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PDEFoam.h
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1// @(#)root/tmva $Id$
2// Author: S. Jadach, Tancredi Carli, Dominik Dannheim, Alexander Voigt
3
4/**********************************************************************************
5 * Project: TMVA - a Root-integrated toolkit for multivariate data analysis *
6 * Package: TMVA *
7 * Classes: PDEFoam *
8 * Web : http://tmva.sourceforge.net *
9 * *
10 * Description: *
11 * Class for PDEFoam object *
12 * *
13 * Authors (alphabetical): *
14 * S. Jadach - Institute of Nuclear Physics, Cracow, Poland *
15 * Tancredi Carli - CERN, Switzerland *
16 * Dominik Dannheim - CERN, Switzerland *
17 * Alexander Voigt - TU Dresden, Germany *
18 * *
19 * Copyright (c) 2008, 2010: *
20 * CERN, Switzerland *
21 * MPI-K Heidelberg, Germany *
22 * *
23 * Redistribution and use in source and binary forms, with or without *
24 * modification, are permitted according to the terms listed in LICENSE *
25 * (http://tmva.sourceforge.net/LICENSE) *
26 **********************************************************************************/
27
28#ifndef ROOT_TMVA_PDEFoam
29#define ROOT_TMVA_PDEFoam
30
31#include <iosfwd>
32#include <cassert>
33
34#include "TH2D.h"
35#include "TObjArray.h"
36#include "TObjString.h"
37#include "TVectorT.h"
38#include "TString.h"
39#include "TMVA/VariableInfo.h"
40#include "TMVA/Timer.h"
41#include "TObject.h"
42#include "TRandom3.h"
43
44namespace TMVA {
45 class PDEFoamCell;
46 class PDEFoamVect;
47 class PDEFoamDensityBase;
48 class PDEFoamKernelBase;
49 class PDEFoam;
50
51 // separation types
52 enum EDTSeparation { kFoam, kGiniIndex, kMisClassificationError,
53 kCrossEntropy, kGiniIndexWithLaplace, kSdivSqrtSplusB };
54
55 // foam types
56 enum EFoamType { kSeparate, kDiscr, kMonoTarget, kMultiTarget, kMultiClass };
57
58 // enum type for possible foam cell values
59 // kValue : cell value who's rms is minimized
60 // kValueError : error on kValue
61 // kValueDensity : kValue / cell volume
62 // kMeanValue : mean sampling value (saved in fIntegral)
63 // kRms : rms of sampling distribution (saved in fDriver)
64 // kRmsOvMean : rms/mean of sampling distribution (saved in
65 // fDriver and fIntegral)
66 // kCellVolume : volume of cell
67 enum ECellValue { kValue, kValueError, kValueDensity, kMeanValue,
68 kRms, kRmsOvMean, kCellVolume };
69}
70
72#include "TMVA/PDEFoamVect.h"
73#include "TMVA/PDEFoamCell.h"
74
75namespace TMVA {
76
77 class PDEFoam : public TObject {
78 protected:
79 // COMPONENTS //
80 //-------------- Input parameters
81 TString fName; // Name of a given instance of the FOAM class
82 Int_t fDim; // Dimension of the integration/simulation space
83 Int_t fNCells; // Maximum number of cells
84 //-------------------
85 Int_t fNBin; // No. of bins in the edge histogram for cell MC exploration
86 Int_t fNSampl; // No. of MC events, when dividing (exploring) cell
87 Int_t fEvPerBin; // Maximum number of effective (wt=1) events per bin
88 //------------------- MULTI-BRANCHING ---------------------
89 Int_t *fMaskDiv; //! [fDim] Dynamic Mask for cell division
90 Int_t *fInhiDiv; //! [fDim] Flags for inhibiting cell division
91 //------------------- GEOMETRY ----------------------------
92 Int_t fNoAct; // Number of active cells
93 Int_t fLastCe; // Index of the last cell
94 PDEFoamCell **fCells; // [fNCells] Array of ALL cells
95 //------------------ M.C. generation----------------------------
96 TObjArray *fHistEdg; // Histograms of wt, one for each cell edge
97 Double_t *fRvec; // [fDim] random number vector from r.n. generator fDim+1 maximum elements
98 //----------- Procedures
99 TRandom3 *fPseRan; // Pointer to user-defined generator of pseudorandom numbers
