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BackgroundHandler Class Reference

#include <BackgroundHandler.h>

Public Member Functions

std::pair< double, double > backgroundFunction (const bool doBackgroundFit, const double *parval, const int resTotNum, const int ires, const bool *resConsidered, const double *ResMass, const double ResHalfWidth[], const int MuonType, const double &mass, const double &eta1, const double &eta2)
 
 BackgroundHandler (const std::vector< int > &identifiers, const std::vector< double > &leftWindowBorders, const std::vector< double > &rightWindowBorders, const double *ResMass, const double *massWindowHalfWidth)
 
bool checkBackgroundWindow (const double &mass, const int iRegion)
 Check if the mass value is inside the given background region. More...
 
void countEventsInAllWindows (const std::vector< std::pair< reco::Particle::LorentzVector, reco::Particle::LorentzVector > > &muonPairs, const double &weight)
 
void initializeParNums ()
 Initialize the parNums to be used in the shifts of parval. More...
 
int regionsParNum ()
 Returns the total number of parameters used for the regions. More...
 
void rescale (std::vector< double > &parBgr, const double *ResMass, const double *massWindowHalfWidth, const std::vector< std::pair< reco::Particle::LorentzVector, reco::Particle::LorentzVector > > &muonPairs, const double &weight=1.)
 
double resMass (const bool doBackgroundFit, const int ires)
 
void setParameters (double *Start, double *Step, double *Mini, double *Maxi, int *ind, TString *parname, const std::vector< double > &parBgr, const std::vector< int > &parBgrOrder, const int muonType)
 Sets initial parameters for all the functions. More...
 
bool unlockParameter (const std::vector< int > &resfind, const unsigned int ipar)
 returns true if the parameter is to be unlocked More...
 
std::pair< double, double > windowBorders (const bool doBackgroundFit, const int ires)
 Returns the appropriate window borders depending on whether the background is being fitted and on the resonance. More...
 
 ~BackgroundHandler ()
 

Private Member Functions

void consistencyCheck (const std::vector< int > &identifiers, const std::vector< double > &leftWindowBorders, const std::vector< double > &rightWindowBorders) const noexcept(false)
 Used to check the consistency of passed parameters. More...
 

Private Attributes

std::vector< MassWindowbackgroundWindow_
 
int parNumsRegions_ [3]
 
int parNumsResonances_ [6]
 
int regToResHW_ [3]
 
std::vector< MassWindowresonanceWindow_
 
int resToReg_ [6]
 

Friends

class TestBackgroundHandler
 

Detailed Description

This class is used to handle the different background functions for the different regions.
It uses the backgroundFunctions defined in Functions.h and the backgroundFunctionService defined in Functions.cc.
More details are in the description of backgroundFunctionBase in Functions.h.

A bool selects if to use the regions functions or the resonances functions.

Definition at line 32 of file BackgroundHandler.h.

Constructor & Destructor Documentation

◆ BackgroundHandler()

BackgroundHandler::BackgroundHandler ( const std::vector< int > &  identifiers,
const std::vector< double > &  leftWindowBorders,
const std::vector< double > &  rightWindowBorders,
const double *  ResMass,
const double *  massWindowHalfWidth 
)

Definition at line 11 of file BackgroundHandler.cc.

References backgroundFunctionService(), backgroundWindow_, consistencyCheck(), mps_fire::i, initializeParNums(), EgHLTOffHistBins_cfi::mass, regToResHW_, resonanceWindow_, and resToReg_.

