下面是我尝试为“Smoothed z-score算法”创建一个Ruby解决方案:

module ThresholdingAlgoMixin
  def mean(array)
    array.reduce(&:+) / array.size.to_f
  end

  def stddev(array)
    array_mean = mean(array)
    Math.sqrt(array.reduce(0.0) { |a, b| a.to_f + ((b.to_f - array_mean) ** 2) } / array.size.to_f)
  end

  def thresholding_algo(lag: 5, threshold: 3.5, influence: 0.5)
    return nil if size < lag * 2
    Array.new(size, 0).tap do |signals|
      filtered = Array.new(self)

      initial_slice = take(lag)
      avg_filter = Array.new(lag - 1, 0.0) + [mean(initial_slice)]
      std_filter = Array.new(lag - 1, 0.0) + [stddev(initial_slice)]
      (lag..size-1).each do |idx|
        prev = idx - 1
        if (fetch(idx) - avg_filter[prev]).abs > threshold * std_filter[prev]
          signals[idx] = fetch(idx) > avg_filter[prev] ? 1 : -1
          filtered[idx] = (influence * fetch(idx)) + ((1-influence) * filtered[prev])
        end

        filtered_slice = filtered[idx-lag..prev]
        avg_filter[idx] = mean(filtered_slice)
        std_filter[idx] = stddev(filtered_slice)
      end
    end
  end
end

以及示例用法:

test_data = [
  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
].extend(ThresholdingAlgoMixin)

puts test_data.thresholding_algo.inspect

# Output: [
#   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
#   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0,
#   0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,
#   1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0
# ]

一种方法是根据以下观察来检测峰:

时间t是一个峰值(y (t) > y (t - 1)) & & ((t) > y (t + 1))

它通过等待上升趋势结束来避免误报。它并不完全是“实时”的，因为它会比峰值差一个dt。灵敏度可以通过要求比较的裕度来控制。在噪声检测和时延检测之间存在一种折衷。 您可以通过添加更多参数来丰富模型:

峰如果y (y (t) - (t-dt) > m) && (y (t) - y (t + dt) > m)

dt和m是控制灵敏度和延时的参数

这是你用上述算法得到的结果:

下面是在python中重现图的代码:

import numpy as np
import matplotlib.pyplot as plt
input = np.array([ 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. ])
signal = (input > np.roll(input,1)) & (input > np.roll(input,-1))
plt.plot(input)
plt.plot(signal.nonzero()[0], input[signal], 'ro')
plt.show()

通过设置m = 0.5，你可以得到一个更清晰的信号，只有一个假阳性:

我在我的机器人项目中需要这样的东西。我想我可以归还Kotlin实现。

/**
* Smoothed zero-score alogrithm shamelessly copied from https://stackoverflow.com/a/22640362/6029703
* Uses a rolling mean and a rolling deviation (separate) to identify peaks in a vector
*
* @param y - The input vector to analyze
* @param lag - The lag of the moving window (i.e. how big the window is)
* @param threshold - The z-score at which the algorithm signals (i.e. how many standard deviations away from the moving mean a peak (or signal) is)
* @param influence - The influence (between 0 and 1) of new signals on the mean and standard deviation (how much a peak (or signal) should affect other values near it)
* @return - The calculated averages (avgFilter) and deviations (stdFilter), and the signals (signals)
*/
fun smoothedZScore(y: List<Double>, lag: Int, threshold: Double, influence: Double): Triple<List<Int>, List<Double>, List<Double>> {
    val stats = SummaryStatistics()
    // the results (peaks, 1 or -1) of our algorithm
    val signals = MutableList<Int>(y.size, { 0 })
    // filter out the signals (peaks) from our original list (using influence arg)
    val filteredY = ArrayList<Double>(y)
    // the current average of the rolling window
    val avgFilter = MutableList<Double>(y.size, { 0.0 })
    // the current standard deviation of the rolling window
    val stdFilter = MutableList<Double>(y.size, { 0.0 })
    // init avgFilter and stdFilter
    y.take(lag).forEach { s -> stats.addValue(s) }
    avgFilter[lag - 1] = stats.mean
    stdFilter[lag - 1] = Math.sqrt(stats.populationVariance) // getStandardDeviation() uses sample variance (not what we want)
    stats.clear()
    //loop input starting at end of rolling window
    (lag..y.size - 1).forEach { i ->
        //if the distance between the current value and average is enough standard deviations (threshold) away
        if (Math.abs(y[i] - avgFilter[i - 1]) > threshold * stdFilter[i - 1]) {
            //this is a signal (i.e. peak), determine if it is a positive or negative signal
            signals[i] = if (y[i] > avgFilter[i - 1]) 1 else -1
            //filter this signal out using influence
            filteredY[i] = (influence * y[i]) + ((1 - influence) * filteredY[i - 1])
        } else {
            //ensure this signal remains a zero
            signals[i] = 0
            //ensure this value is not filtered
            filteredY[i] = y[i]
        }
        //update rolling average and deviation
        (i - lag..i - 1).forEach { stats.addValue(filteredY[it]) }
        avgFilter[i] = stats.getMean()
        stdFilter[i] = Math.sqrt(stats.getPopulationVariance()) //getStandardDeviation() uses sample variance (not what we want)
        stats.clear()
    }
    return Triple(signals, avgFilter, stdFilter)
}

