2.2.5 误差与拟合

误差与拟合的概念

误差（error）是统计样本中某一元素的观测值与其“真值”（未必可直接观测得到）之间的离差的度量。

拟合（fit）是一种构建一个函数曲线，使之最佳地吻合现有数据点的过程。

ROOT 中既可以使用TGraph描述误差并拟合，也可以使用TH类。

注意：拟合操作应在最后进行，如果宏中的marker属性设置在拟合之后，可能会导致拟合失效

误差棒

误差棒（error bar）在统计图表中起到了传达数据不确定性和可靠性的作用，帮助读者更好地理解数据并做出准确的解释和推断。任何测量都会有误差，使用TGraphAsymmErrors()函数绘制误差棒，你甚至可以绘制倾斜的误差棒。

TGraphAsymmErrors *gr = new TGraphAsymmErrors(n,x,y,exl,exh,eyl,eyh)
TGraphAsymmErrors(number of var,x,y,low x,x high,y low,y high)
TGraphAsymmErrors(number of var,x,y,low x,x high,y low, 0)    // 不绘制上误差
auto g = new TGraph2DErrors(np, rx, ry, rz, ex, ey, ez);    // 二维误差棒

设置误差棒的填充颜色：SetFillColor()

设置误差棒的填充风格：SetFillStyle()

设置误差棒的颜色：SetLineColor()

误差棒的样式：

E.g. 包含非对称和倾斜误差棒的图案

{
   auto c45 = new TCanvas("c45","c45",200,10,600,400);
   const Int_t n = 10;
   Double_t x[n]  = {-0.22, 0.05, 0.25, 0.35, 0.5, 0.61,0.7,0.85,0.89,0.95};
   Double_t y[n]  = {1,2.9,5.6,7.4,9,9.6,8.7,6.3,4.5,1};
   Double_t exl[n] = {.05,.1,.07,.07,.04,.05,.06,.07,.08,.05};
   Double_t eyl[n] = {.8,.7,.6,.5,.4,.4,.5,.6,.7,.8};
   Double_t exh[n] = {.02,.08,.05,.05,.03,.03,.04,.05,.06,.03};
   Double_t eyh[n] = {.6,.5,.4,.3,.2,.2,.3,.4,.5,.6};
   Double_t exld[n] = {.0,.0,.0,.0,.0,.0,.0,.0,.0,.0};
   Double_t eyld[n] = {.0,.0,.05,.0,.0,.0,.0,.0,.0,.0};
   Double_t exhd[n] = {.0,.0,.0,.0,.0,.0,.0,.0,.0,.0};
   Double_t eyhd[n] = {.0,.0,.0,.0,.0,.0,.0,.0,.05,.0};
   auto gr = new TGraphBentErrors(n,x,y,exl,exh,eyl,eyh,exld,exhd,eyld,eyhd);
   gr->SetTitle("A graph with bend errors");
   gr->SetMarkerColor(4);
   gr->SetMarkerStyle(21);
   gr->Draw("ALP");
}

包含非对称和倾斜误差棒的图案

误差带

误差带（error band）是在统计图表中用来表示数据的不确定性或误差范围的一种方式，与误差棒类似，但是误差带通常是用来表示连续数据或函数的误差范围。

E.g. 折线误差带和光滑误差带

{
   auto c41 = new TCanvas("c41","c41",200,10,600,300);
   c41->Divide(2,1);
   double x[] = {0, 1, 2, 3, 4};
   double y[] = {0, 2, 4, 1, 3};
   double ex[] = {0.1, 0.2, 0.3, 0.4, 0.5};
   double ey[] = {1, 0.5, 1, 0.5, 1};
   auto ge = new TGraphErrors(5, x, y, ex, ey);
   ge->SetTitle("Errors as a band");
   ge->SetFillColor(4);
   ge->SetFillStyle(3010);
   c41->cd(1);
   ge->Draw("a3");
   c41->cd(2);
   ge->SetFillColor(6);
   ge->SetFillStyle(3005);
   ge->Draw("a4");
}