15  {
16  // : leftWindowFactors_(leftWindowFactors),
17  // rightWindowFactors_(rightWindowFactors)
18 
19  // Define the correspondence between regions and halfWidth to use
20  // Defines also the function type to use (but they are checked to be consistent over a region)
21  regToResHW_[0] = 0; // Region 0 use the one from Z
22  regToResHW_[1] = 3; // Region 1 use the one from Upsilon1S
23  regToResHW_[2] = 5; // Region 2 use the one from J/Psi
24 
25  // Define the correspondence between resonances and regions
26  resToReg_[0] = 0; // Z
27  resToReg_[1] = 1; // Upsilon3S
28  resToReg_[2] = 1; // Upsilon2S
29  resToReg_[3] = 1; // Upsilon1S
30  resToReg_[4] = 2; // Psi2S
31  resToReg_[5] = 2; // J/Psi
32 
33  // Throws cms::Exception("Configuration") in case the parameters are not what is expected
34  consistencyCheck(identifiers, leftWindowBorders, rightWindowBorders);
35 
36  // Build the resonance windows
37  for (unsigned int i = 0; i < 6; ++i) {
38  double mass = ResMass[i];
39  double lowerLimit = mass - massWindowHalfWidth[i];
40  double upperLimit = mass + massWindowHalfWidth[i];
41  resonanceWindow_.push_back(
43  lowerLimit,
44  upperLimit,
45  std::vector<unsigned int>(1, i),
46  backgroundFunctionService(identifiers[resToReg_[i]], lowerLimit, upperLimit)));
47  }
48 
49  // Build the background windows
50  // ----------------------------
51  // Compute the mass center of each region
52  double resMassForRegion[3];
53  resMassForRegion[0] = ResMass[0];
54  resMassForRegion[1] = (ResMass[1] + ResMass[2] + ResMass[3]) / 3;
55  resMassForRegion[2] = (ResMass[4] + ResMass[5]) / 2;
56 
57  // Define which resonance is in which background window
58  std::vector<std::vector<unsigned int> > indexes;
59  indexes.push_back(std::vector<unsigned int>(1, 0));
60  indexes.push_back(std::vector<unsigned int>());
61  for (int i = 1; i <= 3; ++i) {
62  indexes[1].push_back(i);
63  }
64  indexes.push_back(std::vector<unsigned int>());
65  for (int i = 4; i <= 5; ++i) {
66  indexes[2].push_back(i);
67  }
68 
69  unsigned int i = 0;
70  typedef std::vector<unsigned int> indexVec;
71  for (auto const& index : indexes) {
72  // double lowerLimit = resMassForRegion[i] - massWindowHalfWidth[regToResHW_[i]]*leftWindowFactors[i];
73  // double upperLimit = resMassForRegion[i] + massWindowHalfWidth[regToResHW_[i]]*rightWindowFactors[i];
74  // backgroundWindow_.push_back( MassWindow( resMassForRegion[i], lowerLimit, upperLimit, index,
75  // backgroundFunctionService(identifiers[i], lowerLimit, upperLimit ) ) );
76  backgroundWindow_.push_back(
77  MassWindow(resMassForRegion[i],
78  leftWindowBorders[i],
79  rightWindowBorders[i],
80  index,
81  backgroundFunctionService(identifiers[i], leftWindowBorders[i], rightWindowBorders[i])));
82  ++i;
83  }
84  // Initialize the parNums to be used in the shifts of parval
86 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_
void consistencyCheck(const std::vector< int > &identifiers, const std::vector< double > &leftWindowBorders, const std::vector< double > &rightWindowBorders) const noexcept(false)
Used to check the consistency of passed parameters.
backgroundFunctionBase * backgroundFunctionService(const int identifier, const double &lowerLimit, const double &upperLimit)
Service to build the background functor corresponding to the passed identifier.
Definition: Functions.cc:114
void initializeParNums()
Initialize the parNums to be used in the shifts of parval.

◆ ~BackgroundHandler()

BackgroundHandler::~BackgroundHandler ( )

Definition at line 88 of file BackgroundHandler.cc.

88 {}

Member Function Documentation

◆ backgroundFunction()

std::pair< double, double > BackgroundHandler::backgroundFunction ( const bool  doBackgroundFit,
const double *  parval,
const int  resTotNum,
const int  ires,
const bool *  resConsidered,
const double *  ResMass,
const double  ResHalfWidth[],
const int  MuonType,
const double &  mass,
const double &  eta1,
const double &  eta2 
)

Returns the background fraction parameter (parBgr[0], but shifted to the correct function) and the value returned by the background function.
Depending on the value of doBackgroundFit it returns the values for the regions or the resonances.

Definition at line 249 of file BackgroundHandler.cc.

References backgroundWindow_, HLT_2024v14_cff::eta1, HLT_2024v14_cff::eta2, EgHLTOffHistBins_cfi::mass, parNumsRegions_, parNumsResonances_, resonanceWindow_, and resToReg_.

