Normal Bayes 分类器过程详解

OpenCV的机器学习类定义在ml.hpp文件中，基础类是CvStatModel，其他各种分类器从这里继承而来。

今天研究CvNormalBayesClassifier分类器。

1.类定义

在ml.hpp中有以下类定义：

 class CV_EXPORTS_W CvNormalBayesClassifier : public CvStatModel
 {
 public:
     CV_WRAP CvNormalBayesClassifier();
     virtual ~CvNormalBayesClassifier();

     CvNormalBayesClassifier( const CvMat* trainData, const CvMat* responses,
         const CvMat* varIdx=0, const CvMat* sampleIdx=0 );

     virtual bool train( const CvMat* trainData, const CvMat* responses,
         const CvMat* varIdx = 0, const CvMat* sampleIdx=0, bool update=false );

     virtual float predict( const CvMat* samples, CV_OUT CvMat* results=0 ) const;
     CV_WRAP virtual void clear();

     CV_WRAP CvNormalBayesClassifier( const cv::Mat& trainData, const cv::Mat& responses,
                             const cv::Mat& varIdx=cv::Mat(), const cv::Mat& sampleIdx=cv::Mat() );
     CV_WRAP virtual bool train( const cv::Mat& trainData, const cv::Mat& responses,
                        const cv::Mat& varIdx = cv::Mat(), const cv::Mat& sampleIdx=cv::Mat(),
                        bool update=false );
     CV_WRAP virtual float predict( const cv::Mat& samples, CV_OUT cv::Mat* results=0 ) const;

     virtual void write( CvFileStorage* storage, const char* name ) const;
     virtual void read( CvFileStorage* storage, CvFileNode* node );

 protected:
     int     var_count, var_all;
     CvMat*  var_idx;
     CvMat*  cls_labels;
     CvMat** count;
     CvMat** sum;
     CvMat** productsum;
     CvMat** avg;
     CvMat** inv_eigen_values;
     CvMat** cov_rotate_mats;
     CvMat*  c;
 };

2.示例

此类使用方法如下：（引用别人的代码，忘记出处了，非常抱歉这个。。。）

 //openCV中贝叶斯分类器的API函数用法举例
 //运行环境：win7 + VS2005 + openCV2.4.5

 #include "global_include.h"

 using namespace std;
 using namespace cv;

 //10个样本特征向量维数为12的训练样本集，第一列为该样本的类别标签
 double inputArr[10][13] =
 {
      1,0.708333,1,1,-0.320755,-0.105023,-1,1,-0.419847,-1,-0.225806,0,1,
     -1,0.583333,-1,0.333333,-0.603774,1,-1,1,0.358779,-1,-0.483871,0,-1,
      1,0.166667,1,-0.333333,-0.433962,-0.383562,-1,-1,0.0687023,-1,-0.903226,-1,-1,
     -1,0.458333,1,1,-0.358491,-0.374429,-1,-1,-0.480916,1,-0.935484,0,-0.333333,
     -1,0.875,-1,-0.333333,-0.509434,-0.347032,-1,1,-0.236641,1,-0.935484,-1,-0.333333,
     -1,0.5,1,1,-0.509434,-0.767123,-1,-1,0.0534351,-1,-0.870968,-1,-1,
      1,0.125,1,0.333333,-0.320755,-0.406393,1,1,0.0839695,1,-0.806452,0,-0.333333,
      1,0.25,1,1,-0.698113,-0.484018,-1,1,0.0839695,1,-0.612903,0,-0.333333,
      1,0.291667,1,1,-0.132075,-0.237443,-1,1,0.51145,-1,-0.612903,0,0.333333,
      1,0.416667,-1,1,0.0566038,0.283105,-1,1,0.267176,-1,0.290323,0,1
 };

 //一个测试样本的特征向量
 double testArr[]=
 {
     0.25,1,1,-0.226415,-0.506849,-1,-1,0.374046,-1,-0.83871,0,-1
 };


 int _tmain(int argc, _TCHAR* argv[])
 {
     Mat trainData(10, 12, CV_32FC1);//构建训练样本的特征向量
     for (int i=0; i<10; i++)
     {
         for (int j=0; j<12; j++)
         {
             trainData.at<float>(i, j) = inputArr[i][j+1];
         }
     }

