天文学家要求你使用线性模型将数据放入太阳系外信号txt(参见黑板)中。该数据文件的第一列和第二列分别给出ti和vi，即观测时间(天)和相对速度(米/秒)。
运用此模型 f0(t; β0, β1) = β0 sin(ωt) + β1 cos(ωt), β0和β1为参数，ω = 2π/(34.6天)，设X是N × 2矩阵，有一行[sin(ωti)， cos(ωti)]。构造X(作为数组)，并解常规方程 (X^TX)β = X^T y
求出最佳拟合参数β = [β0， β1]^T
绘制数据集和最佳拟合模型。它看起来应该有点像以下的图
import numpy as np
import matplotlib.pyplot as plt
xs,ys = np.loadtxt('extrasolar_signal.txt',unpack = True)
plt.plot(xs,ys, 'g+')
plt.xlabel('$x$')
plt.ylabel('$y$')
A = np.dot(X.transpose(), X)
b = np.dot(X.transpose(), ys)
par = np.linalg.solve(A, b)
print("Beta parameters: ", par)
n = 100
xline = np.linspace(-4.1,4.1,n)
yline = np.zeros(n)
for i in range(m):
yline += par[i]*xline**i
plt.plot(xs, ys, '.');
plt.plot(xline, yline, 'r-', lw=3);
from scipy.optimize import curve_fit
w = 2*np.pi/34.6
def f0(t,b0,b1,w):
return b0*np.sin*w*t + b1*np.cos*w*t
A = np.dot(X.transpose(), X)
b = np.dot(X.transpose(), ys)
par = np.linalg.solve(A, b)
print("Beta parameters: ", par)
n = 100
xline = np.linspace(-4.1,4.1,n)
yline = np.zeros(n)
for i in range(m):
yline += par[i]*xline**i
plt.plot(xs, ys, '.');
plt.plot(xline, yline, 'r-', lw=3);
以下为数据集
0. -0.5462822385491465
0.4 -0.783702451361118
0.8 -0.9089025224257823
1.2000000000000002 -0.7889248051156628
1.6 -0.949226943504489
2. -0.7611050401954805
2.4000000000000004 -0.6650057450647151
2.8000000000000003 -0.8759288670095501
3.2 -0.8656123587289317
3.6 -0.6649140457676437
4. -0.5583916152654299
4.4 -0.6234103572245565
4.800000000000001 -0.4384536670407637
5.2 -0.45475540642816015
5.6000000000000005 -0.2933535893528097
6. -0.41092618504230566
6.4 -0.21640691364542158
6.800000000000001 -0.3233191361448244
7.2 -0.40033772666343287
7.6000000000000005 -0.19619561408794248
8. -0.37232044338162273
8.4 -0.32763639327372057
8.8 -0.07584110402709568
9.200000000000001 -0.07289662739894218
9.600000000000001 -0.3861124072710949
10. 0.0016509499268891892
10.4 -0.18590426894048864
10.8 0.10204635804590274
11.200000000000001 0.09898944148210725
11.600000000000001 -0.16365426432836233
12. 0.022162186215955795
12.4 -0.0873894131756795
12.8 0.14331006792028625
13.200000000000001 -0.07093466063257273
13.600000000000001 -0.08726667971656135
14. 0.10821188257124201
14.4 0.15593856613111876
14.8 0.14897975799398278
15.200000000000001 -0.16871736440162438
15.600000000000001 0.05054500120681124
16. 0.06705897957468973
16.400000000000002 0.1876763504970213
16.8 0.06520926888847367
17.2 0.10134588588772128
17.6 0.26959040983807353
18. 0.2539157102920223
18.400000000000002 0.024874768094809763
18.8 0.143008217569402
19.200000000000003 0.3871661437031221
19.6 0.2807141057341562
20. 0.3884392340985038
20.400000000000002 0.1355900401705764
20.8 0.33355373782689146
21.200000000000003 0.27999830448890783
21.6 0.1672208672103661
22. 0.28611323112030274
22.400000000000002 0.18389979569025794
22.8 0.424354295482916
23.200000000000003 0.4482691277851301
23.6 0.28911625650990647
24. 0.3254114765687105
24.400000000000002 0.3427910094969029
24.8 0.30639615372248097
25.200000000000003 0.3620792422007555
25.6 0.5333743131783425
26. 0.39409808537290963
26.400000000000002 0.5728178137177391
26.8 0.08537572482772388
27.200000000000003 0.28523057343506414
27.6 0.6426259088772276
28. 0.46367151413404734
28.400000000000002 0.45160954743898335
28.8 0.4812533692364098
29.200000000000003 0.6071255451174223
29.6 0.3865498395360522
30. 0.45964468718784435
30.400000000000002 0.32571105852891497
30.8 0.6846011283841505
31.200000000000003 0.5133973059495182
31.6 0.40248598712451333
32. 0.2835389289400173
