特征归一化后 kNN 分类的准确率下降？

Question

我正在对一些数据进行kNN分类。我有按 80/20 的比例随机分割训练集和测试集的数据。我的数据如下所示：

[ [1.0, 1.52101, 13.64, 4.49, 1.1, 71.78, 0.06, 8.75, 0.0, 0.0, 1.0], 
  [2.0, 1.51761, 13.89, 3.6, 1.36, 72.73, 0.48, 7.83, 0.0, 0.0, 2.0],
  [3.0, 1.51618, 13.53, 3.55, 1.54, 72.99, 0.39, 7.78, 0.0, 0.0, 3.0],
  ...
]

矩阵最后一列中的项目是类：1.0、2.0 和 3.0特征标准化

后，我的数据如下所示：

[[-0.5036443480260487, -0.03450760227559746, 0.06723230162846759, 0.23028986544844693, -0.025324623254270005, 0.010553065215338569, 0.0015136367098358505, -0.11291235596166802, -0.05819669234942126, -0.12069793876044387, 1.0], 
[-0.4989050339943617, -0.11566537753097901, 0.010637426608816412, 0.2175704556290625, 0.03073267976659575, 0.05764598316498372, -0.012976783512350588, -0.11815839520204152, -0.05819669234942126, -0.12069793876044387, 2.0],
...
]

我用于标准化的公式：

(X - avg(X)) / (max(X) - min(X))

---

我对K = 1 到 25（仅限奇数）中的每个执行 kNN 分类100次。我记录了所使用的每个K的平均准确度。这是我的结果：

Average accuracy for K=1 after 100 tests with different data split: 98.91313003886198 %   
Average accuracy for K=3 after 100 tests with different data split: 98.11976006170633 %    
Average accuracy for K=5 after 100 tests with different data split: 97.71226079929019 %  
Average accuracy for K=7 after 100 tests with different data split: 97.47493145754373 %    
Average accuracy for K=9 after 100 tests with different data split: 97.16596220947888 %   
Average accuracy for K=11 after 100 tests with different data split: 96.81465365733266 %   
Average accuracy for K=13 after 100 tests with different data split: 95.78772655522567 %    
Average accuracy for K=15 after 100 tests with different data split: 95.23116406332706 %    
Average accuracy for K=17 after 100 tests with different data split: 94.52371789094929 %    
Average accuracy for K=19 after 100 tests with different data split: 93.85285871435981 %   
Average accuracy for K=21 after 100 tests with different data split: 93.26620809747965 %    
Average accuracy for K=23 after 100 tests with different data split: 92.58047022661833 %
Average accuracy for K=25 after 100 tests with different data split: 90.55746523509124 %

但是当我应用特征归一化时，准确率会显着下降。我的具有归一化特征的 kNN 结果：

Average accuracy for K=1 after 100 tests with different data split: 88.56128075154439 % 
Average accuracy for K=3 after 100 tests with different data split: 85.01466511662318 %    
Average accuracy for K=5 after 100 tests with different data split: 83.32096281613967 %    
Average accuracy for K=7 after 100 tests with different data split: 83.09434478900455 %   
Average accuracy for K=9 after 100 tests with different data split: 82.05628926919964 %  
Average accuracy for K=11 after 100 tests with different data split: 79.89732262550343 %   
Average accuracy for K=13 after 100 tests with different data split: 79.60617886853211 %    
Average accuracy for K=15 after 100 tests with different data split: 79.26511126374507 %    
Average accuracy for K=17 after 100 tests with different data split: 77.51457877706329 %   
Average accuracy for K=19 after 100 tests with different data split: 76.97848441605367 %    
Average accuracy for K=21 after 100 tests with different data split: 75.70005919265326 %    
Average accuracy for K=23 after 100 tests with different data split: 76.45758217099551 %   
Average accuracy for K=25 after 100 tests with different data split: 76.16619492431572 %

我的代码中的算法没有逻辑错误，我用简单的数据检查了它。

---

为什么特征归一化后kNN分类准确率下降这么多？我想归一化本身不会降低任何分类的准确率。那么使用特征归一化的目的是什么？

Answer 1

人们普遍误解归一化永远不会降低分类精度。完全可以。

如何 ？

连续的相对值也非常重要。事实上，它们确实确定了特征空间中点的放置。当您进行标准化时，它可能会严重抵消相对位置。这是可以感觉到的，尤其是在 k-NN 分类中，因为它直接针对点之间的距离进行操作。相比之下，它的效果在SVM中感觉不那么强烈，因为在这种情况下，优化过程仍然能够找到一个相当准确的超平面。

您还应该注意，在这里，您使用 avg(X) 进行标准化。因此，考虑特定行的相邻列中的两个点。如果第一个点远低于各自列的平均值，而第二个点远高于各自列的平均值，而在非归一化意义上，它们是非常接近的数值，则距离计算可能会有很大差异。

永远不要指望正常化会创造奇迹。