折线误差带和光滑误差带

选项“4”与选项“3”相似，只是表带是平滑的。 如上图所示，应谨慎使用此选项，因为平滑算法可能会显示一些（巨大的）“弹跳”效果。 在某些情况下，它看起来比选项“3”更好（因为它很光滑），但它可能会具有误导性。svg和其他保存格式在显示上会略有差异，这里只做示意。论坛上关于弹跳的提问与解答https://root-forum.cern.ch/t/confidence-band-on-tf1-fit/34514

多重误差

TMultiErrors是一个类，用于处理多个测量值的误差。它是ROOT数据分析框架中的一部分，用于处理数据的不确定性和误差。

E.g. 绘制多重误差

{
   auto c1 = new TCanvas("c1", "A Simple Graph with multiple y-errors", 200, 10, 700, 500);
   c1->SetGrid();
   c1->GetFrame()->SetBorderSize(12);
   const Int_t np = 5;
   Double_t x[np]       = {0, 1, 2, 3, 4};
   Double_t y[np]       = {0, 2, 4, 1, 3};
   Double_t exl[np]     = {0.3, 0.3, 0.3, 0.3, 0.3};
   Double_t exh[np]     = {0.3, 0.3, 0.3, 0.3, 0.3};
   Double_t eylstat[np] = {1, 0.5, 1, 0.5, 1};
   Double_t eyhstat[np] = {0.5, 1, 0.5, 1, 2};
   Double_t eylsys[np]  = {0.5, 0.4, 0.8, 0.3, 1.2};
   Double_t eyhsys[np]  = {0.6, 0.7, 0.6, 0.4, 0.8};
   auto gme = new TGraphMultiErrors("gme", "TGraphMultiErrors Example", np, x, y, exl, exh, eylstat, eyhstat);
   gme->AddYError(np, eylsys, eyhsys);
   gme->SetMarkerStyle(20);
   gme->SetLineColor(kRed);
   gme->GetAttLine(0)->SetLineColor(kRed);
   gme->GetAttLine(1)->SetLineColor(kBlue);
   gme->GetAttFill(1)->SetFillStyle(0);
   gme->Draw("APS ; Z ; 5 s=0.5");
}

绘制多重误差

多维误差

TGraphMultiErrors的工作方式与TGraphAsymmErrors基本相同。 它有可能定义多个y-Errors的类型/维度。 如果你想同时绘制统计和系统错误，这非常有用。

内置函数拟合

hist.Fit("gaus");

自定义拟合

root 支持用户自定义拟合函数。

#include <fstream>
int fit(){
    TGraph *gr = new TGraph("data1.txt");
    double pi=3.14159265;
    TF1 *f1 = new TF1("f1","[0]*sin([1]*pi*x)*exp([2]*x)+[3]",0,10);

    f1->SetParameter(0,1);    // 第零个参数的参考值
    f1->SetParameter(1,1);
    f1->SetParameter(2,-1);
    f1->SetParameter(3,0.5);
    // be equal in value with below
    // fi->SetParameters(1,1,-1,0.5);
    gr->Fit("f1");
    gr->Draw();

}

root会打印出其拟合参数的精确值：

[zhangzh@node01 test]$ rl fit.c 
root [0] 
Processing fit.c...

****************************************
Minimizer is Minuit / Migrad
Chi2                      =       0.1458
NDf                       =           96
Edm                       =  3.61433e-08
NCalls                    =          123
p0                        =      1.00186   +/-   0.0254347   
p1                        =      1.30098   +/-   0.00369848  
p2                        =    -0.482759   +/-   0.0172724   
p3                        =     0.498504   +/-   0.00393919  
Info in <TCanvas::MakeDefCanvas>:  created default TCanvas with name c1
(int) 0

拟合参数

TF1::SetParLimit(parnumber, parvalue);

TF1::SetParLimits(parvalue1, parvalue2, parvalue3);

