决策树模型(ID3/C4.5/CART)原理和底层代码解读学习笔记

男娘i 2022-10-02 04:54 8阅读 0赞

本文作者：合肥工业大学 管理学院 钱洋 email：1563178220@qq.com 内容可能有不到之处，欢迎交流。  
**未经本人允许禁止转载**  
如下为个人的学习笔记，使用latex编写，再写成文本太麻烦，这里直接截图。

# 个人笔记内容 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 1]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 2]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 3]  
![在这里插入图片描述][2019061710263895.png_pic_center]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 4]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 5]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 6]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 7]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 8]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 9]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 10]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 11]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 12]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 13]

# ID3代码Java解读 #

## 数据组织 ##

**本代码来自于**：[https://github.com/agangotia/ID3][https_github.com_agangotia_ID3]  
在使用编程语言实现算法的过程中，第一步便是要对数据的输入进行组织。如下为所使用的数据格式。其中，训练数据的格式为：  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 14]  
在训练数据集中共有6个属性，最后1列为决策变量。根据数据的格式，读取数据，其核心代码如下：

//读取数据集
    	public void prepareMatrix(String FileNameToRead,
    			int PercentageOfDataToLEarnFrom) { 
    		BufferedReader br = null;
    		try { 
    			br = new BufferedReader(new FileReader(FileNameToRead));
    			int NumberOfColoumns = 0;
    			// Reading Header 读取第一行
    			{ 
    				String line = br.readLine();
    				StringTokenizer st = new StringTokenizer(line);
    
    				boolean notNum = true;
    				while (st.hasMoreElements()) { 
    					if (notNum) {  //为true,添加属性
    						Headers.add((String) st.nextElement());
    						notNum = false;
    					} else { 
    						st.nextElement(); //跳过数组
    						notNum = true;
    					}
    
    				}
    
    			}
    			//最后添加一个类别----此时头共有7个
    			Headers.add("Class");
    			/* * System.out.println("Lets Check Headers"); for(String temp:head){ * System.out.println("\t "+temp); } */
    			NumberOfColoumns = Headers.size();
    			coloumns = NumberOfColoumns;
    			{ // lets read coloumns
    				String line = br.readLine();
    				while (line != null) { 
    					// System.out.print(line);
    					StringTokenizer st = new StringTokenizer(line);
    					// System.out.println("---- Split by space ------");
    					int[] tempCol = new int[NumberOfColoumns];
    					int tempIndex = 0;
    					while (st.hasMoreElements()) { 
    						tempCol[tempIndex++] = Integer.parseInt((String) st
    								.nextElement());
    						// System.out.println(tempCol[tempIndex-1]);
    					}
    					rows.add(tempCol);
    					line = br.readLine();
    				}
    			}
    
    			// Now the truncating part,
    			int rowsAfterTrunc = (int) ((PercentageOfDataToLEarnFrom * (rows
    					.size())) / 100);
    			if (rowsAfterTrunc == rows.size()) { 
    				// do nothing
    			} else { 
    				for (int i = rows.size() - 1; i > rowsAfterTrunc; i--) { 
    					rows.remove(i);
    				}
    			}
    
    			Numrows = rows.size();
    
    		} catch (FileNotFoundException e) { 
    			// TODO Auto-generated catch block
    			e.printStackTrace();
    		} catch (IOException e) { 
    			// TODO Auto-generated catch block
    			e.printStackTrace();
    		} finally { 
    			try { 
    				br.close();
    			} catch (IOException e) { 
    				// TODO Auto-generated catch block
    				e.printStackTrace();
    			}
    		}
    		//数据封装成数组
    		for (int i = 0; i < rows.size(); i++) { 
    			int[] arr = rows.get(i);
    			for (int j = 0; j < arr.length; j++) { 
    				System.out.print(arr[j]);
    				System.out.print("\t");
    			}
    			System.out.println("");
    		}
    		
    	}

从这段代码中，**可以看出怎么将按数字分割的属性和类别添加到Headers集合中**。**另外，ID3算法，需要计算的是每个属性对应的信息增益，因此需要按照列进行组织数据**。这里使用了集合的方式：

