拟合python中的周期图：scipy.interpolate.splrep的参数，曲线方程？

Question

[原始问题]

我需要一个曲线方程，随着时间的推移无限增加，基于以下数据。怎么做到？

[问题更新]

我需要为scipy.interpolate.splrep指定正确的参数。有人可以提供帮助吗？

另外，有没有办法从b样条的coeffs得到一个方程？

[替代问题]

如何使用傅里叶级数中的信号分解来拟合？

这个图似乎是线性图的组合，周期函数pf1增加了四倍，一个更大的周期函数导致pf1一次又一次地无限发生。情节的困难是为什么要问这个问题的原因。

数据：

Time elapsed in sec.    TX + RX Packets
(0,0)
(10,2422)
(20,2902)
(30,2945)
(40,3059)
(50,3097)
(60,4332)
(70,4622)
(80,4708)
(90,4808)
(100,4841)
(110,6081)
(120,6333)
(130,6461)
(140,6561)
(150,6585)
(160,7673)
(170,8091)
(180,8210)
(190,8291)
(200,8338)
(210,8357)
(220,8357)
(230,8414)
(240,8414)
(250,8414)
(260,8414)
(270,8414)
(280,8414)
(290,8471)
(300,8471)
(310,8471)
(320,8471)
(330,8471)
(340,8471)
(350,8471)
(360,8471)
(370,8471)
(380,8471)
(390,8471)
(400,8471)
(410,8471)
(420,8528)
(430,8528)
(440,8528)
(450,8528)
(460,8528)
(470,8528)
(480,8528)
(490,8528)
(500,8528)
(510,9858)
(520,10029)
(530,10129)
(540,10224)
(550,10267)
(560,11440)
(570,11773)
(580,11868)
(590,11968)
(600,12039)
(610,13141)

我的代码：

import numpy as np
import matplotlib.pyplot as plt

points = np.array(
    [(0,0), (10,2422), (20,2902), (30,2945), (40,3059), (50,3097), (60,4332), (70,4622), (80,4708), (90,4808), (100,4841), (110,6081), (120,6333), (130,6461), (140,6561), (150,6585), (160,7673), (170,8091), (180,8210), (190,8291), (200,8338), (210,8357), (220,8357), (230,8414), (240,8414), (250,8414), (260,8414), (270,8414), (280,8414), (290,8471), (300,8471), (310,8471), (320,8471), (330,8471), (340,8471), (350,8471), (360,8471), (370,8471), (380,8471), (390,8471), (400,8471), (410,8471), (420,8528), (430,8528), (440,8528), (450,8528), (460,8528), (470,8528), (480,8528), (490,8528), (500,8528), (510,9858), (520,10029), (530,10129), (540,10224), (550,10267), (560,11440), (570,11773), (580,11868), (590,11968), (600,12039), (610,13141)]
    )
# get x and y vectors
x = points[:,0]
y = points[:,1]

# calculate polynomial
z = np.polyfit(x, y, 3)
print z
f = np.poly1d(z)

# calculate new x's and y's
x_new = np.linspace(x[0], x[-1], 50)
y_new = f(x_new)

plt.plot(x,y,'o', x_new, y_new)
plt.xlim([x[0]-1, x[-1] + 1 ])
plt.show()

我的输出：

我的代码2：

import numpy as N
from scipy.interpolate import splprep, splev

x = N.array([0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420, 430, 440, 450, 460, 470, 480, 490, 500, 510, 520, 530, 540, 550, 560, 570, 580, 590, 600, 610])
y = N.array([0, 2422, 2902, 2945, 3059, 3097, 4332, 4622, 4708, 4808, 4841, 6081, 6333, 6461, 6561, 6585, 7673, 8091, 8210, 8291, 8338, 8357, 8357, 8414, 8414, 8414, 8414, 8414, 8414, 8471, 8471, 8471, 8471, 8471, 8471, 8471, 8471, 8471, 8471, 8471, 8471, 8471, 8528, 8528, 8528, 8528, 8528, 8528, 8528, 8528, 8528, 9858, 10029, 10129, 10224, 10267, 11440, 11773, 11868, 11968, 12039, 13141])

# spline parameters
s=1.0 # smoothness parameter
k=3 # spline order
nest=-1 # estimate of number of knots needed (-1 = maximal)

# find the knot points
tckp,u = splprep([x,y],s=s,k=k,nest=nest,quiet=True,per=1)

# evaluate spline, including interpolated points
xnew,ynew = splev(N.linspace(0,1,400),tckp)

import pylab as P
data,=P.plot(x,y,'bo-',label='data')
fit,=P.plot(xnew,ynew,'r-',label='fit')
P.legend()
P.xlabel('x')
P.ylabel('y')

P.show()

我的输出2：

Answer 1

看起来你有反应动力：

#%%
import numpy as np
from scipy.integrate import odeint
from scipy import optimize
from matplotlib import pyplot as plt
#%%
data = []
with open('data.txt', 'r') as f:
    for line in f:
        data.append(line.strip(' \n ()').split(','))
data = np.array(data,dtype=float)
data = data[0:-1].T

#%%
slope = np.diff(data[1])
index = np.where(slope>1000)
index = np.append(index, len(data[0]) -1 )
plt.plot(data[0],data[1],'.')
plt.plot(data[0,index],data[1,index],'ro')
plt.plot(data[0,1:],np.diff(data[1]))

从这里开始我假设反应从每个标记点（红色）开始。我确信代码可以编写得更清晰，但它是第一个快速而肮脏的黑客。您可以使用scipy curvefit或类似函数来拟合raction常量k

#%%
time = data[0,index]

def model(y,t,k):
    dydt = np.zeros(2)
    dydt[0] = -k*y[0]
    dydt[1] = k*y[0]

    return dydt


def res(k):
    y_hat = []
    t_hat = []
    for i in xrange(len(index) -1):
        '''
        I assume that at every timepoint the reaction is initiated by
        adding y[i + 1] - y[i] new datapackages. Over time they are 
        converted to sent packages. All packages which do not react,
        do not take part in the next cycle.
        '''
        y0 = [data[1, index[i+1]] - data[1, index[i]], 0]
        t0 = data[0, index[i]:index[i+1]]
        y_int,info = odeint(model, y0, t0, args=(k,), full_output = 1 )
        # I am not very happy about the construct below, but could
        # not find a better solution.
        y_hat.append(list(y_int[:,1]))
        t_hat.append(list(t0))
    return y_hat,t_hat

k = 2e-1
y,t = res(k)
''' It may be possible to play with y0[1] in the model in order 
to avoid the construct below. But since I started every reaction at y[1]=0
I have to add the last y value from the last step. This is a bit of a hack,
since data[1, index[i]] is not necessarily the corresponding solution. But hey, It seems to work.
'''
y_hat = [subitem + data[1, index[i]] for i,item in enumerate(y) for subitem in item]
t_hat = [subitem for item in t for subitem in item]
y_hat = np.array(y_hat,dtype=float)
t_hat = np.array(t_hat,dtype=float)


#%%
plt.plot(data[0],data[1],'.')
plt.plot(data[0,index],data[1,index],'ro')
plt.plot(t_hat,y_hat,'go')

另一种方法可能是（在物理上更正确）在每个时间点添加高斯峰的CDF。