更新:到目前为止表现最好的算法是这个。


这个问题探讨了在实时时间序列数据中检测突然峰值的稳健算法。

考虑以下示例数据:

这个数据的例子是Matlab格式的(但这个问题不是关于语言,而是关于算法):

p = [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];

你可以清楚地看到有三个大峰和一些小峰。这个数据集是问题所涉及的时间序列数据集类的一个特定示例。这类数据集有两个一般特征:

有一种具有一般平均值的基本噪声 有很大的“峰值”或“更高的数据点”明显偏离噪声。

让我们假设以下情况:

峰的宽度不能事先确定 峰的高度明显偏离其他值 算法实时更新(因此每个新数据点都会更新)

对于这种情况,需要构造一个触发信号的边值。但是,边界值不能是静态的,必须通过算法实时确定。


我的问题是:什么是实时计算这些阈值的好算法?有没有针对这种情况的特定算法?最著名的算法是什么?


健壮的算法或有用的见解都受到高度赞赏。(可以用任何语言回答:这是关于算法的)


当前回答

原文的附录1:Matlab和R翻译

Matlab代码

function [signals,avgFilter,stdFilter] = ThresholdingAlgo(y,lag,threshold,influence)
% Initialise signal results
signals = zeros(length(y),1);
% Initialise filtered series
filteredY = y(1:lag+1);
% Initialise filters
avgFilter(lag+1,1) = mean(y(1:lag+1));
stdFilter(lag+1,1) = std(y(1:lag+1));
% Loop over all datapoints y(lag+2),...,y(t)
for i=lag+2:length(y)
    % If new value is a specified number of deviations away
    if abs(y(i)-avgFilter(i-1)) > threshold*stdFilter(i-1)
        if y(i) > avgFilter(i-1)
            % Positive signal
            signals(i) = 1;
        else
            % Negative signal
            signals(i) = -1;
        end
        % Make influence lower
        filteredY(i) = influence*y(i)+(1-influence)*filteredY(i-1);
    else
        % No signal
        signals(i) = 0;
        filteredY(i) = y(i);
    end
    % Adjust the filters
    avgFilter(i) = mean(filteredY(i-lag:i));
    stdFilter(i) = std(filteredY(i-lag:i));
end
% Done, now return results
end

例子:

% 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;

% Get results
[signals,avg,dev] = ThresholdingAlgo(y,lag,threshold,influence);

figure; subplot(2,1,1); hold on;
x = 1:length(y); ix = lag+1:length(y);
area(x(ix),avg(ix)+threshold*dev(ix),'FaceColor',[0.9 0.9 0.9],'EdgeColor','none');
area(x(ix),avg(ix)-threshold*dev(ix),'FaceColor',[1 1 1],'EdgeColor','none');
plot(x(ix),avg(ix),'LineWidth',1,'Color','cyan','LineWidth',1.5);
plot(x(ix),avg(ix)+threshold*dev(ix),'LineWidth',1,'Color','green','LineWidth',1.5);
plot(x(ix),avg(ix)-threshold*dev(ix),'LineWidth',1,'Color','green','LineWidth',1.5);
plot(1:length(y),y,'b');
subplot(2,1,2);
stairs(signals,'r','LineWidth',1.5); ylim([-1.5 1.5]);

R代码

ThresholdingAlgo <- function(y,lag,threshold,influence) {
  signals <- rep(0,length(y))
  filteredY <- y[0:lag]
  avgFilter <- NULL
  stdFilter <- NULL
  avgFilter[lag] <- mean(y[0:lag], na.rm=TRUE)
  stdFilter[lag] <- sd(y[0:lag], na.rm=TRUE)
  for (i in (lag+1):length(y)){
    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]
    } else {
      signals[i] <- 0
      filteredY[i] <- y[i]
    }
    avgFilter[i] <- mean(filteredY[(i-lag):i], na.rm=TRUE)
    stdFilter[i] <- sd(filteredY[(i-lag):i], na.rm=TRUE)
  }
  return(list("signals"=signals,"avgFilter"=avgFilter,"stdFilter"=stdFilter))
}

例子:

