从一组坐标中寻找线性回归方位角并没有给出预期的结果

Question

我有一系列坐标，我想为其找到线性回归，这样我就可以找到直线的方位角。我在这组坐标上使用 Swift 算法俱乐部的 Linear Regression algorithm：

51.48163827836369, -0.019464668521006683
51.481654860705667, -0.019452641350085287
51.481674140657908, -0.01943882290242982
51.481690344713748, -0.019423713183982727
51.481705506128442, -0.019419045258473489
51.481722553489625, -0.01940979681751287
51.48173752576799, -0.019412687104136239
51.48174912673462, -0.019409150632213823
51.4817646359283, -0.019389300889997685
51.481779676567427, -0.019388957697628939
51.481792568262044, -0.019402453532393338
51.481804168699682, -0.019415663863242116
51.481822746271966, -0.019423725406568337
51.481838162880258, -0.019428618620622728
51.481855587496689, -0.01942372804705883
51.481867836051975, -0.019430554178484272
51.481883136496599, -0.019432972610502475
51.481899505688553, -0.019425501321734373
51.481914919015246, -0.019424832166464512
51.481932613348015, -0.019457392982985766
51.481949346615657, -0.019472056279255412
51.481968002664601, -0.019458232757642691
51.481986902059589, -0.019446792660346546
51.482003086257393, -0.019433403642779012

使用 darrinward.com 将其添加到地图中，我得到了这条路径，大致向北移动：

因此，为了获得方位，我采用第一个坐标的经度回归（以获得纬度），:

    let regression = linearRegression(longitudes, latitudes)

    let firstRegressionCoordinate = CLLocationCoordinate2D(latitude: regression(firstCoordinate.longitude), longitude:firstCoordinate.longitude)
    let lastRegressionCoordinate = CLLocationCoordinate2D(latitude: regression(lastCoordinate.longitude), longitude: lastCoordinate.longitude)

    return firstRegressionCoordinate.bearing(to: lastRegressionCoordinate)

我的方位角功能正常，并给出了 156.20969º 的方位角。考虑到实际采用的路径，我期望的是接近 0º 或更接近 360º 的数字。

我觉得问题不在于线性回归算法或方位角函数，而在于我要求的坐标以确定方位角。我问错了吗？还有其他我应该做的不同的事情吗？

完整的代码，可以在 playground 中运行：

import CoreLocation

public extension FloatingPoint {
    public var degreesToRadians: Self { return self * .pi / 180 }
    public var radiansToDegrees: Self { return self * 180 / .pi }
}

extension CLLocationCoordinate2D {
    ///Returns the initial bearing of travel to another coordinate
    func bearing(to: CLLocationCoordinate2D) -> CLLocationDegrees {
        let fromLatRadians = latitude.degreesToRadians
        let fromLongRadians = longitude.degreesToRadians

        let toLatRadians = to.latitude.degreesToRadians
        let toLongRadians = to.longitude.degreesToRadians

        let y = sin(toLongRadians - fromLongRadians) * cos(toLatRadians)
        let x = cos(fromLatRadians) * sin(toLatRadians) - sin(fromLatRadians) * cos(toLatRadians) * cos(toLongRadians - fromLongRadians)

        var bearing = atan2(y, x).radiansToDegrees

        bearing = (bearing + 360.0).truncatingRemainder(dividingBy: 360.0)

        return bearing
    }
}

extension Array where Element == CLLocationCoordinate2D {
    func linearRegressionBearing() -> Double {
        var longitudes = [CLLocationDegrees]()
        var latitudes = [CLLocationDegrees]()

        for coordinate in self {
            longitudes.append(coordinate.longitude)
            latitudes.append(coordinate.latitude)
        }

        let regression = linearRegression(longitudes, latitudes)

        let firstCoordinate = CLLocationCoordinate2D(latitude: regression(self.first!.longitude), longitude: self.first!.longitude)
        let lastCoordinate = CLLocationCoordinate2D(latitude: regression(self.last!.longitude), longitude: self.last!.longitude)

        return firstCoordinate.bearing(to: lastCoordinate)
    }
}

// A closed form solution
func average(_ input: [Double]) -> Double {
    return input.reduce(0, +) / Double(input.count)
}