ArrayList<int[]> rows;//rows in matrix

**添加的数据既包含每一列属性也包含最后一列的决策变量**。

## 算法核心代码 ##

/** * */
    package com.qian.id3;
    
    import java.util.ArrayList;
    import java.util.HashMap;
    import java.util.Map;
    /** * @author Anupam Gangotia * Profile::http://en.gravatar.com/gangotia * github::https://github.com/agangotia */
    /** * This class the learner class, which has the function * to learn from the train values, and store it in a Decision tree * */
    public class ID3Learner { 
    	String FileNameToRead;// filename to read the training data set
    	int PercentageOfDataToLEarnFrom;// this is the percent , which shows how
    
    	// many line to be read from input
    
    	// Paramaterized constructor
    	public ID3Learner(String FileName, int percs) { 
    		FileNameToRead = FileName;
    		PercentageOfDataToLEarnFrom = percs;
    	}
    
    	/* * Function :: startLearning, This function, reads the file values into a * MxN Matrix datatype, The data from Matrix is further split across a set * of Training Vectors, and a FinalClass Vector. As our ID3 algortihm takes * set of Training vectors & final class vector as inputs. This * function,internally calls the learnTree Function, which is an * implementation of ID3 algorithm */
    	public TreeNode startLearning() { 
    
    		if (FileNameToRead == null) { 
    			System.out.println("---- Error ------");
    			System.out.println("---- Please Specify test data set ------");
    		}
    
    		if (PercentageOfDataToLEarnFrom < 0) { 
    			System.out.println("---- Error ------");
    			System.out.println("---- Please Specify %correctly ------");
    		}
    
    		MatrixData matrix = new MatrixData();// Prepares a new matrix datatype
    		matrix.prepareMatrix(FileNameToRead, PercentageOfDataToLEarnFrom);// reads
    
    		// 训练数据，不包含决策变量
    		HashMap<String, int[]> setTrainingVector = new HashMap<String, int[]>();
    		// Now i need a set of R training vectors 矩阵的列数 这里只循环了自变量
    		for (int i = 0; i < matrix.coloumns - 1; i++) { // Training Vectors being
    			//按照列存数据
    			int[] trainingVector = new int[matrix.Numrows];
    			matrix.fillArray(trainingVector, i);
    			setTrainingVector.put(matrix.Headers.get(i), trainingVector);
    		}
    
    		// 决策变量
    		int[] FinalClass = new int[matrix.Numrows];
    		matrix.fillArray(FinalClass, matrix.coloumns - 1);// final class vector
    		//初始化TreeNode
    		TreeNode rootNode = new TreeNode();
    		rootNode.setAtrvalue(-1);// since its a root node 种子节点
    
    		// Calling the ID3 implementation algorithm 自变量 因变量 生成树 数据
    		learnTree(setTrainingVector, FinalClass, rootNode, matrix);
    		return rootNode;
    	}
    
    	/* * Function :: startLearning, Recursive Function. AN exact copy of ID3 * algorithm(http://en.wikipedia.org/wiki/ID3_algorithm) This function * generates a decision tree recursively. Parameters: 1.A Hashmap containing * Training Vectors :: HashMap<String,int[]> setTrainingVector 2.A VEctor of * Final class :: int[] FinalClass 3.THe decision tree NOde::TreeNode node * 4.The MAtrix datatype, which is used in constructing vectors of train * data::MatrixData matrix */
    	public void learnTree(HashMap<String, int[]> setTrainingVector,
    			int[] FinalClass, TreeNode node, MatrixData matrix) { 
    
    		// 判断所有的数据对应的类别是否为单一类别
    		if (checkFinalClass(FinalClass, 0)) { // If all examples are 0, Return
    			// the single-node tree Root,
    			// with label = 0.
    			node.fClass = 0;
    			return;
    		} else if (checkFinalClass(FinalClass, 1)) { // If all examples are 1,
    			node.fClass = 1;
    			return;
    		}
    		// 如果只有一个属性的情况
    		if (setTrainingVector.entrySet().size() == 1) { 
    			int cPos = getCountPositives(FinalClass);
    			int cNeg = FinalClass.length - cPos;
    			if (cPos >= cNeg) { 
    				node.fClass = 0;
    				return;
    			} else { 
    				node.fClass = 1;
    				return;
    			}
    		} else { 
    