# Data
y <- c(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)

lag       <- 30
threshold <- 5
influence <- 0

# Run algo with lag = 30, threshold = 5, influence = 0
result <- ThresholdingAlgo(y,lag,threshold,influence)

# Plot result
par(mfrow = c(2,1),oma = c(2,2,0,0) + 0.1,mar = c(0,0,2,1) + 0.2)
plot(1:length(y),y,type="l",ylab="",xlab="") 
lines(1:length(y),result$avgFilter,type="l",col="cyan",lwd=2)
lines(1:length(y),result$avgFilter+threshold*result$stdFilter,type="l",col="green",lwd=2)
lines(1:length(y),result$avgFilter-threshold*result$stdFilter,type="l",col="green",lwd=2)
plot(result$signals,type="S",col="red",ylab="",xlab="",ylim=c(-1.5,1.5),lwd=2)

这段代码(两种语言)将为原始问题的数据产生以下结果:


附录2原答案:Matlab演示代码

(点击创建数据)

function [] = RobustThresholdingDemo()

%% SPECIFICATIONS
lag         = 5;       % lag for the smoothing
threshold   = 3.5;     % number of st.dev. away from the mean to signal
influence   = 0.3;     % when signal: how much influence for new data? (between 0 and 1)
                       % 1 is normal influence, 0.5 is half      
%% START DEMO
DemoScreen(30,lag,threshold,influence);

end

function [signals,avgFilter,stdFilter] = ThresholdingAlgo(y,lag,threshold,influence)
signals = zeros(length(y),1);
filteredY = y(1:lag+1);
avgFilter(lag+1,1) = mean(y(1:lag+1));
stdFilter(lag+1,1) = std(y(1:lag+1));
for i=lag+2:length(y)
    if abs(y(i)-avgFilter(i-1)) > threshold*stdFilter(i-1)
        if y(i) > avgFilter(i-1)
            signals(i) = 1;
        else
            signals(i) = -1;
        end
        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:i));
    stdFilter(i) = std(filteredY(i-lag:i));
end
end

% Demo screen function
function [] = DemoScreen(n,lag,threshold,influence)
figure('Position',[200 100,1000,500]);
subplot(2,1,1);
title(sprintf(['Draw data points (%.0f max)      [settings: lag = %.0f, '...
    'threshold = %.2f, influence = %.2f]'],n,lag,threshold,influence));
ylim([0 5]); xlim([0 50]);
H = gca; subplot(2,1,1);
set(H, 'YLimMode', 'manual'); set(H, 'XLimMode', 'manual');
set(H, 'YLim', get(H,'YLim')); set(H, 'XLim', get(H,'XLim'));
xg = []; yg = [];
for i=1:n
    try
        [xi,yi] = ginput(1);
    catch
        return;
    end
    xg = [xg xi]; yg = [yg yi];
    if i == 1
        subplot(2,1,1); hold on;
        plot(H, xg(i),yg(i),'r.'); 
        text(xg(i),yg(i),num2str(i),'FontSize',7);
    end
    if length(xg) > lag
        [signals,avg,dev] = ...
            ThresholdingAlgo(yg,lag,threshold,influence);
        area(xg(lag+1:end),avg(lag+1:end)+threshold*dev(lag+1:end),...
            'FaceColor',[0.9 0.9 0.9],'EdgeColor','none');
        area(xg(lag+1:end),avg(lag+1:end)-threshold*dev(lag+1:end),...
            'FaceColor',[1 1 1],'EdgeColor','none');
        plot(xg(lag+1:end),avg(lag+1:end),'LineWidth',1,'Color','cyan');
        plot(xg(lag+1:end),avg(lag+1:end)+threshold*dev(lag+1:end),...
            'LineWidth',1,'Color','green');
        plot(xg(lag+1:end),avg(lag+1:end)-threshold*dev(lag+1:end),...
            'LineWidth',1,'Color','green');
        subplot(2,1,2); hold on; title('Signal output');
        stairs(xg(lag+1:end),signals(lag+1:end),'LineWidth',2,'Color','blue');
        ylim([-2 2]); xlim([0 50]); hold off;
    end
    subplot(2,1,1); hold on;
    for j=2:i
        plot(xg([j-1:j]),yg([j-1:j]),'r'); plot(H,xg(j),yg(j),'r.');
        text(xg(j),yg(j),num2str(j),'FontSize',7);
    end
end
end

其他回答

在计算拓扑学中,持久同调的思想导致一个有效的 -快如排序数字-解决方案。它不仅检测峰值,还以一种自然的方式量化峰值的“重要性”,使您能够选择对您重要的峰值。

算法的总结。 在一维设置(时间序列,实值信号)中,算法可以简单地描述为下图:

Think of the function graph (or its sub-level set) as a landscape and consider a decreasing water level starting at level infinity (or 1.8 in this picture). While the level decreases, at local maxima islands pop up. At local minima these islands merge together. One detail in this idea is that the island that appeared later in time is merged into the island that is older. The "persistence" of an island is its birth time minus its death time. The lengths of the blue bars depict the persistence, which is the above mentioned "significance" of a peak.

