c++实现

#include <iostream>
#include <vector>
#include <algorithm>
#include <unordered_map>
#include <cmath>
#include <iterator>
#include <numeric>

using namespace std;

typedef long double ld;
typedef unsigned int uint;
typedef std::vector<ld>::iterator vec_iter_ld;

/**
 * Overriding the ostream operator for pretty printing vectors.
 */
template<typename T>
std::ostream &operator<<(std::ostream &os, std::vector<T> vec) {
    os << "[";
    if (vec.size() != 0) {
        std::copy(vec.begin(), vec.end() - 1, std::ostream_iterator<T>(os, " "));
        os << vec.back();
    }
    os << "]";
    return os;
}

/**
 * This class calculates mean and standard deviation of a subvector.
 * This is basically stats computation of a subvector of a window size qual to "lag".
 */
class VectorStats {
public:
    /**
     * Constructor for VectorStats class.
     *
     * @param start - This is the iterator position of the start of the window,
     * @param end   - This is the iterator position of the end of the window,
     */
    VectorStats(vec_iter_ld start, vec_iter_ld end) {
        this->start = start;
        this->end = end;
        this->compute();
    }

    /**
     * This method calculates the mean and standard deviation using STL function.
     * This is the Two-Pass implementation of the Mean & Variance calculation.
     */
    void compute() {
        ld sum = std::accumulate(start, end, 0.0);
        uint slice_size = std::distance(start, end);
        ld mean = sum / slice_size;
        std::vector<ld> diff(slice_size);
        std::transform(start, end, diff.begin(), [mean](ld x) { return x - mean; });
        ld sq_sum = std::inner_product(diff.begin(), diff.end(), diff.begin(), 0.0);
        ld std_dev = std::sqrt(sq_sum / slice_size);

        this->m1 = mean;
        this->m2 = std_dev;
    }

    ld mean() {
        return m1;
    }

    ld standard_deviation() {
        return m2;
    }

private:
    vec_iter_ld start;
    vec_iter_ld end;
    ld m1;
    ld m2;
};

/**
 * This is the implementation of the Smoothed Z-Score Algorithm.
 * This is direction translation of https://stackoverflow.com/a/22640362/1461896.
 *
 * @param input - input signal
 * @param lag - the lag of the moving window
 * @param threshold - the z-score at which the algorithm signals
 * @param influence - the influence (between 0 and 1) of new signals on the mean and standard deviation
 * @return a hashmap containing the filtered signal and corresponding mean and standard deviation.
 */
unordered_map<string, vector<ld>> z_score_thresholding(vector<ld> input, int lag, ld threshold, ld influence) {
    unordered_map<string, vector<ld>> output;

    uint n = (uint) input.size();
    vector<ld> signals(input.size());
    vector<ld> filtered_input(input.begin(), input.end());
    vector<ld> filtered_mean(input.size());
    vector<ld> filtered_stddev(input.size());

    VectorStats lag_subvector_stats(input.begin(), input.begin() + lag);
    filtered_mean[lag - 1] = lag_subvector_stats.mean();
    filtered_stddev[lag - 1] = lag_subvector_stats.standard_deviation();

    for (int i = lag; i < n; i++) {
        if (abs(input[i] - filtered_mean[i - 1]) > threshold * filtered_stddev[i - 1]) {
            signals[i] = (input[i] > filtered_mean[i - 1]) ? 1.0 : -1.0;
            filtered_input[i] = influence * input[i] + (1 - influence) * filtered_input[i - 1];
        } else {
            signals[i] = 0.0;
            filtered_input[i] = input[i];
        }
        VectorStats lag_subvector_stats(filtered_input.begin() + (i - lag), filtered_input.begin() + i);
        filtered_mean[i] = lag_subvector_stats.mean();
        filtered_stddev[i] = lag_subvector_stats.standard_deviation();
    }

    output["signals"] = signals;
    output["filtered_mean"] = filtered_mean;
    output["filtered_stddev"] = filtered_stddev;

    return output;
};

int main() {
    vector<ld> input = {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;
    ld threshold = 5.0;
    ld influence = 0.0;
    unordered_map<string, vector<ld>> output = z_score_thresholding(input, lag, threshold, influence);
    cout << output["signals"] << endl;
}

下面是@Jean-Paul为Arduino微控制器设计的平滑z分数的C语言实现，用于获取加速度计读数，并判断撞击的方向是来自左边还是右边。这表现得非常好，因为这个设备返回一个反弹信号。这是设备对峰值检测算法的输入-显示了来自右边的冲击，然后是来自左边的冲击。你可以看到最初的峰值然后传感器的振荡。

