这种z-scores方法在峰值检测方面非常有效，也有助于异常值的去除。异常值对话经常讨论每个点的统计价值和变化数据的伦理。

但是，在来自易出错的串行通信或易出错的传感器的重复错误传感器值的情况下，错误或虚假读数中没有统计值。它们需要被识别并移除。

从视觉上看，错误是显而易见的。下图中的直线显示了需要删除的内容。但是用算法识别和消除错误是相当具有挑战性的。z分数效果很好。

下图是通过串行通信从传感器获得的值。偶尔的串行通信错误，传感器错误或两者都导致重复的，明显错误的数据点。

z-score峰值检测器能够在虚假数据点上发出信号，并生成一个干净的结果数据集，同时保留正确数据的特征:

这是一个Python实现的鲁棒峰值检测算法算法。

初始化和计算部分被分开，只有filtered_y数组被保留，它的最大大小等于延迟，因此内存没有增加。(结果与上述答案相同)。 为了绘制图形，还保留了标签数组。

我做了一个github要点。

import numpy as np
import pylab

def init(x, lag, threshold, influence):
    '''
    Smoothed z-score algorithm
    Implementation of algorithm from https://stackoverflow.com/a/22640362/6029703
    '''

    labels = np.zeros(lag)
    filtered_y = np.array(x[0:lag])
    avg_filter = np.zeros(lag)
    std_filter = np.zeros(lag)
    var_filter = np.zeros(lag)

    avg_filter[lag - 1] = np.mean(x[0:lag])
    std_filter[lag - 1] = np.std(x[0:lag])
    var_filter[lag - 1] = np.var(x[0:lag])

    return dict(avg=avg_filter[lag - 1], var=var_filter[lag - 1],
                std=std_filter[lag - 1], filtered_y=filtered_y,
                labels=labels)


def add(result, single_value, lag, threshold, influence):
    previous_avg = result['avg']
    previous_var = result['var']
    previous_std = result['std']
    filtered_y = result['filtered_y']
    labels = result['labels']

    if abs(single_value - previous_avg) > threshold * previous_std:
        if single_value > previous_avg:
            labels = np.append(labels, 1)
        else:
            labels = np.append(labels, -1)

        # calculate the new filtered element using the influence factor
        filtered_y = np.append(filtered_y, influence * single_value
                               + (1 - influence) * filtered_y[-1])
    else:
        labels = np.append(labels, 0)
        filtered_y = np.append(filtered_y, single_value)

    # update avg as sum of the previuos avg + the lag * (the new calculated item - calculated item at position (i - lag))
    current_avg_filter = previous_avg + 1. / lag * (filtered_y[-1]
            - filtered_y[len(filtered_y) - lag - 1])

    # update variance as the previuos element variance + 1 / lag * new recalculated item - the previous avg -
    current_var_filter = previous_var + 1. / lag * ((filtered_y[-1]
            - previous_avg) ** 2 - (filtered_y[len(filtered_y) - 1
            - lag] - previous_avg) ** 2 - (filtered_y[-1]
            - filtered_y[len(filtered_y) - 1 - lag]) ** 2 / lag)  # the recalculated element at pos (lag) - avg of the previuos - new recalculated element - recalculated element at lag pos ....

    # calculate standard deviation for current element as sqrt (current variance)
    current_std_filter = np.sqrt(current_var_filter)

    return dict(avg=current_avg_filter, var=current_var_filter,
                std=current_std_filter, filtered_y=filtered_y[1:],
                labels=labels)

lag = 30
threshold = 5
influence = 0

y = np.array([1,1,1.1,1,0.9,1,1,1.1,1,0.9,1,1.1,1,1,0.9,1,1,1.1,1,1,1,1,1.1,0.9,1,1.1,1,1,0.9,
       1,1.1,1,1,1.1,1,0.8,0.9,1,1.2,0.9,1,1,1.1,1.2,1,1.5,1,3,2,5,3,2,1,1,1,0.9,1,1,3,
       2.6,4,3,3.2,2,1,1,0.8,4,4,2,2.5,1,1,1])

# Run algo with settings from above
result = init(y[:lag], lag=lag, threshold=threshold, influence=influence)

i = open('quartz2', 'r')
for i in y[lag:]:
    result = add(result, i, lag, threshold, influence)

# Plot result
pylab.subplot(211)
pylab.plot(np.arange(1, len(y) + 1), y)
pylab.subplot(212)
pylab.step(np.arange(1, len(y) + 1), result['labels'], color='red',
           lw=2)
pylab.ylim(-1.5, 1.5)
pylab.show()

在信号处理中，峰值检测通常采用小波变换。基本上就是对时间序列数据进行离散小波变换。返回的细节系数中的过零将对应于时间序列信号中的峰值。你会在不同的细节系数水平上检测到不同的峰值振幅，这给了你多层次的分辨率。