100 //---------- working space for CELL exploration -------------
101 Double_t *fAlpha; // [fDim] Internal parameters of the hyperrectangle
102 // --------- PDE-Foam specific variables
103 EFoamType fFoamType; // BACKWARDS COMPATIBILITY: type of foam
104 Double_t *fXmin; // [fDim] minimum for variable transform
105 Double_t *fXmax; // [fDim] maximum for variable transform
106 UInt_t fNElements; // BACKWARDS COMPATIBILITY: number of variables in every cell
107 UInt_t fNmin; // minimal number of events in cell to split cell
108 UInt_t fMaxDepth; // maximum depth of cell tree
109 Float_t fVolFrac; // BACKWARDS COMPATIBILITY: volume fraction (with respect to total phase space
110 Bool_t fFillFoamWithOrigWeights; // BACKWARDS COMPATIBILITY: fill the foam with boost or orig. weights
111 EDTSeparation fDTSeparation; // BACKWARDS COMPATIBILITY: split cells according to decision tree logic
112 Bool_t fPeekMax; // BACKWARDS COMPATIBILITY: peek up cell with max. driver integral for split
113 PDEFoamDensityBase *fDistr; //! distribution of training events
114 Timer *fTimer; //! timer for graphical output
115 TObjArray *fVariableNames;// collection of all variable names
116 mutable MsgLogger* fLogger; //! message logger
117
118 /////////////////////////////////////////////////////////////////
119 // METHODS //
120 /////////////////////////////////////////////////////////////////
121
122 protected:
123 // ---------- TMVA console output
124
125 void OutputGrow(Bool_t finished = false ); // nice TMVA console output
126
127 // ---------- Foam build-up functions
128
129 // Internal foam initialization functions
130 void InitCells(); // Initialisation of all foam cells
131 Int_t CellFill(Int_t, PDEFoamCell*);// Allocates new empty cell and return its index
132 virtual void Explore(PDEFoamCell *Cell); // Exploration of the new cell, determine <wt>, wtMax etc.
133 void Varedu(Double_t [], Int_t&, Double_t&,Double_t&); // Determines the best edge, variance reduction
134 void MakeAlpha(); // Provides random point inside hyperrectangle
135 void Grow(); // build up foam
136 Long_t PeekMax(); // peek cell with max. driver integral
137 Int_t Divide(PDEFoamCell *); // Divide iCell into two daughters; iCell retained, taged as inactive
138 Double_t Eval(Double_t *xRand, Double_t &event_density); // evaluate distribution on point 'xRand'
139
140 // ---------- Cell value access functions
141
142 // low level functions to access a certain cell value
143 Double_t GetCellElement(const PDEFoamCell *cell, UInt_t i) const; // get Element 'i' in cell 'cell'
144 void SetCellElement(PDEFoamCell *cell, UInt_t i, Double_t value); // set Element 'i' in cell 'cell' to value 'value'
145
146 // specific function used during evaluation; determines, whether a cell value is undefined
148
149 // finds cell according to given event variables
150 PDEFoamCell* FindCell(const std::vector<Float_t>&) const ;
151 std::vector<TMVA::PDEFoamCell*> FindCells(const std::vector<Float_t>&) const;
152 std::vector<TMVA::PDEFoamCell*> FindCells(const std::map<Int_t,Float_t>&) const;
153 void FindCells(const std::map<Int_t, Float_t>&, PDEFoamCell*, std::vector<PDEFoamCell*> &) const;
154
155 // get internal density
156 PDEFoamDensityBase* GetDistr() const { assert(fDistr); return fDistr; }
157
158 // Square function (fastest implementation)
159 template<typename T> T Sqr(T x) const { return x*x; }
160
161 PDEFoam(const PDEFoam&); // Copy Constructor NOT USED
162
163 // ---------- Public functions ----------------------------------
164 public:
165 PDEFoam(); // Default constructor (used only by ROOT streamer)
166 PDEFoam(const TString&); // Principal user-defined constructor
167 virtual ~PDEFoam(); // Default destructor
168
169 // ---------- Foam creation functions