Referenced by MuScleFitUtils::massProb().

259  {
260  if (doBackgroundFit) {
261  // Return the values for the region
262  int iReg = resToReg_[ires];
263  // return std::make_pair( parval[parNumsRegions_[iReg]] * backgroundWindow_[iReg].backgroundFunction()->fracVsEta(&(parval[parNumsRegions_[iReg]]), resEta),
264  return std::make_pair(
265  parval[parNumsRegions_[iReg]] *
266  backgroundWindow_[iReg].backgroundFunction()->fracVsEta(&(parval[parNumsRegions_[iReg]]), eta1, eta2),
267  (*(backgroundWindow_[iReg].backgroundFunction()))(&(parval[parNumsRegions_[iReg]]), mass, eta1, eta2));
268  // return std::make_pair( backgroundWindow_[iReg].backgroundFunction()->fracVsEta(&(parval[parNumsRegions_[iReg]]), eta1, eta2),
269  // (*(backgroundWindow_[iReg].backgroundFunction()))( &(parval[parNumsRegions_[iReg]]), mass, eta1, eta2 ) );
270  }
271  // Return the values for the resonance
272  // return std::make_pair( parval[parNumsResonances_[ires]] * resonanceWindow_[ires].backgroundFunction()->fracVsEta(&(parval[parNumsResonances_[ires]]), resEta),
273  // (*(resonanceWindow_[ires].backgroundFunction()))( &(parval[parNumsResonances_[ires]]), mass, resEta ) );
274  return std::make_pair(
275  parval[parNumsResonances_[ires]] *
276  resonanceWindow_[ires].backgroundFunction()->fracVsEta(&(parval[parNumsResonances_[ires]]), eta1, eta2),
277  (*(resonanceWindow_[ires].backgroundFunction()))(&(parval[parNumsResonances_[ires]]), mass, eta1, eta2));
278 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_
std::pair< double, double > backgroundFunction(const bool doBackgroundFit, const double *parval, const int resTotNum, const int ires, const bool *resConsidered, const double *ResMass, const double ResHalfWidth[], const int MuonType, const double &mass, const double &eta1, const double &eta2)

◆ checkBackgroundWindow()

bool BackgroundHandler::checkBackgroundWindow ( const double &  mass,
const int  iRegion 
)
inline

Check if the mass value is inside the given background region.

Definition at line 51 of file BackgroundHandler.h.

References backgroundWindow_, and EgHLTOffHistBins_cfi::mass.

51 { return backgroundWindow_[iRegion].isIn(mass); }
std::vector< MassWindow > backgroundWindow_

◆ consistencyCheck()

void BackgroundHandler::consistencyCheck ( const std::vector< int > &  identifiers,
const std::vector< double > &  leftWindowBorders,
const std::vector< double > &  rightWindowBorders 
) const
privatenoexcept

Used to check the consistency of passed parameters.

Definition at line 306 of file BackgroundHandler.cc.

References Exception.

Referenced by BackgroundHandler().

308  {
309  if (leftWindowBorders.size() != rightWindowBorders.size()) {
310  throw cms::Exception("Configuration")
311  << "BackgroundHandler::BackgroundHandler: leftWindowBorders.size() = " << leftWindowBorders.size()
312  << " != rightWindowBorders.size() = " << rightWindowBorders.size() << std::endl;
313  }
314  if (leftWindowBorders.size() != 3) {
315  throw cms::Exception("Configuration")
316  << "BackgroundHandler::BackgroundHandler: leftWindowBorders.size() = rightWindowBorders.size() = "
317  << leftWindowBorders.size() << " != 3" << std::endl;
318  }
319  if (identifiers.size() != 3) {
320  throw cms::Exception("Configuration")
321  << "BackgroundHandler::BackgroundHandler: identifiers must match the number of regions = 3" << std::endl;
322  }
323 }

◆ countEventsInAllWindows()

void BackgroundHandler::countEventsInAllWindows ( const std::vector< std::pair< reco::Particle::LorentzVector, reco::Particle::LorentzVector > > &  muonPairs,
const double &  weight 
)

Definition at line 280 of file BackgroundHandler.cc.

