假设你的数据来自传感器(所以算法不可能知道未来的任何事情)，

我做了这个算法，它与我在自己的项目中获得的数据非常好。

该算法有2个参数:灵敏度和窗口。

最后，只需一行代码就可以得到你的结果:

detected=data.map((a, b, c) => (a > 0) ? c[b] ** 4 * c[b - 1] ** 3 : -0).map((a, b, c) => a > Math.max(...c.slice(2)) / sensitivity).map((a, b, c) => (b > dwindow) && c.slice(b - dwindow, b).indexOf(a) == -1);

因为我是程序员而不是数学家，所以我不能更好地解释它。但我相信有人可以。

sensitivity = 20; dwindow = 4; data = [1., 1., 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., 1., 1., 1. ]; //data = data.concat(data); //data = data.concat(data); var data1 = [{ name: 'original source', y: data }]; Plotly.newPlot('stage1', data1, { title: 'Sensor data', yaxis: { title: 'signal' } }); filtered = data.map((a, b, c) => (a > 0) ? c[b] ** 4 * c[b - 1] ** 3 : -0); var data2 = [{ name: 'filtered source', y: filtered }]; Plotly.newPlot('stage2', data2, { title: 'Filtered data<br>aₙ = aₙ⁴ * aₙ₋₁³', yaxis: { title: 'signal' } }); dwindow = 6; k = dwindow; detected = filtered.map((a, b, c) => a > Math.max(...c.slice(2)) / sensitivity).map((a, b, c) => (b > k) && c.slice(b - k, b).indexOf(a) == -1) var data3 = [{ name: 'detected peaks', y: detected }]; Plotly.newPlot('stage3', data3, { title: 'Window 6', yaxis: { title: 'signal' } }); dwindow = 10; k = dwindow; detected = filtered.map((a, b, c) => a > Math.max(...c.slice(2)) / 20).map((a, b, c) => (b > k) && c.slice(b - k, b).indexOf(a) == -1) var data4 = [{ name: 'detected peaks', y: detected }]; Plotly.newPlot('stage4', data4, { title: 'Window 10', yaxis: { title: 'signal' } }); <script src="https://cdn.jsdelivr.net/npm/plotly.js@2.16.5/dist/plotly.min.js"></script> <div id="stage1"></div> <div id="stage2"></div> <div id="stage3"></div> <div id="stage4"></div>

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。

下面是在Golang中实现的Smoothed z-score算法(上图)。它假设一个[]int16 (PCM 16bit样本)的切片。你可以在这里找到要点。

/*
Settings (the ones below are examples: choose what is best for your data)
set lag to 5;          # lag 5 for the smoothing functions
set threshold to 3.5;  # 3.5 standard deviations for signal
set influence to 0.5;  # between 0 and 1, where 1 is normal influence, 0.5 is half
*/

// ZScore on 16bit WAV samples
func ZScore(samples []int16, lag int, threshold float64, influence float64) (signals []int16) {
    //lag := 20
    //threshold := 3.5
    //influence := 0.5

    signals = make([]int16, len(samples))
    filteredY := make([]int16, len(samples))
    for i, sample := range samples[0:lag] {
        filteredY[i] = sample
    }
    avgFilter := make([]int16, len(samples))
    stdFilter := make([]int16, len(samples))

    avgFilter[lag] = Average(samples[0:lag])
    stdFilter[lag] = Std(samples[0:lag])

    for i := lag + 1; i < len(samples); i++ {

        f := float64(samples[i])

        if float64(Abs(samples[i]-avgFilter[i-1])) > threshold*float64(stdFilter[i-1]) {
            if samples[i] > avgFilter[i-1] {
                signals[i] = 1
            } else {
                signals[i] = -1
            }
            filteredY[i] = int16(influence*f + (1-influence)*float64(filteredY[i-1]))
            avgFilter[i] = Average(filteredY[(i - lag):i])
            stdFilter[i] = Std(filteredY[(i - lag):i])
        } else {
            signals[i] = 0
            filteredY[i] = samples[i]
            avgFilter[i] = Average(filteredY[(i - lag):i])
            stdFilter[i] = Std(filteredY[(i - lag):i])
        }
    }

    return
}

// Average a chunk of values
func Average(chunk []int16) (avg int16) {
    var sum int64
    for _, sample := range chunk {
        if sample < 0 {
            sample *= -1
        }
        sum += int64(sample)
    }
    return int16(sum / int64(len(chunk)))
}

不需要将极大值与平均值进行比较，还可以将极大值与相邻的最小值进行比较，其中最小值仅定义在噪声阈值之上。 如果局部最大值是>的3倍(或其他置信因子)相邻的最小值，那么这个最大值就是一个峰值。 移动窗口越宽，峰值的确定越准确。 上面使用了以窗口中间为中心的计算， 顺便说一下，而不是在窗口结束时计算(== lag)。