func multiply(_ a: [Double], _ b: [Double]) -> [Double] {
    return zip(a, b).map(*)
}

func linearRegression(_ xs: [Double], _ ys: [Double]) -> (Double) -> Double {
    let sum1 = average(multiply(xs, ys)) - average(xs) * average(ys)
    let sum2 = average(multiply(xs, xs)) - pow(average(xs), 2)
    let slope = sum1 / sum2
    let intercept = average(ys) - slope * average(xs)
    return { x in intercept + slope * x }
}

let coordinates = [
    CLLocationCoordinate2D(latitude: 51.48163827836369, longitude: -0.019464668521006683),
    CLLocationCoordinate2D(latitude: 51.481654860705667, longitude: -0.019452641350085287),
    CLLocationCoordinate2D(latitude: 51.481674140657908, longitude: -0.01943882290242982),
    CLLocationCoordinate2D(latitude: 51.481690344713748, longitude: -0.019423713183982727),
    CLLocationCoordinate2D(latitude: 51.481705506128442, longitude: -0.019419045258473489),
    CLLocationCoordinate2D(latitude: 51.481722553489625, longitude: -0.01940979681751287),
    CLLocationCoordinate2D(latitude: 51.48173752576799, longitude: -0.019412687104136239),
    CLLocationCoordinate2D(latitude: 51.48174912673462, longitude: -0.019409150632213823),
    CLLocationCoordinate2D(latitude: 51.4817646359283, longitude: -0.019389300889997685),
    CLLocationCoordinate2D(latitude: 51.481779676567427, longitude: -0.019388957697628939),
    CLLocationCoordinate2D(latitude: 51.481792568262044, longitude: -0.019402453532393338),
    CLLocationCoordinate2D(latitude: 51.481804168699682, longitude: -0.019415663863242116),
    CLLocationCoordinate2D(latitude: 51.481822746271966, longitude: -0.019423725406568337),
    CLLocationCoordinate2D(latitude: 51.481838162880258, longitude: -0.019428618620622728),
    CLLocationCoordinate2D(latitude: 51.481855587496689, longitude: -0.01942372804705883),
    CLLocationCoordinate2D(latitude: 51.481867836051975, longitude: -0.019430554178484272),
    CLLocationCoordinate2D(latitude: 51.481883136496599, longitude: -0.019432972610502475),
    CLLocationCoordinate2D(latitude: 51.481899505688553, longitude: -0.019425501321734373),
    CLLocationCoordinate2D(latitude: 51.481914919015246, longitude: -0.019424832166464512),
    CLLocationCoordinate2D(latitude: 51.481932613348015, longitude: -0.019457392982985766),
    CLLocationCoordinate2D(latitude: 51.481949346615657, longitude: -0.019472056279255412),
    CLLocationCoordinate2D(latitude: 51.481968002664601, longitude: -0.019458232757642691),
    CLLocationCoordinate2D(latitude: 51.481986902059589, longitude: -0.019446792660346546),
    CLLocationCoordinate2D(latitude: 51.482003086257393, longitude: -0.019433403642779012)
]

coordinates.linearRegressionBearing()

Answer 1

我确认您的轴承功能正常工作。但是，您获得的纬度回归值为 51.45437262420372 - 这是回归线与经度 = 0.0 相交的位置。我认为您需要同时使用回归线的交点和斜率来计算拟合的经纬度值（在您的地图区域内），然后使用这些值找到方位角。

后期补充：这里的一个重要问题是经度坐标与距离的关系不是线性的——经度覆盖的距离随纬度而变化。我认为最简单的解决方案就是将您的 lat,lon 值数组转换为 lat,normalisedLon 值，其中 normalisedLon = lon * cosine(lat).

执行此操作时，从第一个坐标到最后一个坐标的路径角度 = 3.055 度（在北方以东，这似乎是正确的）。

完成此操作后，回归线的斜率实际上将代表您正在寻找的方向，即 heading = atan(slope)。

但是，您的回归代码似乎对我不起作用。它也应该给出非常接近 3 度的结果，但距离很远。

从一组坐标中寻找线性回归方位角并没有给出预期的结果

Finding linear regression bearing from set of coordinates isn't giving desired result

linear-regression

cllocation

ios

cllocationcoordinate2d

swift