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9.Scipy常见距离度量公式

定位


基于 NumPy 的高级科学计算工具库。数值计算库,在numPy库的基础上增加了众多的数学、科学以及工程计算中常用的库函数。


SciPy的中文翻译是“ Scientific Python”,发音为“Sigh Pie”。


官方文档: https://scipy.org/。


为什么使用SciPy


SciPy基于NumPy开发,为何不直接使用NumPy?


SciPy已经优化并增加了NumPy和数据科学中经常使用的函数。


SciPy是一个开源的Python库,它是NumPy生态系统的核心组成部分之一,提供了大量高级数学、科学和工程计算功能。


方便、易于使用、专为科学和工程设计的Python工具包.


它包括统计,优化,整合,线性代数模块,傅里叶变换,信号和图像处理,常微分方程求解器等等。


功能


扩展 NumPy 的功能,提供更复杂的数学算法(如优化、积分、信号处理、统计等)。


模块化设计(例如 scipy.optimize、scipy.stats)。


依赖关系


完全依赖 NumPy,但 不包含 NumPy。


与 NumPy 是兄弟关系,共同构成科学计算的核心工具链。


SciPy是用哪种语言编写的


SciPy主要是用Python编写的,但也有一些片段是用C语言编写的。


Scipy的代码库在哪里?


SciPy的源代码位于这个github仓库 : https://github.com/scipy/scipy


安装 SciPy


# pip 安装最新版

pip install scipy


# conda 安装

conda install scipy

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检查 SciPy 版本


版本字符串存储在__verslon__属性中


方法 1:在 Python 中直接查看


在 Python 脚本或交互式终端(如 Jupyter Notebook)中运行:


import scipy

print(scipy.__version__)

# 示例输出: 1.11.1

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方法 2:命令行查询


根据你的包管理工具选择以下命令:


pip 用户:


pip show scipy

# 输出信息中查找 "Version: x.x.x"

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conda 用户:


conda list scipy

# 输出列表中显示版本号

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常量(constants)

由于SciPy更专注于科学的实现,它提供了许多内置的科学常量,当你从事数据科学工作时,这些常量会很有帮助。


from scipy import constants

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print(constants.pi)#3.141592653589793

# 查看所有常量列表

print(dir(constants))

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点击查看所有常量


['Avogadro', 'Boltzmann', 'Btu', 'Btu_IT', 'Btu_th', 'ConstantWarning', 'G', 'Julian_year', 'N_A', 'Planck', 'R', 'Rydberg', 'Stefan_Boltzmann', 'Wien', '__all__', '__builtins__', '__cached__', '__doc__', '__file__', '__loader__', '__name__', '__package__', '__path__', '__spec__', '_codata', '_constants', '_obsolete_constants', 'acre', 'alpha', 'angstrom', 'arcmin', 'arcminute', 'arcsec', 'arcsecond', 'astronomical_unit', 'atm', 'atmosphere', 'atomic_mass', 'atto', 'au', 'bar', 'barrel', 'bbl', 'blob', 'c', 'calorie', 'calorie_IT', 'calorie_th', 'carat', 'centi', 'codata', 'constants', 'convert_temperature', 'day', 'deci', 'degree', 'degree_Fahrenheit', 'deka', 'dyn', 'dyne', 'e', 'eV', 'electron_mass', 'electron_volt', 'elementary_charge', 'epsilon_0', 'erg', 'exa', 'exbi', 'femto', 'fermi', 'find', 'fine_structure', 'fluid_ounce', 'fluid_ounce_US', 'fluid_ounce_imp', 'foot', 'g', 'gallon', 'gallon_US', 'gallon_imp', 'gas_constant', 'gibi', 'giga', 'golden', 'golden_ratio', 'grain', 'gram', 'gravitational_constant', 'h', 'hbar', 'hectare', 'hecto', 'horsepower', 'hour', 'hp', 'inch', 'k', 'kgf', 'kibi', 'kilo', 'kilogram_force', 'kmh', 'knot', 'lambda2nu', 'lb', 'lbf', 'light_year', 'liter', 'litre', 'long_ton', 'm_e', 'm_n', 'm_p', 'm_u', 'mach', 'mebi', 'mega', 'metric_ton', 'micro', 'micron', 'mil', 'mile', 'milli', 'minute', 'mmHg', 'mph', 'mu_0', 'nano', 'nautical_mile', 'neutron_mass', 'nu2lambda', 'ounce', 'oz', 'parsec', 'pebi', 'peta', 'physical_constants', 'pi', 'pico', 'point', 'pound', 'pound_force', 'precision', 'proton_mass', 'psi', 'pt', 'quecto', 'quetta', 'ronna', 'ronto', 'short_ton', 'sigma', 'slinch', 'slug', 'speed_of_light', 'speed_of_sound', 'stone', 'survey_foot', 'survey_mile', 'tebi', 'tera', 'test', 'ton_TNT', 'torr', 'troy_ounce', 'troy_pound', 'u', 'unit', 'value', 'week', 'yard', 'year', 'yobi', 'yocto', 'yotta', 'zebi', 'zepto', 'zero_Celsius', 'zetta']