References backgroundWindow_, resonanceWindow_, and mps_merge::weight.

Referenced by rescale().

282  {
283  // First reset all the counters
284  for (auto& resonanceWindow : resonanceWindow_) {
285  resonanceWindow.resetCounter();
286  }
287  // Count events in background windows
288  for (auto& backgroundWindow : backgroundWindow_) {
289  backgroundWindow.resetCounter();
290  }
291 
292  // Now count the events in each window
293  std::pair<lorentzVector, lorentzVector> muonPair;
294  for (auto const& muonPair : muonPairs) {
295  // Count events in resonance windows
296  for (auto& resonanceWindow : resonanceWindow_) {
297  resonanceWindow.count((muonPair.first + muonPair.second).mass(), weight);
298  }
299  // Count events in background windows
300  for (auto& backgroundWindow : backgroundWindow_) {
301  backgroundWindow.count((muonPair.first + muonPair.second).mass(), weight);
302  }
303  }
304 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_

◆ initializeParNums()

void BackgroundHandler::initializeParNums ( )

Initialize the parNums to be used in the shifts of parval.

Definition at line 90 of file BackgroundHandler.cc.

References backgroundWindow_, mps_fire::i, parNumsRegions_, parNumsResonances_, and resonanceWindow_.

Referenced by BackgroundHandler().

90  {
91  // Initialize the parNums to be used in the shifts of parval
92  parNumsRegions_[0] = 0;
93  for (unsigned int i = 1; i < backgroundWindow_.size(); ++i) {
94  parNumsRegions_[i] = parNumsRegions_[i - 1] + backgroundWindow_[i - 1].backgroundFunction()->parNum();
95  }
96  parNumsResonances_[0] = parNumsRegions_[2] + backgroundWindow_[2].backgroundFunction()->parNum();
97  for (unsigned int i = 1; i < resonanceWindow_.size(); ++i) {
98  parNumsResonances_[i] = parNumsResonances_[i - 1] + resonanceWindow_[i - 1].backgroundFunction()->parNum();
99  }
100 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_

◆ regionsParNum()

int BackgroundHandler::regionsParNum ( )
inline

Returns the total number of parameters used for the regions.

Definition at line 48 of file BackgroundHandler.h.

References parNumsResonances_.

Referenced by MuScleFitUtils::minimizeLikelihood().

48 { return parNumsResonances_[0]; }

◆ rescale()

void BackgroundHandler::rescale ( std::vector< double > &  parBgr,
const double *  ResMass,
const double *  massWindowHalfWidth,
const std::vector< std::pair< reco::Particle::LorentzVector, reco::Particle::LorentzVector > > &  muonPairs,
const double &  weight = 1. 
)

Computes the rescaled parameters from the regions functions to the resonances functions. It takes into account the difference in intervals and rescales the parameters so that the fraction of events is correctly accounted for.
It uses the list of all muon pairs to compute the number of events in each resonance window.

Definition at line 192 of file BackgroundHandler.cc.

References backgroundWindow_, countEventsInAllWindows(), gather_cfg::cout, parNumsRegions_, parNumsResonances_, resonanceWindow_, and resToReg_.

Referenced by MuScleFitUtils::minimizeLikelihood().