    			/*使用信息增益选择属性*/
    			HashMap<String, Double> attributesGains = new HashMap<String, Double>(); // 存储每个特征对应的信息增益
    			HashMap<String, ArrayList<Integer>> mapAttributesValuesInListUnique = new HashMap<String, ArrayList<Integer>>();// The
    			//计算样本数据的熵
    			double entropyS = getEntropy(FinalClass);// initial entropy
    			//setTrainingVector为自变量，每个属性对应的一列值，通过一个一维数组存储
    			for (Map.Entry entry : setTrainingVector.entrySet()) { 
    				//某属性不同值对应的正类
    				HashMap<Integer, Integer> atrPositive = new HashMap<Integer, Integer>();
    				//某属性不同值对应的负类
    				HashMap<Integer, Integer> atrNegative = new HashMap<Integer, Integer>();
    				ArrayList<Integer> atrUnique = new ArrayList<Integer>();
    				//获取所有的样本的训练集
    				int[] trainingClass = (int[]) entry.getValue();
    				for (int i = 0; i < trainingClass.length; i++) { // NOw finding
    					// individual
    					// entropies
    					addOnlyUnique(atrUnique, trainingClass[i]);
    					if (FinalClass[i] == 0)// its a positive
    					{   //将某一属性下，不同值对应的正类和负类的数量统计出来
    						if (atrPositive.containsKey(trainingClass[i])) { 
    							atrPositive.put(trainingClass[i],
    									atrPositive.get(trainingClass[i]) + 1);
    						} else { 
    							atrPositive.put(trainingClass[i], 1);
    						}
    					} else { // FinalClass is negative
    						if (atrNegative.containsKey(trainingClass[i])) { 
    							atrNegative.put(trainingClass[i],
    									atrNegative.get(trainingClass[i]) + 1);
    						} else { 
    							atrNegative.put(trainingClass[i], 1);
    						}
    
    					}
    				}
    
    				mapAttributesValuesInListUnique.put((String) entry.getKey(),
    						atrUnique);
    				// 针对每个属性，计算熵
    				{ 
    					double gain = entropyS;
    					//每个属性对应的不重复值
    					for (int tempAttr : atrUnique) { 
    						double entropyTemp = 0.0;
    						int positives = 0;
    						int negatives = 0;
    						//获取不重复值对应的正类
    						if (atrPositive.get(tempAttr) != null)
    							positives = atrPositive.get(tempAttr);
    						//获取不重复值对应的负类
    						if (atrNegative.get(tempAttr) != null)
    							negatives = atrNegative.get(tempAttr);
    						double val1 = (double) (positives)
    								/ (positives + negatives);
    						double val2 = (double) (negatives)
    								/ (positives + negatives);
    						//基于公式计算信息熵
    						entropyTemp = -(val1 * log2(val1))
    								- (val2 * log2(val2));
    						//累计计算信息增益值
    						gain = gain
    								- ((((double) positives + negatives) / trainingClass.length) * entropyTemp);
    					}
    					//封装该属性的信息增益值
    					attributesGains.put((String) entry.getKey(), gain);
    				}
    
    			}// loop ends
    
    			/**使用Map排序算法，这里是作者写的， *也可以直接调用 Collections.sort()进行排序 * *找出使得信息增益最大的属性 **/
    			String attributeWithMAxGain = "";
    			double maxGainValue = 0.0;
    			int indexToChoose = 0;
    			for (Map.Entry entry : setTrainingVector.entrySet()) { 
    				double tempGain = attributesGains.get((String) entry.getKey());
    				if (indexToChoose == 0) { 
    					maxGainValue = tempGain;
    					attributeWithMAxGain = (String) entry.getKey();
    					indexToChoose++;
    				}
    				if (tempGain > maxGainValue) { 
    					maxGainValue = tempGain;
    					attributeWithMAxGain = (String) entry.getKey();
    				}
    			}// loop ends
    			//节点添加
    			node.setAttributeName(attributeWithMAxGain);
    			node.setfClass(-1);
    			node.setGain(maxGainValue);
    
    			//下面，开始采用递归的方式向下计算
    			ArrayList<Integer> atrUniqueValuesForAttrMaxGain = mapAttributesValuesInListUnique
    					.get(attributeWithMAxGain);
    			