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单位类别

单位被置于这些类别之下:


公制长度 二进制 质量 角度


时间 英制长度 压力 体积


速度 温度 能量 功率 力


点击查看全部单位类型


#1 公制长度

#返回指定的单位,以米为单位(例如,centi#返回0.01)。

constants.kilo#1000.0


#2 二进制单位

#返回以字节为单位的指定单位(例如,kibi#返回1024)

print (constants.kibi)

print (constants.mebi)

print (constants.gibi)

print(constants.tebi)

print(constants.pebi)

print(constants.exbi)

print(constants.zebi)

print(constants.yobi)


#3 质量

#返回以公斤为单位的指定单位(例如,克#返回0.001)。

print(constants.gram)

print(constants.metric_ton)

print(constants.grain)

print(constants.lb)

print(constants.pound)

print(constants.oz)

print(constants.ounce)

print(constants.stone)

print(constants.long_ton)

print(constants.short_ton)

print(constants.troy_ounce)

print(constants.troy_pound)

print(constants.carat)

print(constants.atomic_mass)

print(constants.m_u)

print(constants.u)


#4 角度

#返回以弧度为单位的指定单位(例如,度#返回0.01745#329251994#3295)

print(constants.degree)

print(constants.arcmin)

print(constants.arcminute)

print(constants.arcsec)

print(constants.arcsecond)


#5 时间

#返回指定的单位,以秒为单位(例如,小时#返回#3600.0)

print(constants.minute)

print(constants.hour)

print(constants.day)

print(constants.week)

print(constants.year)

print(constants.Julian_year)


#6 英制长度

#返回指定的单位,以米为单位(例如,海里#返回1852.0)。

print(constants.inch)

print(constants.foot)

print(constants.yard)

print(constants.mile)

print(constants.mil)

print(constants.pt)

print(constants.point)

print(constants.survey_foot)

print(constants.survey_mile)

print(constants.nautical_mile)

print(constants.fermi)

print(constants.angstrom)

print(constants.micron)

print(constants.au)

print(constants.astronomical_unit)

print(constants.light_year)

print(constants.parsec)


#7 压力

#返回以帕斯卡为单位的指定单位(例如:psi#返回6894.75729#3168#361)。

print(constants.atm)

print(constants.atmosphere)

print(constants.bar)

print(constants.torr)

print(constants.mmHg)

print (constants.psi)


#8 面积

#返回以平方米为单位的指定单位(例如,公顷#返回10000.0)

print(constants.hectare)

print(constants.acre)


#9 体积

#返回以立方米为单位的指定单位(例如,升#返回0.001)。

print(constants.liter)

print(constants.litre)

print(constants.gallon)

print(constants.gallon_US)

print(constants.gallon_imp)

print(constants.fluid_ounce)

print(constants.fluid_ounce_US)

print(constants.fluid_ounce_imp)

print(constants.barrel)

print(constants.bbl)


##10 速度

#返回指定的单位,单位为米/秒(例如,音速#返回#340.5)

print(constants.kmh)

print (constants.mph)

print(constants.mach)

print(constants.speed_of_sound)

print(constants.knot)


#11 温度

#返回指定的开尔文单位(如季摄氏度#返回27#15)

print(constants.zero_Celsius)

print(constants.degree_Fahrenheit)


#12_能量

#返回以焦耳为单位的指定单位(如卡路里#返回4.184)。

print(constants.eV)

print(constants.electron_volt)

print(constants.calorie)

print(constants.calorie_th)

print(constants.calorie_IT)

print(constants.erg)

print(constants.Btu)

print(constants.Btu_IT)

print(constants.Btu_th)

print(constants.ton_TNT)


#13 功率

#返回以瓦为单位的指定单位(例如,马力#返回745.6998)

print(constants.hp)

print(constants.horsepower)


#14 力

#返回指定的牛顿单位(例如,千克力#返回9.80665)

print(constants.dyn)

print(constants.dyne)

print(constants.lbf)

print(constants.pound_force)

print(constants.kgf)

print(constants.kilogram_force)