32.4 0.30208717833464094
32.800000000000004 0.2503493752453388
33.2 0.2898018021224344
33.6 0.09515099754339622
34. -0.20011718696526487
34.4 -0.3750571976984114
34.800000000000004 -0.43371508478312204
35.2 -0.719799032621965
35.6 -0.9004915376393323
36. -1.0075778689532933
36.4 -1.0847746487179042
36.800000000000004 -1.0347158404087944
37.2 -0.7782406698370552
37.6 -0.7013228694375362
38. -0.6999737071538361
38.400000000000006 -0.5417534711626819
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39.2 -0.6252927035832156
39.6 -0.6023207461720973
40. -0.4427931407722867
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40.800000000000004 -0.3047799962996736
41.2 -0.3417332656517241
41.6 -0.40688052553661025
42. -0.2792846817997484
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43.2 -0.1894292487503772
43.6 -0.11481140334502393
44. -0.13884919114085115
44.400000000000006 -0.14197647217998172
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45.2 -0.30106547563085595
45.6 -0.1601079715503942
46. -0.02856438946758659
46.400000000000006 -0.012337329111426557
46.800000000000004 0.12214231419108941
47.2 -0.20972662011023396
47.6 0.08299547427591929
48. 0.14692819172321148
48.400000000000006 0.030999940619061642
48.800000000000004 0.01148690553567826
49.2 0.12558896523923488
49.6 0.21337993717333054
50. 0.05261326571378101
50.400000000000006 0.015171074820157837
50.800000000000004 0.10030943947289518
51.2 0.07566707857393415
51.6 0.2878403012305583
52. 0.06319472140542415
52.400000000000006 0.0696430280349094
52.800000000000004 0.12208331350240692
53.2 0.06359652523747564
53.6 0.31133682129861706
54. 0.14345029247237168
54.400000000000006 0.11571728996734064
54.800000000000004 0.4458110824088779
55.2 0.13603495323086506
55.6 0.17090505396986216
56. 0.4261058399859091
56.400000000000006 0.37980187003159394
56.800000000000004 0.1573904172013318
57.2 0.20380280893732955
57.6 0.17140189170010423
58. 0.38440601207916764
58.400000000000006 0.46088153933955767
58.800000000000004 0.28645206883094193
59.2 0.3817743213824975
59.6 0.5880033160984758
60. 0.26889226016305
60.400000000000006 0.3197292048394207
60.800000000000004 0.5058844619908931
61.2 0.5328303396087486
61.6 0.39809118839584495
62. 0.4790661748830668
62.400000000000006 0.4322866523197192
62.800000000000004 0.5738971870057148
63.2 0.4854131198652305
63.6 0.48251305636966507
64. 0.6405790592698798
64.4 0.40246674211906774
64.8 0.4985131223939545
65.2 0.44244386447180284
65.60000000000001 0.45841714901421676
66. 0.34833181544778136
66.4 0.3979748831459446
66.8 0.458964810348448
67.2 0.5170930019982345
67.60000000000001 0.1759347943583993
68. -0.043156293122121464
68.4 0.09696332441608412
68.8 -0.2558260271758174
69.2 -0.39923232592505825
69.60000000000001 -0.8128179270209015
70. -0.8319290760909005
70.4 -1.080004611307604
70.8 -0.9880242281598463
71.2 -1.0303479098135222
71.60000000000001 -0.8397760052776759
72. -0.9044623385137911
72.4 -0.8567212118283086
72.8 -0.7534653458737495
73.2 -0.6675507741975902
73.60000000000001 -0.5666532372452661
74. -0.7712660454372295
74.4 -0.3974742948276597
74.8 -0.5946877604677512
75.2 -0.4031915593671267
75.60000000000001 -0.2821943186057766
76. -0.466492956801751
76.4 -0.31706573379857794
76.80000000000001 -0.5406061827445913
77.2 -0.2710807545256565
77.60000000000001 -0.34240653068601073
78. -0.32741259669469924
78.4 -0.19460518725123405
78.80000000000001 -0.29598538356708987
79.2 -0.274449627037916
79.60000000000001 -0.12600804723174605
