具有强制零截距的线性回归答案

【问题标题】：Linear regression with a forced zero intercept具有强制零截距的线性回归
【发布时间】：2018-06-11 13:41:58
【问题描述】：

我使用下面的最小二乘法来计算系数：

#Estimate coefficients of linear equation y = a + b*x
    def calc_coefficients(_x, _y):
        x, y = np.mean(_x), np.mean(_y)
        xy = np.mean(_x*_y)
        x2, y2 = np.mean(_x**2), np.mean(_y**2)
        n = len(_x)

        b = (xy - x*y) / (x2 - x**2)
        a = y - b*x
        sig_b = np.sqrt((y2-y**2)/(x2-x**2)-b**2) / np.sqrt(n)
        sig_a = sig_b * np.sqrt(x2 - x**2)

        return a, b, sig_a, sig_b

示例数据：

_x= [(0.009412743,0.014965211,0.013263312,0.013529132,0.009989368,0.013932615,0.020849682,0.010953529,0.003608903,0.007220992,0.012750529,0.021608436,0.031742052,0.022482958,0.021137599,0.018703295,0.021633681,0.019866029,0.020260629,0.034433715,0.009241074,0.012027059)]

_y = 0.294158677,0.359935335,0.313484808,0.301917271,0.169190763,0.486254864,0.305846328,0.347077387,0.188928817,0.422194367,0.41157232,0.39281496,0.497935681,0.34763333,0.281712023,0.352045535,0.339958296,0.395932086,0.359905526,0.450004349,0.395200865,0.365162443)]

但是，我需要 a（y 截距）为零。 (y = bx)。我试过使用：

np.linalg.lstsq(_x, _y)

但我收到此错误：

LinAlgError: 1-dimensional array given. Array must be two-dimensional

为y = bx 拟合数据的最佳方法是什么？

【问题讨论】：

标签： python numpy linear-regression

【解决方案1】：

错误是因为您传递了一个一维数组，它应该是一个形状为 (n, 1) 的二维数组 - 因此，一个具有 1 列的矩阵。你可以只做x.reshape(-1, 1)，但这里有一种方法可以让最小二乘拟合任意一组 x 度数：

import numpy as np
x = np.array([0, 1, 2, 3, 4, 5])
y = np.array([3, 6, 5, 7, 9, 1])
degrees = [1]       # list of degrees of x to use
matrix = np.stack([x**d for d in degrees], axis=-1)   # stack them like columns
coeff = np.linalg.lstsq(matrix, y)[0]    # lstsq returns some additional info we ignore
print("Coefficients", coeff)
fit = np.dot(matrix, coeff)
print("Fitted curve/line", fit)

您传递给lstsq 的矩阵应具有f(x) 形式的列，其中f 贯穿您在模型中允许的项。因此，如果它是一个通用线性模型，您将拥有x**0 和x**1。强制零拦截，它只是x**1。一般来说，这些也不一定是 x 的幂。

度数 = [1] 的输出，模型 y = bx

Coefficients [ 1.41818182]
Fitted curve/line [ 0.          1.41818182  2.83636364  4.25454545  5.67272727  7.09090909]

度数 = [0, 1] 的输出，模型 y = a + bx

Coefficients [ 5.0952381   0.02857143]
Fitted curve/line [ 5.0952381   5.12380952  5.15238095  5.18095238  5.20952381  5.23809524]

【讨论】：