From 5f6b8fc54e2805bd1c109553fc6c1149638dadab Mon Sep 17 00:00:00 2001 From: Olivier Couet <olivier.couet@cern.ch> Date: Fri, 17 May 2013 16:21:29 +0200 Subject: [PATCH] Replace pictures by formulas. Redo picture. 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The Fit -Panel, which is limited, is best for prototyping. The histogram needs to -be drawn in a pad before the Fit Panel is invoked. The method `TH1::Fit` -is more powerful and is used in scripts and programs. +To fit a histogram you can use the Fit Panel on a visible histogram +via the context menu, or you can use the **`TH1`**`::Fit` method. The +Fit Panel, which is limited, is best for prototyping. The histogram +needs to be drawn in a pad before the Fit Panel is invoked. The method +`TH1::Fit` is more powerful and is used in scripts and programs. ## The Fit Method -To fit a histogram programmatically, you can use the `TH1::Fit` method. -Here is the signature of `TH1::Fit` and an explanation of the +To fit a histogram programmatically, you can use the `TH1::Fit` +method. Here is the signature of `TH1::Fit` and an explanation of the parameters: ``` {.cpp} -void Fit(const char *fname, Option_t *option, Option_t *goption, -Axis_t xxmin, Axis_t xxmax) + void Fit(const char *fname, Option_t *option, Option_t *goption, + Axis_t xxmin, Axis_t xxmax) ``` - `*fname: `The name of the fitted function (the model) is passed as @@ -40,9 +40,11 @@ Axis_t xxmin, Axis_t xxmax) - "`W`" Set all weights to 1 for non empty bins; ignore error bars -- "`WW`" Set all weights to 1 including empty bins; ignore error bars +- "`WW`" Set all weights to 1 including empty bins; ignore error + bars -- "`I`" Use integral of function in bin instead of value at bin center +- "`I`" Use integral of function in bin instead of value at bin + center - "`L`" Use log likelihood method (default is chi-square method) @@ -64,18 +66,18 @@ Axis_t xxmin, Axis_t xxmax) function is drawn unless the option "`N`" above is specified. - "`+`" Add this new fitted function to the list of fitted functions - (by default, the previous function is deleted and only the last one - is kept) + (by default, the previous function is deleted and only the last + one is kept) -- "`B`"Use this option when you want to fix one or more parameters and - the fitting function is like `polN`, `expo`, `landau`, `gaus`. +- "`B`"Use this option when you want to fix one or more parameters + and the fitting function is like `polN`, `expo`, `landau`, `gaus`. -- "`LL`"An improved Log Likelihood fit in case of very low statistics - and when bincontentsare not integers. Do not use this option if bin - contents are large (greater than 100). +- "`LL`"An improved Log Likelihood fit in case of very low + statistics and when bincontentsare not integers. Do not use this + option if bin contents are large (greater than 100). -- "`C`"In case of linear fitting, don't calculate the chisquare (saves - time). +- "`C`"In case of linear fitting, don't calculate the chisquare + (saves time). - "`F`"If fitting a `polN`, switch to `Minuit` fitter (by default, `polN` functions are fitted by the linear fitter). @@ -83,11 +85,11 @@ Axis_t xxmin, Axis_t xxmax) - `*goption: `The third parameter is the graphics option that is the same as in the **`TH1`**`::Draw` (see the chapter Draw Options). -- `xxmin`, `xxmax:`Thee fourth and fifth parameters specify the range - over which to apply the fit. +- `xxmin`, `xxmax:`Thee fourth and fifth parameters specify the + range over which to apply the fit. -By default, the fitting function object is added to the histogram and is -drawn in the current pad. +By default, the fitting function object is added to the histogram and +is drawn in the current pad. ## Fit with a Predefined Function @@ -120,8 +122,8 @@ the **`TF1`** object. There are three ways to create a **`TF1`**. ### Creating a TF1 with a Formula -Let's look at the first case. Here we call the **`TF1`** constructor by -giving it the formula: `sin(x)/x`. +Let's look at the first case. Here we call the **`TF1`** constructor +by giving it the formula: `sin(x)/x`. ``` {.cpp} root[] TF1 *f1 = new TF1("f1","sin(x)/x",0,10) @@ -144,7 +146,7 @@ expression. Here we use two parameters: root[] TF1 *f1 = new TF1("f1","[0]*x*sin([1]*x)",-3,3); ``` - + The parameter index is enclosed in square brackets. To set the initial parameters explicitly you can use: @@ -170,8 +172,8 @@ root[] f1->Draw() ### Creating a TF1 with a User Function -The third way to build a **`TF1`** is to define a function yourself and -then give its name to the constructor. A function for a **`TF1`** +The third way to build a **`TF1`** is to define a function yourself +and then give its name to the constructor. A function for a **`TF1`** constructor needs to have this exact signature: ``` {.cpp} @@ -182,98 +184,99 @@ The two parameters are: - `x ` a pointer to the dimension array. Each element contains a dimension. For a 1D histogram only `x[0]` is used, for a 2D - histogram `x[0]` and `x[1]` is used, and for a 3D histogram `x[0]`, - `x[1]`, and `x[2]` are used. For histograms, only 3 dimensions - apply, but this method is also used to fit other objects, for - example an ntuple could have 10 dimensions. + histogram `x[0]` and `x[1]` is used, and for a 3D histogram + `x[0]`, `x[1]`, and `x[2]` are used. For histograms, only 3 + dimensions apply, but this method is also used to fit other + objects, for example an ntuple could have 10 dimensions. - `par` a pointer to the parameters array. This array contains the current values of parameters when it is called by the fitting function. -The following script `$ROOTSYS/tutorials/fit/myfit.C` illustrates how to -fit a 1D histogram with a user-defined function. First we declare the -function. +The following script `$ROOTSYS/tutorials/fit/myfit.C` illustrates how +to fit a 1D histogram with a user-defined function. First we declare +the function. ``` {.cpp} -// define a function with 3 parameters -Double_t fitf(Double_t *x,Double_t *par) { - Double_t arg = 0; - if (par[2]!=0) arg = (x[0] - par[1])/par[2]; - Double_t fitval = par[0]*TMath::Exp(-0.5*arg*arg); - return fitval; -} + // define a function with 3 parameters + Double_t fitf(Double_t *x,Double_t *par) { + Double_t arg = 0; + if (par[2]!=0) arg = (x[0] - par[1])/par[2]; + Double_t fitval = par[0]*TMath::Exp(-0.5*arg*arg); + return fitval; + } ``` Now we use the function: ``` {.cpp} -// this function used fitf to fit a histogram -void fitexample() { - - // open a file and get a histogram - TFile *f = new TFile("hsimple.root"); - TH1F *hpx = (TH1F*)f->Get("hpx"); - - // Create a TF1 object using the function defined above. The last three - // parameters specify the number of parameters for the function. - TF1 *func = new TF1("fit",fitf,-3,3,3); - // set the parameters to the mean and RMS of the histogram - func->SetParameters(500,hpx->GetMean(),hpx->GetRMS()); - - // give the parameters meaningful names - func->SetParNames ("Constant","Mean_value","Sigma"); - - // call TH1::Fit with the name of the TF1 object - hpx->Fit("fit"); -} + // this function used fitf to fit a histogram + void fitexample() { + + // open a file and get a histogram + TFile *f = new TFile("hsimple.root"); + TH1F *hpx = (TH1F*)f->Get("hpx"); + + // Create a TF1 object using the function defined above. + // The last three parameters specify the number of parameters + // for the function. + TF1 *func = new TF1("fit",fitf,-3,3,3); + // set the parameters to the mean and RMS of the histogram + func->SetParameters(500,hpx->GetMean(),hpx->GetRMS()); + + // give the parameters meaningful names + func->SetParNames ("Constant","Mean_value","Sigma"); + + // call TH1::Fit with the name of the TF1 object + hpx->Fit("fit"); + } ``` ## Fixing and Setting Parameters' Bounds Parameters must be initialized before invoking the `Fit` method. The -setting of the parameter initial values is automatic for the predefined -functions: `poln`, `exp`, `gaus`, and `landau`. You can fix one or more -parameters