// @(#)root/mlp:$Id$ // Author: Christophe.Delaere@cern.ch 20/07/03 /************************************************************************* * Copyright (C) 1995-2003, Rene Brun and Fons Rademakers. * * All rights reserved. * * * * For the licensing terms see $ROOTSYS/LICENSE. * * For the list of contributors see $ROOTSYS/README/CREDITS. * *************************************************************************/ /** \class TNeuron This class describes an elementary neuron, which is the basic element for a Neural Network. A network is built connecting neurons by synapses. There are different types of neurons: linear (a+bx), sigmoid (1/(1+exp(-x)), tanh or gaussian. An external function can also be used, together with its derivative. In a Multi Layer Perceptron, the input layer is made of inactive neurons (returning the normalized input) and output neurons are linear. Hidden neurons may be anything, the default being sigmoids. This implementation contains several methods to compute the value, the derivative, the DeDw, ... Values are stored in local buffers. The SetNewEvent() method is there to inform buffered values are outdated. */ #include "TSynapse.h" #include "TNeuron.h" #include "TTree.h" #include "TTreeFormula.h" #include "TFormula.h" #include "TBranch.h" #include "TCanvas.h" #include "TROOT.h" #include "TH1D.h" #include "TRegexp.h" #include "TMath.h" #include "Riostream.h" ClassImp(TNeuron); //////////////////////////////////////////////////////////////////////////////// /// Usual constructor TNeuron::TNeuron(TNeuron::ENeuronType type /*= kSigmoid*/, const char* name /*= ""*/, const char* title /*= ""*/, const char* extF /*= ""*/, const char* extD /*= ""*/ ) :TNamed(name, title) { fpre.SetOwner(false); fpost.SetOwner(false); flayer.SetOwner(false); fWeight = 0.; fNorm[0] = 1.; fNorm[1] = 0.; fType = type; fNewInput = true; fNewValue = true; fNewDeriv = true; fNewDeDw = true; fFormula = 0; fDeDw = 0; fDEDw = 0; fValue = 0; fExtF = 0; fExtD = 0; fIndex = 0; fDerivative = 0; fInput = 0; if(fType==kExternal) { fExtF = (TFormula*)gROOT->FindObject(extF); fExtD = (TFormula*)gROOT->FindObject(extD); } } //////////////////////////////////////////////////////////////////////////////// /// The Sigmoid. /// Fast computation of the values of the sigmoid function. /// Uses values of the function up to the seventh order /// tabulated at 700 points. /// Values were computed in long double precision (16 bytes, /// precision to about 37 digits) on a hp computer. /// Some values were checked with Mathematica. /// Result should be correct to ~ 15 digits (about double /// precision) /// /// From the mlpfit package (J.Schwindling 20-Jul-1999) Double_t TNeuron::Sigmoid(Double_t x) const { static Double_t sigval[7000] = { -3.500000e+01, 6.30511676014698530e-16, 6.30511676014698130e-16, 3.15255838007348670e-16, 1.05085279335782620e-16, 2.62713198339455210e-17, 5.25426396678905190e-18, 8.75710661131491060e-19, 1.25101523018779380e-19, 1.56376903773461590e-20, -3.490000e+01, 6.96823167838580650e-16, 6.96823167838580160e-16, 3.48411583919289640e-16, 1.16137194639762880e-16, 2.90342986599405540e-17, 5.80685973198804710e-18, 9.67809955331319670e-19, 1.38258565047325200e-19, 1.72823206309141090e-20, -3.480000e+01, 7.70108700136546340e-16, 7.70108700136545750e-16, 3.85054350068272280e-16, 1.28351450022757030e-16, 3.20878625056890590e-17, 6.41757250113773230e-18, 1.06959541685626250e-18, 1.52799345265172810e-19, 1.90999181581447200e-20, -3.470000e+01, 8.51101739147945470e-16, 8.51101739147944780e-16, 4.25550869573971650e-16, 1.41850289857990070e-16, 3.54625724644972780e-17, 7.09251449289935940e-18, 1.18208574881652770e-18, 1.68869392688066160e-19, 2.11086740860059710e-20, -3.460000e+01, 9.40612890429916560e-16, 9.40612890429915770e-16, 4.70306445214957000e-16, 1.56768815071651720e-16, 3.91922037679126390e-17, 