寻找简单阈值分类器的多类阈值
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【中文标题】寻找简单阈值分类器的多类阈值【英文标题】：Finding mutliclass thresholds for simple threshold classifier 【发布时间】：2016-06-16 22:59:59 【问题描述】：我有一个函数，它返回一个实例的数值，我稍后使用这个数值将实例分类为三个类别之一。类别是相对可分离的，见下图（三种颜色代表三个不同的类别）。
所以在这里我想要两个阈值，k1 和 k2，这样k1 左边的所有东西都被分类为红色，k2 右边的所有东西都被分类为蓝色，中间的所有东西都被分类为绿色.
我从基于this 解决方案的 Kadane 算法的修改版本开始。我首先按值对所有（颜色，值）元组进行排序，然后生成一个数组，其中所有绿色分类的值都为 1，非绿色为 -1。所以我会得到一个看起来像这样的数组：
[-1, -1, -1, -1, 1, -1, -1, ..., 1, 1, 1, -1, 1, ..., -1, -1, -1, -1]
也就是说，最初有很多 -1（红色），中间有很多绿色，最后大部分是蓝色。现在，通过运行 Kadane 算法，我会得到最优分割吗？
这是我测试的代码：
import java.util.*;
public class Kadanes
private static Color[] correctClasses = new Color[]Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.RED, Color.RED, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.RED, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.RED, Color.BLUE, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.RED, Color.BLUE, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.RED, Color.GREEN, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.BLUE, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.RED, Color.BLUE, Color.GREEN, Color.RED, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.BLUE, Color.RED, Color.GREEN, Color.RED, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.RED, Color.GREEN, Color.RED, Color.BLUE, Color.GREEN, Color.GREEN, Color.BLUE, Color.RED, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.RED, Color.RED, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.RED, Color.RED, Color.BLUE, Color.RED, Color.GREEN, Color.RED, Color.RED, Color.RED, Color.RED, Color.RED, Color.GREEN, Color.BLUE, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.RED, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.RED, Color.RED, Color.GREEN, Color.RED, Color.GREEN, Color.BLUE, Color.RED, Color.RED, Color.GREEN, Color.BLUE, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.RED, Color.RED, Color.RED, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.BLUE, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.BLUE, Color.RED, Color.GREEN, Color.BLUE, Color.GREEN, Color.RED, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.RED, Color.GREEN, Color.BLUE, Color.RED, Color.GREEN, Color.BLUE, Color.BLUE, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.RED, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.BLUE, Color.GREEN, Color.RED, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.RED, Color.GREEN, Color.GREEN, Color.RED, Color.BLUE, Color.BLUE, Color.BLUE, Color.RED, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.BLUE, Color.GREEN, Color.BLUE, Color.RED, Color.BLUE, Color.BLUE, Color.RED, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.GREEN, Color.BLUE, Color.GREEN, Color.BLUE, Color.BLUE, Color.BLUE, Color.RED, Color.BLUE, Color.BLUE, Color.GREEN, Color.GREEN, Color.RED, Color.GREEN, Color.GREEN, Color.BLUE, Color.BLUE, Color.BLUE, Color.BLUE, Color.RED, Color.BLUE, Color.BLUE, Color.BLUE, Color.BLUE, Color.RED, Color.BLUE, Color.RED, Color.BLUE, Color.BLUE, Color.BLUE, Color.BLUE, Color.GREEN, Color.BLUE, Color.BLUE, Color.BLUE, Color.BLUE, Color.RED, Color.RED, Color.RED, Color.BLUE, Color.BLUE, Color.BLUE, Color.BLUE, Color.GREEN;
