神经网络和深度学习-logistic回归

logistic回归

logistic回归常用于分类问题 ，根据输入的x来估计属于不同类的概率为多少，满足分布，即所有类的概率总和为1

在torchvison包中提供了一些常用的数据集，供我们使用，例如：Mnist数据集

Cifar-10数据集，提供了十类的彩色图像

在二分类问题中，我们常常只需要计算一个类的概率就行，因为有概率总和为1的这个限制，一般概率在不确定的区间我们输出不确定，或者直接输出概率，正常输出的就是属于某个类型的概率

在之前的线性模型中，我们输出的是一个实数

$$\hat{y}=wx+b\in \mathbb{R}$$

现在我们想要的分类的输出是一个概率

$$\hat{y}\in [0,1]$$

所以我们需要将实数空间映射到（0-1）区间，我们用到的就是logistic函数（sigmoid函数）

我们也可以看到，但这个函数超过某一个阈值的时候，他的增长或者减少是非常慢的，我们在数学中称为饱和函数。他的导数会越发趋近于0

我们再看一下其他的sigmoid函数，他们都具有极限，以及是单调增函数和饱和函数，当然最出名的还是logistic函数

选择模型

在logistic模型中，我们先是用线性模型，然后经过sigmoid函数，一般在论文中看到这个符号都代表logistic。

$$\sigma (x)$$

损失函数

定义完模型之后我们的损失函数也要进行相应的变换，前面线性模型中，我们希望计算距离的最小化；但是在logistic回归之中我们输出的不在是一个数值了，而是一个分布，表示的是分类的概率

$$\begin{gathered} \hat{y}=P(class=1) \\ (1-\hat{y})=P(class=0) \end{gathered}$$

下面举个例子来表示两个分布之间差异性的大小，当然我们希望这个值越大越好（交叉熵）

在实际的公式中，我们加上负号就会希望损失越小越好，这个是二分类的，我们称为BCE

如果我们需要求小批量的损失，就需要进行求和，在二分类中

• y=1分类中，如果预测y越接近1，则BCE损失越小

• y=0分类中，如果预测y越接近0，则BCE损失越小

在Mini-Batch Loss中我们直接求均值

代码分析

于线性模型的代码进行对比

另外一个不同的地方就是损失函数，现在用的是BCE损失（二分类的交叉熵）

整个代码相较于线性回归就存在这两处的变化

我们在大部分的神经网络中都遵循四部法

下面我们来看一下完整代码

import torch
import torch.nn.functional as F
import numpy as np
import matplotlib.pyplot as plt

# prepare dataset
x_data = torch.Tensor([[1.0], [2.0], [3.0]])
y_data = torch.Tensor([[0], [0], [1]])


# ------------------------------------
# design model using Class
class LogisticRegressionModel(torch.nn.Module):
    def __init__(self):
        super(LogisticRegressionModel, self).__init__()
        self.linear = torch.nn.Linear(1, 1)

    def forward(self, x):
        y_pred = F.sigmoid(self.linear(x))
        return y_pred


model = LogisticRegressionModel()
# -----------------------------------
# construct loss and optimizer
criterion = torch.nn.BCELoss(size_average=False)
optimizer = torch.optim.SGD(model.parameters(), lr=0.01)
# --------------------------------------
# training cycle
for epoch in range(1000):
    y_pred = model(x_data)
    loss = criterion(y_pred, y_data)
    print(epoch, loss.item())

    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

# show
x = np.linspace(0, 10, 200)  # 比如每周学习的时间0-10个小时，采集200个点
x_test = torch.Tensor(x).view((200, 1))  # 相当于numpy中的reshape，变成200行1列的矩阵
y_test = model(x_test)  # 把张量送入模型中
y = y_test.data.numpy()  # 得到numpy的n维数组
plt.plot(x, y)
plt.plot([0, 10], [0.5, 0.5], c='r')
plt.xlabel('Hours')
plt.ylabel('Probability of Pass')
plt.grid()
plt.show()
#  我们可以看到处在Pass和 Not Pass之间的是2.5小时