    			for (int tempAtrUniqueValue : atrUniqueValuesForAttrMaxGain) { 
    				
    				TreeNode NodeChild = new TreeNode();
    				NodeChild.setAtrvalue(tempAtrUniqueValue);// since its a child
    				// node
    				node.getBranches().add(NodeChild);
    				MatrixData matrixChild = matrix.splitMatrix(
    						attributeWithMAxGain, tempAtrUniqueValue);
    				// matrixChild.printMatrix();
    				// calling the algorithm
    				HashMap<String, int[]> setTrainingVectorChild = new HashMap<String, int[]>();
    				// Now i need a set of R training vectors
    				for (int i = 0; i < matrixChild.coloumns - 1; i++) { 
    					int[] trainingVectorChild = new int[matrixChild.Numrows];
    					matrixChild.fillArray(trainingVectorChild, i);
    					setTrainingVectorChild.put(matrixChild.Headers.get(i),
    							trainingVectorChild);
    				}
    				// i need final class vector
    				int[] FinalClassChild = new int[matrixChild.Numrows];
    				matrixChild
    				.fillArray(FinalClassChild, matrixChild.coloumns - 1);
    				learnTree(setTrainingVectorChild, FinalClassChild, NodeChild,
    						matrixChild);
    
    			}
    
    			return;
    
    		}
    
    	}
    
    	// Function:checkFinalClass
    	// Returns True or False
    	// If all the attributes in final class equals valueToChecked returns True
    	public boolean checkFinalClass(int[] FinalClass, int valueToChecked) { 
    		for (int i = 0; i < FinalClass.length; i++) { 
    			if (FinalClass[i] != valueToChecked)
    				return false;
    		}
    		return true;
    	}
    
    	// Function:getCountPositives
    	// Returns the count of positives in final class
    	public int getCountPositives(int[] FinalClass) { 
    		int countPos = 0;
    		for (int i = 0; i < FinalClass.length; i++) { 
    			if (FinalClass[i] == 0)
    				countPos++;
    		}
    		return countPos;
    	}
    
    	// 计算样本数据对应的信息熵
    	public double getEntropy(int[] vector) { 
    		double entropy = 0.0;
    		int positives = 0;
    		int negatives = 0;
    		for (int i = 0; i < vector.length; i++) { 
    			if (vector[i] == 0)// its a positive
    			{ 
    				positives++;
    			} else { // FinalClass is negative
    				negatives++;
    			}
    		}
    		double val1 = (double) (positives) / (positives + negatives);
    		double val2 = (double) (negatives) / (positives + negatives);
    		entropy = -(val1 * log2(val1)) - (val2 * log2(val2));
    		return entropy;
    	}
    
    	// Function:log2
    	// Returns log base 2
    	public static double log2(double num) { 
    		if (num <= 0)
    			return 0.0;
    		return (Math.log(num) / Math.log(2));
    	}
    
    	// Function:addOnlyUnique
    	// Adds a value to the arraylist only if does not exists in the list
    	public void addOnlyUnique(ArrayList<Integer> data, int val) { 
    		if (!data.contains(val))
    			data.add(val);
    	}
    
    }

在上面的代码中，给出了完整的注释。其中，读者需要注意的时，如何使用递归的方法选择树的属性。

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center]: /images/20220113/78401331ea2a420f9f774ecac312eacc.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 1]: /images/20220113/aee81e45622147a482a2040edf39f0cc.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 2]: /images/20220113/99891d90110d428fb24c305795611010.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 3]: /images/20220113/171fd95b0cc3407899b7ac6afbce3731.png
[2019061710263895.png_pic_center]: /images/20220113/de15adb6054645fd98c80a94f8758cf9.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 4]: /images/20220113/fb86fcb6ca1249ff9a84f3dcc6c527aa.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 5]: /images/20220113/a8bc40f1011e4872b532a4e3c6ce20dc.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 6]: /images/20220113/a10b2a21f2224202bb06c0794f5cd19e.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 7]: /images/20220113/c6c5385f7b9645eaae97bd300b12fc61.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 8]: /images/20220113/43d07f8712fd42bb88bdc35045762b02.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 9]: /images/20220113/e1de4f27bf4544f59585463b5a92d6bf.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 10]: /images/20220113/1e7bc500909f4c138a8d2c335e35d516.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 11]: /images/20220113/4e007280ffb14a158bb3364e6d6af75e.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 12]: /images/20220113/f86fb02fd1474f73b17662b09bed167f.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 13]: /images/20220113/14ed14400f994b1595a0618bf5260deb.png
[https_github.com_agangotia_ID3]: https://github.com/agangotia/ID3
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9xaWFueWFuZy1oZnV0LmJsb2cuY3Nkbi5uZXQ_size_16_color_FFFFFF_t_70_pic_center 14]: /images/20220113/3b506468576e477981f34456c93aa980.png