python中三维曲面的最小二乘拟合
我想用一些数据拟合我的曲面方程。我已经尝试了scipy.optimize.leastsq,但由于我无法指定边界,它会给我一个无法使用的结果。我还尝试了scipy.optimize.least_squares,但它给了我一个错误:python中三维曲面的最小二乘拟合,python,least-squares,data-fitting,Python,Least Squares,Data Fitting,我想用一些数据拟合我的曲面方程。我已经尝试了scipy.optimize.leastsq,但由于我无法指定边界,它会给我一个无法使用的结果。我还尝试了scipy.optimize.least_squares,但它给了我一个错误: ValueError: too many values to unpack 我的方程式是: f(x,y,z)=(x-A+y-B)/2+sqrt(((x-A-y+B)/2)^2+C*z^2) 应找到参数A、B、C,以便当以下点用于x、y、z时,上述方程尽可能接近于零:
ValueError: too many values to unpack
我的方程式是:
f(x,y,z)=(x-A+y-B)/2+sqrt(((x-A-y+B)/2)^2+C*z^2)
应找到参数A、B、C,以便当以下点用于x、y、z时,上述方程尽可能接近于零:
[
[-0.071, -0.85, 0.401],
[-0.138, -1.111, 0.494],
[-0.317, -0.317, -0.317],
[-0.351, -2.048, 0.848]
]
边界将是A>0,B>0,C>1
我怎样才能得到这样的健康呢?python中最好的工具是什么。我搜索了有关如何拟合3d曲面的示例,但大多数涉及函数拟合的示例都是关于直线或平面拟合的。我编辑了此答案,以提供一个更一般的示例,说明如何使用scipy的general optimize.minimize方法以及scipy的optimize.least_squares方法解决此问题
首先让我们设置问题:
import numpy as np
import scipy.optimize
# ===============================================
# SETUP: define common compoments of the problem
def our_function(coeff, data):
"""
The function we care to optimize.
Args:
coeff (np.ndarray): are the parameters that we care to optimize.
data (np.ndarray): the input data
"""
A, B, C = coeff
x, y, z = data.T
return (x - A + y - B) / 2 + np.sqrt(((x - A - y + B) / 2) ** 2 + C * z ** 2)
# Define some training data
data = np.array([
[-0.071, -0.85, 0.401],
[-0.138, -1.111, 0.494],
[-0.317, -0.317, -0.317],
[-0.351, -2.048, 0.848]
])
# Define training target
# This is what we want the target function to be equal to
target = 0
# Make an initial guess as to the parameters
# either a constant or random guess is typically fine
num_coeff = 3
coeff_0 = np.ones(num_coeff)
# coeff_0 = np.random.rand(num_coeff)
这不是严格意义上的最小二乘法,但是像这样的东西怎么样? 这个解决方案就像是向问题扔一把大锤。使用SVD解算器可能有一种使用最小二乘法更有效地获得解决方案的方法,但如果您只是在寻找答案,scipy.optimize.minimize将为您找到一个答案
# ===============================================
# FORMULATION #1: a general minimization problem
# Here the bounds and error are all specified within the general objective function
def general_objective(coeff, data, target):
"""
General function that simply returns a value to be minimized.
The coeff will be modified to minimize whatever the output of this function
may be.