#include <stdio.h>
#include <math.h>
#include <string.h>


#define SAMPLE_LENGTH 1000

float stddev(float data[], int len);
float mean(float data[], int len);
void thresholding(float y[], int signals[], int lag, float threshold, float influence);


void thresholding(float y[], int signals[], int lag, float threshold, float influence) {
    memset(signals, 0, sizeof(int) * SAMPLE_LENGTH);
    float filteredY[SAMPLE_LENGTH];
    memcpy(filteredY, y, sizeof(float) * SAMPLE_LENGTH);
    float avgFilter[SAMPLE_LENGTH];
    float stdFilter[SAMPLE_LENGTH];

    avgFilter[lag - 1] = mean(y, lag);
    stdFilter[lag - 1] = stddev(y, lag);

    for (int i = lag; i < SAMPLE_LENGTH; i++) {
        if (fabsf(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];
        } else {
            signals[i] = 0;
        }
        avgFilter[i] = mean(filteredY + i-lag, lag);
        stdFilter[i] = stddev(filteredY + i-lag, lag);
    }
}

float mean(float data[], int len) {
    float sum = 0.0, mean = 0.0;

    int i;
    for(i=0; i<len; ++i) {
        sum += data[i];
    }

    mean = sum/len;
    return mean;


}

float stddev(float data[], int len) {
    float the_mean = mean(data, len);
    float standardDeviation = 0.0;

    int i;
    for(i=0; i<len; ++i) {
        standardDeviation += pow(data[i] - the_mean, 2);
    }

    return sqrt(standardDeviation/len);
}

int main() {
    printf("Hello, World!\n");
    int lag = 100;
    float threshold = 5;
    float influence = 0;
    float 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,       2.6,4,3,3.2,2,1,1,0.8,4,4,2,2.5,1,1,1,1.2,1,1.5,1,3,2,5,3,2,1,1,1,0.9,1,1,3,
       2.6,4,3,3.2,2,1,1,0.8,4,4,2,2.5,1,1,1}

    int signal[SAMPLE_LENGTH];

    thresholding(y, signal,  lag, threshold, influence);

    return 0;
}

她的结果是影响= 0

不是很好，但这里的影响力= 1

这很好。

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。

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:

我想把我的Julia算法实现提供给其他人。要点可以在这里找到

using Statistics
using Plots
function SmoothedZscoreAlgo(y, lag, threshold, influence)
    # Julia implimentation of http://stackoverflow.com/a/22640362/6029703
    n = length(y)
    signals = zeros(n) # init signal results
    filteredY = copy(y) # init filtered series
    avgFilter = zeros(n) # init average filter
    stdFilter = zeros(n) # init std filter
    avgFilter[lag - 1] = mean(y[1:lag]) # init first value
    stdFilter[lag - 1] = std(y[1:lag]) # init first value

    for i in range(lag, stop=n-1)
        if abs(y[i] - avgFilter[i-1]) > threshold*stdFilter[i-1]
            if y[i] > avgFilter[i-1]
                signals[i] += 1 # postive signal
            else
                signals[i] += -1 # negative signal
            end
            # Make influence lower
            filteredY[i] = influence*y[i] + (1-influence)*filteredY[i-1]
        else
            signals[i] = 0
            filteredY[i] = y[i]
        end
        avgFilter[i] = mean(filteredY[i-lag+1:i])
        stdFilter[i] = std(filteredY[i-lag+1:i])
    end
    return (signals = signals, avgFilter = avgFilter, stdFilter = stdFilter)
end


# Data
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]

# Settings: lag = 30, threshold = 5, influence = 0
lag = 30
threshold = 5
influence = 0

results = SmoothedZscoreAlgo(y, lag, threshold, influence)
upper_bound = results[:avgFilter] + threshold * results[:stdFilter]
lower_bound = results[:avgFilter] - threshold * results[:stdFilter]
x = 1:length(y)

yplot = plot(x,y,color="blue", label="Y",legend=:topleft)
yplot = plot!(x,upper_bound, color="green", label="Upper Bound",legend=:topleft)
yplot = plot!(x,results[:avgFilter], color="cyan", label="Average Filter",legend=:topleft)
yplot = plot!(x,lower_bound, color="green", label="Lower Bound",legend=:topleft)
signalplot = plot(x,results[:signals],color="red",label="Signals",legend=:topleft)
plot(yplot,signalplot,layout=(2,1),legend=:topleft)