带有验证图的示例项目可以在github上找到。

@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;
  }
}

@Jean-Paul算法的Perl实现。

#!/usr/bin/perl

use strict;
use Data::Dumper;

sub mean {
    my $data = shift;
    my $sum = 0;
    my $mean_val = 0;
    for my $item (@$data) {
        $sum += $item;
    }
    $mean_val = $sum / (scalar @$data) if @$data;
    return $mean_val;
}

sub variance {
    my $data = shift;
    my $variance_val = 0;
    my $mean_val = mean($data);
    my $sum = 0;
    for my $item (@$data) {
        $sum += ($item - $mean_val)**2;
    }
    $variance_val = $sum / (scalar @$data) if @$data;
    return $variance_val;
}

sub std {
    my $data = shift;
    my $variance_val = variance($data);
    return sqrt($variance_val);
}

# @param y - The input vector to analyze
# @parameter lag - The lag of the moving window
# @parameter threshold - The z-score at which the algorithm signals
# @parameter influence - The influence (between 0 and 1) of new signals on the mean and standard deviation
sub thresholding_algo {
    my ($y, $lag, $threshold, $influence) = @_;

    my @signals = (0) x @$y;
    my @filteredY = @$y;
    my @avgFilter = (0) x @$y;
    my @stdFilter = (0) x @$y;

    $avgFilter[$lag - 1] = mean([@$y[0..$lag-1]]);
    $stdFilter[$lag - 1] = std([@$y[0..$lag-1]]);

    for (my $i=$lag; $i <= @$y - 1; $i++) {
        if (abs($y->[$i] - $avgFilter[$i-1]) > $threshold * $stdFilter[$i-1]) {
            if ($y->[$i] > $avgFilter[$i-1]) {
                $signals[$i] = 1;
            } else {
                $signals[$i] = -1;
            }

            $filteredY[$i] = $influence * $y->[$i] + (1 - $influence) * $filteredY[$i-1];
            $avgFilter[$i] = mean([@filteredY[($i-$lag)..($i-1)]]);
            $stdFilter[$i] = std([@filteredY[($i-$lag)..($i-1)]]);
        }
        else {
            $signals[$i] = 0;
            $filteredY[$i] = $y->[$i];
            $avgFilter[$i] = mean([@filteredY[($i-$lag)..($i-1)]]);
            $stdFilter[$i] = std([@filteredY[($i-$lag)..($i-1)]]);
        }
    }

    return {
        signals => \@signals,
        avgFilter => \@avgFilter,
        stdFilter => \@stdFilter
    };
}

my $y = [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];

my $lag = 30;
my $threshold = 5;
my $influence = 0;

my $result = thresholding_algo($y, $lag, $threshold, $influence);

print Dumper $result;

另外，这个算法对我来说也很好…

sensitivity = 4; dwindow = 4; k = dwindow; 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>=Math.max(...c.slice(b-k,b))?a**3:0); var data2 = [{ name: 'filtered source', y: filtered }]; Plotly.newPlot('stage2', data2, { title: 'Filtered data<br>aₙ = aₙ³', 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: 'Maximum in a window of 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: 'Maximum in a window of 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>