170
171 void Initialize() {} // initialize the PDEFoam
172 void FillBinarySearchTree( const Event* ev ); // fill event into BST
173 void Create(); // build-up foam
174
175 // function to fill created cell with given value
176 virtual void FillFoamCells(const Event* ev, Float_t wt);
177
178 // remove all cell elements
179 void ResetCellElements();
180
181 // function to call after foam is grown
182 virtual void Finalize() {}
183
184 // ---------- Getters and Setters
185
186 void SetDim(Int_t kDim); // Sets dimension of cubical space
187 void SetnCells(Long_t nCells){fNCells =nCells;} // Sets maximum number of cells
188 void SetnSampl(Long_t nSampl){fNSampl =nSampl;} // Sets no of MC events in cell exploration
189 void SetnBin(Int_t nBin){fNBin = nBin;} // Sets no of bins in histograms in cell exploration
190 void SetEvPerBin(Int_t EvPerBin){fEvPerBin =EvPerBin;} // Sets max. no. of effective events per bin
191 void SetInhiDiv(Int_t, Int_t ); // Set inhibition of cell division along certain edge
192 void SetDensity(PDEFoamDensityBase *dens) { fDistr = dens; }
193
194 // coverity[ -tainted_data_return ]
195 Int_t GetTotDim() const {return fDim; } // Get total dimension
196 TString GetFoamName() const {return fName; } // Get name of foam
197 UInt_t GetNActiveCells() const {return fNoAct;} // returns number of active cells
198 UInt_t GetNInActiveCells() const {return GetNCells()-GetNActiveCells();} // returns number of not active cells
199 UInt_t GetNCells() const {return fNCells;} // returns number of cells
200 PDEFoamCell* GetRootCell() const {return fCells[0];} // get pointer to root cell
201
202 // Getters and Setters for user cut options
203 void SetNmin(UInt_t val) { fNmin=val; }
204 UInt_t GetNmin() { return fNmin; }
205 void SetMaxDepth(UInt_t maxdepth) { fMaxDepth = maxdepth; }
206 UInt_t GetMaxDepth() const { return fMaxDepth; }
207
208 // Getters and Setters for foam boundaries
209 void SetXmin(Int_t idim, Double_t wmin);
210 void SetXmax(Int_t idim, Double_t wmax);
211 Double_t GetXmin(Int_t idim) const {return fXmin[idim];}
212 Double_t GetXmax(Int_t idim) const {return fXmax[idim];}
213
214 // Getters and Setters for variable names
215 void AddVariableName(const char *s) { AddVariableName(new TObjString(s)); }
217 TObjString* GetVariableName(Int_t idx) {return dynamic_cast<TObjString*>(fVariableNames->At(idx));}
218
219 // Delete the fDistr object, which contains the binary search
220 // tree
222
223 // ---------- Transformation functions for event variables into foam boundaries
224 // reason: foam always has boundaries [0, 1]
225
226 Float_t VarTransform(Int_t idim, Float_t x) const; // transform [xmin, xmax] --> [0, 1]
227 std::vector<Float_t> VarTransform(const std::vector<Float_t> &invec) const;
228 Float_t VarTransformInvers(Int_t idim, Float_t x) const; // transform [0, 1] --> [xmin, xmax]
229 std::vector<Float_t> VarTransformInvers(const std::vector<Float_t> &invec) const;
230
231 // ---------- Debug functions
232
233 void CheckAll(Int_t); // Checks correctness of the entire data structure in the FOAM object
234 void PrintCell(Long_t iCell=0); // Print content of cell
235 void PrintCells(); // Prints content of all cells
236
237 // Message logger
238 MsgLogger& Log() const { return *fLogger; }
239
240 // ---------- Foam projection methods
241
242 // project foam to two-dimensional histogram
243 virtual TH2D* Project2(Int_t idim1, Int_t idim2, ECellValue cell_value=kValue,
244 PDEFoamKernelBase *kernel=NULL, UInt_t nbin=50);
245
246 // Project one-dimensional foam to a 1-dim histogram
247 TH1D* Draw1Dim(ECellValue cell_value, Int_t nbin, PDEFoamKernelBase *kernel=NULL);
248
249 // Generates C++ code (root macro) for drawing foam with boxes (only 2-dim!)
