scipy#
稀疏矩阵#
import numpy as np
from scipy import sparse
# 1️⃣ 记录非零元素的行索引
row = np.array([0, 2, 2, 4]) # 3 在第0行，4 在第2行，5 在第2行，6 在第4行
# 2️⃣ 记录非零元素的列索引
col = np.array([2, 0, 3, 2]) # 3 在第2列，4 在第0列，5 在第3列，6 在第2列
# 3️⃣ 记录非零元素的值
data = np.array([3, 4, 5, 6]) # 这些索引位置对应的值
# 4️⃣ 生成 5x5 的稀疏矩阵
matrix = sparse.coo_matrix((data, (row, col)), shape=(5, 5))
# 打印矩阵
print(matrix.toarray()) # 转换回普通的二维数组看下
print(matrix.row)
print(matrix.col)
print(matrix.data)
[[0 0 3 0 0]
[0 0 0 0 0]
[4 0 0 5 0]
[0 0 0 0 0]
[0 0 6 0 0]]
[0 2 2 4]
[2 0 3 2]
[3 4 5 6]
概率分布#
norm为例
from scipy.stats import norm
# help
print(norm.__doc__)
A normal continuous random variable.
The location (``loc``) keyword specifies the mean.
The scale (``scale``) keyword specifies the standard deviation.
As an instance of the `rv_continuous` class, `norm` object inherits from it
a collection of generic methods (see below for the full list),
and completes them with details specific for this particular distribution.
Methods
-------
rvs(loc=0, scale=1, size=1, random_state=None)
Random variates.
pdf(x, loc=0, scale=1)
Probability density function.
logpdf(x, loc=0, scale=1)
Log of the probability density function.
cdf(x, loc=0, scale=1)
Cumulative distribution function.
logcdf(x, loc=0, scale=1)
Log of the cumulative distribution function.
sf(x, loc=0, scale=1)
Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(x, loc=0, scale=1)
Log of the survival function.
ppf(q, loc=0, scale=1)
Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, loc=0, scale=1)
Inverse survival function (inverse of ``sf``).
moment(order, loc=0, scale=1)
Non-central moment of the specified order.
stats(loc=0, scale=1, moments='mv')
Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(loc=0, scale=1)
(Differential) entropy of the RV.
fit(data)
Parameter estimates for generic data.
See `scipy.stats.rv_continuous.fit <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#scipy.stats.rv_continuous.fit>`__ for detailed documentation of the
keyword arguments.
expect(func, args=(), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
Expected value of a function (of one argument) with respect to the distribution.
median(loc=0, scale=1)
Median of the distribution.
mean(loc=0, scale=1)
Mean of the distribution.
var(loc=0, scale=1)
Variance of the distribution.
std(loc=0, scale=1)
Standard deviation of the distribution.
interval(confidence, loc=0, scale=1)
Confidence interval with equal areas around the median.
Notes
-----
The probability density function for `norm` is:
.. math::
f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}
for a real number :math:`x`.
The probability density above is defined in the "standardized" form. To shift
and/or scale the distribution use the ``loc`` and ``scale`` parameters.
Specifically, ``norm.pdf(x, loc, scale)`` is identically
equivalent to ``norm.pdf(y) / scale`` with
``y = (x - loc) / scale``. Note that shifting the location of a distribution
does not make it a "noncentral" distribution; noncentral generalizations of
some distributions are available in separate classes.
Examples
--------
>>> import numpy as np
>>> from scipy.stats import norm
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> mean, var, skew, kurt = norm.stats(moments='mvsk')
Display the probability density function (``pdf``):
>>> x = np.linspace(norm.ppf(0.01),
... norm.ppf(0.99), 100)
>>> ax.plot(x, norm.pdf(x),
... 'r-', lw=5, alpha=0.6, label='norm pdf')
Alternatively, the distribution object can be called (as a function)
to fix the shape, location and scale parameters. This returns a "frozen"
RV object holding the given parameters fixed.
Freeze the distribution and display the frozen ``pdf``:
>>> rv = norm()
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = norm.ppf([0.001, 0.5, 0.999])
>>> np.allclose([0.001, 0.5, 0.999], norm.cdf(vals))
True
Generate random numbers:
>>> r = norm.rvs(size=1000)
And compare the histogram:
>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
# 属性 方法
rv = norm()
print(dir(rv))
['__class__', '__delattr__', '__dict__', '__dir__', '__doc__', '__eq__', '__format__', '__ge__', '__getattribute__', '__getstate__', '__gt__', '__hash__', '__init__', '__init_subclass__', '__le__', '__lt__', '__module__', '__ne__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__sizeof__', '__str__', '__subclasshook__', '__weakref__', 'a', 'args', 'b', 'cdf', 'dist', 'entropy', 'expect', 'interval', 'isf', 'kwds', 'logcdf', 'logpdf', 'logsf', 'mean', 'median', 'moment', 'pdf', 'ppf', 'random_state', 'rvs', 'sf', 'stats', 'std', 'support', 'var']
主要方法#
rvs：随机变量 随机采样
pdf：概率密度函数。 （单点处的概率是0，但可以讨论区间上的概率,即pdf区间积分. pdf值表示一点处的可能性密度)
cdf：累积分布函数
sf：生存函数 (1-CDF)
ppf：百分点函数（CDF 的逆函数）
isf：逆生存函数（SF 的逆函数）
stats：返回均值、方差、（Fisher 的）偏度或（Fisher 的）峰度
moment：分布的非中心矩
norm.cdf(0)
np.float64(0.5)
norm.cdf([-1, 0, 1]) # 多个点的cdf
array([0.15865525, 0.5 , 0.84134475])
norm.mean()
np.float64(0.0)
norm.std()
np.float64(1.0)
norm.var()
np.float64(1.0)
norm.ppf(0.5) # 寻找对应分位点
np.float64(0.0)
norm.pdf(0)
np.float64(0.3989422804014327)
生成分布随机变量#
from numpy.random import default_rng
rng = default_rng()
norm.rvs(size = 5, random_state= rng)
array([ 1.54168013, 0.63091442, 0.63829445, -0.40356971, -1.61088559])
移位和缩放#
使用均值和标准差。
norm.stats(loc=3, scale=4)
(np.float64(3.0), np.float64(16.0))
形状参数#
gamma类似分布
from scipy.stats import gamma
gamma(a=1, scale=2.)
<scipy.stats._distn_infrastructure.rv_continuous_frozen at 0x1f2f3719d00>
zscore 标准分数，（常在正态分布）#
衡量一数值在总体中相对位置。
$$ z = \frac{x - \mu}{\sigma} $$
假设一个班的数学成绩均值是 70，标准差是 10。你考了 85 分：$z = \frac{85 - 70}{10} = 1.5$.表示你比平均水平高了 1.5 个标准差。