参数获得

TF1::GetChisquare()

TF1::GetNDF()

TF1::GetParameter(int i)

TF1::GetParError(int i)

TF1::GetParErrorLow(int i),GetParErrorUp(int i)

多Graph拟合

合并拟合

用不同的marker表征不同的实验结果，但x轴范围不一致，那么拟合哪一段的数据最为准确？

使用TMultiGraph::TMultiGraph()函数，将多个 graph 合并，既能保留不同 marker 的个性，也能让所有数据点参与拟合。

分别拟合

E.g. 多条拟合线绘制在一张graph上

void multigraph()
{
   gStyle->SetOptFit();
   auto c1 = new TCanvas("c1","multigraph",700,500);
   c1->SetGrid();
 
   // draw a frame to define the range
   auto mg = new TMultiGraph();
 
   // create first graph
   const Int_t n1 = 10;
   Double_t px1[] = {-0.1, 0.05, 0.25, 0.35, 0.5, 0.61,0.7,0.85,0.89,0.95};
   Double_t py1[] = {-1,2.9,5.6,7.4,9,9.6,8.7,6.3,4.5,1};
   Double_t ex1[] = {.05,.1,.07,.07,.04,.05,.06,.07,.08,.05};
   Double_t ey1[] = {.8,.7,.6,.5,.4,.4,.5,.6,.7,.8};
   auto gr1 = new TGraphErrors(n1,px1,py1,ex1,ey1);
   gr1->SetMarkerColor(kBlue);
   gr1->SetMarkerStyle(21);
 
   gr1->Fit("gaus","q");
   auto func1 = (TF1 *) gr1->GetListOfFunctions()->FindObject("gaus");
   func1->SetLineColor(kBlue);
 
   mg->Add(gr1);
 
   // create second graph
   const Int_t n2 = 10;
   Float_t x2[]  = {-0.28, 0.005, 0.19, 0.29, 0.45, 0.56,0.65,0.80,0.90,1.01};
   Float_t y2[]  = {2.1,3.86,7,9,10,10.55,9.64,7.26,5.42,2};
   Float_t ex2[] = {.04,.12,.08,.06,.05,.04,.07,.06,.08,.04};
   Float_t ey2[] = {.6,.8,.7,.4,.3,.3,.4,.5,.6,.7};
   auto gr2 = new TGraphErrors(n2,x2,y2,ex2,ey2);
   gr2->SetMarkerColor(kRed);
   gr2->SetMarkerStyle(20);
 
   gr2->Fit("pol5","q");
   auto func2 = (TF1 *) gr2->GetListOfFunctions()->FindObject("pol5");
   func2->SetLineColor(kRed);
   func2->SetLineStyle(2);
 
   mg->Add(gr2);
 
   mg->Draw("ap");
 
   //force drawing of canvas to generate the fit TPaveStats
   c1->Update();
 
   auto stats1 = (TPaveStats*) gr1->GetListOfFunctions()->FindObject("stats");
   auto stats2 = (TPaveStats*) gr2->GetListOfFunctions()->FindObject("stats");
 
   if (stats1 && stats2) {
      stats1->SetTextColor(kBlue);
      stats2->SetTextColor(kRed);
      stats1->SetX1NDC(0.12); stats1->SetX2NDC(0.32); stats1->SetY1NDC(0.82);
      stats2->SetX1NDC(0.72); stats2->SetX2NDC(0.92); stats2->SetY1NDC(0.75);
      c1->Modified();
   }
}

多条拟合线绘制在一张graph上

含本底拟合

E.g. 含本底拟合

#include <Fit/Fitter.h>
#include <Fit/BinData.h>
#include <Fit/Chi2FCN.h>
#include <TH1.h>
#include <Math/WrappedMultiTF1.h>
#include <HFitInterface.h>
#include <TCanvas.h>
#include <TStyle.h>
 
// definition of shared parameter
// background function
int iparB[2] = {
   0, // exp amplitude in B histo
   2  // exp common parameter
};
 