197  {
198  countEventsInAllWindows(muonPairs, weight);
199 
200  // Loop on all regions and on all the resonances of each region and compute the background fraction
201  // for each resonance window.
202  unsigned int iRegion = 0;
203  for (auto const& backgroundWindow : backgroundWindow_) {
204  // Iterator pointing to the first parameter of this background function in the full set of parameters
205  std::vector<double>::const_iterator parBgrIt = (parBgr.begin() + parNumsRegions_[iRegion]);
206  TF1* backgroundFunctionForIntegral = backgroundWindow.backgroundFunction()->functionForIntegral(parBgrIt);
207  // WARNING: this expects the background fraction parameter to be parBgr[0] for all the background functions.
208  double kOld = *parBgrIt;
209  double Nbw = backgroundWindow.events();
210  double Ibw = backgroundFunctionForIntegral->Integral(backgroundWindow.lowerBound(), backgroundWindow.upperBound());
211 
212  // index is the index of the resonance in the background window
213  for (unsigned int index : *(backgroundWindow.indexes())) {
214  // First set all parameters of the resonance window background function to those of the corresponding region
215  for (int iPar = 0; iPar < resonanceWindow_[index].backgroundFunction()->parNum(); ++iPar) {
216  parBgr[parNumsResonances_[index] + iPar] = parBgr[parNumsRegions_[resToReg_[index]] + iPar];
217  }
218  // Estimated fraction of events in the resonance window
219  double Irw = backgroundFunctionForIntegral->Integral(resonanceWindow_[index].lowerBound(),
220  resonanceWindow_[index].upperBound());
221  double Nrw = resonanceWindow_[index].events();
222 
223  // Ibw is 1 (to avoid effects from approximation errors we set it to 1 and do not include it in the computation).
224  if (Nrw != 0)
225  parBgr[parNumsResonances_[index]] = kOld * Nbw / Nrw * Irw;
226  else
227  parBgr[parNumsResonances_[index]] = 0.;
228 
229  // Protect against fluctuations of number of events which could cause the fraction to go above 1.
230  if (parBgr[parNumsResonances_[index]] > 1.)
231  parBgr[parNumsResonances_[index]] = 1.;
232 
233  double kNew = parBgr[parNumsResonances_[index]];
234  std::cout << "For resonance = " << index << std::endl;
235  std::cout << "backgroundWindow.lowerBound = " << backgroundWindow.lowerBound() << std::endl;
236  std::cout << "backgroundWindow.upperBound = " << backgroundWindow.upperBound() << std::endl;
237  std::cout << "parNumsResonances_[" << index << "] = " << parNumsResonances_[index] << std::endl;
238  std::cout << "Nbw = " << Nbw << ", Ibw = " << Ibw << std::endl;
239  std::cout << "Nrw = " << Nrw << ", Irw = " << Irw << std::endl;
240  std::cout << "k = " << kOld << ", k' = " << parBgr[parNumsResonances_[index]] << std::endl;
241  std::cout << "background fraction in background window = Nbw*k = " << Nbw * kOld << std::endl;
242  std::cout << "background fraction in resonance window = Nrw*k' = " << Nrw * kNew << std::endl;
243  }
244  ++iRegion;
245  delete backgroundFunctionForIntegral;
246  }
247 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_
Definition: weight.py:1
void countEventsInAllWindows(const std::vector< std::pair< reco::Particle::LorentzVector, reco::Particle::LorentzVector > > &muonPairs, const double &weight)

◆ resMass()

double BackgroundHandler::resMass ( const bool  doBackgroundFit,
const int  ires 
)

Returns the appropriate resMass value depending on whether the background is being fitted and on the resonance.
The resMass used for the region is the mean of the mass of the corresponding resonances, so for the Z is the same Z mass, for the Upsilons is the arithmetic mean of the Upsilon masses and the same for the J/Psi and Psi2S region.

Definition at line 182 of file BackgroundHandler.cc.

References backgroundWindow_, resonanceWindow_, and resToReg_.

Referenced by MuScleFitUtils::massProb().

182  {
183  if (doBackgroundFit) {
184  // Fitting the background: use the regions
185  return backgroundWindow_[resToReg_[ires]].mass();
186  } else {
187  // Not fitting the background: use the resonances
188  return resonanceWindow_[ires].mass();
189  }
190 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_

◆ setParameters()

void BackgroundHandler::setParameters ( double *  Start,
double *  Step,
double *  Mini,
double *  Maxi,
int *  ind,
TString *  parname,
const std::vector< double > &  parBgr,
const std::vector< int > &  parBgrOrder,
const int  muonType 
)

Sets initial parameters for all the functions.

Definition at line 102 of file BackgroundHandler.cc.

References backgroundWindow_, parNumsRegions_, parNumsResonances_, resonanceWindow_, and edm::shift.

Referenced by MuScleFitUtils::minimizeLikelihood().