     Mat trainResponse(10, 1, CV_32FC1);//构建训练样本的类别标签
     for (int i=0; i<10; i++)
     {
         trainResponse.at<float>(i, 0) = inputArr[i][0];
     }

     CvNormalBayesClassifier nbc;
     bool trainFlag = nbc.train(trainData, trainResponse);//进行贝叶斯分类器训练
     if (trainFlag)
     {
         cout<<"train over..."<<endl;
         nbc.save("normalBayes.txt");
     }
     else
     {
         cout<<"train error..."<<endl;
         system("pause");
         exit(-1);
     }


     CvNormalBayesClassifier testNbc;
     testNbc.load("normalBayes.txt");

     Mat testSample(1, 12, CV_32FC1);//构建测试样本
     for (int i=0; i<12; i++)
     {
         testSample.at<float>(0, i) = testArr[i];
     }

     float flag = testNbc.predict(testSample);//进行测试
     cout<<"flag = "<<flag<<endl;

     system("pause");
     return 0;
 }

3.步骤

两步走：

1.调用train函数训练分类器；

2.调用predict函数，判定测试样本的类别。

以上示例代码还延时了怎样使用save和load函数，使得训练好的分类器可以保存在文本中。

4.初始化

接下来，看CvNormalBayesClassifier类的无参数初始化：

 CvNormalBayesClassifier::CvNormalBayesClassifier()
 {
     var_count = var_all = 0;
     var_idx = 0;
     cls_labels = 0;
     count = 0;
     sum = 0;
     productsum = 0;
     avg = 0;
     inv_eigen_values = 0;
     cov_rotate_mats = 0;
     c = 0;
     default_model_name = "my_nb";
 }

还有另一种带参数的初始化形式：

 CvNormalBayesClassifier::CvNormalBayesClassifier(
     const CvMat* _train_data, const CvMat* _responses,
     const CvMat* _var_idx, const CvMat* _sample_idx )
 {
     var_count = var_all = 0;
     var_idx = 0;
     cls_labels = 0;
     count = 0;
     sum = 0;
     productsum = 0;
     avg = 0;
     inv_eigen_values = 0;
     cov_rotate_mats = 0;
     c = 0;
     default_model_name = "my_nb";

     train( _train_data, _responses, _var_idx, _sample_idx );
 }

可见，带参数形式糅合了类的初始化和train函数。

另外，以Mat参数形式的对应函数版本，功能是一致的，只不过为了体现2.0以后版本的C++特性罢了。如下：

 CV_WRAP CvNormalBayesClassifier( const cv::Mat& trainData, const cv::Mat& responses,
                         const cv::Mat& varIdx=cv::Mat(), const cv::Mat& sampleIdx=cv::Mat() );
 CV_WRAP virtual bool train( const cv::Mat& trainData, const cv::Mat& responses,
                    const cv::Mat& varIdx = cv::Mat(), const cv::Mat& sampleIdx=cv::Mat(),
                    bool update=false );
 CV_WRAP virtual float predict( const cv::Mat& samples, CV_OUT cv::Mat* results=0 ) const;

5.训练

下面开始分析train函数，分析CvMat格式参数的train函数，即：

 bool train( const CvMat* trainData, const CvMat* responses,const CvMat* varIdx = 0, const CvMat* sampleIdx=0, bool update=false );

在进入该函数之前，还要先回头看看CvNormalBayesClassifier类有哪些数据成员：

 protected:
     int     var_count, var_all; //每个样本的特征维数、即变量数目，或者说trainData的列数目(在varIdx=0时)
     CvMat*  var_idx;        //特征子集的索引，可能特征数目为100，但是只用其中一部分训练
     CvMat*  cls_labels;     //类别数目
     CvMat** count;      //count[0...(classNum-1)]，每个元素是一个CvMat(rows=1,cols=var_count)指针，代表训练数据中每一类的某个特征的数目
     CvMat** sum;        //sum[0...(classNum-1)]，每个元素是一个CvMat(rows=1,cols=var_count)指针，代表训练数据中每一类的某个特征的累加和
     CvMat** productsum;     //productsum[0...(classNum-1)]，每个元素是一个CvMat(rows=cols=var_count)指针,存储类内特征相关矩阵
     CvMat** avg;        //avg[0...(classNum-1)]，每个元素是一个CvMat(rows=1,cols=var_count)指针，代表训练数据中每一类的某个特征的平均值
     CvMat** inv_eigen_values;//inv_eigen_values[0...(classNum-1)]，每个元素是一个CvMat(rows=1,cols=var_count)指针，代表训练数据中每一类的某个特征的特征值的倒数
     CvMat** cov_rotate_mats;    //特征变量的协方差矩阵经过SVD奇异值分解后得到的特征向量矩阵
     CvMat*  c;

