variableLog2ParamComplexity.cpp

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 *   Copyright (c) 2005-2025 by                                             *
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 *       - Christophe GONZALES(_at_AMU)                                     *
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 *       - Pierre-Henri WUILLEMIN(_at_LIP6)                                 *
 *       - Christophe GONZALES(_at_AMU)                                     *
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#include <agrum/base/core/math/variableLog2ParamComplexity.h>
 
#ifdef GUM_NO_INLINE
#  include <agrum/base/core/math/variableLog2ParamComplexity_inl.h>
#endif   // GUM_NO_INLINE
 
namespace gum {

  double VariableLog2ParamComplexity::log2Cnr(const std::size_t r, const double n) {
    // we know that c_n^1 = 1 for all values of n
    // in addition, c_0^r = 1 for all values of r
    // finally, it is easy to see that c_1^r = r for all r
    if (r == std::size_t(1)) return 0.0;         // log2(1)
    if (n == 0.0) return 0.0;                    // log2(1)
    if (n == 1.0) return std::log2((double)r);   // log2(r)
 
    if (n < 0.0) {
      GUM_ERROR(OutOfBounds,
                "In the penalty of the fNML score, n must be greater "
                    << "than or equal to 0. But, here, n = " << n);
    }
 
    if (n < VariableLog2ParamComplexityCTableNSize) {
      // check if we can find the value we look for in precomputed table
      // ScorefNMLVariableLog2ParamComplexity
      std::size_t xn = (std::size_t)n;
      if (r - 2 < VariableLog2ParamComplexityCTableRSize) {
        return VariableLog2ParamComplexityCTable[r - 2][xn];
      } else {
        // try to find the value in the cache
        if (_use_cache_) {
          try {
            return _cache_[std::pair< std::size_t, double >{r, n}];
          } catch (NotFound const&) {}
        }
 
        // use Equation (13) of the paper to compute the value of cnr:
        // C_n^r = C_n^{r-1} + (n / (r-2)) C_n^{r-2}
        // as we handle only log2's of C_n^r, we have the following:
        // let k_r be such that C_n^{r-2} = k_r * C_n^{r-1}
        // log2 ( C_n^r ) = log2 ( C_n^{r-1} + k_r * (n/(r-2)) * C_n^{r-1} )
        //                = log2 ( C_n^{r-1} ) + log2 ( 1 + k_r * (n/(r-2)) )
        // as  k_r = C_n^{r-2} / C_n^{r-1}, we have that
        // log2(k_r) = log2 ( C_n^{r-2} ) - log2 ( C_n^{r-1} )
        // so, k_r = exp ( (log2(cn_^{r-2}) - log2(C_n^{r-1})) * log(2) )
        // now, let q_r = 1 + k_r * (n/(r-2)), then
        // C_n^r = C_n^{r-1} * q_r, or, equivalently,
        // log2(C_n^r) = log2(C_n^{r-1}) + log2(q_r)
        // Now, we can use the same method to compute C_n^{r+1}:
        // k_{r+1}   = C_n^{r-1} / C_n^r = 1 / q_r
        // q_{r+1}   = 1 + k_{r+1} * (n/(r-1))
        // C_n^{r+1} = C_n^r * q_{r+1}
        double log2Cnr1 = VariableLog2ParamComplexityCTable[3][xn];   // log(C_n^5)
        double log2Cnr2 = VariableLog2ParamComplexityCTable[2][xn];   // log(C_n^4)
        double log2Cnr  = 0.0;
        double k_r      = std::exp((log2Cnr2 - log2Cnr1) * M_LN2);
        double q_r      = 1.0 + k_r * n / (6.0 - 2.0);                // we first compute C_n^6
        for (std::size_t i = std::size_t(6); i <= r; ++i) {
          log2Cnr  = log2Cnr1 + std::log2(q_r);
          log2Cnr1 = log2Cnr;
          k_r      = 1.0 / q_r;
          q_r      = 1.0 + k_r * (n / (i - 1.0));
        }
 