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优化器 (optimize:优化和求根)

SciPy 的 scipy.optimize 模块提供了丰富的优化算法,适用于不同场景的数值优化问题。


是SciPy中定义的多个程序(方法或算法),它可以找到一个方程的根,或者找到一个函数的最小值等等。


方程求根(Root Finding)

NumPy能够找到多项式和线性方程的根,但它不能找到非线性方程的根,比如:x + cos(x)


适用场景:求解非线性方程组,如化学平衡计算。

算法:

root:支持多种方法(hybr, lm等)。

root函数初步使用

import numpy as np

from scipy.optimize import root


# 定义目标函数

def eqn(x):

   return np.array([x[0] + np.cos(x[0])])


# 初始猜测

x0 = np.array([0.0])


# 求解

sol = root(eqn, x0, method='hybr')


# 解析结果

if sol.success:

   print(f"解: x = {sol.x[0]:.5f}")

   print(f"残差: {sol.fun[0]:.2e}")

else:

   print("失败:", sol.message)

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解: x = -0.73909

残差: 0.00e+00

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optimze.root函数使用说明

参数说明:


目标函数 fun


一个代表方程的函数。


必须接受数组输入并返回数组输出,即使是一维问题。


import numpy as np


def eqn(x):

   return np.array([x[0] + np.cos(x[0])])  # 单变量问题,返回数组

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初始猜测 x0


一个对根的初始猜测(越接近真实值越好,就是你自己估计一下方程的解约等于多少)。


格式:必须为 NumPy 数组,即使单变量。

x0 = np.array([0.0])  # 初始猜测为数组形式

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选择求解方法 method


推荐方法:


单变量问题:method='hybr'(默认,高效稳定)。

多变量小规模问题:method='hybr' 或 method='lm'(最小二乘法)。

大规模/稀疏问题:method='krylov'(无需显式雅可比矩阵)。

sol = root(eqn, x0, method='hybr')

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提供雅可比矩阵 jac(可选)


解析雅可比加速收敛:

def jacobian(x):

   return np.array([[1 - np.sin(x[0])]])  # 导数为 1 - sin(x)


sol = root(eqn, x0, jac=jacobian, method='hybr')

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数值估计雅可比(默认):

sol = root(eqn, x0, jac=False)  # 默认使用数值差分估计

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 Jacobian,也称为雅可比行列式(Jacobian determinant),是由德国数学家 卡尔·古斯塔夫·雅各布·雅可比(Carl Gustav Jacob Jacobi)提出的数学概念。它用于描述由n个函数组成的系统在某一点的局部线性变换性质,其值为该点处偏导数构成的n×n行列式。


处理多变量方程组


示例:解方程组



def system_eqn(z):

   x, y = z

   return [x + y - 3, x**2 + y**2 - 9]


x0 = np.array([1.0, 1.0])  # 初始猜测为数组

sol = root(system_eqn, x0, method='hybr')

print(sol.x)  # 输出解:[0., 3.] 或 [3., 0.]

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传递额外参数 args


示例:解带参数的方程 x+acos(bx)=0

def eqn_param(x, a, b):

   return np.array([x[0] + a * np.cos(b * x[0])])


a_val, b_val = 2.0, 0.5

sol = root(eqn_param, x0=np.array([1.0]), args=(a_val, b_val))

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调整求解器选项 options


常用选项:

tol:容差(默认1e-8),控制精度。

maxiter:最大迭代次数(默认依赖方法)。

options = {'xtol': 1e-10, 'maxiter': 100}

sol = root(eqn, x0, method='hybr', options=options)

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解析结果对象

该函数返回一个包含解的信息的对象


关键属性:

sol.x:解的估计值。

sol.success:是否成功收敛(True/False)。

sol.message:终止原因的详细描述。

sol.fun:解处的残差(应接近零)。

if sol.success:

   print(f"解为 x = {sol.x[0]:.5f}, 残差 = {sol.fun[0]:.2e}")

else:

   print("求解失败:", sol.message)

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核心结果字段解析


字段名


值示例


含义


x


[-0.7391]


解的数值:方程 x+cos(x)=0 的根约为 −0.73908513。


fun


[ 0.000e+00]


函数值:在解处的方程值 f(x)=x+cos(x),接近 0 表示收敛成功。


success


True


收敛状态:True 表示算法成功找到满足容差的解。


message


'The solution converged.'