效率。 在对函数值进行排序之后,找到一个在线性时间内运行的实现并不难——实际上它是一个单一的、简单的循环。因此,这种实现在实践中应该是快速的,也很容易实现。

参考文献 一篇关于整个故事的文章和对持久同调(计算代数拓扑中的一个领域)动机的引用可以在这里找到: https://www.sthu.org/blog/13-perstopology-peakdetection/index.html

原文的附录1:Matlab和R翻译

Matlab代码

function [signals,avgFilter,stdFilter] = ThresholdingAlgo(y,lag,threshold,influence)
% Initialise signal results
signals = zeros(length(y),1);
% Initialise filtered series
filteredY = y(1:lag+1);
% Initialise filters
avgFilter(lag+1,1) = mean(y(1:lag+1));
stdFilter(lag+1,1) = std(y(1:lag+1));
% Loop over all datapoints y(lag+2),...,y(t)
for i=lag+2:length(y)
    % If new value is a specified number of deviations away
    if abs(y(i)-avgFilter(i-1)) > threshold*stdFilter(i-1)
        if y(i) > avgFilter(i-1)
            % Positive signal
            signals(i) = 1;
        else
            % Negative signal
            signals(i) = -1;
        end
        % Make influence lower
        filteredY(i) = influence*y(i)+(1-influence)*filteredY(i-1);
    else
        % No signal
        signals(i) = 0;
        filteredY(i) = y(i);
    end
    % Adjust the filters
    avgFilter(i) = mean(filteredY(i-lag:i));
    stdFilter(i) = std(filteredY(i-lag:i));
end
% Done, now return results
end

例子:

% 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;

% Get results
[signals,avg,dev] = ThresholdingAlgo(y,lag,threshold,influence);

figure; subplot(2,1,1); hold on;
x = 1:length(y); ix = lag+1:length(y);
area(x(ix),avg(ix)+threshold*dev(ix),'FaceColor',[0.9 0.9 0.9],'EdgeColor','none');
area(x(ix),avg(ix)-threshold*dev(ix),'FaceColor',[1 1 1],'EdgeColor','none');
plot(x(ix),avg(ix),'LineWidth',1,'Color','cyan','LineWidth',1.5);
plot(x(ix),avg(ix)+threshold*dev(ix),'LineWidth',1,'Color','green','LineWidth',1.5);
plot(x(ix),avg(ix)-threshold*dev(ix),'LineWidth',1,'Color','green','LineWidth',1.5);
plot(1:length(y),y,'b');
subplot(2,1,2);
stairs(signals,'r','LineWidth',1.5); ylim([-1.5 1.5]);

R代码

ThresholdingAlgo <- function(y,lag,threshold,influence) {
  signals <- rep(0,length(y))
  filteredY <- y[0:lag]
  avgFilter <- NULL
  stdFilter <- NULL
  avgFilter[lag] <- mean(y[0:lag], na.rm=TRUE)
  stdFilter[lag] <- sd(y[0:lag], na.rm=TRUE)
  for (i in (lag+1):length(y)){
    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]
    } else {
      signals[i] <- 0
      filteredY[i] <- y[i]
    }
    avgFilter[i] <- mean(filteredY[(i-lag):i], na.rm=TRUE)
    stdFilter[i] <- sd(filteredY[(i-lag):i], na.rm=TRUE)
  }
  return(list("signals"=signals,"avgFilter"=avgFilter,"stdFilter"=stdFilter))
}

例子:

# Data
y <- c(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)

lag       <- 30
threshold <- 5
influence <- 0

# Run algo with lag = 30, threshold = 5, influence = 0
result <- ThresholdingAlgo(y,lag,threshold,influence)

# Plot result
par(mfrow = c(2,1),oma = c(2,2,0,0) + 0.1,mar = c(0,0,2,1) + 0.2)
plot(1:length(y),y,type="l",ylab="",xlab="") 
lines(1:length(y),result$avgFilter,type="l",col="cyan",lwd=2)
lines(1:length(y),result$avgFilter+threshold*result$stdFilter,type="l",col="green",lwd=2)
lines(1:length(y),result$avgFilter-threshold*result$stdFilter,type="l",col="green",lwd=2)
plot(result$signals,type="S",col="red",ylab="",xlab="",ylim=c(-1.5,1.5),lwd=2)