        // if we use a cache, update it
        if (_use_cache_) { _cache_.insert(std::pair< std::size_t, double >{r, n}, log2Cnr); }
 
        return log2Cnr;
      }
    } else {
      // try to find the value in the cache
      if (_use_cache_) {
        try {
          return _cache_[std::pair< std::size_t, double >{r, n}];
        } catch (NotFound const&) {}
      }
 
      // compute the corrected Szpankowski approximation of cn2 (see the
      // documentation of constants cst1, cst2, cst3 in the ScorefNML header)
      double log2Cnr1 = 0.5 * std::log2(n) + _cst1_ + _cst2_ / std::sqrt(n) + _cst3_ / n;
      if (r == std::size_t(2)) return log2Cnr1;
 
      // the value of log2(cn1), which is always equal to 0
      double log2Cnr2 = 0.0;
 
      // use Equation (13) of the paper to compute the value of cnr
      // (see the detail of the formulas in the above if statement)
      double k_r     = std::exp((log2Cnr2 - log2Cnr1) * M_LN2);
      double q_r     = 1.0 + k_r * n / (3.0 - 2.0);   // we first compute C_n^3
      double log2Cnr = 0.0;
      for (std::size_t i = std::size_t(3); i <= r; ++i) {
        log2Cnr  = log2Cnr1 + std::log2(q_r);
        log2Cnr1 = log2Cnr;
        k_r      = 1.0 / q_r;
        q_r      = 1.0 + k_r * (n / (i - 1.0));
      }
 
      // if we use a cache, update it
      if (_use_cache_) { _cache_.insert(std::pair< std::size_t, double >{r, n}, log2Cnr); }
 
      return log2Cnr;
    }
  }

  void VariableLog2ParamComplexity::CnrToFile(const std::string& filename) {
    // save all the value of cn2
    std::vector< long double > cn2_table(VariableLog2ParamComplexityCTableNSize);
    cn2_table[0] = 1;
    cn2_table[1] = 2;
 
    // for every value of n less than Szpankowski_threshold, we compute the
    // value of C_n^2 and write it into cn2_table
    GammaLog2 gamma_log2;
    for (double n = 2; n < VariableLog2ParamComplexityCTableNSize; ++n) {
      // here, note that, in  Silander, Roos, Kontkanen and Myllymaki (2007)
      // "Factorized Normalized Maximum Likelihood Criterion for Learning
      // Bayesian network Structures" paper, there is an uppercase N in the
      // formula, but this should be a lowercase n. In addition, we will loop
      // only on h=1 to n-1 and add to 2.0 the value computed to take into
      // account of h=0 and h=n.
      long double cn2 = 2;
      for (double h = 1; h < n; ++h) {
        long double elt = (gamma_log2(n + 1) - gamma_log2(h + 1) - gamma_log2((n - h) + 1)) * M_LN2
                        + h * std::log(h / n) + (n - h) * std::log((n - h) / n);
        cn2 += std::exp(elt);
      }
 
      // const double logCn2 = (double) std::log2 ( cn2 );
 
      cn2_table[int(n)] = cn2;
    }
 
    // write the header of the output file
    std::ofstream outfile(filename);
    if (!outfile.is_open()) { GUM_ERROR(IOError, "It is impossible to open file " << filename) }
    outfile.precision(20);
    outfile << "namespace gum {\n\n";
    /*
      outfile << "  // the size in r of VariableLog2ParamComplexityCTable\n";
      outfile << "  const std::size_t VariableLog2ParamComplexityCTableRSize = "
      << "4;\n\n";
      outfile << "  // the size in n of VariableLog2ParamComplexityCTable\n";
      outfile << "  const std::size_t VariableLog2ParamComplexityCTableNSize = "
      << VariableLog2ParamComplexityCTableNSize << ";\n\n";
    */
    outfile << "  // the CTable cache for log2(Cnr), n < " << VariableLog2ParamComplexityCTableNSize
            << " and r in {2,3,4,5}\n";
    outfile << "  const double VariableLog2ParamComplexityCTable[4]["
            << VariableLog2ParamComplexityCTableNSize << "] = {\n";
 