private static double[] predictedValues = new double[]0.0, 0.34, 2.0, 2.67, 7.53, -0.04, 2.0, -3.55, 3.78, 0.33, 3.0, -0.21, 1.41, -0.37, 0.84, 3.94, 8.34, 0.0, -1.39, 3.0, -1.63, 0.0, 3.0, 1.26, 0.0, 0.0, 0.0, 0.0, 0.61, 0.0, 3.34, 0.57, -1.05, 0.63, 0.0, 0.71, 0.0, 2.34, -0.41, -1.77, 3.0, 0.62, 0.93, 1.55, 2.0, 8.0, -1.55, 5.75, 0.0, 0.0, -0.25, 0.0, 1.0, 10.51, 0.0, 0.47, 0.78, -1.08, -1.51, 1.0, 1.0, 0.0, 4.33, -0.6, 0.37, 6.0, 1.16, -4.07, 2.0, 0.91, -0.05, 1.78, 0.0, 0.0, 0.0, 0.0, 0.0, 1.64, 1.55, 4.44, 2.78, 1.47, 3.75, 0.0, 7.59, 0.0, 0.94, 2.46, -0.23, -0.2, 0.0, 0.39, -2.31, 3.0, -1.15, 2.0, -0.76, -1.33, 0.0, 0.61, 0.77, -1.77, -1.08, 0.0, -3.2, 3.46, 1.0, 0.0, 0.0, 3.33, 0.0, 0.0, 2.81, 0.0, 0.0, 0.0, 3.0, 0.0, -0.88, 1.65, -1.09, -0.35, 0.0, 0.0, 5.0, 0.0, 2.88, -0.72, 0.87, 7.0, 7.48, -1.98, 1.0, 1.11, 4.0, 1.53, 0.0, 8.07, 1.54, 4.23, 0.0, -0.73, 6.61, 0.07, 0.0, -4.32, -1.77, 2.05, -1.08, 4.3, 1.61, 2.96, 3.0, 0.0, 3.66, 0.0, 0.0, 0.05, -0.77, -1.0, 0.0, 5.43, 2.12, -1.55, 2.3, 0.0, 3.6, 0.0, 0.0, -10.21, 2.0, 0.55, -0.63, 0.0, 1.0, 0.0, 0.0, 1.28, 3.0, 0.0, 0.44, 1.27, 2.12, 2.17, 1.76, -1.9, 5.42, 1.0, 3.76, -3.55, -0.82, 0.0, 0.11, -1.7, -0.33, 0.0, 0.0, -2.01, 0.0, 3.52, 2.0, 6.0, 0.92, 7.22, 0.0, 0.0, 0.0, 0.0, 0.36, -1.77, 0.0, -3.32, -0.91, 2.69, -0.86, -0.27, 3.28, -1.02, 0.41, -0.6, 2.61, 0.0, 0.36, 0.0, 0.91, 0.0, -2.82, 0.0, -1.77, 0.0, -0.33, 3.94, -2.55, 8.0, 3.29, 2.7, -4.4, 9.0, 0.0, 2.81, -0.23, -2.51, 2.0, -0.19, 0.0, 0.0, 0.0, 0.8, 8.33, 0.0, 0.59, 0.0, 0.41, 0.0, 0.8, 1.7, 3.27, 0.0, 0.34, -1.83, 0.0, -1.0, 0.29, 3.71, -0.44, -0.59, 1.25, 2.3, -1.56, 0.0, 6.21, -0.68, 0.0, 0.0, -0.3, 0.0, 1.0, 0.86, 0.0, 0.0, 0.0, 0.0, 0.41, 1.91, -0.17, -0.77, 1.0, 3.0, 2.0, 3.0, -0.71, 0.0, 0.62, 0.0, 2.54, 1.14, 0.0, 0.0, 3.27, 0.0, 0.96, -0.33, 0.0, 0.0, 1.91, -0.2, 0.0, 0.0, 0.6, 0.0, -0.82, 1.0, -0.54, 6.52, -2.48, 2.0, 0.0, 0.0, 1.61, 0.0, 0.0, 0.0, -0.17, 0.0, 1.0, -5.36, 2.73, 0.0, 7.97, 3.67, 0.0, -0.88, 0.93, 0.0, 3.0, -1.03, -0.64, 2.78, 0.0, 1.0, 3.0, 0.0, 0.46, 0.0, -0.63, 0.0, 4.0, 4.0, 1.61, 0.0, 0.0, 1.07, 0.0, 1.0, 18.39, -1.82, 0.0, 0.86, -0.42, -1.77, -0.61, 0.0, 0.68, -3.13, 0.53, 0.0, 3.0, 0.0, 2.47, 0.0, -1.74, 5.31, 0.0, 0.3, 0.0, 0.0, 4.0, 1.0, 0.64, 1.0, 0.0, -1.77, 3.31, -1.77, -0.43, -3.55, 0.94, 8.59, 0.0, 1.81, 3.69, -1.77, -0.32, 0.0, 3.0, 1.93, -1.47, 1.0, 3.21, 0.0, 0.0, 0.0, 0.33, 0.0, 0.0, -0.39, 0.0, 1.0, 0.0, 1.98, 0.0, 0.0, 7.45, 0.72, 0.34, 0.0, 0.35, 0.0, -2.74, 0.28, 4.0, 3.0, -0.91, -4.43, 0.0, 2.28, 3.0, -2.5, -2.66, 2.0, -0.66, 3.0, 11.06, 1.43, 3.0, 0.0, -0.79, 6.3, 0.94, 3.92, -4.43, 5.14, -2.35, 8.83, 1.04, 2.6, 5.0, 3.72;
private static List<Tuple> previousResults = new ArrayList<>();
static
for(int i=0; i<correctClasses.length; i++)
previousResults.add(new Tuple(correctClasses[i], predictedValues[i]));
public static void main(String[] args)
double[] exampleThresholds = new double[]-1.65, 1.65;