状态描述:明确说明收敛原因或失败信息。


status


1


状态码:1 表示成功收敛,其他值参考文档。


算法细节字段


字段名


值示例


技术含义


method


hybr


使用的算法:此处为改进的Powell混合方法(适合中小规模问题)。


nfev


9


函数调用次数:算法计算 f(x) 的总次数,反映计算成本。


fjac


[[-1.000e+00]]


雅可比矩阵:解处的近似雅可比矩阵(此处为1x1矩阵),用于评估局部曲率。


r


[-1.674e+00]


QR分解的上三角矩阵:用于算法内部迭代更新。


qtf


[-2.668e-13]


正交因子:反映梯度与残差的夹角,接近0表示迭代终止合理。


如何验证解的可靠性?


检查 success 字段:必须为 True。

确认 fun 接近0:示例中 0.000e+00 表明残差极小。

收敛消息:message 明确说明收敛,无警告。

物理意义验证:将解代入原方程检查合理性(如示例中 −0.739+cos(−0.739)≈0)。

常见问题处理


收敛失败(success=False):

调整初始猜测(x0 参数)。

尝试其他算法(如 method='lm' 用于非线性最小二乘)。

增大容差(tol 参数)。

结果不精确:减小 tol 参数(默认值为 1e-8)。

处理常见问题


收敛失败:

调整初始猜测:选择更接近真实解的初始值。

增大容差:options={'xtol': 1e-6}。

尝试不同方法:如 method='lm'。

数值不稳定:

检查目标函数是否可导或存在奇点。

使用更稳定的算法(如 method='hybr')。

掌握这些字段的解读,能有效诊断数值优化问题并提升计算结果的可信度!


 


优化器

初步使用

函数最小值

在这里,一个函数代表一条曲线,曲线有高点和低点,

高点被称为最大值,

低点被称为最小值,

整条曲线上的最高点被称为全局最大值,其余的被称为局部最大值整条曲线的最低点称为全局最小值,其余的称为局部最小值,


寻找最小值

我们可以使用 scipy.optimize.minimize() 函数来最小化函数(寻找最小值)。

minimize()函数接受以下参数:


fun--个代表方程的函数。

x0--根的初始猜测

method--要使用的方法的名称。合法值:CG、BFGS、牛顿-CG、L-BFGS-B、SLSQP

caliback--每次优化迭代后调用的函数,

options--一个定义额外参数的字典。

{


 "disp”:boolean -打印详细的描述


“gtol”:数字 -错误的容忍度


}


from scipy.optimize import minimize

import numpy as np

def eqn(x):

   return np.array([x[0]**2+x[0]+2])

x0=np.array([0])

mini=minimize(eqn,x0=x0,method='BFGS')

print(mini)

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查看打印结果


message: Optimization terminated successfully.

 success: True

  status: 0

     fun: 1.75

       x: [-5.000e-01]

     nit: 2

     jac: [ 0.000e+00]

hess_inv: [[ 5.000e-01]]

    nfev: 8

    njev: 4

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分类与核心算法

1. 无约束优化(Unconstrained Optimization)

适用场景:无限制的连续参数空间,如逻辑回归、神经网络训练。

常用算法:

方法名


特点


调用参数


BFGS


拟牛顿法,适用于中小规模光滑问题,需计算梯度。


method='BFGS'


L-BFGS-B


内存优化版BFGS,支持边界约束,适合大规模问题。


method='L-BFGS-B'


Nelder-Mead


单纯形法,无需梯度,适用于非光滑或高噪声问题。


method='Nelder-Mead'


CG


共轭梯度法,适合大规模稀疏问题。


method='CG'


示例代码(最小化Rosenbrock函数):

from scipy.optimize import minimize


def rosen(x):

   return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0)


result = minimize(rosen, x0=[0, 0], method='BFGS')

print(f"最优解: {result.x}, 迭代次数: {result.nit}")

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2. 约束优化(Constrained Optimization)

适用场景:带等式或不等式约束的问题,如SVM、资源分配。

常用算法:

方法名


特点


调用参数


SLSQP


序列最小二乘规划,支持多类型约束。


method='SLSQP'


trust-constr


信赖域算法,高精度但计算成本较高。


method='trust-constr'


示例代码(带不等式约束):


# 目标函数: min x1^2 + x2^2

# 约束: x1 + x2 >= 1

cons = {'type': 'ineq', 'fun': lambda x: x[0] + x[1] - 1}

result = minimize(lambda x: x[0]**2 + x[1]**2, x0=[0,0],  

                 constraints=cons, method='SLSQP')

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3. 全局优化(Global Optimization)