7.83844075358240950e-18, 1.30640679226369560e-18, 1.86629541751945290e-19, 2.33286927189903510e-20, -3.450000e+01, 1.03953801167022100e-15, 1.03953801167021980e-15, 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-2.84937296312450620e-20, 3.450000e+01, 9.99999999999999000e-01, 1.03953801167021980e-15, -5.19769005835108810e-16, 1.73256335278368870e-16, -4.33140838195918600e-17, 8.66281676391822790e-18, -1.44380279398632340e-18, 2.06257541998031890e-19, -2.57821927497502700e-20, 3.460000e+01, 9.99999999999999110e-01, 9.40612890429909060e-16, -4.70306445214953640e-16, 1.56768815071650610e-16, -3.91922037679123620e-17, 7.83844075358235410e-18, -1.30640679226368620e-18, 1.86629541751943390e-19, -2.33286927189902040e-20, 3.470000e+01, 9.99999999999999110e-01, 8.51101739147944780e-16, -4.25550869573971700e-16, 1.41850289857990070e-16, -3.54625724644972780e-17, 7.09251449289935940e-18, -1.18208574881652830e-18, 1.68869392688066110e-19, -2.11086740860057180e-20, 3.480000e+01, 9.99999999999999220e-01, 7.70108700136545650e-16, -3.85054350068272230e-16, 1.28351450022757000e-16, -3.20878625056890530e-17, 6.41757250113773230e-18, -1.06959541685626230e-18, 1.52799345265172830e-19, -1.90999181581443530e-20, 3.490000e+01, 9.99999999999999330e-01, 6.96823167838575130e-16, -3.48411583919287070e-16, 1.16137194639762020e-16, -2.90342986599403450e-17, 5.80685973198800470e-18, -9.67809955331312550e-19, 1.38258565047323660e-19, -1.72823206309143410e-20 }; Int_t i = (Int_t)(10*x+350.5); if(i<0) return TMath::Exp(x); if(i>699) return 1; Double_t* basePoint = sigval+(10*i); Double_t res = x - basePoint[0]; res = basePoint[1] + res * ( basePoint[2] + res * ( basePoint[3] + res * ( basePoint[4] + res * ( basePoint[5] + res * ( basePoint[6] + res * ( basePoint[7] + res * ( basePoint[8] ))))))); /* if(x<-15) return TMath::Exp(x); if(x> 15) return 1; return (1/(1+TMath::Exp(-x))); */ return res; } //////////////////////////////////////////////////////////////////////////////// /// The Derivative of the Sigmoid. Double_t TNeuron::DSigmoid(Double_t x) const { //if (x>2.1972246) return x/(1+TMath::Exp(x)); //Double_t a = Sigmoid(x); //return a-a*a; if (TMath::Abs(x) > 5) return TMath::Exp(-TMath::Abs(x)); Double_t expmx = TMath::Exp(-x); return expmx / ((1 + expmx) * (1 + expmx)); } //////////////////////////////////////////////////////////////////////////////// /// Adds a synapse to the neuron as an input /// This method is used by the TSynapse while /// connecting two neurons. void TNeuron::AddPre(TSynapse * pre) { fpre.AddLast(pre); if (fpre.GetEntriesFast() == fpre.GetSize()) fpre.Expand(2 * fpre.GetSize()); } //////////////////////////////////////////////////////////////////////////////// /// Adds a synapse to the neuron as an output /// This method is used by the TSynapse while /// connecting two neurons. void TNeuron::AddPost(TSynapse * post) { fpost.AddLast(post); if (fpost.GetEntriesFast() == fpost.GetSize()) fpost.Expand(2 * fpost.GetSize()); } //////////////////////////////////////////////////////////////////////////////// /// Tells a neuron which neurons form its layer (including itself). /// This is needed for self-normalizing functions, like Softmax. void TNeuron::AddInLayer(TNeuron * nearP) { flayer.AddLast(nearP); if (flayer.GetEntriesFast() == flayer.GetSize()) flayer.Expand(2 * flayer.GetSize()); } //////////////////////////////////////////////////////////////////////////////// /// Returns the neuron type. TNeuron::ENeuronType TNeuron::GetType() const { return fType; } //////////////////////////////////////////////////////////////////////////////// /// Sets a formula that can be used to make the neuron an input. /// The formula is automatically normalized to mean=0, RMS=1. /// This normalisation is used by GetValue() (input neurons) /// and GetError() (output neurons) TTreeFormula* TNeuron::UseBranch(TTree* input, const