    // write the values of Cn2:
    outfile << "      { ";
    bool first = true;
    for (const auto cn2: cn2_table) {
      if (first) first = false;
      else outfile << ",\n        ";
      const double logCn2 = (double)std::log2(cn2);
      outfile << logCn2;
    }
    outfile << " },\n";
 
    // write the values of cn3, which are equal to cn2 + n
    outfile << "      { ";
    for (std::size_t i = std::size_t(0); i < VariableLog2ParamComplexityCTableNSize; ++i) {
      if (i > std::size_t(0)) outfile << ",\n        ";
      const double logCn3 = (double)std::log2(cn2_table[i] + i);
      outfile << logCn3;
    }
    outfile << " },\n";
 
    // write the values of cn4, which are equal to cn2 * (1 + n/2) + n
    outfile << "      { ";
    for (std::size_t i = std::size_t(0); i < VariableLog2ParamComplexityCTableNSize; ++i) {
      if (i > std::size_t(0)) outfile << ",\n        ";
      const double logCn4 = (double)std::log2(cn2_table[i] * (1.0 + i / 2.0) + i);
      outfile << logCn4;
    }
    outfile << " },\n";
 
    // write the values of cn5, which are equal to cn2 * (1 + 5n/6) + n + n^2/3
    outfile << "      { ";
    for (std::size_t i = std::size_t(0); i < VariableLog2ParamComplexityCTableNSize; ++i) {
      if (i > std::size_t(0)) outfile << ",\n        ";
      const double logCn5
          = (double)std::log2(cn2_table[i] * (1.0 + 5.0 * i / 6.0) + i + i * i / 3.0);
      outfile << logCn5;
    }
    outfile << " }\n";
 
    // write the footer and close the file
    outfile << "  };\n\n";
    outfile << "} /* namespace gum */\n";
    outfile.close();
  }
 