110  {
111  std::vector<double>::const_iterator parBgrIt = parBgr.begin();
112  std::vector<int>::const_iterator parBgrOrderIt = parBgrOrder.begin();
113  // Set the parameters for the regions only if this is not a rescaling
114  for (int iReg = 0; iReg < 3; ++iReg) {
115  int shift = parNumsRegions_[iReg];
116  backgroundWindow_[iReg].backgroundFunction()->setParameters(&(Start[shift]),
117  &(Step[shift]),
118  &(Mini[shift]),
119  &(Maxi[shift]),
120  &(ind[shift]),
121  &(parname[shift]),
122  parBgrIt + shift,
123  parBgrOrderIt + shift,
124  muonType);
125  }
126  for (int iRes = 0; iRes < 6; ++iRes) {
127  // parNumsResonances is already shifted for the regions parameters
128  int shift = parNumsResonances_[iRes];
129  resonanceWindow_[iRes].backgroundFunction()->setParameters(&(Start[shift]),
130  &(Step[shift]),
131  &(Mini[shift]),
132  &(Maxi[shift]),
133  &(ind[shift]),
134  &(parname[shift]),
135  parBgrIt + shift,
136  parBgrOrderIt + shift,
137  muonType);
138  }
139 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_
static unsigned int const shift

◆ unlockParameter()

bool BackgroundHandler::unlockParameter ( const std::vector< int > &  resfind,
const unsigned int  ipar 
)

returns true if the parameter is to be unlocked

Definition at line 141 of file BackgroundHandler.cc.

References parNumsRegions_, parNumsResonances_, and MuScleFitGenFilter_cfi::resfind.

Referenced by MuScleFitUtils::minimizeLikelihood().

141  {
142  // parNumsRegions_ are shifted: [1] contains the number of parameters for 0 and so on.
143  if (ipar < unsigned(parNumsRegions_[1]) && resfind[0] > 0) {
144  return true;
145  }
146  if (ipar >= unsigned(parNumsRegions_[1]) && ipar < unsigned(parNumsRegions_[2]) &&
147  (resfind[1] > 0 || resfind[2] > 0 || resfind[3] > 0)) {
148  return true;
149  }
150  // The first of parNumsResonances_ has the sum of parNums of the regions.
151  if (ipar >= unsigned(parNumsRegions_[2]) && ipar < unsigned(parNumsResonances_[0]) &&
152  (resfind[4] > 0 || resfind[5] > 0)) {
153  return true;
154  }
155  return false;
156 }

◆ windowBorders()

std::pair< double, double > BackgroundHandler::windowBorders ( const bool  doBackgroundFit,
const int  ires 
)

Returns the appropriate window borders depending on whether the background is being fitted and on the resonance.

Definition at line 170 of file BackgroundHandler.cc.

References backgroundWindow_, resonanceWindow_, and resToReg_.

Referenced by MuScleFitUtils::computeWeight(), MuScleFitUtils::massProb(), and MuScleFitUtils::minimizeLikelihood().

170  {
171  if (doBackgroundFit) {
172  // Fitting the background: use the regions
173  return std::make_pair(backgroundWindow_[resToReg_[ires]].lowerBound(),
174  backgroundWindow_[resToReg_[ires]].upperBound());
175  } else {
176  // Not fitting the background: use the resonances
177  // return std::make_pair(1.,1.);
178  return std::make_pair(resonanceWindow_[ires].lowerBound(), resonanceWindow_[ires].upperBound());
179  }
180 }
std::vector< MassWindow > backgroundWindow_
std::vector< MassWindow > resonanceWindow_

Friends And Related Function Documentation

◆ TestBackgroundHandler

friend class TestBackgroundHandler
friend

Definition at line 34 of file BackgroundHandler.h.

Member Data Documentation

◆ backgroundWindow_

std::vector<MassWindow> BackgroundHandler::backgroundWindow_
private

◆ parNumsRegions_

int BackgroundHandler::parNumsRegions_[3]
private

◆ parNumsResonances_

int BackgroundHandler::parNumsResonances_[6]
private

◆ regToResHW_

int BackgroundHandler::regToResHW_[3]
private

Definition at line 120 of file BackgroundHandler.h.

Referenced by BackgroundHandler().

◆ resonanceWindow_

std::vector<MassWindow> BackgroundHandler::resonanceWindow_
private

◆ resToReg_

int BackgroundHandler::resToReg_[6]
private