80. -0.023851583103075544
80.4 -0.19711257253457726
80.80000000000001 -0.25355325905702014
81.2 -0.14202573944496455
81.60000000000001 -0.005505476692198655
82. -0.01007757352545851
82.4 -0.27423396530714594
82.80000000000001 0.05774683759947964
83.2 -0.09628081995743432
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84. -0.0846046097999839
84.4 0.1150230372096937
84.80000000000001 0.00018716335145349539
85.2 -0.022823370411284058
85.60000000000001 -0.03137259399756902
86. 0.011054934507681594
86.4 -0.02655844915242281
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87.2 0.07913254678138325
87.60000000000001 0.2300278285313324
88. 0.21479961123956673
88.4 0.20944251357249155
88.80000000000001 0.18348117381816614
89.2 0.2574212926057774
89.60000000000001 0.41126513377888513
90. 0.12305959029805205
90.4 0.1638351308839136
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91.60000000000001 0.3631498591790082
92. 0.1078738865005095
92.4 0.3454088263034277
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93.2 0.3043549492065313
93.60000000000001 0.3363562785095267
94. 0.2612311629090467
94.4 0.26030193696870574
94.80000000000001 0.44800148917546334
95.2 0.42499185674750584
95.60000000000001 0.4429400564887961
96. 0.5415513671999952
96.4 0.3423197257336098
96.80000000000001 0.33418612244379925
97.2 0.4968735700779301
97.60000000000001 0.5924748360892591
98. 0.6147045551005446
98.4 0.5659946440638943
98.80000000000001 0.5462887976749852
99.2 0.5608357659259421
99.60000000000001 0.5062835898828245
100. 0.5676729761918977
100.4 0.47168711800682717
100.80000000000001 0.3366713565408654
101.2 0.5129243199308822
101.60000000000001 0.4708176758882343
102. 0.5427855549276028
102.4 0.11936031480780501
102.80000000000001 0.23424575082343574
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104. -0.4928384630473328
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105.2 -0.9735619511653839
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106. -0.8958602997771499
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107.2 -0.7819640958844551
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108. -0.6541210481036482
108.4 -0.5214383138802767
108.80000000000001 -0.6003153297645414
109.2 -0.5787779769802132
109.60000000000001 -0.3480504032971965
110. -0.5325703531094098
110.4 -0.40142888927534914
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111.2 -0.37808976333664585
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112. -0.1360650372313087
112.4 -0.4268937164048808
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113.2 -0.34549384224190594
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118. -0.1070181087744724
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119.60000000000001 0.07731983501122021
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171.20000000000002 0.5233041913949565
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172.8 -0.17953490141744538
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193.60000000000002 0.15998539631512884
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195.20000000000002 0.24731560455929016
195.60000000000002 0.35284292515800403
196. 0.2675357804985887
196.4 0.21555440816274485
196.8 0.005006448281731468
197.20000000000002 0.5238287163387917
197.60000000000002 0.41429219557707203
198. 0.4173298692386332
198.4 0.3727151012376218
198.8 0.46631327699492664
199.20000000000002 0.39089889533974337
199.60000000000002 0.46922445398441637
200. 0.19261170005827852