  // the CTable cache for log2(Cnr), n < 1000 and r in {2,3,4,5}
  const double VariableLog2ParamComplexityCTable[4][1000] = {{0,
                                                              1,
                                                              1.3219280948873624038,
                                                              1.5305148703230577745,
                                                              1.6865005853516363832,
                                                              1.8116353900270942834,
                                                              1.9163583844844809256,
                                                              2.0065271644294311315,
                                                              2.0857704966116465961,
                                                              2.1564986340920730967,
                                                              2.2203965250090043071,
                                                              2.2786902190668105739,
                                                              2.3322990996918790962,
                                                              2.3819333135364253451,
                                                              2.4281502505698502326,
                                                              2.4713970635580784219,
                                                              2.5120389433034655546,
                                                              2.5503761743106676541,
                                                              2.5866596296997594884,
                                                              2.6211016700234877064,
                                                              2.6538820078470202191,
                                                              2.6851559155058404293,
                                                              2.7150583520600020115,
                                                              2.7437053059117562093,
                                                              2.7711991437760832291,
                                                              2.79763156718914896,
                                                              2.8230804530606610214,
                                                              2.8476192305148533279,
                                                              2.8713103180418357852,
                                                              2.8942112722722521667,
                                                              2.9163733091011145859,
                                                              2.9378445477276624587,
                                                              2.9586657568594088019,
                                                              2.9788760126291720098,
                                                              2.9985108275544174461,
                                                              3.0176020678569406286,
                                                              3.0361794852617554064,
                                                              3.054270010943741287,
                                                              3.0718997050205953947,
                                                              3.0890904999196213687,
                                                              3.1058645897056522678,
                                                              3.1222418647242431433,
                                                              3.1382404560408532213,
                                                              3.1538784964406412215,
                                                              3.1691716096623245846,
                                                              3.1841343571506448562,
                                                              3.1987813380348359971,
                                                              3.2131256939662806715,
                                                              3.2271791995449077284,
                                                              3.2409529086398065623,
                                                              3.2544605816423333167,
                                                              3.2677107412971055922,
                                                              3.2807131309745982684,
                                                              3.2934769361591986758,
                                                              3.3060108350527364074,
                                                              3.3183230373618513021,
                                                              3.3304213181985549141,
                                                              3.3423130485925671707,
                                                              3.3540052232282806521,
                                                              3.365504485776890764,
                                                              3.3768171521001946189,
                                                              3.3879492315483643594,
                                                              3.3989064465446894303,
                                                              3.4096942506232235104,
                                                              3.4203178450640367991,
                                                              3.4307821942586769204,
                                                              3.4410920399192130859,
                                                              3.4512519142354931567,
                                                              3.4612661520716652142,
                                                              3.4711389022871026455,
                                                              3.4808741382541863452,
                                                              3.4904756676427037831,
                                                              3.4999471415291836252,
                                                              3.5092920628885035939,
                                                              3.5185137945163771356,
                                                              3.5276155664278530111,
                                                              3.5366004827734669469,
                                                              3.5454715283100086687,
                                                              3.5542315744600920802,
                                                              3.5628833849918311039,
                                                              3.5714296213456182549,
                                                              3.5798728476357393191,
                                                              3.5882155353493616623,
                                                              3.5964600677650082616,
                                                              3.6046087441100200799,
                                                              3.6126637834754928846,
                                                              3.6206273285060115441,
                                                              3.6285014488783944309,
                                                              3.6362881445853285634,
                                                              3.6439893490354289263,
                                                              3.6516069319827231254,
                                                              3.6591427022967493166,
                                                              3.6665984105827660322,
                                                              3.6739757516632027823,
                                                              3.6812763669277042133,
                                                              3.6885018465600203363,
                                                              3.6956537316512108049,
                                                              3.7027335162031129734,
                                                              3.709742649032294004,
                                                              3.7166825355772776796,
                                                              3.7235545396163005627,
                                                              3.7303599849017525791,
                                                              3.7371001567142370092,
                                                              3.7437763033413591351,
                                                              3.7503896374878165076,
                                                              3.7569413376170515129,
                                                              3.7634325492308957628,
                                                              3.769864386090051589,
                                                              3.776237931378247481,
                                                              3.7825542388137973759,
                                                              3.7888143337102198061,
                                                              3.7950192139907077404,
                                                              3.8011698511566636149,
                                                              3.8072671912138411621,
                                                              3.813312155558833183,
                                                              3.8193056418268742647,
                                                              3.8252485247035634686,
                                                              3.8311416567014151546,
                                                              3.8369858689048319533,
                                                              3.8427819716839515252,
                                                              3.8485307553783489887,
                                                              3.8542329909524455367,
                                                              3.8598894306260795872,
                                                              3.8655008084775124289,
                                                              3.8710678410228189783,
                                                              3.876591227773807713,
                                                              3.882071651772597054,
                                                              3.8875097801044717905,
                                                              3.8929062643933862731,
                                                              3.8982617412758329145,