请注意，最大值必须被视为信号之前的增加 之后下降。

根据@Jean-Paul提出的解决方案，我用c#实现了他的算法

public class ZScoreOutput
{
    public List<double> input;
    public List<int> signals;
    public List<double> avgFilter;
    public List<double> filtered_stddev;
}

public static class ZScore
{
    public static ZScoreOutput StartAlgo(List<double> input, int lag, double threshold, double influence)
    {
        // init variables!
        int[] signals = new int[input.Count];
        double[] filteredY = new List<double>(input).ToArray();
        double[] avgFilter = new double[input.Count];
        double[] stdFilter = new double[input.Count];

        var initialWindow = new List<double>(filteredY).Skip(0).Take(lag).ToList();

        avgFilter[lag - 1] = Mean(initialWindow);
        stdFilter[lag - 1] = StdDev(initialWindow);

        for (int i = lag; i < input.Count; i++)
        {
            if (Math.Abs(input[i] - avgFilter[i - 1]) > threshold * stdFilter[i - 1])
            {
                signals[i] = (input[i] > avgFilter[i - 1]) ? 1 : -1;
                filteredY[i] = influence * input[i] + (1 - influence) * filteredY[i - 1];
            }
            else
            {
                signals[i] = 0;
                filteredY[i] = input[i];
            }

            // Update rolling average and deviation
            var slidingWindow = new List<double>(filteredY).Skip(i - lag).Take(lag+1).ToList();

            var tmpMean = Mean(slidingWindow);
            var tmpStdDev = StdDev(slidingWindow);

            avgFilter[i] = Mean(slidingWindow);
            stdFilter[i] = StdDev(slidingWindow);
        }

        // Copy to convenience class 
        var result = new ZScoreOutput();
        result.input = input;
        result.avgFilter       = new List<double>(avgFilter);
        result.signals         = new List<int>(signals);
        result.filtered_stddev = new List<double>(stdFilter);

        return result;
    }

    private static double Mean(List<double> list)
    {
        // Simple helper function! 
        return list.Average();
    }

    private static double StdDev(List<double> values)
    {
        double ret = 0;
        if (values.Count() > 0)
        {
            double avg = values.Average();
            double sum = values.Sum(d => Math.Pow(d - avg, 2));
            ret = Math.Sqrt((sum) / (values.Count() - 1));
        }
        return ret;
    }
}

使用示例:

var input = new List<double> {1.0, 1.0, 1.1, 1.0, 0.9, 1.0, 1.0, 1.1, 1.0, 0.9, 1.0,
    1.1, 1.0, 1.0, 0.9, 1.0, 1.0, 1.1, 1.0, 1.0, 1.0, 1.0, 1.1, 0.9, 1.0, 1.1, 1.0, 1.0, 0.9,
    1.0, 1.1, 1.0, 1.0, 1.1, 1.0, 0.8, 0.9, 1.0, 1.2, 0.9, 1.0, 1.0, 1.1, 1.2, 1.0, 1.5, 1.0,
    3.0, 2.0, 5.0, 3.0, 2.0, 1.0, 1.0, 1.0, 0.9, 1.0, 1.0, 3.0, 2.6, 4.0, 3.0, 3.2, 2.0, 1.0,
    1.0, 0.8, 4.0, 4.0, 2.0, 2.5, 1.0, 1.0, 1.0};

int lag = 30;
double threshold = 5.0;
double influence = 0.0;

var output = ZScore.StartAlgo(input, lag, threshold, influence);

@Jean-Paul Smoothed Z Score算法的Dart版本:

class SmoothedZScore {
  int lag = 5;
  num threshold = 10;
  num influence = 0.5;

  num sum(List<num> a) {
    num s = 0;
    for (int i = 0; i < a.length; i++) s += a[i];
    return s;
  }

  num mean(List<num> a) {
    return sum(a) / a.length;
  }

  num stddev(List<num> arr) {
    num arrMean = mean(arr);
    num dev = 0;
    for (int i = 0; i < arr.length; i++) dev += (arr[i] - arrMean) * (arr[i] - arrMean);
    return sqrt(dev / arr.length);
  }

  List<int> smoothedZScore(List<num> y) {
    if (y.length < lag + 2) {
      throw 'y data array too short($y.length) for given lag of $lag';
    }

    // init variables
    List<int> signals = List.filled(y.length, 0);
    List<num> filteredY = List<num>.from(y);
    List<num> leadIn = y.sublist(0, lag);

    var avgFilter = List<num>.filled(y.length, 0);
    var stdFilter = List<num>.filled(y.length, 0);
    avgFilter[lag - 1] = mean(leadIn);
    stdFilter[lag - 1] = stddev(leadIn);

    for (var i = lag; i < y.length; i++) {
      if ((y[i] - avgFilter[i - 1]).abs() > (threshold * stdFilter[i - 1])) {
        signals[i] = y[i] > avgFilter[i - 1] ? 1 : -1;
        // make influence lower
        filteredY[i] = influence * y[i] + (1 - influence) * filteredY[i - 1];
      } else {
        signals[i] = 0; // no signal
        filteredY[i] = y[i];
      }

      // adjust the filters
      List<num> yLag = filteredY.sublist(i - lag, i);
      avgFilter[i] = mean(yLag);
      stdFilter[i] = stddev(yLag);
    }

    return signals;
  }
}