本系列文章的第 1 部分结尾处提到了简单线性回归（Simple Linear Regression）类中缺少的三个元素。在本文中，作者 Paul Meagher 用基于 PHP 的概率函数弥补了这些缺陷，演示了如何将输出方法集成到 SimpleLinearRegression 类中并创建了图形输出。他通过构建数据研究工具解决了这些问题，该工具旨在深层次地研究中小规模的数据集所包含的信息。（在第 1 部分中，作者演示了如何用 PHP 作为实现语言来开发和实现简单线性回归算法包的核心部分。）

var PiD2 = pi() / 2 

<?php 

 // Distribution.php 

 // Copyright John Pezullo 
 // Released under same terms as PHP. 
 // PHP Port and OO'fying by Paul Meagher 

 class Distribution { 

  function doCommonMath($q, $i, $j, $b) { 
       
   $zz = 1;  
   $z  = $zz;  
   $k  = $i;  
       
       
   while($k <= $j) {  
        $zz = $zz * $q * $k / ($k - $b);  
        $z  = $z + $zz;  
        $k  = $k + 2;  
   } 
   return $z; 
  } 
       
  function getStudentT($t, $df) {   

   $t  = abs($t);  
   $w  = $t  / sqrt($df);  
   $th = atan($w); 
       
   if ($df == 1) {  
    return 1 - $th / (pi() / 2);  
   } 
     
   $sth = sin($th);  
   $cth = cos($th); 
     
   if( ($df % 2) ==1 ) {  
    return 
      1 - ($th + $sth * $cth * $this->doCommonMath($cth * $cth, 2, $df - 3, -1)) 
                         / (pi()/2); 
   } else { 
    return 1 - $sth * $this->doCommonMath($cth * $cth, 1, $df - 3, -1);  
   } 
     
  } 
     
  function getInverseStudentT($p, $df) {  
       
   $v =  0.5;  
   $dv = 0.5;  
   $t  = 0; 
       
   while($dv > 1e-6) {  
    $t = (1 / $v) - 1;  
    $dv = $dv / 2;  
    if ( $this->getStudentT($t, $df) > $p) {  
     $v = $v - $dv; 
    } else {  
     $v = $v + $dv; 
    }  
   } 
   return $t; 
  } 
     

  function getFisherF($f, $n1, $n2) { 
   // implemented but not shown     
  } 

  function getInverseFisherF($p, $n1, $n2) {  
   // implemented but not shown     
  } 

 } 
 ?> 

 

<?php 

  // HTML.php 

  // Copyright 2003, Paul Meagher 
  // Distributed under GPL   

  include_once "slr/SimpleLinearRegression.php"; 

  class SimpleLinearRegressionHTML extends SimpleLinearRegression { 

    function SimpleLinearRegressionHTML($X, $Y, $conf_int) { 
      SimpleLinearRegression::SimpleLinearRegression($X, $Y, $conf_int); 
    } 

    function showTableSummary($x_name, $y_name) { } 
       
    function showAnalysisOfVariance() { } 

    function showParameterEstimates() { } 

    function showFormula($x_name, $y_name) { } 

    function showRValues() {} 
  } 

  ?> 

无论是对于只有最少代码的“以快捷但不恰当方式获得的”图形，还是对于需要非常细粒度控制的复杂专业图形，JpGraph 都可以使它们的绘制变得简单。JpGraph 同样适用于科学和商业类型的图形。

<?php 

  // Snippet extracted from explore.php script 

  include ("jpgraph/jpgraph.php"); 
  include ("jpgraph/jpgraph_scatter.php"); 
  include ("jpgraph/jpgraph_line.php"); 

  // Create the graph 
  $graph = new Graph(300,200,'auto'); 
  $graph->SetScale("linlin"); 

  // Setup title   
  $graph->title->Set("$title"); 
  $graph->img->SetMargin(50,20,20,40);    
  $graph->xaxis->SetTitle("$x_name","center"); 
  $graph->yaxis->SetTitleMargin(30);      
  $graph->yaxis->title->Set("$y_name");  