by specifying the "B" option when calling the `Fit` method. -When a function is not predefined, the fit parameters must be -initialized to some value as close as possible to the expected values -before calling the fit function. +setting of the parameter initial values is automatic for the +predefined functions: `poln`, `exp`, `gaus`, and `landau`. You can fix +one or more parameters by specifying the "B" option when calling the +`Fit` method. When a function is not predefined, the fit parameters +must be initialized to some value as close as possible to the expected +values before calling the fit function. To set bounds for one parameter, use `TF1::SetParLimits`: ``` {.cpp} -func->SetParLimits(0,-1,1); + func->SetParLimits(0,-1,1); ``` -When the lower and upper limits are equal, the parameter is fixed. Next -two statements fix parameter 4 at 10. +When the lower and upper limits are equal, the parameter is fixed. +Next two statements fix parameter 4 at 10. ``` {.cpp} -func->SetParameter(4,10); -func->SetParLimits(4,10,10); + func->SetParameter(4,10); + func->SetParLimits(4,10,10); ``` However, to fix a parameter to 0, one must call the `FixParameter` function: ``` {.cpp} -func->SetParameter(4,0); -func->FixParameter(4,0); + func->SetParameter(4,0); + func->FixParameter(4,0); ``` Note that you are not forced to set the limits for all parameters. For example, if you fit a function with 6 parameters, you can: ``` {.cpp} -func->SetParameters(0,3.1,1.e-6,-1.5,0,100); -func->SetParLimits(3,-10,4); -func->FixParameter(4,0); + func->SetParameters(0,3.1,1.e-6,-1.5,0,100); + func->SetParLimits(3,-10,4); + func->FixParameter(4,0); ``` With this setup, parameters 0`->`2 can vary freely, parameter 3 has -boundaries [-10, 4] with initial value -1.5, and parameter 4 is fixed to -0. +boundaries [-10, 4] with initial value -1.5, and parameter 4 is fixed +to 0. ## Fitting Sub Ranges @@ -295,32 +298,33 @@ You can also specify a range in the call to `TH1::Fit`: root[] hist->Fit("f1","","",-2,2) ``` -See macros `$ROOTSYS/tutorials/fit/myfit.C` and `multifit.C `as more +See macros `$ROOTSYS/tutorials/fit/myfit.C` and `multifit.C` as more completed examples. ## The Fit Panel -To display the Fit Panel right click on a -histogram to pop up the context menu, and then select the menu entry Fit -Panel. + + +To display the Fit Panel right click on a histogram to pop up the +context menu, and then select the menu entry Fit Panel. -The new Fit Panel GUI is available in ROOT v5.14. Its goal is to replace -the old Fit Panel and to provide more user friendly way for performing, -exploring and comparing fits. +The new Fit Panel GUI is available in ROOT v5.14. Its goal is to +replace the old Fit Panel and to provide more user friendly way for +performing, exploring and comparing fits. -By design, this user interface is planned to contain two tabs: "General" -and "Minimization". Currently, the "General" tab provides user interface -elements for setting the fit function, fit method and different fit, -draw, print options. +By design, this user interface is planned to contain two tabs: +"General" and "Minimization". Currently, the "General" tab provides +user interface elements for setting the fit function, fit method and +different fit, draw, print options. The new fit panel is a modeless dialog, i.e. when opened, it does not prevent users from interacting with other windows. Its first prototype is a singleton application. When the Fit Panel is activated, users can -select an object for fitting in the usual way, i.e. by left-mouse click -on it. If the selected object is suitable for fitting, the fit panel is -connected with this object and users can perform fits by setting -different parameters and options. +select an object for fitting in the usual way, i.e. by left-mouse +click on it. If the selected object is suitable for fitting, the fit +panel is connected with this object and users can perform fits by +setting different parameters and options. ### Function Choice and Settings @@ -338,13 +342,13 @@ between functions: *Conv* - convolution (will be implemented in the future). -Users can enter the function expression into the text entry field below -the 鈥楶redefined' combo box. The entered string is checked after the -Enter key was pressed and an error message shows up, if the function -string is not accepted. +Users can enter the function expression into the text entry field +below the 鈥楶redefined' combo box. The entered string is checked after +the Enter key was pressed and an error message shows up, if the +function string is not accepted. -鈥�*Set Parameters*' button opens a dialog for parameters settings, which -will be explaned later. +鈥�*Set Parameters*' button opens a dialog for parameters settings, +which will be explaned later. ### Fitter Settings @@ -354,20 +358,20 @@ Chi-square and Binned Likelihood. The default one is Chi-square. The Binned Likelihood is recomended for bins with low statistics. *鈥楲inear Fit' check button* sets the use of Linear fitter when is -selected. Otherwise the minimization is done by Minuit, i.e. fit option -"`F`" is applied. The Linear fitter can be selected only for functions -linears in parameters (for example - `polN)`. +selected. Otherwise the minimization is done by Minuit, i.e. fit +option "`F`" is applied. The Linear fitter can be selected only for +functions linears in parameters (for example - `polN)`. *鈥楻obust' number entry* sets the robust value when fitting graphs. -*鈥楴o Chi-square' check button* switch On/Off the fit option "`C`" - do -not calculate Chi-square (for Linear fitter). +*鈥楴o Chi-square' check button* switch On/Off the fit option "`C`" - +do not calculate Chi-square (for Linear fitter). -*鈥業ntegral' check button* switch On/Off the option "`I`" - use integral -of function instead of value in bin center. +*鈥業ntegral' check button* switch On/Off the option "`I`" - use +integral of function instead of value in bin center. -*鈥楤est Errors'* sets On/Off the option "`E`" - better errors estimation -by using Minos technique. +*鈥楤est Errors'* sets On/Off the option "`E`" - better errors +estimation by using Minos technique. *鈥楢ll weights = 1'* sets On/Off the option "`W`"- all weights set to 1 excluding empty bins; error bars ignored. @@ -378,9 +382,9 @@ equal to 1 including empty bins; error bars ignored. *鈥楿se range'* sets On/Off the option "`R`" - fit only data within the specified function range. Sliders settings are used if this option is set to On. Users can change the function range values by pressing the -left mouse button near to the left/right slider edges. It is possible to -change both values simultaneously by pressing the left mouse button near -to the slider center and moving it to a new position. +left mouse button near to the left/right slider edges. It is possible +to change both values simultaneously by pressing the left mouse button +near to the slider center and moving it to a new position. *鈥業mprove fit results'* sets On/Off the option "`M`"- after minimum is found, search for a new one. @@ -389,11 +393,11 @@ found, search for a new one. without deleting the previous one. When fitting a histogram, the function is attached to the histogram's list of functions. By default, the previously fitted function is deleted and replaced with the most -recent one, so the list only contains one function. Setting this option -to On will add the newly fitted function to the existing list of -functions for the histogram. Note that the fitted functions are saved -with the histogram when it is written to a ROOT file. By default, the -function is drawn on the pad displaying the histogram. +recent one, so the list only contains one function. Setting this +option to On will add the newly fitted function to the existing list +of functions for the histogram. Note that the fitted functions are +saved with the histogram when it is written to a ROOT file. By +default, the function is drawn on the pad displaying the histogram. ### Draw Options @@ -410,8 +414,8 @@ function and do not draw it. ### Print Options -This set of options specifies the amount of feedback printed on the root -command line after performed fits. +This set of options specifies the amount of feedback printed on the +root command line after performed fits. *鈥榁erbose'* - prints fit results after each iteration. @@ -425,8 +429,8 @@ command line after performed fits. *Fit button* - performs a fit taking different option settings via the Fit Panel interface. -*Reset* - sets the GUI elements and related fit settings to the default -ones. +*Reset* - sets the GUI elements and related fit settings to the +default ones. *Close* - closes the Fit panel window. @@ -434,10 +438,10 @@ ones. The script for this example is `$ROOTSYS/tutorials/fit/multifit.C`. It -shows how to use several Gaussian functions with different parameters on -separate sub ranges of the same histogram. To use a Gaussian, or any -other ROOT built in function, on a sub range you need to define a new -**`TF1`**. Each is 'derived' from the canned function `gaus`. +shows how to use several Gaussian functions with different parameters +on separate sub ranges of the same histogram. To use a Gaussian, or +any other ROOT built in function, on a sub range you need to define a +new **`TF1`**. Each is 'derived' from the canned function `gaus`.  @@ -445,138 +449,143 @@ functions](pictures/03000062.png) First, four TF1 objects are created - one for each sub-range: ``` {.cpp} -g1 = new TF1("m1","gaus",85,95); -g2 = new TF1("m2","gaus",98,108); -g3 = new TF1("m3","gaus",110,121); -// The total is the sum of the three, each has 3 parameters -total = new TF1("mstotal","gaus(0)+gaus(3)+gaus(6)",85,125); + g1 = new TF1("m1","gaus",85,95); + g2 = new TF1("m2","gaus",98,108); + g3 = new TF1("m3","gaus",110,121); + // The total is the sum of the three, each has 3 parameters + total = new TF1("mstotal","gaus(0)+gaus(3)+gaus(6)",85,125); ``` Next, we fill a histogram with bins defined in the array x. ``` {.cpp} -// Create a histogram and set it's contents -h = new TH1F("g1","Example of several fits in subranges",np,85,134); -h->SetMaximum(7); -for (int i=0; i<np; i++) { - h->SetBinContent(i+1,x[i]); -} -// Define the parameter array for the total function -Double_t par[9]; + // Create a histogram and set it's contents + h = new TH1F("g1","Example of several fits in subranges", + np,85,134); + h->SetMaximum(7); + for (int i=0; i<np; i++) { + h->SetBinContent(i+1,x[i]); + } + // Define the parameter array for the total function + Double_t par[9]; ``` -When fitting simple functions, such as a Gaussian, the initial values of -the parameters are automatically computed by ROOT. In the more -complicated case of the sum of 3 Gaussian functions, the initial values -of parameters must be set. In this particular case, the initial values -are taken from the result of the individual fits. The use of the "+" -sign is explained below: +When fitting simple functions, such as a Gaussian, the initial values +of the parameters are automatically computed by ROOT. In the more +complicated case of the sum of 3 Gaussian functions, the initial +values of parameters must be set. In this particular case, the initial +values are taken from the result of the individual fits. The use of +the "+" sign is explained below: ``` {.cpp} -// Fit each function and add it to the list of functions -h->Fit(g1,"R"); -h->Fit(g2,"R+"); -h->Fit(g3,"R+"); - -// Get the parameters from the fit -g1->GetParameters(&par[0]); -g2->GetParameters(&par[3]); -g3->GetParameters(&par[6]); - -// Use the parameters on the sum -total->SetParameters(par); -h->Fit(total,"R+"); + // Fit each function and add it to the list of functions + h->Fit(g1,"R"); + h->Fit(g2,"R+"); + h->Fit(g3,"R+"); + + // Get the parameters from the fit + g1->GetParameters(&par[0]); + g2->GetParameters(&par[3]); + g3->GetParameters(&par[6]); + + // Use the parameters on the sum + total->SetParameters(par); + h->Fit(total,"R+"); ``` ## Adding Functions to the List -The example `$ROOTSYS/tutorials/fit/multifit.C` also illustrates how to -fit several functions on the same histogram. By default a Fit command -deletes the previously fitted function in the histogram object. You can -specify the option "+" in the second parameter to add the newly fitted -function to the existing list of functions for the histogram. +The example `$ROOTSYS/tutorials/fit/multifit.C` also illustrates how +to fit several functions on the same histogram. By default a Fit +command deletes the previously fitted function in the histogram +object. You can specify the option "+" in the second parameter to add +the newly fitted function to the existing list of functions for the +histogram. ``` {.cpp} root[] hist->Fit("f1","+","",-2,2) ``` -Note that the fitted function(s) are saved with the histogram when it is -written to a ROOT file. +Note that the fitted function(s) are saved with the histogram when it +is written to a ROOT file. ## Combining Functions You can combine functions to fit a histogram with their sum as it is illustrated in the macro `FitDemo.C` -(`$ROOTSYS/tutorials/fit/FittingDemo.C)`. We have a function that is the -combination of a background and Lorentzian peak. Each function +(`$ROOTSYS/tutorials/fit/FittingDemo.C)`. We have a function that is +the combination of a background and Lorentzian peak. Each function contributes 3 parameters: - +$$ +y(E) = a_{1} + a_{2}E + a_{3}E^{2} + \frac{A_{p}(\frac{G}{2p})} +{(E-m)^{2} + (\frac{G}{2})^2 } +$$ + + BackgroundLorentzian Peak -`par[0]` = `par[0]` = - +`par[0]` = $a_{1}$ `par[0]` = $A_{p}$ -`par[1]` = `par[1]` = - +`par[1]` = $a_{2}$ `par[1]` = $G$ -`par[2]` = `par[2]` = - +`par[2]` = $a_{3}$ `par[2]` = $m$ The combination function (`fitFunction`) has six parameters: `fitFunction = background(x,par) + LorentzianPeak(x,&par[3])` -`par[0]=` `par[1]=` -`par[2]=` `par[3]=` -`par[4]=` ` par[5]=` +`par[0]=`$a_{1}$ `par[1]=`$a_{2}$ `par[2]=`$a_{3}$ `par[3]=`$A_{p}$ +`par[4]=`$G$ `par[5]=`$m$ This script creates a histogram and fits it with the combination of two functions. First we define the two functions and the combination function: ``` {.cpp} -// Quadratic background function -Double_t background(Double_t *x, Double_t *par) { -return par[0] + par[1]*x[0] + par[2]*x[0]*x[0]; -} -// Lorentzian Peak function</code> -Double_t lorentzianPeak(Double_t *x, Double_t *par) { -return (0.5*par[0]*par[1]/TMath::Pi()) / TMath::Max(1.e-10, -(x[0]-par[2])*(x[0]-par[2])+ .25*par[1]*par[1]); -} - -// Sum of background and peak function -Double_t fitFunction(Double_t *x, Double_t *par) { -return background(x,par) + lorentzianPeak(x,&par[3]); -} -void FittingDemo() { -// bevington exercise by P. Malzacher, modified by R. Brun -const int nBins = 60; -Stat_t data[nBins] = { 6, 1,10,12, 6,13,23,22,15,21, -23,26,36,25,27,35,40,44,66,81, -75,57,48,45,46,41,35,36,53,32, -40,37,38,31,36,44,42,37,32,32, -43,44,35,33,33,39,29,41,32,44, -26,39,29,35,32,21,21,15,25,15}; -TH1F *histo = new TH1F("example_9_1", -"Lorentzian Peak on Quadratic Background",60,0,3); - -for(int i=0; i < nBins; i++) { -// we use these methods to explicitly set the content - // and error instead of using the fill method. -histo->SetBinContent(i+1,data[i]); -histo->SetBinError(i+1,TMath::Sqrt(data[i])); -} + // Quadratic background function + Double_t background(Double_t *x, Double_t *par) { + return par[0] + par[1]*x[0] + par[2]*x[0]*x[0]; + } + + // Lorentzian Peak function + Double_t lorentzianPeak(Double_t *x, Double_t *par) { + return (0.5*par[0]*par[1]/TMath::Pi()) / TMath::Max(1.e-10, + (x[0]-par[2])*(x[0]-par[2])+ .25*par[1]*par[1]); + } + + // Sum of background and peak function + Double_t fitFunction(Double_t *x, Double_t *par) { + return background(x,par) + lorentzianPeak(x,&par[3]); + } + + void FittingDemo() { + // bevington exercise by P. Malzacher, modified by R. Brun + const int nBins = 60; + Stat_t data[nBins] = { 6, 1,10,12, 6,13,23,22,15,21, + 23,26,36,25,27,35,40,44,66,81, + 75,57,48,45,46,41,35,36,53,32, + 