"""
# Constraints to keep coeff above 0
if np.any(coeff < 0):
# If any constraint is violated return infinity
return np.inf
# The function we care about
prediction = our_function(coeff, data)
# (optional) L2 regularization to keep coeff small
# (optional) reg_amount = 0.0
# (optional) reg = reg_amount * np.sqrt((coeff ** 2).sum())
losses = (prediction - target) ** 2
# (optional) losses += reg
# Return the average squared error
loss = losses.sum()
return loss
general_result = scipy.optimize.minimize(general_objective, coeff_0,
method='Nelder-Mead',
args=(data, target))
# Test what the squared error of the returned result is
coeff = general_result.x
general_output = our_function(coeff, data)
print('====================')
print('general_result =\n%s' % (general_result,))
print('---------------------')
print('general_output = %r' % (general_output,))
print('====================')
我在文档中发现,要使其适应实际的最小二乘法,只需指定计算残差的函数
# ===============================================
# FORMULATION #2: a special least squares problem
# Here all that is needeed is a function that computes the vector of residuals
# the optimization function takes care of the rest
def least_squares_residuals(coeff, data, target):
"""
Function that returns the vector of residuals between the predicted values
and the target value. Here we want each predicted value to be close to zero
"""
A, B, C = coeff
x, y, z = data.T
prediction = our_function(coeff, data)
vector_of_residuals = (prediction - target)
return vector_of_residuals
# Here the bounds are specified in the optimization call
bound_gt = np.full(shape=num_coeff, fill_value=0, dtype=np.float)
bound_lt = np.full(shape=num_coeff, fill_value=np.inf, dtype=np.float)
bounds = (bound_gt, bound_lt)
lst_sqrs_result = scipy.optimize.least_squares(least_squares_residuals, coeff_0,
args=(data, target), bounds=bounds)
# Test what the squared error of the returned result is
coeff = lst_sqrs_result.x
lst_sqrs_output = our_function(coeff, data)
print('====================')
print('lst_sqrs_result =\n%s' % (lst_sqrs_result,))
print('---------------------')
print('lst_sqrs_output = %r' % (lst_sqrs_output,))
print('====================')
这里的输出是:
====================
lst_sqrs_result =
active_mask: array([ 0, -1, 0])
cost: 6.197329866927735e-05
fun: array([ 0.00518416, -0.00564099, -0.00710112, 0.00385024])
grad: array([ -4.61826888e-09, 3.70771396e-03, 1.26659198e-09])
jac: array([[-0.72611025, -0.27388975, 0.13653112],
[-0.74479565, -0.25520435, 0.1644325 ],
[-0.35777232, -0.64222767, 0.11601263],
[-0.77338046, -0.22661953, 0.27104366]])
message: '`gtol` termination condition is satisfied.'
nfev: 13
njev: 13
optimality: 4.6182688779976278e-09
status: 1
success: True
x: array([ 2.46392438e-01, 5.39025298e-17, 1.71555150e+00])
---------------------
lst_sqrs_output = array([ 0.00518416, -0.00564099, -0.00710112, 0.00385024])
====================
你写的代码给了我正确的结果!然而,我真的不明白你是怎么强加边界条件的?哎呀。我忘了做那件事。¯\_(ツ)_/“”。我认为这是可行的,因为我最初的猜测是使用正值,L2正则化鼓励系数收缩,所以当得到一个好的答案时,它就停止了。尽管该方法非常通用,所以可以包括if-np.any(coeff<0):返回np.inf,这将施加约束。啊,好的,这是有意义的。谢谢!我知道了最小二乘法实现的工作原理。您指定的函数需要计算残差。在这种情况下,可以显式指定边界。
====================
lst_sqrs_result =
active_mask: array([ 0, -1, 0])
cost: 6.197329866927735e-05
fun: array([ 0.00518416, -0.00564099, -0.00710112, 0.00385024])
grad: array([ -4.61826888e-09, 3.70771396e-03, 1.26659198e-09])
jac: array([[-0.72611025, -0.27388975, 0.13653112],
[-0.74479565, -0.25520435, 0.1644325 ],
[-0.35777232, -0.64222767, 0.11601263],
[-0.77338046, -0.22661953, 0.27104366]])
message: '`gtol` termination condition is satisfied.'
nfev: 13
njev: 13
optimality: 4.6182688779976278e-09
status: 1
success: True
x: array([ 2.46392438e-01, 5.39025298e-17, 1.71555150e+00])
---------------------
lst_sqrs_output = array([ 0.00518416, -0.00564099, -0.00710112, 0.00385024])
====================