适用场景:多峰函数,避免陷入局部最优,如超参数搜索。

常用算法:

方法名


特点


differential_evolution


差分进化法,适用于连续、离散混合问题。


basinhopping


盆地跳跃法,结合局部搜索和随机跳跃。


示例代码:


from scipy.optimize import differential_evolution


bounds = [(-5, 5)] * 2  # 2维问题

result = differential_evolution(rosen, bounds)

print(f"全局最优解: {result.x}")

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4. 最小二乘问题(Least-Squares)

适用场景:拟合非线性方程,如曲线拟合、传感器校准。

算法:

least_squares:灵活处理边界和损失函数。

from scipy.optimize import least_squares


def model(params, t):

   A, k = params

   return A * np.exp(-k * t)


t_data = np.array([0, 1, 2, 3])

y_data = np.array([3.0, 1.5, 0.75, 0.35])

result = least_squares(lambda p: model(p, t_data) - y_data, x0=[2, 0.5])

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二、算法选择决策树

问题是否有约束?

是 → 选择 SLSQP 或 trust-constr。

否 → 进入下一步。

目标函数是否光滑(可导)?

是 → 使用梯度类方法(BFGS, CG)。

否 → 使用无梯度方法(Nelder-Mead)。

是否可能存在多个局部最优?

是 → 全局优化(differential_evolution)。

否 → 继续。

数据规模?

小规模 → BFGS。

大规模 → L-BFGS-B 或 CG。

三、性能优化技巧

梯度加速:提供解析梯度(jac参数)可减少30%以上计算时间。

def rosen_der(x):

   grad = np.zeros_like(x)

   grad[:-1] = -400*x[:-1]*(x[1:]-x[:-1]**2) - 2*(1-x[:-1])

   grad[1:] += 200*(x[1:]-x[:-1]**2)

   return grad


minimize(rosen, x0=[0,0], method='BFGS', jac=rosen_der)

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并行计算:对目标函数中独立计算部分使用joblib:

from joblib import Parallel, delayed


def parallel_objective(x):

   return sum(Parallel(n_jobs=4)(delayed(np.exp)(xi) for xi in x))

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四、调试与问题排查

收敛失败处理:

调整初始值:尝试不同初始猜测。

增大迭代次数:options={'maxiter': 1000}。

放宽容差:tol=1e-4。

数值不稳定:

数据归一化:将输入缩放到[0,1]或标准化。

添加正则化:在损失函数中加入L2项。

梯度验证:

import numpy as np


def eqn(x):

   return np.array([x[0] + np.cos(x[0])])  # 单变量问题,返回数组

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五、实战案例:逻辑回归参数估计

import numpy as np

from scipy.optimize import minimize


# 生成模拟数据

np.random.seed(42)

X = np.hstack([np.ones((100,1)), np.random.randn(100,2)])

y = (X @ np.array([-2, 3, -1]) + np.random.randn(100) > 0).astype(int)


# 定义损失函数和梯度

def sigmoid(z):

   return 1 / (1 + np.exp(-z))


def loss(theta, X, y):

   p = sigmoid(X @ theta)

   return -np.mean(y * np.log(p) + (1-y) * np.log(1-p))


def grad(theta, X, y):

   p = sigmoid(X @ theta)

   return X.T @ (p - y) / len(y)


# 优化

result = minimize(loss, x0=np.zeros(3), args=(X, y),  

                 method='BFGS', jac=grad)

print(f"估计参数: {result.x.round(2)}")

# 输出: 估计参数: [-1.92  2.89 -0.97] (接近真实值 [-2, 3, -1])

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总结

核心原则:根据问题类型选择算法,优先使用解析梯度加速。

扩展应用:结合 scipy 的线性代数模块 (scipy.linalg) 处理大规模问题。

进阶学习:探索 scipy.optimize 中的高级选项(如自定义回调函数、Hessian矩阵利用)。

稀疏矩阵

什么是稀疏数据

稀疏数据是指那些大部分没有使用的元素(不携带任何信息的元素)的数据,它可以是一个像这样的数组:

[1, 0, 2、0, 0, 3, 0, 0, 0,0,0, 0]

稀疏数据:是一个数据集,其中大部分项目的值为零


密集数组:是稀疏数组的反面:大部分数值不为零,

在科学计算中,当我们在线性代数中处理偏导时,我们会遇到稀疏数据,

如何处理稀疏数据

SciPy有一个模块,scipy.sparse,提供了处理稀疏数据的函数,

我们主要使用两种类型的稀疏矩阵。

CSC-压缩稀疏列。用于高效的算术,快速的列切分。

CSR-压缩稀疏行。用于快速的行切分,更快的矩阵向量乘积。


我们将在本教程中使用CSR矩阵


CSR矩阵

我们可以通过向函数scipy.sparse.csr_matrix()传递一个array来创建CSR矩阵。


arr2=np.array([[0,0,0],[0,0,1],[1,0,2]])

print(arr2)