这些数据成员，怎样使用呢？在train函数中见分晓：

 bool CvNormalBayesClassifier::train( const CvMat* _train_data, const CvMat* _responses,
                                     const CvMat* _var_idx, const CvMat* _sample_idx, bool update )
 {
     const float min_variation = FLT_EPSILON;
     bool result = false;
     CvMat* responses   = 0;
     const float** train_data = 0;
     CvMat* __cls_labels = 0;
     CvMat* __var_idx = 0;
     CvMat* cov = 0;

     CV_FUNCNAME( "CvNormalBayesClassifier::train" );

     __BEGIN__;

     int cls, nsamples = 0, _var_count = 0, _var_all = 0, nclasses = 0;
     int s, c1, c2;
     const int* responses_data;

     //1.整理训练数据
     CV_CALL( cvPrepareTrainData( 0,
         _train_data, CV_ROW_SAMPLE, _responses, CV_VAR_CATEGORICAL,
         _var_idx, _sample_idx, false, &train_data,
         &nsamples, &_var_count, &_var_all, &responses,
         &__cls_labels, &__var_idx ));

     if( !update )   //如果是初始训练数据
     {
         const size_t mat_size = sizeof(CvMat*);
         size_t data_size;

         clear();

         var_idx = __var_idx;
         cls_labels = __cls_labels;
         __var_idx = __cls_labels = 0;
         var_count = _var_count;
         var_all = _var_all;

         nclasses = cls_labels->cols;
         data_size = nclasses*6*mat_size;

         CV_CALL( count = (CvMat**)cvAlloc( data_size ));
         memset( count, 0, data_size );          //count[cls]存储第cls类每个属性变量个数

         sum             = count      + nclasses;//sum[cls]存储第cls类每个属性取值的累加和
         productsum      = sum        + nclasses;//productsum[cls]存储第cls类的协方差矩阵的乘积项sum(XiXj)，cov(Xi,Xj)=sum(XiXj)-sum(Xi)E(Xj)
         avg             = productsum + nclasses;//avg[cls]存储第cls类的每个变量均值
         inv_eigen_values= avg        + nclasses;//inv_eigen_values[cls]存储第cls类的协方差矩阵的特征值
         cov_rotate_mats = inv_eigen_values         + nclasses;//存储第cls类的矩阵的特征值对应的特征向量

         CV_CALL( c = cvCreateMat( 1, nclasses, CV_64FC1 ));

         for( cls = 0; cls < nclasses; cls++ )    //对所有类别
         {
             CV_CALL(count[cls]            = cvCreateMat( 1, var_count, CV_32SC1 ));
             CV_CALL(sum[cls]              = cvCreateMat( 1, var_count, CV_64FC1 ));
             CV_CALL(productsum[cls]       = cvCreateMat( var_count, var_count, CV_64FC1 ));
             CV_CALL(avg[cls]              = cvCreateMat( 1, var_count, CV_64FC1 ));
             CV_CALL(inv_eigen_values[cls] = cvCreateMat( 1, var_count, CV_64FC1 ));
             CV_CALL(cov_rotate_mats[cls]  = cvCreateMat( var_count, var_count, CV_64FC1 ));
             CV_CALL(cvZero( count[cls] ));
             CV_CALL(cvZero( sum[cls] ));
             CV_CALL(cvZero( productsum[cls] ));
             CV_CALL(cvZero( avg[cls] ));
             CV_CALL(cvZero( inv_eigen_values[cls] ));
             CV_CALL(cvZero( cov_rotate_mats[cls] ));
         }
     }
     else    //如果是更新训练数据
     {
         // check that the new training data has the same dimensionality etc.
         if( _var_count != var_count || _var_all != var_all || !((!_var_idx && !var_idx) ||
             (_var_idx && var_idx && cvNorm(_var_idx,var_idx,CV_C) < DBL_EPSILON)) )
             CV_ERROR( CV_StsBadArg,
             "The new training data is inconsistent with the original training data" );

         if( cls_labels->cols != __cls_labels->cols ||
             cvNorm(cls_labels, __cls_labels, CV_C) > DBL_EPSILON )
             CV_ERROR( CV_StsNotImplemented,
             "In the current implementation the new training data must have absolutely "
             "the same set of class labels as used in the original training data" );

         nclasses = cls_labels->cols;
     }

     responses_data = responses->data.i;
     CV_CALL( cov = cvCreateMat( _var_count, _var_count, CV_64FC1 ));