40,37,38,31,36,44,42,37,32,32, + 43,44,35,33,33,39,29,41,32,44, + 26,39,29,35,32,21,21,15,25,15}; + TH1F *histo = new TH1F("example_9_1", + "Lorentzian Peak on Quadratic Background",60,0,3); + + for(int i=0; i < nBins; i++) { + // we use these methods to explicitly set the content + // and error instead of using the fill method. + histo->SetBinContent(i+1,data[i]); + histo->SetBinError(i+1,TMath::Sqrt(data[i])); + } // create a TF1 with the range from 0 to 3 and 6 parameters -TF1 *fitFcn = new TF1("fitFcn",fitFunction,0,3,6); + TF1 *fitFcn = new TF1("fitFcn",fitFunction,0,3,6); // first try without starting values for the parameters - // this defaults to 1 for each param. -histo->Fit("fitFcn"); + // this defaults to 1 for each param. + histo->Fit("fitFcn"); // this results in an ok fit for the polynomial function however // the non-linear part (Lorentzian ``` @@ -591,11 +600,11 @@ For another example see: One or more objects (typically a **`TF1`**\*) can be added to the list of functions (`fFunctions`) associated to each histogram. A call to -`TH1::Fit` adds the fitted function to this list. Given a histogram `h`, -one can retrieve the associated function with: +`TH1::Fit` adds the fitted function to this list. Given a histogram +`h`, one can retrieve the associated function with: ``` {.cpp} -TF1 *myfunc = h->GetFunction("myfunc"); + TF1 *myfunc = h->GetFunction("myfunc"); ``` ## Access to the Fit Parameters and Results @@ -603,8 +612,8 @@ TF1 *myfunc = h->GetFunction("myfunc"); If the histogram (or graph) is made persistent, the list of associated functions is also persistent. Retrieve a pointer to the function with -the **`TH1::GetFunction()` method. Then you can retrieve the fit -parameters from the function (**`TF1`**) with calls such as:** +the **`TH1`**::GetFunction()` method. Then you can retrieve the fit +parameters from the function (**`TF1`**) with calls such as: ``` {.cpp} root[] TF1 *fit = hist->GetFunction(function_name); @@ -620,13 +629,13 @@ root[] Double_t e1 = fit->GetParError(0); By default, for each bin, the sum of weights is computed at fill time. One can also call `TH1::Sumw2` to force the storage and computation of -the sum of the square of weights per bin. If Sumw2 has been called, the -error per bin is computed as the `sqrt` `(`sum of squares of weights`)`; -otherwise, the error is set equal to the `sqrt` `(bin content`). To -return the error for a given bin number, do: +the sum of the square of weights per bin. If Sumw2 has been called, +the error per bin is computed as the `sqrt(sum of squares of +weights)`; otherwise, the error is set equal to the +`sqrt(bin content)`. To return the error for a given bin number, do: ``` {.cpp} -Double_t error = h->GetBinError(bin); + Double_t error = h->GetBinError(bin); ``` Empty bins are excluded in the fit when using the Chi-square fit method. @@ -640,28 +649,25 @@ You can change the statistics box to display the fit parameters with the `TStyle::SetOptFit(mode)` method. This parameter has four digits: `mode = pcev` (`default = 0111`) -- `p`= 1print probability - -- `c` = 1print Chi-square/number of degrees of freedom - -- `e`= 1print errors (if `e=1`, `v` must be 1) - -- `v`= 1print name/values of parameters +- `p` = 1 print probability +- `c` = 1 print Chi-square/number of degrees of freedom +- `e` = 1 print errors (if `e=1`, `v` must be 1) +- `v` = 1 print name/values of parameters For example, to print the fit probability, parameter names/values, and errors, use: ``` {.cpp} -gStyle->SetOptFit(1011); + gStyle->SetOptFit(1011); ``` ## The Minimization Package -This package was originally written in FORTRAN by Fred James and part of -`PACKLIB` (patch D506). It has been converted to a C++ class by Rene -Brun. The current implementation in C++ is a straightforward conversion -of the original FORTRAN version. The main changes are: +This package was originally written in FORTRAN by Fred James and part +of `PACKLIB` (patch D506). It has been converted to a C++ class by +Ren茅 Brun. The current implementation in C++ is a straightforward +conversion of the original FORTRAN version. The main changes are: - The variables in the various `Minuit` labeled common blocks have been changed to the **`TMinuit`** class data members @@ -680,13 +686,13 @@ of the original FORTRAN version. The main changes are: - Constructors and destructor have been added - Instead of passing the `FCN` function in the argument list, the - addresses of this function is stored as pointer in the data members - of the class. This is by far more elegant and flexible in an - interactive environment. The member function `SetFCN` can be used to - define this pointer + addresses of this function is stored as pointer in the data + members of the class. This is by far more elegant and flexible in + an interactive environment. The member function `SetFCN` can be + used to define this pointer -- The ROOT static function `Printf` is provided to replace all format - statements and to print on currently defined output file +- The ROOT static function `Printf` is provided to replace all + format statements and to print on currently defined output file - The derived class **`TMinuitOld`** contains obsolete routines from the FORTRAN based version @@ -702,27 +708,28 @@ of the original FORTRAN version. The main changes are: **`TMinuit`**`::GetPlot().` For example: ``` {.cpp} -h->Fit("gaus"); -gMinuit->Command("SCAn 1"); -TGraph *gr = (TGraph*)gMinuit->GetPlot(); -gr->SetMarkerStyle(21); -gr->Draw("alp"); + h->Fit("gaus"); + gMinuit->Command("SCAn 1"); + TGraph *gr = (TGraph*)gMinuit->GetPlot(); + gr->SetMarkerStyle(21); + gr->Draw("alp"); ``` - To set `Minuit` in no graphics mode, call ``` {.cpp} -gMinuit->SetGraphicsMode(kFALSE); + gMinuit->SetGraphicsMode(kFALSE); ``` ### Basic Concepts of Minuit -The `Minuit` package acts on a multi parameter FORTRAN function to which -one must give the generic name `FCN`. In the ROOT implementation, the -function `FCN` is defined via the `Minuit` `SetFCN` member function when -a `HistogramFit` command is invoked. The value of `FCN` will in general -depend on one or more variable parameters. +The `Minuit` package acts on a multi parameter FORTRAN function to +which one must give the generic name `FCN`. In the ROOT +implementation, the function `FCN` is defined via the `Minuit` +`SetFCN` member function when a `HistogramFit` command is invoked. The +value of `FCN` will in general depend on one or more variable +parameters. To take a simple example, in case of ROOT histograms (classes **`TH1C`**, **`TH1S`**, **`TH1F`**, **`TH1D`**) the `Fit` function @@ -747,9 +754,9 @@ so that the internal value `Pint` can take on any value, while the external value Pext can take on values only between the lower limit a and the ext upper limit b. Since the transformation is necessarily non-linear, it would transform a nice linear problem into a nasty -non-linear one, which is the reason why limits should be avoided if not -necessary. In addition, the transformation does require some computer -time, so it slows down the computation a little bit, and more +non-linear one, which is the reason why limits should be avoided if +not necessary. In addition, the transformation does require some +computer time, so it slows down the computation a little bit, and more importantly, it introduces additional numerical inaccuracy into the problem in addition to what is introduced in the numerical calculation of the FCN value. The effects of non-linearity and numerical round off @@ -765,14 +772,15 @@ so Minuit does a transformation of the error matrix (and the ''parabolic'' parameter errors) when there are parameter limits. Users should however realize that the transformation is only a linear approximation, and that it cannot give a meaningful result if one or -more parameters is very close to a limit, where partial `Pext/`partial -`Pint!