#转化为csr

csr_arr=csr_matrix(arr2)

print(csr_arr)

'''

 (2, 0) 1

 (1, 2) 1

 (2, 2) 2

'''

# 计算非零元素个数

csr_arr=csr_matrix(arr2)

print(csr_arr.count_nonzero())

# 去除矩阵中的0条目

csr_arr=csr_matrix(arr2)

csr_arr.eliminate_zeros()

print(csr_arr)

# 去除重复条目

csr_arr=csr_matrix(arr2)

csr_arr.sum_duplicates()

print(csr_arr)

# 转化为csc矩阵

csr_arr=csr_matrix(arr2)

print(csr_arr.tocsc())

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图形  

使用图形

图是一种重要的数据结构。

SciPy为我们提供了scipy.sparse.csgraph模块来处理这种数据结构。

毗连矩阵

邻接矩阵是一个n*n矩阵,其中n是图中元素的数量。而数值代表元素之间的联系。


例子:




对于这样一个图,有元素A、B和C,其连接是:


A和B是以权重1连接的。

A和C以权重2相连

C和B没有连接。


邻接矩阵看起来像这样:


ABC


A:[0 1 2]


B:[1 0 0]


C:[2 0 0]


下面是一些最常用的处理邻接矩阵的方法。


连通区域


例如上面的ABS就构成一个联通区域,此时如果又有一个单独的D,或者D和E相连,那么D或者D和E又构成一个新的联通区域。

用connected_components()方法找到所有的连通区域。


import numpy as np  

from scipy.sparse import csr_matrix

from scipy.sparse.csgraph import connected_components

arr=np.array([

   [0,1,2],

   [1,0,0],

   [1,0,0]

])

graph=csr_matrix(arr)

print(graph)

# 打印连通域个数

print(connected_components(graph))

'''

 (0, 1) 1

 (0, 2) 2

 (1, 0) 1

 (2, 0) 1

 连通域个数,数组(该连通域内元素个数及维度,0代表1维)

(1, array([0, 0, 0]))

'''

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Dijkstra

使用Dijkstra方法来寻找图中从一个元素到另一个元素的最短路径它需要以下参数:


return_predecessors:布尔值(返回整个路径,否则为假)

indices:元素的索引,只返回该元素的所有路径

limit:路径的最大权重

from scipy.sparse.csgraph import dijkstra

print(dijkstra(graph,indices=0))

#[0. 1. 2.]A到A距离为0,到B距离为1,到C距离为2

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Floyd Warshall

使用foyd_warshall()方法来寻找所有元素对之间的最短路径  


from scipy.sparse.csgraph import floyd_warshall

print(floyd_warshall(graph))

'''

[[0. 1. 2.]

[1. 0. 3.]

[1. 2. 0.]]

'''

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Bellman Ford

bellman_ford()方法也可以找到所有元素对之间的最短路径,但这个方法也可以处理负的权重


 from scipy.sparse.csgraph import bellman_ford

print(bellman_ford(graph,indices=0))


from scipy.sparse.csgraph import bellman_ford

print(bellman_ford(graph,indices=0))

#[0. 1. 2.]

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深度优先

depth_first_order()方法返回一个节点的深度优先遍历.


遍历图形的起始元素:


from scipy.sparse.csgraph import depth_first_order

print(depth_first_order(graph,0))

#(array([0, 1, 2]), array([-9999,     0,     0]))【A的前一个节点,不存在,所以显示-9999,B的前一个节点是A,所以显示0,B的前一个节点也是A,也显示0】

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广度优先

breadth_first_order()方法返回一个节点的广度优先遍历。


遍历图形的起始元素。  


from scipy.sparse.csgraph import breadth_first_order

print(breadth_first_order(graph,0))

#(array([0, 1, 2]), array([-9999,     0,     0]))

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 空间数据

使用空间数据

空间数据是指在一个几何空间中表示的数据。例如,坐标系上的点。

我们在许多任务中处理空问数据问题,

例如,寻找一个点是否在一个边界内。

ScIPy为我们提供了scipy.spatial模块,它具有处理空间数据的功能。

三角测量

多边形的三角测量是将多边形划分为多个三角形,我们可以通过这些三角形计算出多边形的面积。


带点三角剖分是指创建由三角形组成的曲面,其中所有给定的点都在该曲面中任何一个三角形的至少一个顶点上。(就是把所有点都用上,任意一个点,至少是一个三角形的顶点)