使用实时流的Python版本(不会在每个新数据点到达时重新计算所有数据点)。您可能想要调整类函数返回的内容—对于我的目的，我只需要信号。

import numpy as np


class real_time_peak_detection():
    def __init__(self, array, lag, threshold, influence):
        self.y = list(array)
        self.length = len(self.y)
        self.lag = lag
        self.threshold = threshold
        self.influence = influence
        self.signals = [0] * len(self.y)
        self.filteredY = np.array(self.y).tolist()
        self.avgFilter = [0] * len(self.y)
        self.stdFilter = [0] * len(self.y)
        self.avgFilter[self.lag - 1] = np.mean(self.y[0:self.lag]).tolist()
        self.stdFilter[self.lag - 1] = np.std(self.y[0:self.lag]).tolist()

    def thresholding_algo(self, new_value):
        self.y.append(new_value)
        i = len(self.y) - 1
        self.length = len(self.y)
        if i < self.lag:
            return 0
        elif i == self.lag:
            self.signals = [0] * len(self.y)
            self.filteredY = np.array(self.y).tolist()
            self.avgFilter = [0] * len(self.y)
            self.stdFilter = [0] * len(self.y)
            self.avgFilter[self.lag] = np.mean(self.y[0:self.lag]).tolist()
            self.stdFilter[self.lag] = np.std(self.y[0:self.lag]).tolist()
            return 0

        self.signals += [0]
        self.filteredY += [0]
        self.avgFilter += [0]
        self.stdFilter += [0]

        if abs(self.y[i] - self.avgFilter[i - 1]) > (self.threshold * self.stdFilter[i - 1]):

            if self.y[i] > self.avgFilter[i - 1]:
                self.signals[i] = 1
            else:
                self.signals[i] = -1

            self.filteredY[i] = self.influence * self.y[i] + \
                (1 - self.influence) * self.filteredY[i - 1]
            self.avgFilter[i] = np.mean(self.filteredY[(i - self.lag):i])
            self.stdFilter[i] = np.std(self.filteredY[(i - self.lag):i])
        else:
            self.signals[i] = 0
            self.filteredY[i] = self.y[i]
            self.avgFilter[i] = np.mean(self.filteredY[(i - self.lag):i])
            self.stdFilter[i] = np.std(self.filteredY[(i - self.lag):i])

        return self.signals[i]

下面是一个基于Groovy回答的实际Java实现。(我知道已经发布了Groovy和Kotlin实现，但对于像我这样只做Java的人来说，弄清楚如何在其他语言和Java之间转换真的很麻烦)。

(结果与他人图表相匹配)

算法实现

import java.util.ArrayList;
import java.util.Collections;
import java.util.HashMap;
import java.util.List;

import org.apache.commons.math3.stat.descriptive.SummaryStatistics;

public class SignalDetector {

    public HashMap<String, List> analyzeDataForSignals(List<Double> data, int lag, Double threshold, Double influence) {

        // init stats instance
        SummaryStatistics stats = new SummaryStatistics();

        // the results (peaks, 1 or -1) of our algorithm
        List<Integer> signals = new ArrayList<Integer>(Collections.nCopies(data.size(), 0));

        // filter out the signals (peaks) from our original list (using influence arg)
        List<Double> filteredData = new ArrayList<Double>(data);

        // the current average of the rolling window
        List<Double> avgFilter = new ArrayList<Double>(Collections.nCopies(data.size(), 0.0d));

        // the current standard deviation of the rolling window
        List<Double> stdFilter = new ArrayList<Double>(Collections.nCopies(data.size(), 0.0d));

        // init avgFilter and stdFilter
        for (int i = 0; i < lag; i++) {
            stats.addValue(data.get(i));
        }
        avgFilter.set(lag - 1, stats.getMean());
        stdFilter.set(lag - 1, Math.sqrt(stats.getPopulationVariance())); // getStandardDeviation() uses sample variance
        stats.clear();

        // loop input starting at end of rolling window
        for (int i = lag; i < data.size(); i++) {

            // if the distance between the current value and average is enough standard deviations (threshold) away
            if (Math.abs((data.get(i) - avgFilter.get(i - 1))) > threshold * stdFilter.get(i - 1)) {

                // this is a signal (i.e. peak), determine if it is a positive or negative signal
                if (data.get(i) > avgFilter.get(i - 1)) {
                    signals.set(i, 1);
                } else {
                    signals.set(i, -1);
                }

                // filter this signal out using influence
                filteredData.set(i, (influence * data.get(i)) + ((1 - influence) * filteredData.get(i - 1)));
            } else {
                // ensure this signal remains a zero
                signals.set(i, 0);
                // ensure this value is not filtered
                filteredData.set(i, data.get(i));
            }