// signal + background function
int iparSB[5] = {
   1, // exp amplitude in S+B histo
   2, // exp common parameter
   3, // Gaussian amplitude
   4, // Gaussian mean
   5  // Gaussian sigma
};
 
// Create the GlobalCHi2 structure
 
struct GlobalChi2 {
   GlobalChi2(ROOT::Math::IMultiGenFunction &f1, ROOT::Math::IMultiGenFunction &f2) : fChi2_1(&f1), fChi2_2(&f2) {}
 
   // parameter vector is first background (in common 1 and 2)
   // and then is signal (only in 2)
   double operator()(const double *par) const
   {
      double p1[2];
      for (int i = 0; i < 2; ++i)
         p1[i] = par[iparB[i]];
 
      double p2[5];
      for (int i = 0; i < 5; ++i)
         p2[i] = par[iparSB[i]];
 
      return (*fChi2_1)(p1) + (*fChi2_2)(p2);
   }
 
   const ROOT::Math::IMultiGenFunction *fChi2_1;
   const ROOT::Math::IMultiGenFunction *fChi2_2;
};
 
void combinedFit()
{
 
   TH1D *hB = new TH1D("hB", "histo B", 100, 0, 100);
   TH1D *hSB = new TH1D("hSB", "histo S+B", 100, 0, 100);
 
   TF1 *fB = new TF1("fB", "expo", 0, 100);
   fB->SetParameters(1, -0.05);
   hB->FillRandom("fB");
 
   TF1 *fS = new TF1("fS", "gaus", 0, 100);
   fS->SetParameters(1, 30, 5);
 
   hSB->FillRandom("fB", 2000);
   hSB->FillRandom("fS", 1000);
 
   // perform now global fit
 
   TF1 *fSB = new TF1("fSB", "expo + gaus(2)", 0, 100);
 
   ROOT::Math::WrappedMultiTF1 wfB(*fB, 1);
   ROOT::Math::WrappedMultiTF1 wfSB(*fSB, 1);
 
   ROOT::Fit::DataOptions opt;
   ROOT::Fit::DataRange rangeB;
   // set the data range
   rangeB.SetRange(10, 90);
   ROOT::Fit::BinData dataB(opt, rangeB);
   ROOT::Fit::FillData(dataB, hB);
 
   ROOT::Fit::DataRange rangeSB;
   rangeSB.SetRange(10, 50);
   ROOT::Fit::BinData dataSB(opt, rangeSB);
   ROOT::Fit::FillData(dataSB, hSB);
 
   ROOT::Fit::Chi2Function chi2_B(dataB, wfB);
   ROOT::Fit::Chi2Function chi2_SB(dataSB, wfSB);
 
   GlobalChi2 globalChi2(chi2_B, chi2_SB);
 
   ROOT::Fit::Fitter fitter;
 
   const int Npar = 6;
   double par0[Npar] = {5, 5, -0.1, 100, 30, 10};
 
   // create before the parameter settings in order to fix or set range on them
   fitter.Config().SetParamsSettings(6, par0);
   // fix 5-th parameter
   fitter.Config().ParSettings(4).Fix();
   // set limits on the third and 4-th parameter
   fitter.Config().ParSettings(2).SetLimits(-10, -1.E-4);
   fitter.Config().ParSettings(3).SetLimits(0, 10000);
   fitter.Config().ParSettings(3).SetStepSize(5);
 
   fitter.Config().MinimizerOptions().SetPrintLevel(0);
   fitter.Config().SetMinimizer("Minuit2", "Migrad");
 
   // fit FCN function directly
   // (specify optionally data size and flag to indicate that is a chi2 fit)
   fitter.FitFCN(6, globalChi2, nullptr, dataB.Size() + dataSB.Size(), true);
   ROOT::Fit::FitResult result = fitter.Result();
   result.Print(std::cout);
 