这段代码(两种语言)将为原始问题的数据产生以下结果:


附录2原答案:Matlab演示代码

(点击创建数据)

function [] = RobustThresholdingDemo()

%% SPECIFICATIONS
lag         = 5;       % lag for the smoothing
threshold   = 3.5;     % number of st.dev. away from the mean to signal
influence   = 0.3;     % when signal: how much influence for new data? (between 0 and 1)
                       % 1 is normal influence, 0.5 is half      
%% START DEMO
DemoScreen(30,lag,threshold,influence);

end

function [signals,avgFilter,stdFilter] = ThresholdingAlgo(y,lag,threshold,influence)
signals = zeros(length(y),1);
filteredY = y(1:lag+1);
avgFilter(lag+1,1) = mean(y(1:lag+1));
stdFilter(lag+1,1) = std(y(1:lag+1));
for i=lag+2:length(y)
    if abs(y(i)-avgFilter(i-1)) > threshold*stdFilter(i-1)
        if y(i) > avgFilter(i-1)
            signals(i) = 1;
        else
            signals(i) = -1;
        end
        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:i));
    stdFilter(i) = std(filteredY(i-lag:i));
end
end

% Demo screen function
function [] = DemoScreen(n,lag,threshold,influence)
figure('Position',[200 100,1000,500]);
subplot(2,1,1);
title(sprintf(['Draw data points (%.0f max)      [settings: lag = %.0f, '...
    'threshold = %.2f, influence = %.2f]'],n,lag,threshold,influence));
ylim([0 5]); xlim([0 50]);
H = gca; subplot(2,1,1);
set(H, 'YLimMode', 'manual'); set(H, 'XLimMode', 'manual');
set(H, 'YLim', get(H,'YLim')); set(H, 'XLim', get(H,'XLim'));
xg = []; yg = [];
for i=1:n
    try
        [xi,yi] = ginput(1);
    catch
        return;
    end
    xg = [xg xi]; yg = [yg yi];
    if i == 1
        subplot(2,1,1); hold on;
        plot(H, xg(i),yg(i),'r.'); 
        text(xg(i),yg(i),num2str(i),'FontSize',7);
    end
    if length(xg) > lag
        [signals,avg,dev] = ...
            ThresholdingAlgo(yg,lag,threshold,influence);
        area(xg(lag+1:end),avg(lag+1:end)+threshold*dev(lag+1:end),...
            'FaceColor',[0.9 0.9 0.9],'EdgeColor','none');
        area(xg(lag+1:end),avg(lag+1:end)-threshold*dev(lag+1:end),...
            'FaceColor',[1 1 1],'EdgeColor','none');
        plot(xg(lag+1:end),avg(lag+1:end),'LineWidth',1,'Color','cyan');
        plot(xg(lag+1:end),avg(lag+1:end)+threshold*dev(lag+1:end),...
            'LineWidth',1,'Color','green');
        plot(xg(lag+1:end),avg(lag+1:end)-threshold*dev(lag+1:end),...
            'LineWidth',1,'Color','green');
        subplot(2,1,2); hold on; title('Signal output');
        stairs(xg(lag+1:end),signals(lag+1:end),'LineWidth',2,'Color','blue');
        ylim([-2 2]); xlim([0 50]); hold off;
    end
    subplot(2,1,1); hold on;
    for j=2:i
        plot(xg([j-1:j]),yg([j-1:j]),'r'); plot(H,xg(j),yg(j),'r.');
        text(xg(j),yg(j),num2str(j),'FontSize',7);
    end
end
end

如果你的数据在一个数据库表中,这里是一个简单的z-score算法的SQL版本:

with data_with_zscore as (
    select
        date_time,
        value,
        value / (avg(value) over ()) as pct_of_mean,
        (value - avg(value) over ()) / (stdev(value) over ()) as z_score
    from {{tablename}}  where datetime > '2018-11-26' and datetime < '2018-12-03'
)


-- select all
select * from data_with_zscore 

-- select only points greater than a certain threshold
select * from data_with_zscore where z_score > abs(2)

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。