250 void RootPlot2dim( const TString& filename, TString opt,
251 Bool_t createCanvas = kTRUE, Bool_t colors = kTRUE );
252
253 // ---------- Foam evaluation functions
254
255 // get cell value for a given event
256 virtual Float_t GetCellValue( const std::vector<Float_t>& xvec, ECellValue cv, PDEFoamKernelBase* );
257
258 // get cell values for a given (incomplete) event vector
259 virtual std::vector<Float_t> GetCellValue( const std::map<Int_t,Float_t>& xvec, ECellValue cv );
260
261 // get cell value stored in a foam cell
262 virtual Float_t GetCellValue( const PDEFoamCell* cell, ECellValue cv );
263
264 // ---------- friend classes
265 friend class PDEFoamKernelBase;
267 friend class PDEFoamKernelLinN;
268 friend class PDEFoamKernelGauss;
269
270 // ---------- ROOT class definition
271 ClassDef(PDEFoam,7) // Tree of PDEFoamCells
272 }; // end of PDEFoam
273
274} // namespace TMVA
275
276// ---------- Inline functions
277
278//_____________________________________________________________________
280{
281 // transform variable x from [xmin, xmax] --> [0, 1]
282 return (x-fXmin[idim])/(fXmax[idim]-fXmin[idim]);
283}
284
285//_____________________________________________________________________
286inline std::vector<Float_t> TMVA::PDEFoam::VarTransform(const std::vector<Float_t> &invec) const
287{
288 // transform vector invec from [xmin, xmax] --> [0, 1]
289 std::vector<Float_t> outvec;
290 for(UInt_t i=0; i<invec.size(); i++)
291 outvec.push_back(VarTransform(i, invec.at(i)));
292 return outvec;
293}
294
295//_____________________________________________________________________
297{
298 // transform variable x from [0, 1] --> [xmin, xmax]
299 return x*(fXmax[idim]-fXmin[idim]) + fXmin[idim];
300}
301
302//_____________________________________________________________________
303inline std::vector<Float_t> TMVA::PDEFoam::VarTransformInvers(const std::vector<Float_t> &invec) const
304{
305 // transform vector invec from [0, 1] --> [xmin, xmax]
306 std::vector<Float_t> outvec;
307 for(UInt_t i=0; i<invec.size(); i++)
308 outvec.push_back(VarTransformInvers(i, invec.at(i)));
309 return outvec;
310}
311
312#endif
int Int_t
Definition: RtypesCore.h:41
unsigned int UInt_t
Definition: RtypesCore.h:42
long Long_t
Definition: RtypesCore.h:50
bool Bool_t
Definition: RtypesCore.h:59
double Double_t
Definition: RtypesCore.h:55
float Float_t
Definition: RtypesCore.h:53
const Bool_t kTRUE
Definition: RtypesCore.h:87
#define ClassDef(name, id)
Definition: Rtypes.h:326
Color * colors
Definition: X3DBuffer.c:21
1-D histogram with a double per channel (see TH1 documentation)}
Definition: TH1.h:614
2-D histogram with a double per channel (see TH1 documentation)}
Definition: TH2.h:292
ostringstream derivative to redirect and format output
Definition: MsgLogger.h:59
This is an abstract class, which provides an interface for a PDEFoam density estimator.
This class is the abstract kernel interface for PDEFoam.
This PDEFoam kernel estimates a cell value for a given event by weighting all cell values with a gaus...
This PDEFoam kernel estimates a cell value for a given event by weighting with cell values of the nea...
This class is a trivial PDEFoam kernel estimator.
Implementation of PDEFoam.
Definition: PDEFoam.h:77
void AddVariableName(const char *s)
Definition: PDEFoam.h:215
void SetNmin(UInt_t val)
Definition: PDEFoam.h:203
void Varedu(Double_t[], Int_t &, Double_t &, Double_t &)
Internal subprogram used by Create.
Definition: PDEFoam.cxx:571
Double_t GetCellElement(const PDEFoamCell *cell, UInt_t i) const
Returns cell element i of cell 'cell'.
Definition: PDEFoam.cxx:1417
Bool_t fPeekMax
Definition: PDEFoam.h:112
void DeleteBinarySearchTree()
Delete the foam's density estimator, which contains the binary search tree.
Definition: PDEFoam.cxx:1667
TObjString * GetVariableName(Int_t idx)
Definition: PDEFoam.h:217
void SetMaxDepth(UInt_t maxdepth)
Definition: PDEFoam.h:205
void MakeAlpha()
Internal subprogram used by Create.