   TCanvas *c1 = new TCanvas("Simfit", "Simultaneous fit of two histograms", 10, 10, 700, 700);
   c1->Divide(1, 2);
   c1->cd(1);
   gStyle->SetOptFit(1111);
 
   fB->SetFitResult(result, iparB);
   fB->SetRange(rangeB().first, rangeB().second);
   fB->SetLineColor(kBlue);
   hB->GetListOfFunctions()->Add(fB);
   hB->Draw();
 
   c1->cd(2);
   fSB->SetFitResult(result, iparSB);
   fSB->SetRange(rangeSB().first, rangeSB().second);
   fSB->SetLineColor(kRed);
   hSB->GetListOfFunctions()->Add(fSB);
   hSB->Draw();
}

含本底拟合

拟合误差

拟合误差是用来描述实际观测值与拟合模型（如线性回归、曲线拟合等）之间的差异或偏差，即拟合模型对真实数据的逼近程度。

拟合误差通常以各种统计指标来表示，比如残差平方和（residual sum of squares）、平均绝对误差（mean absolute error）、均方根误差（root mean square error）等。

E.g. 拟合误差棒和拟合误差带

#include "TGraphErrors.h"
#include "TGraph2DErrors.h"
#include "TCanvas.h"
#include "TF2.h"
#include "TH1.h"
#include "TVirtualFitter.h"
#include "TRandom.h"
 
void ConfidenceIntervals()
{
   TCanvas *myc = new TCanvas("myc","Confidence intervals on the fitted function",1000, 500);
   myc->Divide(3,1);
 
//### 1. A graph
   //Create and fill a graph
   int ngr = 100;
   TGraph *gr = new TGraph(ngr);
   gr->SetName("GraphNoError");
   double x, y;
   int i;
   for (i=0; i<ngr; i++){
      x = gRandom->Uniform(-1, 1);
      y = -1 + 2*x + gRandom->Gaus(0, 1);
      gr->SetPoint(i, x, y);
   }
   //Create the fitting function
   TF1 *fpol = new TF1("fpol", "pol1", -1, 1);
   fpol->SetLineWidth(2);
   gr->Fit(fpol, "Q");
 
   /*Create a TGraphErrors to hold the confidence intervals*/
   TGraphErrors *grint = new TGraphErrors(ngr);
   grint->SetTitle("Fitted line with .95 conf. band");
   for (i=0; i<ngr; i++)
      grint->SetPoint(i, gr->GetX()[i], 0);
   /*Compute the confidence intervals at the x points of the created graph*/
   (TVirtualFitter::GetFitter())->GetConfidenceIntervals(grint);
   //Now the "grint" graph contains function values as its y-coordinates
   //and confidence intervals as the errors on these coordinates
   //Draw the graph, the function and the confidence intervals
   myc->cd(1);
   grint->SetLineColor(kRed);
   grint->Draw("ap");
   gr->SetMarkerStyle(5);
   gr->SetMarkerSize(0.7);
   gr->Draw("psame");
 
//### 2. A histogram
   myc->cd(2);
   //Create, fill and fit a histogram
   int nh=5000;
   TH1D *h = new TH1D("h",
      "Fitted Gaussian with .95 conf.band", 100, -3, 3);
   h->FillRandom("gaus", nh);
   TF1 *f = new TF1("fgaus", "gaus", -3, 3);
   f->SetLineWidth(2);
   h->Fit(f, "Q");
   h->Draw();
 
   /*Create a histogram to hold the confidence intervals*/
   TH1D *hint = new TH1D("hint","Fitted Gaussian with .95 conf.band", 100, -3, 3);
   (TVirtualFitter::GetFitter())->GetConfidenceIntervals(hint,0.68);
   //Now the "hint" histogram has the fitted function values as the
   //bin contents and the confidence intervals as bin errors
   hint->SetStats(false);
   hint->SetFillColor(2);
   hint->Draw("e3 same");
 