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51

训练数据如下

0 2.661794424057007
1 2.6548328399658203
2 2.6481523513793945
3 2.6417336463928223
4 2.635560989379883
5 2.6296184062957764
6 2.6238908767700195
7 2.618364095687866
8 2.6130247116088867
9 2.607860565185547
10 2.6028599739074707
11 2.5980114936828613
12 2.5933053493499756
13 2.5887317657470703
14 2.5842809677124023
15 2.5799460411071777
16 2.5757179260253906
17 2.571589708328247
18 2.567554235458374
19 2.563605785369873
20 2.5597376823425293
21 2.5559449195861816
22 2.5522217750549316
23 2.5485639572143555
24 2.5449671745300293
25 2.54142689704895
26 2.5379390716552734
27 2.534501075744629
28 2.5311083793640137
29 2.5277585983276367
30 2.524448871612549
31 2.521176338195801
32 2.5179386138916016
33 2.5147337913513184
34 2.51155948638916
35 2.508413553237915
36 2.5052943229675293
37 2.5022006034851074
38 2.4991297721862793
39 2.496081590652466
40 2.493053674697876
41 2.4900457859039307
42 2.487056255340576
43 2.484083890914917
44 2.481127977371216
45 2.4781875610351562
46 2.47526216506958
47 2.472350597381592
48 2.469452142715454
49 2.466566324234009
50 2.4636926651000977
51 2.460829973220825
52 2.4579784870147705
53 2.4551374912261963
54 2.4523062705993652
55 2.4494845867156982
56 2.4466724395751953
57 2.44386887550354
58 2.4410741329193115
59 2.4382872581481934
60 2.4355084896087646
61 2.4327375888824463
62 2.42997407913208
63 2.427217721939087
64 2.424468517303467
65 2.4217259883880615
66 2.41899037361145
67 2.4162614345550537
68 2.413538932800293
69 2.4108221530914307
70 2.408112049102783
71 2.4054079055786133
72 2.402709484100342
73 2.400017023086548
74 2.3973302841186523
75 2.3946492671966553
76 2.3919732570648193
77 2.38930344581604
78 2.386638879776001
79 2.383979558944702
80 2.3813254833221436
81 2.378676652908325
82 2.376033067703247
83 2.373394727706909
84 2.3707616329193115
85 2.368133068084717
86 2.3655097484588623
87 2.362891435623169
88 2.3602781295776367
89 2.3576698303222656
90 2.3550662994384766
91 2.3524675369262695
92 2.3498737812042236
93 2.3472845554351807
94 2.344700336456299
95 2.342120885848999
96 2.3395462036132812
97 2.3369760513305664
98 2.3344106674194336
99 2.331850290298462
100 2.329294204711914
101 2.326742649078369
102 2.3241958618164062
103 2.3216540813446045
104 2.3191165924072266
105 2.3165836334228516
106 2.3140552043914795
107 2.3115315437316895
108 2.3090124130249023
109 2.306497573852539
110 2.303987503051758
111 2.3014819622039795
112 2.298980712890625
113 2.2964842319488525
114 2.293992042541504
115 2.291504383087158
116 2.2890212535858154
117 2.2865426540374756
118 2.2840683460235596
119 2.2815985679626465
120 2.2791330814361572
121 2.27667236328125
122 2.2742156982421875
123 2.271763324737549
124 2.269315719604492
125 2.2668724060058594
126 2.2644333839416504
127 2.2619986534118652
128 2.259568214416504
129 2.2571427822113037
130 2.25472092628479
131 2.2523038387298584
132 2.2498910427093506
133 2.2474822998046875
134 2.2450783252716064
135 2.242678642272949
136 2.2402827739715576
137 2.237891435623169
138 2.235504388809204
139 2.233121395111084
140 2.230743169784546
141 2.2283689975738525
142 2.225998878479004
143 2.223633289337158
144 2.2212717533111572
145 2.21891450881958
146 2.216561794281006
147 2.2142128944396973
148 2.2118682861328125