Definition: PDEFoam.cxx:638
Timer * fTimer
distribution of training events
Definition: PDEFoam.h:114
TObjArray * fVariableNames
timer for graphical output
Definition: PDEFoam.h:115
Float_t fVolFrac
Definition: PDEFoam.h:109
UInt_t GetMaxDepth() const
Definition: PDEFoam.h:206
virtual Float_t GetCellValue(const std::vector< Float_t > &xvec, ECellValue cv, PDEFoamKernelBase *)
This function finds the cell, which corresponds to the given untransformed event vector 'xvec' and re...
Definition: PDEFoam.cxx:1017
void OutputGrow(Bool_t finished=false)
message logger
Definition: PDEFoam.cxx:1461
UInt_t GetNInActiveCells() const
Definition: PDEFoam.h:198
UInt_t GetNCells() const
Definition: PDEFoam.h:199
Double_t GetXmax(Int_t idim) const
Definition: PDEFoam.h:212
void SetCellElement(PDEFoamCell *cell, UInt_t i, Double_t value)
Set cell element i of cell to value.
Definition: PDEFoam.cxx:1433
void SetDensity(PDEFoamDensityBase *dens)
Definition: PDEFoam.h:192
Float_t VarTransformInvers(Int_t idim, Float_t x) const
Definition: PDEFoam.h:296
Double_t * fRvec
Definition: PDEFoam.h:97
Float_t VarTransform(Int_t idim, Float_t x) const
Definition: PDEFoam.h:279
Int_t Divide(PDEFoamCell *)
Internal subprogram used by Create.
Definition: PDEFoam.cxx:715
Bool_t fFillFoamWithOrigWeights
Definition: PDEFoam.h:110
MsgLogger & Log() const
Definition: PDEFoam.h:238
EDTSeparation fDTSeparation
Definition: PDEFoam.h:111
void PrintCell(Long_t iCell=0)
Prints geometry of and elements of 'iCell', as well as relations to parent and daughter cells.
Definition: PDEFoam.cxx:896
void ResetCellElements()
Remove the cell elements from all cells.
Definition: PDEFoam.cxx:972
void Grow()
Internal subprogram used by Create.
Definition: PDEFoam.cxx:768
MsgLogger * fLogger
Definition: PDEFoam.h:116
Double_t Eval(Double_t *xRand, Double_t &event_density)
Internal subprogram.
Definition: PDEFoam.cxx:748
Int_t fNBin
Definition: PDEFoam.h:85
void PrintCells()
Prints geometry of ALL cells of the FOAM.
Definition: PDEFoam.cxx:943
void SetXmax(Int_t idim, Double_t wmax)
set upper foam bound in dimension idim
Definition: PDEFoam.cxx:277
PDEFoamDensityBase * GetDistr() const
Definition: PDEFoam.h:156
Double_t * fXmin
Definition: PDEFoam.h:104
virtual ~PDEFoam()
Default destructor.
Definition: PDEFoam.cxx:185
PDEFoam()
Default constructor for streamer, user should not use it.
Definition: PDEFoam.cxx:104
TRandom3 * fPseRan
Definition: PDEFoam.h:99
void SetEvPerBin(Int_t EvPerBin)
Definition: PDEFoam.h:190
PDEFoamCell * GetRootCell() const
Definition: PDEFoam.h:200
void SetnBin(Int_t nBin)
Definition: PDEFoam.h:189
PDEFoamDensityBase * fDistr
Definition: PDEFoam.h:113
TH1D * Draw1Dim(ECellValue cell_value, Int_t nbin, PDEFoamKernelBase *kernel=NULL)
Draws 1-dimensional foam (= histogram)
Definition: PDEFoam.cxx:1219
virtual Bool_t CellValueIsUndefined(PDEFoamCell *)
Returns true, if the value of the given cell is undefined.
Definition: PDEFoam.cxx:991
virtual void FillFoamCells(const Event *ev, Float_t wt)
This function fills a weight 'wt' into the PDEFoam cell, which corresponds to the given event 'ev'.
Definition: PDEFoam.cxx:956
void Create()
Basic initialization of FOAM invoked by the user.
Definition: PDEFoam.cxx:293
TObjArray * fHistEdg
Definition: PDEFoam.h:96
void SetInhiDiv(Int_t, Int_t)
This can be called before Create, after setting kDim It defines which variables are excluded in the p...
Definition: PDEFoam.cxx:803
Int_t * fInhiDiv
[fDim] Dynamic Mask for cell division
Definition: PDEFoam.h:90
void InitCells()
Internal subprogram used by Create.