=0`. Therefore, it is recommended that: +more parameters is very close to a limit, where +$\frac{\partial Pext}{\partial Pint} \neq 0$. Therefore, it is +recommended that: - Limits on variable parameters should be used only when needed in order to prevent the parameter from taking on unphysical values -- When a satisfactory minimum has been found using limits, the limits - should then be removed if possible, in order to perform or +- When a satisfactory minimum has been found using limits, the + limits should then be removed if possible, in order to perform or re-perform the error analysis without limits ### How to Get the Right Answer from Minuit @@ -781,8 +789,8 @@ more parameters is very close to a limit, where partial `Pext/`partial `Minuit` offers the user a choice of several minimization algorithms. The `MIGRAD` algorithm is in general the best minimized for nearly all functions. It is a variable-metric method with inexact line search, a -stable metric updating scheme, and checks for positive-definiteness. Its -main weakness is that it depends heavily on knowledge of the first +stable metric updating scheme, and checks for positive-definiteness. +Its main weakness is that it depends heavily on knowledge of the first derivatives, and fails miserably if they are very inaccurate. If parameter limits are needed, in spite of the side effects, then the @@ -798,17 +806,18 @@ more parameters is near its limit at the minimum, this may be because the true minimum is indeed at a limit, or it may be because the minimized has become ''blocked'' at a limit. This may normally happen only if the parameter is so close to a limit (internal value at an odd -multiple of `#((pi)/(2))` that `Minuit` prints a warning to this effect -when it prints the parameter values. The minimized can become blocked at -a limit, because at a limit the derivative seen by the minimized -`partial F/partial Pint` is zero no matter what the real derivative -`partial F/partial Pext` is. - -`((partial F)/(partial Pint)) = ` - -`((partial F)/(partial Pext))((partial Pext)/(partial Pint)) =` - -` ((partial F)/(partial Pext)) = 0 ` +multiple of $\pm \frac{\pi}{2}$ that `Minuit` prints a warning to this +effect when it prints the parameter values. The minimized can become +blocked at a limit, because at a limit the derivative seen by the +minimized $\frac{\partial F}{\partial Pint}$ is zero no matter what the real +derivative $\frac{\partial F}{\partial Pext}$ is. + +$$ +\left(\frac{\partial F}{\partial Pint}\right) = +\left(\frac{\partial F}{\partial Pext}\right) +\left(\frac{\partial Pext}{\partial Pint}\right) = +\left(\frac{\partial F}{\partial Pext}\right) = 0 +$$ #### Getting the Right Parameter Errors with Limits @@ -830,15 +839,16 @@ assuming they are accurate. #### Statistical Interpretation -For discussion of basic concepts, such as the meaning of the elements of -the error matrix, or setting of exact confidence levels see the +For discussion of basic concepts, such as the meaning of the elements +of the error matrix, or setting of exact confidence levels see the articles: - F.James. Determining the statistical Significance of experimental - Results. Technical Report DD/81/02 and CERN Report 81-03, CERN, 1981 + Results. Technical Report DD/81/02 and CERN Report 81-03, CERN, + 1981 -- W.T.Eadie, D.Drijard, F.James, M.Roos, and B.Sadoulet. *Statistical - Methods in Experimental Physics. North-Holland, 1971* +- W.T.Eadie, D.Drijard, F.James, M.Roos, and B.Sadoulet. + *Statistical Methods in Experimental Physics. North-Holland, 1971* ### Reliability of Minuit Error Estimates @@ -852,15 +862,16 @@ quite accurate, unless there has been a problem. `Minuit`, when it prints out error values, also gives some indication of how reliable it thinks they are. For example, those marked `CURRENT GUESS ERROR` are only working values not to be believed, and `APPROXIMATE ERROR` means -that they have been calculated but there is reason to believe that they -may not be accurate. +that they have been calculated but there is reason to believe that +they may not be accurate. -If no mitigating adjective is given, then at least `Minuit` believes the -errors are accurate, although there is always a small chance that -`Minuit` has been fooled. Some visible signs that `Minuit` may have been -fooled: +If no mitigating adjective is given, then at least `Minuit` believes +the errors are accurate, although there is always a small chance that +`Minuit` has been fooled. Some visible signs that `Minuit` may have +been fooled: -- Warning messages produced during the minimization or error analysis +- Warning messages produced during the minimization or error + analysis - Failure to find new minimum @@ -869,10 +880,10 @@ fooled: - Correlation coefficients exactly equal to zero, unless some parameters are known to be uncorrelated with the others -- Correlation coefficients very close to one (greater than 0.99). This - indicates both an exceptionally difficult problem, and one which has - been badly parameterized so that individual errors are not very - meaningful because they are so highly correlated +- Correlation coefficients very close to one (greater than 0.99). + This indicates both an exceptionally difficult problem, and one + which has been badly parameterized so that individual errors are + not very meaningful because they are so highly correlated - Parameter at limit. This condition, signaled by a `Minuit` warning message, may make both the function minimum and parameter errors @@ -880,19 +891,20 @@ fooled: errors with limits' The best way to be absolutely sure of the errors is to use -''independent'' calculations and compare them, or compare the calculated -errors with a picture of the function. Theoretically, the covariance -matrix for a ''physical'' function must be positive-definite at the -minimum, although it may not be so for all points far away from the -minimum, even for a well-determined physical problem. Therefore, if -`MIGRAD` reports that it has found a non-positive-definite covariance -matrix, this may be a sign of one or more of the following: +''independent'' calculations and compare them, or compare the +calculated errors with a picture of the function. Theoretically, the +covariance matrix for a ''physical'' function must be +positive-definite at the minimum, although it may not be so for all +points far away from the minimum, even for a well-determined physical +problem. Therefore, if `MIGRAD` reports that it has found a +non-positive-definite covariance matrix, this may be a sign of one or +more of the following: #### A Non-physical Region -On its way to the minimum, `MIGRAD` may have traversed a region that has -unphysical behavior, which is of course not a serious problem as long as -it recovers and leaves such a region. +On its way to the minimum, `MIGRAD` may have traversed a region that +has unphysical behavior, which is of course not a serious problem as +long as it recovers and leaves such a region. #### An Underdetermined Problem @@ -900,22 +912,22 @@ If the matrix is not positive-definite even at the minimum, this may mean that the solution is not well defined, for example that there are more unknowns than there are data points, or that the parameterization of the fit contains a linear dependence. If this is the case, then -`Minuit` (or any other program) cannot solve your problem uniquely. The -error matrix will necessarily be largely meaningless, so the user must -remove the under determinedness by reformulating the parameterization. -`Minuit` cannot do this itself. +`Minuit` (or any other program) cannot solve your problem uniquely. +The error matrix will necessarily be largely meaningless, so the user +must remove the under determinedness by reformulating the +parameterization. `Minuit` cannot do this itself. #### Numerical Inaccuracies -It is possible that the apparent lack of positive-definiteness is due to -excessive round off errors in numerical calculations (in the user +It is possible that the apparent lack of positive-definiteness is due +to excessive round off errors in numerical