            // update rolling average and deviation
            for (int j = i - lag; j < i; j++) {
                stats.addValue(filteredData.get(j));
            }
            avgFilter.set(i, stats.getMean());
            stdFilter.set(i, Math.sqrt(stats.getPopulationVariance()));
            stats.clear();
        }

        HashMap<String, List> returnMap = new HashMap<String, List>();
        returnMap.put("signals", signals);
        returnMap.put("filteredData", filteredData);
        returnMap.put("avgFilter", avgFilter);
        returnMap.put("stdFilter", stdFilter);

        return returnMap;

    } // end
}

主要方法

import java.text.DecimalFormat;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.List;

public class Main {

    public static void main(String[] args) throws Exception {
        DecimalFormat df = new DecimalFormat("#0.000");

        ArrayList<Double> data = new ArrayList<Double>(Arrays.asList(1d, 1d, 1.1d, 1d, 0.9d, 1d, 1d, 1.1d, 1d, 0.9d, 1d,
                1.1d, 1d, 1d, 0.9d, 1d, 1d, 1.1d, 1d, 1d, 1d, 1d, 1.1d, 0.9d, 1d, 1.1d, 1d, 1d, 0.9d, 1d, 1.1d, 1d, 1d,
                1.1d, 1d, 0.8d, 0.9d, 1d, 1.2d, 0.9d, 1d, 1d, 1.1d, 1.2d, 1d, 1.5d, 1d, 3d, 2d, 5d, 3d, 2d, 1d, 1d, 1d,
                0.9d, 1d, 1d, 3d, 2.6d, 4d, 3d, 3.2d, 2d, 1d, 1d, 0.8d, 4d, 4d, 2d, 2.5d, 1d, 1d, 1d));

        SignalDetector signalDetector = new SignalDetector();
        int lag = 30;
        double threshold = 5;
        double influence = 0;

        HashMap<String, List> resultsMap = signalDetector.analyzeDataForSignals(data, lag, threshold, influence);
        // print algorithm params
        System.out.println("lag: " + lag + "\t\tthreshold: " + threshold + "\t\tinfluence: " + influence);

        System.out.println("Data size: " + data.size());
        System.out.println("Signals size: " + resultsMap.get("signals").size());

        // print data
        System.out.print("Data:\t\t");
        for (double d : data) {
            System.out.print(df.format(d) + "\t");
        }
        System.out.println();

        // print signals
        System.out.print("Signals:\t");
        List<Integer> signalsList = resultsMap.get("signals");
        for (int i : signalsList) {
            System.out.print(df.format(i) + "\t");
        }
        System.out.println();

        // print filtered data
        System.out.print("Filtered Data:\t");
        List<Double> filteredDataList = resultsMap.get("filteredData");
        for (double d : filteredDataList) {
            System.out.print(df.format(d) + "\t");
        }
        System.out.println();

        // print running average
        System.out.print("Avg Filter:\t");
        List<Double> avgFilterList = resultsMap.get("avgFilter");
        for (double d : avgFilterList) {
            System.out.print(df.format(d) + "\t");
        }
        System.out.println();

        // print running std
        System.out.print("Std filter:\t");
        List<Double> stdFilterList = resultsMap.get("stdFilter");
        for (double d : stdFilterList) {
            System.out.print(df.format(d) + "\t");
        }
        System.out.println();

        System.out.println();
        for (int i = 0; i < signalsList.size(); i++) {
            if (signalsList.get(i) != 0) {
                System.out.println("Point " + i + " gave signal " + signalsList.get(i));
            }
        }
    }
}