Definition: PDEFoam.cxx:357
void AddVariableName(TObjString *s)
Definition: PDEFoam.h:216
Int_t GetTotDim() const
Definition: PDEFoam.h:195
void Initialize()
Definition: PDEFoam.h:171
Double_t * fAlpha
Definition: PDEFoam.h:101
virtual void Finalize()
Definition: PDEFoam.h:182
PDEFoamCell * FindCell(const std::vector< Float_t > &) const
Find cell that contains 'xvec' (in foam coordinates [0,1]).
Definition: PDEFoam.cxx:1081
UInt_t fNElements
Definition: PDEFoam.h:106
Int_t * fMaskDiv
Definition: PDEFoam.h:89
void FillBinarySearchTree(const Event *ev)
Insert event to internal foam's density estimator PDEFoamDensityBase.
Definition: PDEFoam.cxx:1658
virtual TH2D * Project2(Int_t idim1, Int_t idim2, ECellValue cell_value=kValue, PDEFoamKernelBase *kernel=NULL, UInt_t nbin=50)
Project foam variable idim1 and variable idim2 to histogram.
Definition: PDEFoam.cxx:1271
UInt_t fMaxDepth
Definition: PDEFoam.h:108
Int_t fNCells
Definition: PDEFoam.h:83
PDEFoamCell ** fCells
Definition: PDEFoam.h:94
virtual void Explore(PDEFoamCell *Cell)
Internal subprogram used by Create.
Definition: PDEFoam.cxx:435
void SetXmin(Int_t idim, Double_t wmin)
set lower foam bound in dimension idim
Definition: PDEFoam.cxx:266
UInt_t GetNmin()
Definition: PDEFoam.h:204
std::vector< TMVA::PDEFoamCell * > FindCells(const std::vector< Float_t > &) const
Find all cells, that contain txvec.
Definition: PDEFoam.cxx:1166
UInt_t GetNActiveCells() const
Definition: PDEFoam.h:197
Int_t fDim
Definition: PDEFoam.h:82
UInt_t fNmin
Definition: PDEFoam.h:107
T Sqr(T x) const
Definition: PDEFoam.h:159
Int_t fLastCe
Definition: PDEFoam.h:93
void SetnCells(Long_t nCells)
Definition: PDEFoam.h:187
Int_t CellFill(Int_t, PDEFoamCell *)
Internal subprogram used by Create.
Definition: PDEFoam.cxx:390
Long_t PeekMax()
Internal subprogram used by Create.
Definition: PDEFoam.cxx:652
void RootPlot2dim(const TString &filename, TString opt, Bool_t createCanvas=kTRUE, Bool_t colors=kTRUE)
Debugging tool which plots the cells of a 2-dimensional PDEFoam as rectangles in C++ format readable ...
Definition: PDEFoam.cxx:1512
TString fName
Definition: PDEFoam.h:81
void SetnSampl(Long_t nSampl)
Definition: PDEFoam.h:188
Int_t fNoAct
[fDim] Flags for inhibiting cell division
Definition: PDEFoam.h:92
Double_t * fXmax
Definition: PDEFoam.h:105
TString GetFoamName() const
Definition: PDEFoam.h:196
Double_t GetXmin(Int_t idim) const
Definition: PDEFoam.h:211
void CheckAll(Int_t)
User utility, miscellaneous and debug.
Definition: PDEFoam.cxx:823
Int_t fNSampl
Definition: PDEFoam.h:86
void SetDim(Int_t kDim)
Sets dimension of cubical space.
Definition: PDEFoam.cxx:251
Int_t fEvPerBin
Definition: PDEFoam.h:87
EFoamType fFoamType
Definition: PDEFoam.h:103
Timing information for training and evaluation of MVA methods.
Definition: Timer.h:58
An array of TObjects.
Definition: TObjArray.h:37
void Add(TObject *obj)
Definition: TObjArray.h:74
TObject * At(Int_t idx) const
Definition: TObjArray.h:166
Collectable string class.
Definition: TObjString.h:28
Mother of all ROOT objects.
Definition: TObject.h:37
Random number generator class based on M.
Definition: TRandom3.h:27
Basic string class.
Definition: TString.h:131
Double_t x[n]
Definition: legend1.C:17
double T(double x)
Definition: ChebyshevPol.h:34
static constexpr double s
create variable transformations