     //2.处理训练数据，计算每一类的
     // process train data (count, sum , productsum)
     for( s = 0; s < nsamples; s++ )
     {
         cls = responses_data[s];
         int* count_data = count[cls]->data.i;
         double* sum_data = sum[cls]->data.db;
         double* prod_data = productsum[cls]->data.db;
         const float* train_vec = train_data[s];

         for( c1 = 0; c1 < _var_count; c1++, prod_data += _var_count )
         {
             double val1 = train_vec[c1];
             sum_data[c1] += val1;
             count_data[c1]++;
             for( c2 = c1; c2 < _var_count; c2++ )
                 prod_data[c2] += train_vec[c2]*val1;
         }
     }

     //计算每一类的每个属性平均值、协方差矩阵
     // calculate avg, covariance matrix, c
     for( cls = 0; cls < nclasses; cls++ )    //对每一类
     {
         double det = 1;
         int i, j;
         CvMat* w = inv_eigen_values[cls];
         int* count_data = count[cls]->data.i;
         double* avg_data = avg[cls]->data.db;
         double* sum1 = sum[cls]->data.db;

         cvCompleteSymm( productsum[cls], 0 );

         for( j = 0; j < _var_count; j++ )    //计算当前类别cls的每个变量属性值的平均值
         {
             int n = count_data[j];
             avg_data[j] = n ? sum1[j] / n : 0.;
         }

         count_data = count[cls]->data.i;
         avg_data = avg[cls]->data.db;
         sum1 = sum[cls]->data.db;

         for( i = 0; i < _var_count; i++ )//计算当前类别cls的变量协方差矩阵，矩阵大小为_var_count * _var_count，注意协方差矩阵对称。
         {
             double* avg2_data = avg[cls]->data.db;
             double* sum2 = sum[cls]->data.db;
             double* prod_data = productsum[cls]->data.db + i*_var_count;
             double* cov_data = cov->data.db + i*_var_count;
             double s1val = sum1[i];
             double avg1 = avg_data[i];
             int _count = count_data[i];

             for( j = 0; j <= i; j++ )
             {
                 double avg2 = avg2_data[j];
                 double cov_val = prod_data[j] - avg1 * sum2[j] - avg2 * s1val + avg1 * avg2 * _count;
                 cov_val = (_count > 1) ? cov_val / (_count - 1) : cov_val;
                 cov_data[j] = cov_val;
             }
         }

         CV_CALL( cvCompleteSymm( cov, 1 ));
         CV_CALL( cvSVD( cov, w, cov_rotate_mats[cls], 0, CV_SVD_U_T ));
         CV_CALL( cvMaxS( w, min_variation, w ));
         for( j = 0; j < _var_count; j++ )
             det *= w->data.db[j];

         CV_CALL( cvDiv( NULL, w, w ));
         c->data.db[cls] = det > 0 ? log(det) : -700;
     }

     result = true;

     __END__;

     if( !result || cvGetErrStatus() < 0 )
         clear();

     cvReleaseMat( &cov );
     cvReleaseMat( &__cls_labels );
     cvReleaseMat( &__var_idx );
     cvFree( &train_data );

     return result;
 }

训练部分就此完成。

6.预测

下面看用于预测的predict函数的实现代码：

 float CvNormalBayesClassifier::predict( const CvMat* samples, CvMat* results ) const
 {
     float value = 0;

     if( !CV_IS_MAT(samples) || CV_MAT_TYPE(samples->type) != CV_32FC1 || samples->cols != var_all )
         CV_Error( CV_StsBadArg,
         "The input samples must be 32f matrix with the number of columns = var_all" );

     if( samples->rows > 1 && !results )
         CV_Error( CV_StsNullPtr,
         "When the number of input samples is >1, the output vector of results must be passed" );