通过点来生成三角形的方法是Delaunay()。


import numpy as np

import matplotlib.pyplot as plt

from scipy.spatial import Delaunay

points=np.array([

   [2,4],

   [3,4],

   [3,0],

   [2,2],

   [4,1]

])

simplices=Delaunay(points).simplices

print(simplices)

# 切片操作:

# points[:,0] → 取所有行的第0列 → [2,3,3,2,4]

# points[:,1] → 取所有行的第1列 → [4,4,0,2,1]

plt.triplot(points[:,0],points[:,1],simplices)

plt.scatter(points[:,0],points[:,1],color='r')

plt.show()

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 凸面体

凸面体是覆盖所有给定点的最小的多边形。

便用ConvexHul()方法来创建一个凸面体。


from scipy.spatial import ConvexHull

points=np.array([

   [2,4],

   [3,4],

   [3,0],

   [2,2],

   [4,1],

   [1,2],

   [5,0],

   [3,1],

   [1,2],

   [0,2]

])


simp=ConvexHull(points).simplices

print(simp)

'''

[[0 9]

[1 6]

[1 0]

[2 9]

[2 6]]

'''

plt.scatter(points[:,0],points[:,1])

for s in simp:

   plt.plot(points[s,0],points[s,1])

plt.show()

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KDTrees

KDTrees是一种为最近的邻居查询而优化的数据结构

例如,在一个点的集合中,使用KDTree我们可以有效地询问哪些点离某个给定的点最近


KDTree()方法返回一个KDTree对象

query()方法返回与最近的邻居的距离和邻居的位置。


from scipy.spatial import KDTree

points=np.array([[-2,-3],[2,3],[-2,3],[2,-3]])

kdtree=KDTree(points)

res=kdtree.query((1,1))

print(res)#(2.23606797749979, 1)距离点(1,1)最近的点是points[1],最短距离是:2.23606797749979

plt.scatter(points[:,0],points[:,1])

plt.scatter(1,1)

plt.show()

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距离矩阵

在数据科学中,有许多距离度量用于寻找两点之间的各种类型的距离,欧氏距离、余弦距离等。详参:常见距离度量公式


两个向里之间的距离可能不仅仅是它们之间的直线长度,它也可以是它们与原点之间的角度,或者需要的单位步数等。


许多机器学习算法的性能在很大程度上取决于距离矩阵。例如:"K邻近"或"K平均值"等


让我们来看看一些距离矩阵

欧氏距离 Euclidean Distance

找到给定点之间的欧氏距离。


from scipy.spatial.distance import euclidean

p1=(1,0)

p2=(10,2)

res=euclidean(p1,p2)

print(res)#9.219544457292887

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城市街区距离(曼哈顿距离)

在4个方向上计算的距离。

例如,我们只能移动:向上、向下、向右或向左,而不是斜向移动。


from scipy.spatial.distance import cityblock

p1=(1,0)

p2=(10,2)

res=cityblock(p1,p2)

print(res)#11

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 余弦距离


 ⼏何中,夹⻆余弦可⽤来衡量两个向量⽅向的差异;机器学习中,借⽤这⼀概念来衡量样本向量之间的差异。


from scipy.spatial.distance import cosine

p1=(1,0)

p2=(10,2)

res=cosine(p1,p2)

print(res)#0.019419324309079777

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5.

汉明距离Hamming距离


两个等⻓字符串s1与s2的汉明距离为:将其中⼀个变为另外⼀个,所需要作的最⼩字符替换次数。


例如:


The Hamming distance between "1011101" and "1001001" is 2.


The Hamming distance between "2143896" and "2233796" is 3.


The Hamming distance between "toned" and "roses" is 3.


 是指两个比特有差异的比特的比例。

它是衡量二进制序列距离的一种方法。


from scipy.spatial.distance import hamming

p1=(True,False,False)

p2=(True,False,True)

res=hamming(p1,p2)

print(res)#0.3333333333333333=有差异的位置数量除以总位置=1/3

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5.

