peak detection for growing time series using Swift

Question

Would anyone have a good algorithm to measure peaks in growing time series data using Swift (v3)? So, detect peaks as the data is streaming in.

E.g. a Swift version

Answer 1

Translation of smooth z-score algo into Swift

Well, to quickly help you out: here is a translation of the algo into Swift: Demo in Swift Sandbox

Warning: I am by no means a swift programmer, so there could be mistakes in there!

Also note that I have turned off negative signals, as for OP's purpose we only want positive signals.

Swift code:

import Glibc // or Darwin/ Foundation/ Cocoa/ UIKit (depending on OS)

// Function to calculate the arithmetic mean
func arithmeticMean(array: [Double]) -> Double {
    var total: Double = 0
    for number in array {
        total += number
    }
    return total / Double(array.count)
}

// Function to calculate the standard deviation
func standardDeviation(array: [Double]) -> Double
{
    let length = Double(array.count)
    let avg = array.reduce(0, {$0 + $1}) / length
    let sumOfSquaredAvgDiff = array.map { pow($0 - avg, 2.0)}.reduce(0, {$0 + $1})
    return sqrt(sumOfSquaredAvgDiff / length)
}

// Function to extract some range from an array
func subArray<T>(array: [T], s: Int, e: Int) -> [T] {
    if e > array.count {
        return []
    }
    return Array(array[s..<min(e, array.count)])
}

// Smooth z-score thresholding filter
func ThresholdingAlgo(y: [Double],lag: Int,threshold: Double,influence: Double) -> ([Int],[Double],[Double]) {

    // Create arrays
    var signals   = Array(repeating: 0, count: y.count)
    var filteredY = Array(repeating: 0.0, count: y.count)
    var avgFilter = Array(repeating: 0.0, count: y.count)
    var stdFilter = Array(repeating: 0.0, count: y.count)

    // Initialise variables
    for i in 0...lag-1 {
        signals[i] = 0
        filteredY[i] = y[i]
    }

    // Start filter
    avgFilter[lag-1] = arithmeticMean(array: subArray(array: y, s: 0, e: lag-1))
    stdFilter[lag-1] = standardDeviation(array: subArray(array: y, s: 0, e: lag-1))

    for i in lag...y.count-1 {
        if abs(y[i] - avgFilter[i-1]) > threshold*stdFilter[i-1] {
            if y[i] > avgFilter[i-1] {
                signals[i] = 1      // Positive signal
            } else {
                // Negative signals are turned off for this application
                //signals[i] = -1       // Negative signal
            }
            filteredY[i] = influence*y[i] + (1-influence)*filteredY[i-1]
        } else {
            signals[i] = 0          // No signal
            filteredY[i] = y[i]
        }
        // Adjust the filters
        avgFilter[i] = arithmeticMean(array: subArray(array: filteredY, s: i-lag, e: i))
        stdFilter[i] = standardDeviation(array: subArray(array: filteredY, s: i-lag, e: i))
    }

    return (signals,avgFilter,stdFilter)
}

// Demo
let samples = [0.01, -0.02, -0.02, 0.01, -0.01, -0.01, 0.00, 0.10, 0.31,
  -0.10, -0.73, -0.68, 0.21, 1.22, 0.67, -0.59, -1.04, 0.06, 0.42, 0.07, 
  0.03, -0.18, 0.11, -0.06, -0.02, 0.16, 0.21, 0.03, -0.68, -0.89, 0.18, 
  1.31, 0.66, 0.07, -1.62, -0.16, 0.67, 0.19, -0.42, 0.23, -0.05, -0.01,
  0.03, 0.06, 0.27, 0.15, -0.50, -1.18, 0.11, 1.30, 0.93, 0.16, -1.32, 
  -0.10, 0.55, 0.23, -0.03, -0.23, 0.16, -0.04, 0.01, 0.12, 0.35, -0.38,
  -1.11, 0.07, 1.46, 0.61, -0.68, -1.16, 0.29, 0.54, -0.05, 0.02, -0.01,
  0.12, 0.23, 0.29, -0.75, -0.95, 0.11, 1.51, 0.70, -0.30, -1.48, 0.13,
  0.50, 0.18, -0.06, -0.01, -0.02, 0.03, -0.02, 0.06, 0.03, 0.03, 0.02,
  -0.01, 0.01, 0.02, 0.01]

// Run filter
let (signals,avgFilter,stdFilter) = ThresholdingAlgo(y: samples, lag: 10, threshold: 3, influence: 0.2)
// Print output to console
print("\nOutput: \n ")
for i in 0...signals.count - 1 {
    print("Data point \(i)\t\t sample: \(samples[i]) \t signal: \(signals[i])\n")
}

// Raw data for creating a plot in Excel
print("\n \n Raw data for creating a plot in Excel: \n ")
for i in 0...signals.count - 1 {
    print("\(i+1)\t\(samples[i])\t\(signals[i])\t\(avgFilter[i])\t\(stdFilter[i])\n")
}

With the result for the sample data (for lag = 10, threshold = 3, influence = 0.2):

Update

You can improve the performance of the algorithm by using different values for the lag of the mean and the standard deviation. E.g.:

// Smooth z-score thresholding filter
func ThresholdingAlgo(y: [Double], lagMean: Int, lagStd: Int, threshold: Double, influenceMean: Double, influenceStd: Double) -> ([Int],[Double],[Double]) {

    // Create arrays
    var signals   = Array(repeating: 0, count: y.count)
    var filteredYmean = Array(repeating: 0.0, count: y.count)
    var filteredYstd = Array(repeating: 0.0, count: y.count)
    var avgFilter = Array(repeating: 0.0, count: y.count)
    var stdFilter = Array(repeating: 0.0, count: y.count)

    // Initialise variables
    for i in 0...lagMean-1 {
        signals[i] = 0
        filteredYmean[i] = y[i]
        filteredYstd[i] = y[i]
    }

    // Start filter
    avgFilter[lagMean-1] = arithmeticMean(array: subArray(array: y, s: 0, e: lagMean-1))
    stdFilter[lagStd-1] = standardDeviation(array: subArray(array: y, s: 0, e: lagStd-1))

    for i in max(lagMean,lagStd)...y.count-1 {
        if abs(y[i] - avgFilter[i-1]) > threshold*stdFilter[i-1] {
            if y[i] > avgFilter[i-1] {
                signals[i] = 1      // Positive signal
            } else {
                signals[i] = -1       // Negative signal
            }
            filteredYmean[i] = influenceMean*y[i] + (1-influenceMean)*filteredYmean[i-1]
            filteredYstd[i] = influenceStd*y[i] + (1-influenceStd)*filteredYstd[i-1]
        } else {
            signals[i] = 0          // No signal
            filteredYmean[i] = y[i]
            filteredYstd[i] = y[i]
        }
        // Adjust the filters
        avgFilter[i] = arithmeticMean(array: subArray(array: filteredYmean, s: i-lagMean, e: i))
        stdFilter[i] = standardDeviation(array: subArray(array: filteredYstd, s: i-lagStd, e: i))
    }

    return (signals,avgFilter,stdFilter)
}

Then using for example let (signals,avgFilter,stdFilter) = ThresholdingAlgo(y: samples, lagMean: 10, lagStd: 100, threshold: 2, influenceMean: 0.5, influenceStd: 0.1) can give a lot better results:

DEMO