//### 3. A 2d graph
   //Create and fill the graph
   int ngr2 = 100;
   double z, rnd, e=0.3;
   TGraph2D *gr2 = new TGraph2D(ngr2);
   gr2->SetName("Graph2DNoError");
   TF2  *f2 = new TF2("f2","1000*(([0]*sin(x)/x)*([1]*sin(y)/y))+250",-6,6,-6,6);
   f2->SetParameters(1,1);
   for (i=0; i<ngr2; i++){
      f2->GetRandom2(x,y);
      // Generate a random number in [-e,e]
      rnd = 2*gRandom->Rndm()*e-e;
      z = f2->Eval(x,y)*(1+rnd);
      gr2->SetPoint(i,x,y,z);
   }
   //Create a graph with errors to store the intervals
   TGraph2DErrors *grint2 = new TGraph2DErrors(ngr2);
   for (i=0; i<ngr2; i++)
      grint2->SetPoint(i, gr2->GetX()[i], gr2->GetY()[i], 0);
 
   //Fit the graph
   f2->SetParameters(0.5,1.5);
   gr2->Fit(f2, "Q");
   /*Compute the confidence intervals*/
   (TVirtualFitter::GetFitter())->GetConfidenceIntervals(grint2);
   //Now the "grint2" graph contains function values as z-coordinates
   //and confidence intervals as their errors
   //draw
   myc->cd(3);
   f2->SetNpx(30);
   f2->SetNpy(30);
   f2->SetFillColor(kBlue);
   f2->Draw("surf4");
   grint2->SetNpx(20);
   grint2->SetNpy(20);
   grint2->SetMarkerStyle(24);
   grint2->SetMarkerSize(0.7);
   grint2->SetMarkerColor(kRed);
   grint2->SetLineColor(kRed);
   grint2->Draw("E0 same");
   grint2->SetTitle("Fitted 2d function with .95 error bars");
 
   myc->cd();
}

拟合误差棒和拟合误差带

拟合框

设置直方图拟合框TStyle::SetOptFit(pcev)。

p = 1；打印概率
c = 1; 打印卡方/自由度数
e = 1; 打印误差（如果 e = 1，v 必须为 1）
v = 1；打印参数名称/值

gStyle->SetOptFit(1)表示“默认值”，等价于gStyle->SetOptFit(111)，gStyle->SetOptFit(0)表示关闭面板。

拟合框位置

适用于打开多个拟合面板

TPave::SetX1 (x1)设置拟合框的左下角的值为x1。

TPave::SetX1NDC(newx1)设置拟合框相对位置，取值范围(0,1)。

TPave::SetX2NDC(newx2); TPave::SetY1NDC(newx2);TPave::SetY2NDC(newx2);同理。

拟合参数名

修改参数名称可以让你的拟合结果更加清晰，TF1::SetParNames("parname", "", "parname")最多支持10位的参数名称重制，空字符串会被跳过。

或另外单独控制：TF1::SetParName(number, parname)。

参数范围

TF1::SetParLimits(number, parmin, parmax)

Associated errors

By default, for each bin, the sum of weights is computed at fill time. You 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, use:

double error = h->GetBinError(bin);

Empty bins are excluded in the fit when using the Chi-square fit method. When fitting an histogram representing counts (that is with Poisson statistics) it is recommended to use the Log-Likelihood method (option L or WL), particularly in case of low statistics.

参数返回

// TFitResultPtr contains only the fit status.
int fitStatus = hist->Fit(myFunction);

// TFitResultPtr contains the TFitResult.
TFitResultPtr r = hist->Fit(myFunction,"S");

// Access the covariance matrix.
TMatrixDSym cov = r->GetCovarianceMatrix();

// Retrieve the fit chi2.
double chi2 = r->Chi2();

// Retrieve the value for the parameter 0.
double par0 = r->Parameter(0);

// Retrieve the error for the parameter 0.
double err0 = r->ParError(0);

// Print the full information of the fit including covariance matrix.
r->Print("V");

// Store the result in a ROOT file.
r->Write();

---

参考

https://root.cern.ch/doc/master/classTMultiGraph.html