the modified data. This suggests the modified SHARP parameters have great potential to identify the eruptive ARs.

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Here, the application of the random forest algorithm with a single predictor variable in the above context does not involve many predictors at a time but only one predictor. This does not use the full power of the random forest algorithm. So, results of this numerical application of random forests are not exhaustive and a multiparameter random forest analysis will be done in a future work.