1.numpy.random.normal

numpy.random.normal

numpy.random.normal(loc=0.0, scale=1.0, size=None)

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [R250]

, is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution 

[R250].

Parameters:	loc : float Mean (“centre”) of the distribution. scale : float Standard deviation (spread or “width”) of the distribution. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

See also

scipy.stats.distributions.norm

probability density function, distribution or cumulative density function, etc.

Notes

The probability density for the Gaussian distribution is

where  is the mean and  the standard deviation. The square of the standard deviation, , is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at  and  [R250]). This implies that numpy.random.normal is more likely to return samples lying close to the mean, rather than those far away.

References

[R250]

(1, 2, 3, 4) P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.

Examples

Draw samples from the distribution:

>>>

>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)

Verify the mean and the variance:

>>>

>>> abs(mu - np.mean(s)) < 0.01
True

>>>

>>> abs(sigma - np.std(s, ddof=1)) < 0.01
True

Display the histogram of the samples, along with the probability density function:

>>>

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
...          linewidth=2, color='r')
>>> plt.show()

import numpy as np
np.random.randn(2,3)

array([[ 0.59941534,  1.0991949 ,  1.36316028],
       [-0.01979197,  1.30783162, -0.69808199]])

意思是從標準正太分佈中隨機抽取。

3.scipy.optimize.leastsq

最小二乘法

import numpy as np
from scipy.optimize import leastsq


#待擬合的函式，x是變數，p是引數
def fun(x, p):
    a, b = p
    return a*x + b


#計算真實資料和擬合數據之間的誤差，p是待擬合的引數，x和y分別是對應的真實資料
def residuals(p, x, y):
    return fun(x, p) - y


#一組真實資料，在a=2, b=1的情況下得出
x1 = np.array([1, 2, 3, 4, 5, 6], dtype=float)
y1 = np.array([3, 5, 7, 9, 11, 13], dtype=float)


#呼叫擬合函式，第一個引數是需要擬合的差值函式，第二個是擬合初始值，第三個是傳入函式的其他引數
r = leastsq(residuals, [1, 1], args=(x1, y1))


#列印結果，r[0]儲存的是擬合的結果，r[1]、r[2]代表其他資訊
print r[0]


執行之後，擬合結果是


[2. 1.]


但是在這次實際的使用過程中，我擬合的函式不是這樣簡單的，其中的一個難點是待擬合函式是一個分段函式，需要判斷自變數的值，然後給出不同的函式方程式，舉個例子, 這樣一個分段函式:當x > 3時，y = ax + b, 當x <= 3 時，y = ax – b, 用Python程式碼寫一下：


def fun(x, p):
    a, b = p
    if (x > 3):
        return a*x + b
    else:
        return a*x - b


如果我們還是使用原來的差值函式進行擬合，會得到這樣的錯誤：


ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()


原因很簡單，我們現在的fun函式只能計算單個值了，如果傳入的還是一個array，自然就會報錯。那麼怎麼辦呢？我也很鬱悶，於是在scipy的maillist裡尋求幫助, 外國牛牛們都很熱心，很快就指出了問題。其實是我對於差值函式理解錯了，leastsq函式所要傳入的差值函式需要返回的其實是一個array, 於是我們可以這樣修改差值函式：


def residuals(p, x, y):
    temp = np.array([0,0,0,0,0,0],dtype=float)
    for i in range(0, len(x)):
        temp[i] = fun(x[i], p)
    return temp - y

4.

import numpy as np  #慣例
import scipy as sp  #慣例
from scipy.optimize import leastsq #這裡就是我們要使用的最小二乘的函式
import pylab as pl
 
m = 9  #多項式的次數
 
def real_func(x):
    return np.sin(2*np.pi*x) #sin(2 pi x)
 
def fake_func(p, x):
    f = np.poly1d(p) #多項式分佈的函式
    return f(x)
 
#殘差函式
def residuals(p, y, x):
    return y - fake_func(p, x)
 
#隨機選了9個點，作為x
x = np.linspace(0, 1, 9)
#畫圖的時候需要的“連續”的很多個點
x_show = np.linspace(0, 1, 1000)
 
y0 = real_func(x)
#加入正態分佈噪音後的y
y1 = [np.random.normal(0, 0.1) + y for y in y0]
 
#先隨機產生一組多項式分佈的引數
p0 = np.random.randn(m)


plsq = leastsq(residuals, p0, args=(y1, x))
 
print ('Fitting Parameters ：', plsq[0]) #輸出擬合引數
 
pl.plot(x_show, real_func(x_show), label='real')
pl.plot(x_show, fake_func(plsq[0], x_show), label='fitted curve')
pl.plot(x, y1, 'bo', label='with noise')
pl.legend()
pl.show()