calculations (in the user function), or not enough precision. This is unlikely in general, but becomes more likely if the number of free parameters is very large, or if the parameters are badly scaled (not all of the same order of magnitude), and correlations are large. In any case, whether the non-positive-definiteness is real or only numerical is largely -irrelevant, since in both cases the error matrix will be unreliable and -the minimum suspicious. +irrelevant, since in both cases the error matrix will be unreliable +and the minimum suspicious. #### An Ill-posed Problem @@ -923,9 +935,9 @@ For questions of parameter dependence, see the discussion above on positive-definiteness. Possible other mathematical problems are the following: -- Excessive numerical round off - be especially careful of exponential - and factorial functions which get big very quickly and lose - accuracy. +- Excessive numerical round off - be especially careful of + exponential and factorial functions which get big very quickly and + lose accuracy. - Starting too far from the solution - the function may have unphysical local minima, especially at infinity in some variables. @@ -934,61 +946,93 @@ following: FUMILI is used to minimize Chi-square function or to search maximum of -likelihood function. Experimentally measured -values are fitted with theoretical -functions, where  -are coordinates, and  - vector of parameters. +likelihood function. Experimentally measured values $F_{i}$ are fitted +with theoretical +functions $f_{i}(\vec{x_{i}},\vec{\theta})$, where $\vec{x_{i}}$ are +coordinates, and $\vec{\theta}$ - vector of parameters. For better convergence Chi-square function has to be the following form - +$$ +\frac{\chi^2}{2} = \frac{1}{2} \sum_{i=1}^{n} +\left(\frac{f_{i}(\vec{x_{i}},\vec{\theta}) - F_{i}} +{\sigma_{i}}\right)^{2} +$$ -where  are errors of the measured function. -The minimum condition is: +where $\sigma_{i}$ are errors of the measured function. The minimum +condition is: - +$$ +\frac{\partial \chi^{2}}{\partial \theta_{i}} = +\sum_{j=1}^{n} +\frac{1}{\sigma_{j}^{2}} . \frac{\partial f_{i}}{\partial \theta_{i}} +\left[ (\vec{x_{j}},\vec{\theta}) + - F_{j}\right] = 0, i = 1 ... m +$$ -where `m `is the quantity of parameters. Expanding left part of this +where $m$ is the quantity of parameters. Expanding left part of this equation over parameter increments and retaining only linear terms one gets - +$$ +\left(\frac{\partial \chi^{2}}{\theta_{i}}\right) +_{\theta = \vec{\theta}^{0}} + +\sum_{k} +\left(\frac{\partial^{2} \chi^{2}}{\partial \theta_{i} +\partial \theta_{k}}\right) +_{\theta = \vec{\theta}^{0}} . (\theta_{k} - \theta_{k}^{0}) = 0 +$$ -here  is some initial value of parameters. In +here $\vec{\theta}^{0}$ is some initial value of parameters. In general case: - +$$ +{\partial^2\chi^2\over\partial\theta_i\partial\theta_k}= +\sum^n_{j=1}{1\over\sigma^2_j} +{\partial f_j\over\theta_i} {\partial f_j\over\theta_k} + +\sum^n_{j=1}{(f_j - F_j)\over\sigma^2_j}\cdot +{\partial^2f_j\over\partial\theta_i\partial\theta_k} +$$ + +In FUMILI algorithm for second derivatives of Chi-square approximate +expression is used when last term in previous equation is discarded. +It is often done, not always wittingly, and sometimes causes troubles, +for example, if user wants to limit parameters with positive values by +writing down $\theta_i^2$ instead of $\theta_i$. FUMILI will fail if +one tries minimize $\chi^2 = g^2(\vec\theta)$ where g is arbitrary +function. -In FUMILI algorithm for second derivatives of Chi-square approximate -expression is used when last term in previous equation is discarded. It -is often done, not always wittingly, and sometimes causes troubles, for -example, if user wants to limit parameters with positive values by -writing down  instead of -. FUMILI will fail if one tries minimize - where an arbitrary -function is. Approximate value is: +Approximate value is: - +$$ +{\partial^2\chi^2\over\partial\theta_i\partial\theta_k}\approx Z_{ik}= +\sum^n_{j=1}{1\over\sigma^2_j}{\partial f_j\over\theta_i} +{\partial f_j\over\theta_k} +$$ Then the equations for parameter increments are: - +$$ +\left(\partial\chi^2\over\partial\theta_i\right)_ +{\vec\theta={\vec\theta}^0} +\sum_k Z_{ik}\cdot(\theta_k-\theta^0_k) = + 0, \qquad i=1\ldots m +$$ Remarkable feature of algorithm is the technique for step restriction. -For an initial value of parameter  a -parallelepiped is built with the center at - and axes parallel to coordinate axes -. The lengths of parallelepiped sides along --axis is , where - is such a value that the functions - are quasi-linear all over the parallelepiped. +For an initial value of parameter ${\vec\theta}^0$ a parallelepiped +$P_0$ is built with the center at ${\vec\theta}^0$ and axes parallel +to coordinate axes $\theta_i$. The lengths of parallelepiped sides +along i-th axis is $2b_i$, where $b_i$ is such a value that the +functions $f_j(\vec\theta)$ are quasi-linear all over the +parallelepiped. + FUMILI takes into account simple linear inequalities in the form: - -They form parallelepiped  -( may be deformed by -). Very similar step formulae are used in -FUMILI for negative logarithm of the likelihood function with the same -idea - linearization of functional argument. +$$ \theta_i^{\rm min}\le\theta_i\le\theta^{\rm max}_i$$ + +They form parallelepiped $P$ ($P_0$ may be deformed by $P$). Very +similar step formulae are used in FUMILI for negative logarithm of +the likelihood function with the same idea - linearization of function +argument. ## Neural Networks @@ -1011,39 +1055,42 @@ networks studies. The multilayer perceptron is a simple feed-forward network with the -following structure showed on the left. +following structure showed on the left. + + It is made of neurons characterized by a bias and weighted links in -between - let's call those links synapses. The input neurons receive the -inputs, normalize them and forward them to the first hidden layer. Each -neuron in any subsequent layer first computes a linear combination of -the outputs of the previous layer. The output of the neuron is then -function of that combination with f being linear for output neurons or a -sigmoid for hidden layers. +between - let's call those links synapses. The input neurons receive +the inputs, normalize them and forward them to the first hidden layer. +Each neuron in any subsequent layer first computes a linear +combination of the outputs of the previous layer. The output of the +neuron is then function of that combination with f being linear for +output neurons or a sigmoid for hidden layers. Such a structure is very useful because of two theorems: 1- A linear combination of `sigmoids` can approximate any continuous function. -2- Trained with `output=1` for the signal and 0 for the background, the -approximated function of inputs `X` is the probability of signal, +2- Trained with `output=1` for the signal and 0 for the background, +the approximated function of inputs `X` is the probability of signal, knowing `X`. ### Learning Methods -The aim of all learning methods is to minimize the total error on a set -of weighted examples. The error is defined as the sum in quadrate, +The aim of all learning methods is to minimize the total error on a +set of weighted examples. The error is defined as the sum in quadrate, divided by two, of the error on each individual output neuron. In all methods implemented in this library, one needs to compute the first derivative of that error with respect to the weights. Exploiting the -well-known properties of the derivative, one can express this derivative -as the product of the local partial derivative by the weighted sum of -the outputs derivatives (for a neuron) or as the product of the input -value with the local partial derivative of the output neuron (for a -synapse). This computation is called "back-propagation of the errors". -Six learning methods are implemented. +well-known properties of the derivative, one can express