Precision/Recall和ROC曲线

【Precision/Recall的基本概念】

转载自http://www.zhizhihu.com/html/y2010/2137.html。

查准率和查全率是信息检索效率评价的两个定量指标，不仅可以用来评价每次检索的准确性和全面性，也是在信息检索系统评价中衡量系统检索性能的重要方面。

（1）查准率（Precision ratio，简称为P），是指检出的相关文献数占检出文献总数的百分比。查准率反映检索准确性，其补数就是误检率。

查准率＝（检索出的相关信息量/检索出的信息总量）*100%

（2）查全率（Recall ratio，简称为R），是指检出的相关文献数占系统中相关文献总数的百分比。查全率反映检索全面性，其补数就是漏检率。

查全率＝（检索出的相关信息量/系统中的相关信息总量）*100%

【Precision/Recall曲线的来源】

要评价信息检索系统的性能水平，就必须在一个检索系统中进行多次检索。每进行一次检索，都计算其查准率和查全率，并以此作为坐标值，在平面坐标图上标示出来。通过大量的检索，就可以得到检索系统的性能曲线。
Precision/Recall曲线一般是以每一次计算的查全率为横坐标，每一次计算的查准率为纵坐标。如下图所示：

该图是由100次检索得到的，由图可知：在查全率和查准率之间存在着相反的相互依赖关系–如果提高输出的查全率，就会降低其查准率

【ROC曲线】

ROC曲线的横坐标为false positive rate，纵坐标为true positive rate。
其中

false positive rate=搜索出的不相关信息量/系统中的不相关信息量

true positive rate=搜索到的相关信息量/系统中的相关信息量

如下图所示：

该曲线也是由100次检索得到的，由图可知，false positive rate与true positive rate之间是正相关的。

【Precision/Recall和ROC曲线的一个小实例】

功能函数代码：

function [prec, tpr, fpr, thresh] = prec_rec(score, target, varargin)
% PREC_REC - Compute and plot precision/recall and ROC curves.
%
%   PREC_REC(SCORE,TARGET), where SCORE and TARGET are equal-sized vectors,
%   and TARGET is binary, plots the corresponding precision-recall graph
%   and the ROC curve.
%
%   Several options of the form PREC_REC(...,'OPTION_NAME', OPTION_VALUE)
%   can be used to modify the default behavior.
%      - 'instanceCount': Usually it is assumed that one line in the input
%                         data corresponds to a single sample. However, it
%                         might be the case that there are a total of N
%                         instances with the same SCORE, out of which
%                         TARGET are classified as positive, and (N -
%                         TARGET) are classified as negative. Instead of
%                         using repeated samples with the same SCORE, we
%                         can summarize these observations by means of this
%                         option. Thus it requires a vector of the same
%                         size as TARGET.
%      - 'numThresh'    : Specify the (maximum) number of score intervals.
%                         Generally, splits are made such that each
%                         interval contains about the same number of sample
%                         lines.
%      - 'holdFigure'   : [0,1] draw into the current figure, instead of
%                         creating a new one.
%      - 'style'        : Style specification for plot command.
%      - 'plotROC'      : [0,1] Explicitly specify if ROC curve should be
%                         plotted.
%      - 'plotPR'       : [0,1] Explicitly specify if precision-recall curve
%                         should be plotted.
%      - 'plotBaseline' : [0,1] Plot a baseline of the random classifier.
%
%   By default, when output arguments are specified, as in
%         [PREC, TPR, FPR, THRESH] = PREC_REC(...),
%   no plot is generated. The arguments are the score thresholds, along
%   with the respective precisions, true-positive, and false-positive
%   rates.


optargin = size(varargin, 2);
stdargin = nargin - optargin;  

if stdargin < 2
 error('at least 2 arguments required');
end

%%  parse optional arguments（解析可选参数）
num_thresh = -1;
hold_fig = 0;
% 无输出时，画出Precision/Recall和ROC曲线
plot_roc = (nargout <= 0);  
plot_pr  = (nargout <= 0);
instance_count = -1;
style = '';
plot_baseline = 1;

i = 1;
while (i <= optargin)
 if (strcmp(varargin{i}, 'numThresh'))
    if (i >= optargin)
    error('argument required for %s', varargin{i});
    else
    num_thresh = varargin{i+1};
    i = i + 2; 
    end
 elseif (strcmp(varargin{i}, 'style'))
    if (i >= optargin)
    error('argument required for %s', varargin{i});
    else
    style = varargin{i+1};
    i = i + 2;
    end
 elseif (strcmp(varargin{i}, 'instanceCount'))
    if (i >= optargin)
    error('argument required for %s', varargin{i});
    else
    instance_count = varargin{i+1};
    i = i + 2;
    end
 elseif (strcmp(varargin{i}, 'holdFigure'))
    if (i >= optargin)
    error('argument required for %s', varargin{i});
    else
       if ~isempty(get(0,'CurrentFigure'))
       hold_fig = varargin{i+1};
       end
    i = i + 2;
    end
 elseif (strcmp(varargin{i}, 'plotROC'))
    if (i >= optargin)
    error('argument required for %s', varargin{i});
    else
    plot_roc = varargin{i+1};
    i = i + 2;
    end
 elseif (strcmp(varargin{i}, 'plotPR'))
    if (i >= optargin)
    error('argument required for %s', varargin{i});
    else
    plot_pr = varargin{i+1};
    i = i + 2;
    end
 elseif (strcmp(varargin{i}, 'plotBaseline'))
    if (i >= optargin)
    error('argument required for %s', varargin{i});
    else
    plot_baseline = varargin{i+1};
    i = i + 2;
    end
 elseif (~ischar(varargin{i}))
    error('only two numeric arguments required');
 else
 error('unknown option: %s', varargin{i});
 end
end


