Scipy：optimize.fmin和optimize.leastsq之间的区别

Question

scipy的optimize.fmin和optimize.leastsq有什么区别？它们在这个示例页面中使用方式似乎基本相同。唯一的区别是，leastsq实际上会自行计算平方和（正如其名称所示），而使用fmin时必须手动计算。除此之外，这两个函数是否等效？

- gandi2223

Answer 1

不同的算法在底层运行。

fmin使用单纯形法; leastsq使用最小二乘拟合。

- duffymo

---

谢谢你，duffymo。那么，选择最小化算法的最佳方法是什么？我已经尝试过optimize.leastsq和optimize.fmin_slsqp，但在某些情况下，结果略有不同。是否有一种“科学”的方法来选择正确的例程，还是只是试错，看哪个对于给定的数据集效果最好？ - gandi2223

试错和判断。在每种情况下可能没有唯一的“正确”答案。 - duffymo

Answer 2

仅补充一些信息，我正在开发一个模块来拟合双指数函数，leastsq和minimize之间的时间差似乎几乎是100倍。请查看下面的代码以获取更多详细信息。

我使用了一个双指数曲线，它是两个指数的和，模型函数有4个参数要拟合。S，f，D_star和D。

所有默认拟合参数都已使用

S [f e^(-x * D_star) + (1 - f) e^(-x * D)]

('Time taken for minimize:', 0.011617898941040039)
('Time taken for leastsq :', 0.0003180503845214844)

所使用的代码：

import numpy as np
from scipy.optimize import minimize, leastsq
from time import time


def ivim_function(params, bvals):
    """The Intravoxel incoherent motion (IVIM) model function.

        S(b) = S_0[f*e^{(-b*D\*)} + (1-f)e^{(-b*D)}]

        S_0, f, D\* and D are the IVIM parameters.

    Parameters
    ----------
        params : array
                parameters S0, f, D_star and D of the model

        bvals : array
                bvalues

    References
    ----------
    .. [1] Le Bihan, Denis, et al. "Separation of diffusion
               and perfusion in intravoxel incoherent motion MR
               imaging." Radiology 168.2 (1988): 497-505.
    .. [2] Federau, Christian, et al. "Quantitative measurement
               of brain perfusion with intravoxel incoherent motion
               MR imaging." Radiology 265.3 (2012): 874-881.
    """
    S0, f, D_star, D = params
    S = S0 * (f * np.exp(-bvals * D_star) + (1 - f) * np.exp(-bvals * D))
    return S


def _ivim_error(params, bvals, signal):
    """Error function to be used in fitting the IVIM model
    """
    return (signal - ivim_function(params, bvals))


def sum_sq(params, bvals, signal):
    """Sum of squares of the errors. This function is minimized"""
    return np.sum(_ivim_error(params, bvals, signal)**2)

x0 = np.array([100., 0.20, 0.008, 0.0009])
bvals = np.array([0., 10., 20., 30., 40., 60., 80., 100.,
                  120., 140., 160., 180., 200., 220., 240.,
                  260., 280., 300., 350., 400., 500., 600.,
                  700., 800., 900., 1000.])
data = ivim_function(x0, bvals)

optstart = time()
opt = minimize(sum_sq, x0, args=(bvals, data))
optend = time()
time_taken = optend - optstart
print("Time taken for opt:", time_taken)


lstart = time()
lst = leastsq(_ivim_error,
              x0,
              args=(bvals, data),)
lend = time()
time_taken = lend - lstart
print("Time taken for leastsq :", time_taken)

print('Parameters estimated using minimize :', opt.x)
print('Parameters estimated using leastsq :', lst[0])

- Shahnawaz Ahmed