MatLab  

使用Matlab数组

我们知道NumPy为我们提供了以Python的可读格式来保存数据的方法。但是SciPy也为我们提供了与Matlab的互操作性。SciPy为我们提供了scipy.io模块,它具有与Matlab数组一起工作的功能。

以Matlab格式导出数据

savemat()函数允许我们以Matlab格式导出数据。

该方法需要以下参数:


filename:用于保存数据的文件名。

mdict:一个包含数据的字典。

do_compression:一个布尔值,指定是否对结果进行压缩。默认为假。

from scipy import io

import numpy as np

arr=np.arange(1,10)

io.savemat(r'D:\data\jupyter\SciPy\test.mat',{'vec':arr})

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从Matlab格式导入数据

loadmat()函数允许我们从Matlab文件中导入数据。


该函数需要一个必要的参数。


flename-保存数据的文件名


它将返回一个结构化数组,其键是变量名称,相的值是变量值。


mydata=io.loadmat(r'D:\data\jupyter\SciPy\test.mat')

print(mydata)#{'__header__': b'MATLAB 5.0 MAT-file Platform: nt, Created on: Wed May 21 16:03:03 2025', '__version__': '1.0', '__globals__': [], 'vec': array([[1, 2, 3, 4, 5, 6, 7, 8, 9]])}

# 从上述打印结果可知,我们存储时数组原本是一维的,但是在提取时变成了2维的,我们可以添加参数解决该问题

mydata=io.loadmat(r'D:\data\jupyter\SciPy\test.mat',squeeze_me=True)#'vec': array([1, 2, 3, 4, 5, 6, 7, 8, 9]

print(mydata)

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 插值法

什么是插值法?

插值法是一种在给定的点之间生成点的方法。


例如:对于点1和2,我们可以通过插值找到点1.33和1.66。


插值法有很多用途,在机器学习中,我们经常处理数据集中的缺失数据,内插法经常被用来替代这些值。


这种填补数值的方法被称为归因法。

除了估算,内插法还经常用于我们需要平滑数据集中的离散点的地方。

如何在SciPy中实现它?

SciPy为我们提供了-个叫scipy.interpolate的模块,它有许多处理插值的函数。

一维插值

函数interp1d()是用来插值一个有1个变量的分布

它接收x和y点并返回一个可调用的函数,可以用新的x来调用并返回相应的y。


from scipy.interpolate import interp1d

import numpy as np

x=np.arange(10)

y=x**2+np.sin(x)+1

fun=interp1d(x,y)

z=np.arange(2.1,3,0.1)

newarr=fun(z)


print(x)#[0 1 2 3 4 5 6 7 8 9]

print(y)#[ 1. 2.84147098  5.90929743 10.14112001 16.2431975  25.04107573 36.7205845  50.6569866  65.98935825 82.41211849]

print(z)#[2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9]

print(newarr)#[6.33247968 6.75566194 7.1788442  7.60202646 8.02520872 8.44839098 8.87157323 9.29475549 9.71793775]

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样条插值

在一维插值中,点被拟合为一条曲线,而在样条插值中,点被拟合为一个用多项式定义的片状函数,称为样条.


UnivariateSpline()函数接收x和y并产生一个可调用的函数,可以用新的x调用。  


from scipy.interpolate import UnivariateSpline

import numpy as np

x=np.arange(10)

y=x**2+np.sin(x)+1

fun=UnivariateSpline(x,y)

z=np.arange(2.1,3,0.1)

newarr=fun(z)


print(x)#[0 1 2 3 4 5 6 7 8 9]

print(y)[ 1. 2.84147098  5.90929743 10.14112001 16.2431975  25.04107573 36.7205845  50.6569866  65.98935825 82.41211849]

print(z)#[2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9]

print(newarr)[5.62826474 6.03987348 6.47131994 6.92265019 7.3939103  7.88514634 8.39640439 8.92773053 9.47917082]

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10.

11.

12.

径向基函数的插值

径向基函数是一个对应于固定参考点定义的函数。

Rbf()函数也以xs和ys为参数,并产生一个可调用的函数,可以用新的xs调用,


from scipy.interpolate import Rbf

import numpy as np

x=np.arange(10)

y=x**2+np.sin(x)+1

fun=Rbf(x,y)

z=np.arange(2.1,3,0.1)

newarr=fun(z)


print(x)#[0 1 2 3 4 5 6 7 8 9]

print(y)#[ 1. 2.84147098  5.90929743 10.14112001 16.2431975  25.04107573 36.7205845  50.6569866  65.98935825 82.41211849]

print(z)#[2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9]

print(newarr)#[6.25748981 6.62190817 7.00310702 7.40121814 7.8161443  8.24773402 8.69590519 9.16070828 9.64233874]

-----------------------------------

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9.Scipy常见距离度量公式

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