this +derivative as the product of the local partial derivative by the +weighted sum of the outputs derivatives (for a neuron) or as the +product of the input value with the local partial derivative of the +output neuron (for a synapse). This computation is called +"back-propagation of the errors". Six learning methods are +implemented. #### Stochastic Minimization @@ -1051,7 +1098,17 @@ This is the most trivial learning method. The Robbins-Monro stochastic approximation is applied to multilayer perceptrons. The weights are updated after each example according to the formula: -  +$$ +w_{ij}(t+1) = w_{ij}(t) + \Delta w_{ij}(t) +$$ + +with: + +$$ +\Delta w_{ij}(t) = - \eta \left( +\frac{\partial e_p}{\partial w_{ij}} + \delta \right) + +\epsilon \Delta w_{ij}(t-1) +$$ The parameters for this method are `Eta`, `EtaDecay`, `Delta` and `Epsilon`. @@ -1073,71 +1130,76 @@ precision = slower search. A value `Tau=3` seems reasonable. Weights are set to the minimum along the line defined by the conjugate gradient. Parameters are `Tau` and `Reset`, which defines the epochs -where the direction is reset to the steepest descent (estimated by using -the Polak-Ribiere formula). +where the direction is reset to the steepest descent (estimated by +using the Polak-Ribiere formula). #### Conjugate Gradients With the Fletcher-Reeves Updating Formula Weights are set to the minimum along the line defined by the conjugate gradient. Parameters are `Tau` and `Reset`, which defines the epochs -where the direction is reset to the steepest descent (estimated by using -the Fletcher-Reeves formula). +where the direction is reset to the steepest descent (estimated by +using the Fletcher-Reeves formula). #### The Broyden, Fletcher, Goldfarb, Shanno (BFGS) Method -It implies the computation of a `NxN` matrix, but seems more powerful at -least for less than 300 weights. Parameters are `Tau` and `Reset`, which -defines the epochs where the direction is reset to the steepest descent. +It implies the computation of a `NxN` matrix, but seems more powerful +at least for less than 300 weights. Parameters are `Tau` and `Reset`, +which defines the epochs where the direction is reset to the steepest +descent. ### Using the Network Neural network are build from a set of "samples". A sample is a set of -values defining the inputs and the corresponding output that the network -should ideally provide. In ROOT this is a **`TTree`** entry. The first -thing to be decided is the network layout. This layout is described in a -string where the layers are separated by semicolons. The input/output -layers are defined by giving the expression for each neuron, separated -by comas. Hidden layers are just described by the number of neurons. +values defining the inputs and the corresponding output that the +network should ideally provide. In ROOT this is a **`TTree`** entry. +The first thing to be decided is the network layout. This layout is +described in a string where the layers are separated by semicolons. +The input/output layers are defined by giving the expression for each +neuron, separated by comas. Hidden layers are just described by the +number of neurons. In addition, input and output layer formulas can be preceded by '@' (e.g. "@out") if one wants to normalize the corresponding value. Also, if the string ends with '`!`', output neurons are set up for -classification, i.e. with a sigmoid (1 neuron) or softmax (more neurons) -activation function. +classification, i.e. with a sigmoid (1 neuron) or softmax (more +neurons) activation function. Many questions on the good usage of neural network, including rules of dumb to determine the best network topology are addressed at <ftp://ftp.sas.com/pub/neural/FAQ.html> ``` {.cpp} -// a simple network: 2 inputs, 10 hidden and 1 normalized output neuron -TMultiLayerPerceptron network("r,z:10:@Br",tree); + // a simple network: 2 inputs, 10 hidden and 1 normalized + // output neuron + TMultiLayerPerceptron network("r,z:10:@Br",tree); ``` -Expressions are evaluated as for `TTree::Draw()`. Input and outputs are -taken from the **`TTree`** associated with the network. This **`TTree`** -can be given as argument of the constructor or defined later with -`TMultiLayerPerceptron::SetData()`. Events can also be weighted. The -weight expression can be given in the constructor or set later with the -method `SetWeight()` of the class **`TMultiLayerPerceptron`**. Two -datasets must be defined before learning the network: a training dataset -that is used when minimizing the error, and a test dataset that will -avoid bias. Those two datasets can be build aside and then given to the -network, or can be build from a standard expression. By default, half of -the events are put in both datasets. +Expressions are evaluated as for `TTree::Draw()`. Input and outputs +are taken from the **`TTree`** associated with the network. This +**`TTree`** can be given as argument of the constructor or defined +later with `TMultiLayerPerceptron::SetData()`. Events can also be +weighted. The weight expression can be given in the constructor or set +later with the method `SetWeight()` of the class +**`TMultiLayerPerceptron`**. Two datasets must be defined before +learning the network: a training dataset that is used when minimizing +the error, and a test dataset that will avoid bias. Those two datasets +can be build aside and then given to the network, or can be build from +a standard expression. By default, half of the events are put in both +datasets. ``` {.cpp} -// a more complex 4:8:1 network -// the ptsumf branch is used as weigh; default event lists are explicit -TMultiLayerPerceptron network("m,pt,acol,acopl:8:type","pt",tree, -"Entry$%2","Entry$/2"); + // a more complex 4:8:1 network + // the ptsumf branch is used as weigh; + // default event lists are explicit + TMultiLayerPerceptron network("m,pt,acol,acopl:8:type","pt",tree, + "Entry$%2","Entry$/2"); ``` The method `TMultiLayerPerceptron::SetLearningMethod()` defines the learning method. Learning methods are: -``` {.cpp} +``` TMultiLayerPerceptron::kStochastic, TMultiLayerPerceptron::kBatch, TMultiLayerPerceptron::kSteepestDescent, @@ -1148,22 +1210,23 @@ TMultiLayerPerceptron::kBFGS // default The training can start with `TMultiLayerPerceptron::Train(Int_t nepoch,Option_t* options).` The -first argument is the number of epochs while option is a string that can -contain "`text`" (simple text output), "`graph`" (evaluating graphical -training curves), "`update` `=` `X`" (step for the text/graph output -update) or "`+`" (will skip the randomization and start from the -previous values). All combinations are available. +first argument is the number of epochs while option is a string that +can contain "`text`" (simple text output), "`graph`" (evaluating +graphical training curves), "`update` `=` `X`" (step for the +text/graph output update) or "`+`" (will skip the randomization and +start from the previous values). All combinations are available. ``` {.cpp} -network.Train(1000,"text,graph,update=10"); //full output every 10 epochs -network.Train(100,"text,+"); //100 more epochs - //starts with existing weights + network.Train(1000,"text,graph,update=10"); // full output every + // 10 epochs + network.Train(100,"text,+"); // 100 more epochs + //starts with existing weights ``` The weights can be saved to a file (`DumpWeights`) and then reloaded (`LoadWeights`) to a new compatible network. The output can also be -evaluated (`Evaluate`) for a given output neuron and an array of double -input parameters or the network can be exported (`Export`) as a +evaluated (`Evaluate`) for a given output neuron and an array of +double input parameters or the network can be exported (`Export`) as a standalone code. Up to now, this is only as a C++ or PYTHON class, but other languages could be implemented. @@ -1172,82 +1235,87 @@ other languages could be implemented. An example of how to use **`TMultiLayerPerceptron`** is the macro `mlpHiggs.C` in \$ROOTSYS/tutorials. Using