     if( results )
     {
         if( !CV_IS_MAT(results) || (CV_MAT_TYPE(results->type) != CV_32FC1 &&
         CV_MAT_TYPE(results->type) != CV_32SC1) ||
         (results->cols != 1 && results->rows != 1) ||
         results->cols + results->rows - 1 != samples->rows )
         CV_Error( CV_StsBadArg, "The output array must be integer or floating-point vector "
         "with the number of elements = number of rows in the input matrix" );
     }

     const int* vidx = var_idx ? var_idx->data.i : 0;

     cv::parallel_for(cv::BlockedRange(0, samples->rows), predict_body(c, cov_rotate_mats, inv_eigen_values, avg, samples,
                                                                       vidx, cls_labels, results, &value, var_count
     ));

     return value;
 }

可以发现，预测部分核心代码是：

 cv::parallel_for(cv::BlockedRange(0, samples->rows), predict_body(c, cov_rotate_mats, inv_eigen_values, avg, samples,
                                                                       vidx, cls_labels, results, &value, var_count));

parallel_for是用于并行支持的，可能会调用tbb模块。predict_body则是一个结构体，内部的()符号被重载，实现预测功能。其完整定义如下：

 //predict函数调用predict_body结构体的(）符号重载函数，实现基于贝叶斯的分类
 struct predict_body
 {
     predict_body(CvMat* _c, CvMat** _cov_rotate_mats, CvMat** _inv_eigen_values, CvMat** _avg,
                 const CvMat* _samples, const int* _vidx, CvMat* _cls_labels,
                 CvMat* _results, float* _value, int _var_count1)
     {
         c = _c;
         cov_rotate_mats = _cov_rotate_mats;
         inv_eigen_values = _inv_eigen_values;
         avg = _avg;
         samples = _samples;
         vidx = _vidx;
         cls_labels = _cls_labels;
         results = _results;
         value = _value;
         var_count1 = _var_count1;
     }

     CvMat* c;
     CvMat** cov_rotate_mats;
     CvMat** inv_eigen_values;
     CvMat** avg;
     const CvMat* samples;
     const int* vidx;
     CvMat* cls_labels;

     CvMat* results;
     float* value;
     int var_count1;

     void operator()( const cv::BlockedRange& range ) const
     {

         int cls = -1;
         int rtype = 0, rstep = 0;
         int nclasses = cls_labels->cols;
         int _var_count = avg[0]->cols;

         if (results)
         {
             rtype = CV_MAT_TYPE(results->type);
             rstep = CV_IS_MAT_CONT(results->type) ? 1 : results->step/CV_ELEM_SIZE(rtype);
         }
         // allocate memory and initializing headers for calculating
         cv::AutoBuffer<double> buffer(nclasses + var_count1);
         CvMat diff = cvMat( 1, var_count1, CV_64FC1, &buffer[0] );

         for(int k = range.begin(); k < range.end(); k += 1 )//对于每个输入测试样本
         {
             int ival;
             double opt = FLT_MAX;

             for(int i = 0; i < nclasses; i++ )   //对于每一类别，计算其似然概率
             {

                 double cur = c->data.db[i];
                 CvMat* u = cov_rotate_mats[i];
                 CvMat* w = inv_eigen_values[i];

                 const double* avg_data = avg[i]->data.db;
                 const float* x = (const float*)(samples->data.ptr + samples->step*k);

                 // cov = u w u'  -->  cov^(-1) = u w^(-1) u'
                 for(int j = 0; j < _var_count; j++ ) //计算特征相对于均值的偏移
                     diff.data.db[j] = avg_data[j] - x[vidx ? vidx[j] : j];

                 cvGEMM( &diff, u, 1, 0, 0, &diff, CV_GEMM_B_T );
                 for(int j = 0; j < _var_count; j++ )//计算特征的联合概率
                 {
                     double d = diff.data.db[j];
                     cur += d*d*w->data.db[j];
                 }

                 if( cur < opt )  //找到分类概率最大的
                 {
                     cls = i;
                     opt = cur;
                 }
                 // probability = exp( -0.5 * cur )

             }//for(int i = 0; i < nclasses; i++ )

             ival = cls_labels->data.i[cls];
             if( results )
             {
                 if( rtype == CV_32SC1 )
                     results->data.i[k*rstep] = ival;
                 else
                     results->data.fl[k*rstep] = (float)ival;
             }
             if( k == 0 )
                 *value = (float)ival;

         }//for(int k = range.begin()...

     }//void operator()...
 };