double[] thresholds = getThreshold();
System.out.println(Arrays.toString(thresholds));
System.out.println("Example threshold accuracy: " + getAccuracy(exampleThresholds));
System.out.println("Optimal threshold accuracy: " + getAccuracy(thresholds));
private static double[] getThreshold()
Collections.sort(previousResults, Collections.reverseOrder());
int max_so_far = 0;
int max_ending_here = 0;
int max_start_index = 0;
int startIndex = 0;
int max_end_index = -1;
for(int i = 0; i < previousResults.size(); i++)
int currentElementScore = (previousResults.get(i).correct == Color.GREEN ? 1 : -1);
if(max_ending_here + currentElementScore < 0)
startIndex = i+1;
max_ending_here = 0;
else
max_ending_here += currentElementScore;
if(max_ending_here > max_so_far)
max_so_far = max_ending_here;
max_start_index = startIndex;
max_end_index = i;
double lowThreshold = getAvgValue(max_start_index-1, max_start_index);
double highThreshold = getAvgValue(max_end_index, max_end_index+1);
return new double[]lowThreshold, highThreshold;
private static double getAccuracy(double[] thresholds)
int numCorrectlyClassified = 0;
for(int i=0; i<correctClasses.length; i++)
Color predictedClassification = classify(predictedValues[i], thresholds[0], thresholds[1]);
if(predictedClassification == correctClasses[i])
numCorrectlyClassified++;
return (double) numCorrectlyClassified / correctClasses.length;
private static Color classify(double value, double lowThresh, double highThresh)
if(value < lowThresh) return Color.RED;
if(value > highThresh) return Color.BLUE;
return Color.GREEN;
private static double getAvgValue(int index1, int index2)
if(index1 < 0)
return Double.NEGATIVE_INFINITY;
else if (index2 >= previousResults.size())
return Double.POSITIVE_INFINITY;
return (previousResults.get(index1).predicted + previousResults.get(index2).predicted) / 2;
static class Tuple implements Comparable<Tuple>
private Color correct;
private double predicted;
Tuple(Color correct, double predicted)
this.correct = correct;
this.predicted = predicted;
public String toString()
return "[" + correct.name() + ", " + predicted + "]";
@Override
public int compareTo(Tuple o)
double diff = o.predicted - predicted;
return diff != 0 ? (int) Math.signum(diff) : correct.compareTo(o.correct);
enum Color
BLUE, GREEN, RED
我得到的输出是：
[0.0, 0.0]
Example threshold accuracy: 0.5602678571428571
Optimal threshold accuracy: 0.49107142857142855
所以它找到的最佳阈值刚好在 0.0 范围内，我只输入了一个性能更好的快速示例阈值。是实现错误还是无法使用 Kadane 的算法来解决这个简单的问题，如果不是，我可以使用哪种算法？
【问题讨论】：
我认为您需要定义“最优”的含义。 @GordonLinoff 最大精度 【参考方案1】：你不能使用 Kadane 算法来解决这个问题，因为它优化了果岭的数量。 假设你有类似的东西：
1, 1, -1, 1, -1, 1,.., -1, 1, 1, 1
该算法希望最大化总和并取第一个到一个和最后一个 3，因为中间部分的总和仅为 -1。
因此，我将使用 N logN 算法来查找阈值。 首先对数组进行排序。 预计算绿色、红色和蓝色的数量介于 0 和 x 之间的部分计数数组（对于所有 x）。 使用第一个阈值遍历已排序的数组，并使用准确度作为指导指标，对第二个阈值进行二分搜索。要计算准确性，您可以使用预先计算的部分计数。您需要查看指标增加的方式。
可能有一些极端情况。如果您负担得起 N^2，只需尝试所有阈值并使用预先计算的数组来加快评估速度。
【讨论】：
我认为 Kadane 的算法会起作用的原因是因为范围有些可分离，红色和蓝色之间几乎没有重叠，所以优化绿色的数量不会也优化其他的吗？我不太明白你的第二段，Kadane选择了一个范围，所以即使中间部分的总和为0，但开始部分和结束部分> 0，所以它会包括整个范围？ @Limon 是的，我的意思是因为中间部分是 0，所以解决方案将包括整个范围，这可能不是您想要的。以上是关于寻找简单阈值分类器的多类阈值的主要内容，如果未能解决你的问题，请参考以下文章