char* formula) { if (fFormula) delete fFormula; // Set the formula // One checks for {}: defined as instance in TMultiLayerPerceptron // so that EvalInstance can be used. This allows to use arrays as // input, also in complex calculation, without detailed string parsing. TRegexp re("{[0-9]+}$"); TString f(formula); Ssiz_t len = f.Length(); Ssiz_t pos = re.Index(f,&len); if(pos==-1 || len<3) fFormula = new TTreeFormula(Form("NF%lu",(ULong_t)this),formula,input); else { TString newformula(formula,pos); TString val = f(pos+1,len-2); fFormula = new TTreeFormula(Form("NF%lu",(ULong_t)this), (const char*) newformula,input); fIndex = val.Atoi(); f = newformula; } // Computes the default normalization TH1D tmp("tmpb", "tmpb", 1, -FLT_MAX, FLT_MAX); input->Draw(Form("%s>>tmpb",(const char*)f),"","goff"); fNorm[0] = tmp.GetRMS(); if(fNorm[0]<1e-15) fNorm[0]=1.; fNorm[1] = tmp.GetMean(); // Check the dimensionality if(fFormula->GetNdata()>1 && fIndex==0) Warning("TNeuron::UseBranch()","all indices in arrays must be specified, otherwise the first element will be assumed."); return fFormula; } //////////////////////////////////////////////////////////////////////////////// /// Returns the formula value. Double_t TNeuron::GetBranch() const { Double_t branch = fFormula->EvalInstance(fIndex); if (TMath::IsNaN(branch)) branch = 0.; return branch; } //////////////////////////////////////////////////////////////////////////////// /// Returns neuron input Double_t TNeuron::GetInput() const { if (!fNewInput) { return fInput; } ((TNeuron*)this)->fNewInput = false; Double_t input = 0.0; Int_t nEntries = fpre.GetEntriesFast(); if (nEntries) { input = fWeight; for (Int_t i = 0; i < nEntries; i++) { TSynapse *preSynapse = (TSynapse*)fpre.UncheckedAt(i); input += preSynapse->GetValue(); } } return (((TNeuron*)this)->fInput = input); } //////////////////////////////////////////////////////////////////////////////// /// Computes the output using the appropriate function and all /// the weighted inputs, or uses the branch as input. /// In that case, the branch normalisation is also used. Double_t TNeuron::GetValue() const { if (!fNewValue) { return fValue; } ((TNeuron*)this)->fNewValue = false; Int_t nentries = fpre.GetEntriesFast(); if (!nentries) { Double_t branch = GetBranch(); return (((TNeuron*)this)->fValue = (branch - fNorm[1]) / fNorm[0]); } else { Double_t input = GetInput(); Double_t value = 0; switch (fType) { case TNeuron::kOff: { value = 0; break; } case TNeuron::kLinear: { value = input; break; } case TNeuron::kSigmoid: { value = Sigmoid(input); break; } case TNeuron::kTanh: { value = TMath::TanH(input); break; } case TNeuron::kGauss: { value = TMath::Exp(-input * input); break; } case TNeuron::kSoftmax: { Double_t normalization = 0.0; for (Int_t i = 0; i < flayer.GetEntriesFast(); i++) { // this is not very efficient, as the same normalization // is calculated once per neuron instead of just once normalization += TMath::Exp(((TNeuron*) flayer.UncheckedAt(i))->GetInput()); } if (normalization > 0.0) { // exp(a_k) / sum_k'[ exp(a_k') ] value = TMath::Exp(input) / normalization; } else { // all neurons have output = 0 ? value = 1.0 / flayer.GetEntriesFast(); } break; } case TNeuron::kExternal: { value = fExtF->Eval(input); break; } } return (((TNeuron*)this)->fValue = value); } } //////////////////////////////////////////////////////////////////////////////// /// computes the derivative for the appropriate function /// at the working point Double_t TNeuron::GetDerivative() const { if (!fNewDeriv) return fDerivative; ((TNeuron*)this)->fNewDeriv = false; Int_t nentries = fpre.GetEntriesFast(); Double_t input = fWeight; for (Int_t i=0;iGetValue(); } Double_t derivative = 0; switch (fType) { case TNeuron::kOff: { derivative = 0; break; } case TNeuron::kLinear: { derivative = 1; break; } case TNeuron::kSigmoid: { derivative = DSigmoid(input); break; } case TNeuron::kTanh: { derivative = (1 - (TMath::TanH(input) * TMath::TanH(input))); break; } case TNeuron::kGauss: { derivative = (-2) * input * TMath::Exp(-input * input); break; } case TNeuron::kSoftmax: { // actually this is not the derivative, as the Softmax function needs // exp(activation) of all the neurons in the layer for the normalization. // it will be smartly taken care of in GetDeDw() derivative = GetValue(); break; } case TNeuron::kExternal: { derivative = fExtD->Eval(input); break; } } return (((TNeuron*)this)->fDerivative = derivative); } //////////////////////////////////////////////////////////////////////////////// /// Computes the error for output neurons. /// Returns 0 for other neurons. Double_t TNeuron::GetError() const { if (!fpost.GetEntriesFast()) return (GetValue() - (GetBranch() - fNorm[1]) / fNorm[0]); return 0; } //////////////////////////////////////////////////////////////////////////////// /// Computes the normalized target pattern for output neurons. /// Returns 0 for other neurons. Double_t TNeuron::GetTarget() const { if (!fpost.GetEntriesFast()) return ((GetBranch() - fNorm[1]) / fNorm[0]); return 0; } //////////////////////////////////////////////////////////////////////////////// /// Computes the derivative of the error wrt the neuron weight. Double_t TNeuron::GetDeDw() const { if (!fNewDeDw) return fDeDw; ((TNeuron*)this)->fNewDeDw = false; // Output units DeDw is just the error for these cases: // - linear output, with sum-of-squares error // - sigmoid output, with binary cross-entropy // - softmax output, with 1-of-c cross entropy Int_t nEntries = fpost.GetEntriesFast(); if (nEntries == 0) { // output neuron ((TNeuron*)this)->fDeDw = GetError(); } else { // hidden nuron ((TNeuron*)this)->fDeDw = 0.0; if (fType != TNeuron::kSoftmax) { // non-softmax for (Int_t i = 0; i < nEntries; i++) { TSynapse *postSynapse = (TSynapse*)fpost.UncheckedAt(i); ((TNeuron*)this)->fDeDw += postSynapse->GetWeight() * postSynapse->GetPost()->GetDeDw(); } } else { // softmax derivative can be taken care of correcting the forward weight for (Int_t i = 0; i < nEntries; i++) { TSynapse *postSynapse = (TSynapse*)fpost.UncheckedAt(i); ((TNeuron*)this)->fDeDw += (postSynapse->GetWeight() - postSynapse->GetPost()->GetInput()) * postSynapse->GetPost()->GetDeDw(); } } ((TNeuron*)this)->fDeDw *= GetDerivative(); } return fDeDw; } //////////////////////////////////////////////////////////////////////////////// /// Uses the branch type to force an external value. void TNeuron::ForceExternalValue(Double_t value) { fNewValue = false; fValue = (value - fNorm[1]) / fNorm[0]; } //////////////////////////////////////////////////////////////////////////////// /// Sets the normalization variables. /// Any input neuron will return (branch-mean)/RMS. /// When UseBranch is called, mean and RMS are automatically set /// to the actual branch mean and RMS. void TNeuron::SetNormalisation(Double_t mean, Double_t RMS) { fNorm[0] = RMS; fNorm[1] = mean; } //////////////////////////////////////////////////////////////////////////////// /// Sets the neuron weight to w. /// The neuron weight corresponds to the bias in the /// linear combination of the inputs. void TNeuron::SetWeight(Double_t w) { fWeight = w; } //////////////////////////////////////////////////////////////////////////////// /// Inform the neuron that inputs of the network have changed, /// so that the buffered values have to be recomputed. void TNeuron::SetNewEvent() const { ((TNeuron*)this)->fNewInput = true; ((TNeuron*)this)->fNewValue = true; ((TNeuron*)this)->fNewDeriv = true; ((TNeuron*)this)->fNewDeDw = true; } //////////////////////////////////////////////////////////////////////////////// /// Sets the derivative of the total error wrt the neuron weight. void TNeuron::SetDEDw(Double_t in) { fDEDw = in; }