  $graph->title->SetFont(FF_FONT1,FS_BOLD); 

  // make sure that the X-axis is always at the 
  // bottom at the plot and not just at Y=0 which is 
  // the default position   
  $graph->xaxis->SetPos('min'); 

  // Create the scatter plot with some nice colors 
  $sp1 = new ScatterPlot($slr->Y, $slr->X); 
  $sp1->mark->SetType(MARK_FILLEDCIRCLE); 
  $sp1->mark->SetFillColor("red"); 
  $sp1->SetColor("blue"); 
  $sp1->SetWeight(3); 
  $sp1->mark->SetWidth(4); 

  // Create the regression line 
  $lplot = new LinePlot($slr->PredictedY, $slr->X); 
  $lplot->SetWeight(2); 
  $lplot->SetColor('navy'); 

  // Add the pltos to the line 
  $graph->Add($sp1); 
  $graph->Add($lplot); 

  // ... and stroke 
  $graph_name = "temp/test.png"; 
  $graph->Stroke($graph_name); 
  ?> 
  <img src=><?php echo $graph_name ?>' vspace='15'> 

  ?>

<?php 

  // explore.php 

  if (!empty($x_values)) { 
    $X    = explode(",", $x_values); 
    $numX = count($X); 
  }   

  if (!empty($y_values)) { 
    $Y    = explode(",", $y_values); 
    $numY = count($Y); 
  }   

  // display entry data entry form if variables not set 

  if ( (empty($title)) OR (empty($x_name)) OR (empty($x_values)) OR  
       (empty($y_name)) OR (empty($conf_int)) OR (empty($y_values)) OR  
       ($numX != $numY) ) {          

    // Omitted code for displaying entry form 
     
  } else { 
     
    include_once "slr/SimpleLinearRegressionHTML.php";                       
    $slr = new SimpleLinearRegressionHTML($X, $Y, $conf_int);    

    echo "<h2>$title</h2>"; 
     
    $slr->showTableSummary($x_name, $y_name); 
    echo "<br><br>"; 
     
    $slr->showAnalysisOfVariance();   
    echo "<br><br>"; 

    $slr->showParameterEstimates($x_name, $y_name);  
    echo "<br>"; 

    $slr->showFormula($x_name, $y_name); 
    echo "<br><br>"; 

    $slr->showRValues($x_name, $y_name); 
    echo "<br>"; 

    include ("jpgraph/jpgraph.php"); 
    include ("jpgraph/jpgraph_scatter.php"); 
    include ("jpgraph/jpgraph_line.php");   
                       
    // The code for displaying the graphics is inline in the 
    // explore.php script.  The code for these two line plots 
    // finishes off the script: 
     
    // Omitted code for displaying scatter plus line plot 
    // Omitted code for displaying residuals plot 
     
  } 

  ?>

phpmath/
    
    burnout_study.php    
    explore.php
    fire_study.php  
    navbar.php   
      
    dist/
      Distribution.php  
      fisher.php
      student.php
      source.php
      
    jpgraph/  
      etc...
    
    slr/ 
      SimpleLinearRegression.php 
      SimpleLinearRegressionHTML.php
      
    temp/

 

使某个全局 PHP_MATH 变量有权访问非 Web 根目录位置，并且
确保在所有需要或包括的文件路径前面加上这个已定义的常量作为前缀。

*如果想从您的方程以及可以使用的其它计算公式中略去截距，则何时可以这样做
*何时以及如何使用幂、对数和其它转换来对数据进行线性化，以便用简单线性回归来对该数据建模
*可以用来评估您的建模假设的充分性，并可以更清晰地洞察数据中的模式的其它可视化方法

L. C. Hamilton（1992年）。Regression with Graphics。加州 Pacific Grove：Brooks/Cole Publishing Company。
J Neter、M.H. Kutner 和 W Wasserman W（1990 年）。Applied Linear Regression Models（第 3 版）。芝加哥 Irwin。
E. J. Pedhazur（1982 年）。Multiple regression in behavioral research。纽约州，纽约市：Holt，Rinehart and Winston。