结果

lag: 30     threshold: 5.0      influence: 0.0
Data size: 74
Signals size: 74
Data:           1.000   1.000   1.100   1.000   0.900   1.000   1.000   1.100   1.000   0.900   1.000   1.100   1.000   1.000   0.900   1.000   1.000   1.100   1.000   1.000   1.000   1.000   1.100   0.900   1.000   1.100   1.000   1.000   0.900   1.000   1.100   1.000   1.000   1.100   1.000   0.800   0.900   1.000   1.200   0.900   1.000   1.000   1.100   1.200   1.000   1.500   1.000   3.000   2.000   5.000   3.000   2.000   1.000   1.000   1.000   0.900   1.000   1.000   3.000   2.600   4.000   3.000   3.200   2.000   1.000   1.000   0.800   4.000   4.000   2.000   2.500   1.000   1.000   1.000   
Signals:        0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   1.000   0.000   1.000   1.000   1.000   1.000   1.000   0.000   0.000   0.000   0.000   0.000   0.000   1.000   1.000   1.000   1.000   1.000   1.000   0.000   0.000   0.000   1.000   1.000   1.000   1.000   0.000   0.000   0.000   
Filtered Data:  1.000   1.000   1.100   1.000   0.900   1.000   1.000   1.100   1.000   0.900   1.000   1.100   1.000   1.000   0.900   1.000   1.000   1.100   1.000   1.000   1.000   1.000   1.100   0.900   1.000   1.100   1.000   1.000   0.900   1.000   1.100   1.000   1.000   1.100   1.000   0.800   0.900   1.000   1.200   0.900   1.000   1.000   1.100   1.200   1.000   1.000   1.000   1.000   1.000   1.000   1.000   1.000   1.000   1.000   1.000   0.900   1.000   1.000   1.000   1.000   1.000   1.000   1.000   1.000   1.000   1.000   0.800   0.800   0.800   0.800   0.800   1.000   1.000   1.000   
Avg Filter:     0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   1.003   1.003   1.007   1.007   1.003   1.007   1.010   1.003   1.000   0.997   1.003   1.003   1.003   1.000   1.003   1.010   1.013   1.013   1.013   1.010   1.010   1.010   1.010   1.010   1.007   1.010   1.010   1.003   1.003   1.003   1.007   1.007   1.003   1.003   1.003   1.000   1.000   1.007   1.003   0.997   0.983   0.980   0.973   0.973   0.970   
Std filter:     0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.060   0.060   0.063   0.063   0.060   0.063   0.060   0.071   0.073   0.071   0.080   0.080   0.080   0.077   0.080   0.087   0.085   0.085   0.085   0.083   0.083   0.083   0.083   0.083   0.081   0.079   0.079   0.080   0.080   0.080   0.077   0.077   0.075   0.075   0.075   0.073   0.073   0.063   0.071   0.080   0.078   0.083   0.089   0.089   0.086   

Point 45 gave signal 1
Point 47 gave signal 1
Point 48 gave signal 1
Point 49 gave signal 1
Point 50 gave signal 1
Point 51 gave signal 1
Point 58 gave signal 1
Point 59 gave signal 1
Point 60 gave signal 1
Point 61 gave signal 1
Point 62 gave signal 1
Point 63 gave signal 1
Point 67 gave signal 1
Point 68 gave signal 1
Point 69 gave signal 1
Point 70 gave signal 1

我认为delica的Python回答器有一个bug。我不能评论他的帖子，因为我没有代表来做这件事，编辑队列已经满了，所以我可能不是第一个注意到它的人。

avgFilter[lag - 1]和stdFilter[lag - 1]在init中设置，然后在lag == i时再次设置，而不是改变[lag]值。这个结果使得第一个信号总是1。

以下是带有轻微修正的代码:

import numpy as np

class real_time_peak_detection():
    def __init__(self, array, lag, threshold, influence):
        self.y = list(array)
        self.length = len(self.y)
        self.lag = lag
        self.threshold = threshold
        self.influence = influence
        self.signals = [0] * len(self.y)
        self.filteredY = np.array(self.y).tolist()
        self.avgFilter = [0] * len(self.y)
        self.stdFilter = [0] * len(self.y)
        self.avgFilter[self.lag - 1] = np.mean(self.y[0:self.lag]).tolist()
        self.stdFilter[self.lag - 1] = np.std(self.y[0:self.lag]).tolist()

    def thresholding_algo(self, new_value):
        self.y.append(new_value)
        i = len(self.y) - 1
        self.length = len(self.y)
        if i < self.lag:
            return 0
        elif i == self.lag:
            self.signals = [0] * len(self.y)
            self.filteredY = np.array(self.y).tolist()
            self.avgFilter = [0] * len(self.y)
            self.stdFilter = [0] * len(self.y)
            self.avgFilter[self.lag] = np.mean(self.y[0:self.lag]).tolist()
            self.stdFilter[self.lag] = np.std(self.y[0:self.lag]).tolist()
            return 0

        self.signals += [0]
        self.filteredY += [0]
        self.avgFilter += [0]
        self.stdFilter += [0]

        if abs(self.y[i] - self.avgFilter[i - 1]) > self.threshold * self.stdFilter[i - 1]:
            if self.y[i] > self.avgFilter[i - 1]:
                self.signals[i] = 1
            else:
                self.signals[i] = -1

            self.filteredY[i] = self.influence * self.y[i] + (1 - self.influence) * self.filteredY[i - 1]
            self.avgFilter[i] = np.mean(self.filteredY[(i - self.lag):i])
            self.stdFilter[i] = np.std(self.filteredY[(i - self.lag):i])
        else:
            self.signals[i] = 0
            self.filteredY[i] = self.y[i]
            self.avgFilter[i] = np.mean(self.filteredY[(i - self.lag):i])
            self.stdFilter[i] = np.std(self.filteredY[(i - self.lag):i])

        return self.signals[i]