%% 数据处理
[nx,ny]=size(score);
if (nx~=1 && ny~=1)
 error('first argument must be a vector');
end

[mx,my]=size(target);
if (mx~=1 && my~=1)
 error('second argument must be a vector');
end
 %convert to colume vector
score  =  score(:);
target = target(:);

if (length(target) ~= length(score))
 error('score and target must have same length');
end

if (instance_count == -1)
 % set default for total instances
 instance_count = ones(length(score),1);
 target = max(min(target(:),1),0); % ensure binary target
else
 if numel(instance_count)==1
 % scalar
 instance_count = instance_count * ones(length(target), 1);
 end
 [px,py] = size(instance_count);
 if (px~=1 && py~=1)
 error('instance count must be a vector');
 end
 instance_count = instance_count(:);
 if (length(target) ~= length(instance_count))
 error('instance count must have same length as target');
 end
 target = min(instance_count, target);
end

 % set default for number of thresholds 
if num_thresh < 0
 score_uniq = unique(score);
 num_thresh = min(length(score_uniq), 100);
end
% set thresholds 
qvals = (1:(num_thresh-1))/num_thresh;
thresh = [min(score) quantile(score,qvals)];
% remove identical bins
thresh = sort(unique(thresh),2,'descend');
%计算总的target中1的个数和0的个数
total_target = sum(target);
total_neg = sum(instance_count - target);

prec = zeros(length(thresh),1);
tpr  = zeros(length(thresh),1);
fpr  = zeros(length(thresh),1);
for i = 1:length(thresh)    //进行多次检索
  %找出score值大于阈值的那部分数据target
 idx     = (score >= thresh(i));
 fpr(i)  = sum(instance_count(idx) - target(idx)); %计算这部分target中等于0的数目
 tpr(i)  = sum(target(idx));  %计算这部分target中等于1的数目
  % 查准率=搜索到的相关信息量/搜索到的所有信息量
 prec(i) = sum(target(idx)) / sum(instance_count(idx));  
end
%  查全率=搜索到的相关信息量/系统中的相关信息量
tpr=tpr / total_target;  
% 代价=搜索到的不相关信息量/系统中的不相关信息量
fpr = fpr / total_neg;    

if (plot_pr || plot_roc)
 % draw
 if (~hold_fig)
 figure
 if (plot_pr)
 if (plot_roc)
 subplot(1,2,1);
 end

 if (plot_baseline)
 target_ratio = total_target / (total_target + total_neg);  
 plot([0 1], [target_ratio target_ratio], 'k');
 end

 hold on
 hold all

 plot([0; tpr], [1 ; prec], style); % add pseudo point to complete curve
 xlabel('recall');
 ylabel('precision');
 title('precision-recall graph');
 end
 if (plot_roc)
 if (plot_pr)
 subplot(1,2,2);
 end

 if (plot_baseline)
 plot([0 1], [0 1], 'k');
 end

 hold on;
 hold all;

 plot([0; fpr], [0; tpr], style); % add pseudo point to complete curve

 xlabel('false positive rate');
 ylabel('true positive rate');
 title('roc curve');
 %axis([0 1 0 1]);
 if (plot_roc && plot_pr)
 % double the width
 rect = get(gcf,'pos');
 rect(3) = 2 * rect(3);
 set(gcf,'pos',rect);
 end
 end

 else
 if (plot_pr)
 if (plot_roc)
 subplot(1,2,1);
 end
 plot([0; tpr],[1 ; prec], style); % add pseudo point to complete curve
 end

 if (plot_roc)
 if (plot_pr)
 subplot(1,2,2);
 end
 plot([0; fpr], [0; tpr], style);
 end
 end
end

创建具体实例：

clear;
clc;
x1 = rand(1000, 1);
y1 = round(x1 + 0.5*(rand(1000,1) - 0.5));  %取整
prec_rec(x1, y1);  % 向量y1代表系统中的信息量，y1的值只能为0或1，y1取值为1代表相关信息量，否则为不相关信息量
% x2 = rand(1000,1);
% y2 = round(x2 + 0.75 * (rand(1000,1)-0.5));
% prec_rec(x2, y2, 'holdFigure', 1);
% legend('baseline','x1/y1','x2/y2','Location','SouthEast');