                                                              3.9035768328586790687,
                                                              3.9088521471579533717,
                                                              3.9140882785229149654,
                                                              3.9192858080418018218,
                                                              3.9244453039363653701,
                                                              3.9295673219370383933,
                                                              3.9346524056491860577,
                                                              3.9397010869028807889,
                                                              3.9447138860910029301,
                                                              3.9496913124958692443,
                                                              3.9546338646043004417,
                                                              3.9595420304092820984,
                                                              3.9644162877048079174,
                                                              3.9692571043686029064,
                                                              3.9740649386334756521,
                                                              3.9788402393527797507,
                                                              3.9835834462536925393,
                                                              3.9882949901852819963,
                                                              3.9929752933534228276,
                                                              3.9976247695539166571,
                                                              4.0022438243928535329,
                                                              4.0068328555018473125,
                                                              4.0113922527467709145,
                                                              4.0159223984285752351,
                                                              4.0204236674779094685,
                                                              4.0248964276438972121,
                                                              4.0293410396769857584,
                                                              4.0337578575052832619,
                                                              4.0381472284054762412,
                                                              4.0425094931694394162,
                                                              4.0468449862648130377,
                                                              4.0511540359901010433,
                                                              4.0554369646269510241,
                                                              4.0596940885851129366,
                                                              4.0639257185455752719,
                                                              4.0681321595976314143,
                                                              4.0723137113727130298,
                                                              4.0764706681735560778,
                                                              4.0806033191015051997,
                                                              4.0847119481760802984,
                                                              4.0887968344559588729,
                                                              4.0928582521525260773,
                                                              4.096896470742770191,
                                                              4.1009117550762148596,
                                                              4.1049043654819339366,
                                                              4.1088745578704353179,
                                                              4.1128225838338998699,
                                                              4.1167486907416792619,
                                                              4.1206531218352182577,
                                                              4.1245361163190610299,
                                                              4.1283979094510820929,
                                                              4.132238732626686506,
                                                              4.1360588134663576199,
                                                              4.139858375893271436,
                                                              4.1436376402167542921,
                                                              4.1473968232080631324,
                                                              4.1511361381755982336,
                                                              4.1548557950390989646,
                                                              4.1585560004008677026,
                                                              4.1622369576153950277,
                                                              4.1658988668573400105,
                                                              4.1695419251878274025,
                                                              4.1731663266171183935,
                                                              4.1767722621704104213,
                                                              4.1803599199458023605,
                                                              4.1839294851754313953,
                                                              4.1874811402837543994,
                                                              4.1910150649428796754,
                                                              4.1945314361289192107,
                                                              4.1980304281754907691,
                                                              4.2015122128245518951,
                                                              4.2049769592809953522,
                                                              4.2084248342581513924,
                                                              4.2118560020307898029,
                                                              4.2152706244788999967,
                                                              4.2186688611367610946,
                                                              4.2220508692363925007,
                                                              4.225416803753091699,
                                                              4.2287668174469059679,
                                                              4.2321010609056335383,
                                                              4.2354196825864427467,
                                                              4.2387228288547360577,
                                                              4.2420106440234812695,
                                                              4.2452832703925196256,
                                                              4.2485408482849811307,
                                                              4.2517835160841324083,
                                                              4.2550114102685414608,
                                                              4.2582246654469715352,
                                                              4.26142341439412764,
                                                              4.2646077880808599403,
                                                              4.2677779157089421602,
                                                              4.2709339247424109587,
                                                              4.274075940938657503,
                                                              4.2772040883774593567,
                                                              4.2803184894933243854,
                                                              4.2834192651025713161,
                                                              4.2865065344308401762,
                                                              4.289580415144497394,
                                                              4.2926410233744620726,
                                                              4.2956884737434544164,
                                                              4.2987228793938232485,
                                                              4.3017443520103322285,
                                                              4.3047530018470325786,
                                                              4.3077489377515956193,
                                                              4.3107322671875394349,
                                                              4.3137030962596352168,
                                                              4.316661529734781233,
                                                              4.3196076710668069865,
                                                              4.3225416224157466871,
                                                              4.3254634846717516794,
                                                              4.3283733574735760996,
                                                              4.3312713392326411821,
                                                              4.3341575271484868992,
                                                              4.3370320172331373598,
                                                              4.3398949043284797966,
                                                              4.3427462821229729784,
                                                              4.3455862431756866471,
                                                              4.3484148789292014214,
                                                              4.3512322797302420696,
                                                              4.3540385348456256409,
                                                              4.356833732481850241,
                                                              4.3596179597991531196,
                                                              4.3623913029289527188,
                                                              4.3651538469910677875,
                                                              4.3679056761079779747,
                                                              4.3706468734213030913,
                                                              4.3733775211061569621,
                                                              4.3760977003871506241,
                                                              4.3788074915526937758,
                                                              4.3815069739687659833,
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} /* namespace gum */