some standard simulated -information that could have been obtained at `LEP`, a neural network is -build, which can make the difference between `WW` events and events +information that could have been obtained at `LEP`, a neural network +is build, which can make the difference between `WW` events and events containing a Higgs boson. Starting with a **`TFile`** containing two -**`TTree`**s: one for the signal, the other for the background, a simple -script is used: +**`TTree`**s: one for the signal, the other for the background, a +simple script is used: ``` {.cpp} -void mlpHiggs(Int_t ntrain=100) -{ if (!gROOT->GetClass("TMultiLayerPerceptron")) -gSystem->Load("libMLP"); - // prepare inputs - the 2 trees are merged into one, and a "type" - // branch, equal to 1 for the signal and 0 for the background is added -TFile input("mlpHiggs.root"); -TTree *signal = (TTree *)input.Get("sig_filtered"); -TTree *background = (TTree *)input.Get("bg_filtered"); -TTree *simu = new TTree("MonteCarlo","Filtered Monte Carlo Events"); -... + void mlpHiggs(Int_t ntrain=100) { + if (!gROOT->GetClass("TMultiLayerPerceptron")) + gSystem->Load("libMLP"); + // prepare inputs - the 2 trees are merged into one, and a + // "type" branch, equal to 1 for the signal and 0 for the + // background is added + TFile input("mlpHiggs.root"); + TTree *signal = (TTree *)input.Get("sig_filtered"); + TTree *background = (TTree *)input.Get("bg_filtered"); + TTree *simu = new TTree("MonteCarlo", + "Filtered Monte Carlo Events"); + ... ``` -Since the input is a **`TTree`** and we are starting from two different -**`TTree`**s (with different names), they are first merged into one, and -a "`type`" branch is added, that says whether there is a signal or a -background event. Those irrelevant details are skipped here. +Since the input is a **`TTree`** and we are starting from two +different **`TTree`**s (with different names), they are first merged +into one, and a "`type`" branch is added, that says whether there is +a signal or a background event. Those irrelevant details are skipped +here. ``` {.cpp} -... -TMultiLayerPerceptron *mlp = new TMultiLayerPerceptron("msumf,ptsumf, acolin, -acopl:8:type","ptsumf",simu,"Entry$%2","Entry$/2"); -mlp->Train(ntrain, "text,graph,update=10"); + ... + TMultiLayerPerceptron *mlp = new TMultiLayerPerceptron( + "msumf,ptsumf, acolin, acopl:8:type","ptsumf",simu, + "Entry$%2","Entry$/2"); + mlp->Train(ntrain, "text,graph,update=10"); ``` -The neural network is instantiated and trained. "`ptsumf`" is used as a -weight, and the standard event lists are explicit. The network that is -then build has four input neurons, eight additional ones in the only -hidden layer and one single output neuron. +The neural network is instantiated and trained. "`ptsumf`" is used as +a weight, and the standard event lists are explicit. The network that +is then build has four input neurons, eight additional ones in the +only hidden layer and one single output neuron. ``` {.cpp} - // Use the NN to plot the results for each sample -TH1F *bg = new TH1F("bgh","NN output",50,-.5,1.5); -TH1F *sig = new TH1F("sigh","NN output",50,-.5,1.5); -bg->SetDirectory(0); -sig->SetDirectory(0); -Double_t params[4]; -for (i = 0; i < background->GetEntries(); i++) { -background->GetEntry(i); -params[0] = msumf; params[1] = ptsumf; -params[2] = acolin; params[3] = acopl; -bg->Fill(mlp->Evaluate(0,params)); -} -for (i = 0; i < signal->GetEntries(); i++) { -signal->GetEntry(i); -params[0] = msumf; -params[1] = ptsumf; -params[2] = acolin; -params[3] = acopl; -sig->Fill(mlp->Evaluate(0,params)); -} -TCanvas *cv = new TCanvas("NNout_cv","Neural net output"); -bg->SetFillStyle(3008); -bg->SetFillColor(kBlue); -sig->SetFillStyle(3003); -sig->SetFillColor(kRed); -bg->SetStats(0); -sig->SetStats(0); -bg->Draw(); -sig->Draw("same"); -TLegend *legend = new TLegend(.75,.80,.95,.95); -legend->AddEntry(bg,"Background(WW)"); -legend->AddEntry(sig,"Signal(Higgs)"); -legend->Draw(); + // Use the NN to plot the results for each sample + TH1F *bg = new TH1F("bgh","NN output",50,-.5,1.5); + TH1F *sig = new TH1F("sigh","NN output",50,-.5,1.5); + bg->SetDirectory(0); + sig->SetDirectory(0); + Double_t params[4]; + for (i = 0; i < background->GetEntries(); i++) { + background->GetEntry(i); + params[0] = msumf; params[1] = ptsumf; + params[2] = acolin; params[3] = acopl; + bg->Fill(mlp->Evaluate(0,params)); + } + for (i = 0; i < signal->GetEntries(); i++) { + signal->GetEntry(i); + params[0] = msumf; + params[1] = ptsumf; + params[2] = acolin; + params[3] = acopl; + sig->Fill(mlp->Evaluate(0,params)); + } + TCanvas *cv = new TCanvas("NNout_cv","Neural net output"); + bg->SetFillStyle(3008); + bg->SetFillColor(kBlue); + sig->SetFillStyle(3003); + sig->SetFillColor(kRed); + bg->SetStats(0); + sig->SetStats(0); + bg->Draw(); + sig->Draw("same"); + TLegend *legend = new TLegend(.75,.80,.95,.95); + legend->AddEntry(bg,"Background(WW)"); + legend->AddEntry(sig,"Signal(Higgs)"); + legend->Draw(); ``` The neural net output is then used to display the final difference -between background and signal events. The next figure shows this plot. +between background and signal events. The figure "The neural net +output" shows this plot. - + As it can be seen, this is a quite efficient technique. As mentioned earlier, neural networks are also used for fitting function. For some @@ -1258,8 +1326,8 @@ well as the radial and `z` components of the `B` field. One wants to fit those distributions with a function in order to plug them into the `Geant` simulation code. Polynomial fits could be tried, but it seems difficult to reach the desired precision over the full -range. One could also use a `spline` interpolation between known points. -In all cases, the resulting field would not be `C`-infinite. +range. One could also use a `spline` interpolation between known +points. In all cases, the resulting field would not be `C`-infinite. An example of output (for Br) is shown. First the initial function can be seen as the target. Then, the resulting (normalized) neural net diff --git a/docbook/users-guide/Trees.md b/docbook/users-guide/Trees.md index 5a50683a8f8..cf37edcb080 100644 --- a/docbook/users-guide/Trees.md +++ b/docbook/users-guide/Trees.md @@ -2950,7 +2950,7 @@ can be grown by using the "`>>+`" syntax. For example to add the entries, with exactly 600 tracks: ``` {.cpp} -root[] T->Draw(">>+ myList","fNtrack == 600"</code>, "entrylist"); +root[] T->Draw(">>+ myList","fNtrack == 600", "entrylist"); ``` If the `Draw` command generates duplicate entries, they are not added to @@ -3017,7 +3017,7 @@ called `myList`. We get the list from the current directory and assign it to a variable list. ``` {.cpp} -root[] T->Draw(">>myList","fNtrack > 600"</code>,"entrylist"); +root[] T->Draw(">>myList","fNtrack > 600","entrylist"); root[]TEntryList ``` @@ -3064,7 +3064,7 @@ all operation in a **`TEntryList`** for a **`TChain`** are on the **`TTree`** level. This is illustrated by the following example: ``` {.cpp} -root[] TEntryList *list1 = new TEntryList("list1",</code>"list1"); +root[] TEntryList *list1 = new TEntryList("list1","list1"); root[] list1->SetTree("tree1","file1") root[] list1->Enter(0); root[] list1->Enter(2); @@ -3269,7 +3269,7 @@ Loop through the first 10 entries and print the values of `fNtrack`: ``` {.cpp} root[] for (int i = 0; i < 10; i++) -root[] cout << a[i] << " " << endl // need an endl to see the values</code> +root[] cout << a[i] << " " << endl // need an endl to see the values 594 597 606 595 604 610 604 602 603 596 ``` diff --git a/docbook/users-guide/pictures/02000030.png b/docbook/users-guide/pictures/02000030.png new file mode 100644 index 0000000000000000000000000000000000000000..3395b13cf2bc342a26c629d586e9923dcaee206b GIT binary patch literal 35320 zcmb5VbyOSO7dDD(p_JlMEVvaX6bgkP!JQzXxRXL~ZE<(EqQ#x!?heJBVu9kN6ffmV z-~0XU-}h!EYi6CSnKNt7KF{9!+0TwpSCuEgqryW$K_LJufHYB1&}mRm(8O`jkR!jl 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