149 2.2095279693603516
150 2.2071919441223145
151 2.204859733581543
152 2.2025320529937744
153 2.2002084255218506
154 2.1978890895843506
155 2.1955738067626953
156 2.1932625770568848
157 2.190955638885498
158 2.188652992248535
159 2.186354160308838
160 2.1840596199035645
161 2.181769371032715
162 2.179482936859131
163 2.1772005558013916
164 2.1749227046966553
165 2.1726484298706055
166 2.1703786849975586
167 2.1681127548217773
168 2.165850877761841
169 2.1635935306549072
170 2.16133975982666
171 2.159090042114258
172 2.1568443775177
173 2.1546030044555664
174 2.1523654460906982
175 2.150132179260254
176 2.147902488708496
177 2.145677089691162
178 2.1434555053710938
179 2.141238212585449
180 2.1390247344970703
181 2.136815309524536
182 2.1346096992492676
183 2.1324081420898438
184 2.1302103996276855
185 2.128016948699951
186 2.1258273124694824
187 2.1236414909362793
188 2.121459484100342
189 2.119281768798828
190 2.117107629776001
191 2.1149375438690186
192 2.11277174949646
193 2.1106090545654297
194 2.1084506511688232
195 2.1062963008880615
196 2.1041455268859863
197 2.101999282836914
198 2.09985613822937
199 2.097717046737671
200 2.0955817699432373
201 2.0934503078460693
202 2.091322898864746
203 2.0891990661621094
204 2.0870790481567383
205 2.084963083267212
206 2.082850933074951
207 2.080742359161377
208 2.0786375999450684
209 2.0765368938446045
210 2.0744400024414062
211 2.0723466873168945
212 2.0702567100524902
213 2.0681710243225098
214 2.066089153289795
215 2.0640106201171875
216 2.061936140060425
217 2.0598654747009277
218 2.057797908782959
219 2.055734634399414
220 2.0536749362945557
221 2.051619052886963
222 2.0495665073394775
223 2.047518014907837
224 2.045473098754883
225 2.0434317588806152
226 2.041393995285034
227 2.039360523223877
228 2.037329912185669
229 2.0353033542633057
230 2.033280372619629
231 2.0312609672546387
232 2.029245376586914
233 2.027233123779297
234 2.025224447250366
235 2.023219585418701
236 2.0212182998657227
237 2.0192205905914307
238 2.0172266960144043
239 2.0152359008789062
240 2.013248920440674
241 2.011265754699707
242 2.0092859268188477
243 2.007309675216675
244 2.0053367614746094
245 2.0033679008483887
246 2.001401901245117
247 1.9994397163391113
248 1.997481346130371
249 1.9955259561538696
250 1.9935743808746338
251 1.991626262664795
252 1.9896817207336426
253 1.987740397453308
254 1.9858028888702393
255 1.9838688373565674
256 1.9819378852844238
257 1.980010747909546
258 1.9780869483947754
259 1.9761667251586914
260 1.9742497205734253
261 1.9723362922668457
262 1.970426082611084
263 1.968519687652588
264 1.9666162729263306
265 1.9647164344787598
266 1.9628199338912964
267 1.9609270095825195
268 1.959037184715271
269 1.957151174545288
270 1.955268144607544
271 1.9533886909484863
272 1.9515125751495361
273 1.9496397972106934
274 1.9477702379226685
275 1.945904016494751
276 1.9440416097640991
277 1.9421820640563965
278 1.9403254985809326
279 1.9384729862213135
280 1.936623454093933
281 1.934777021408081
282 1.9329341650009155
283 1.931094765663147
284 1.9292583465576172
285 1.9274253845214844
286 1.9255955219268799
287 1.9237689971923828
288 1.921945571899414
289 1.9201254844665527
290 1.918308973312378
291 1.9164950847625732
292 1.9146850109100342
293 1.9128779172897339
294 1.9110740423202515
295 1.9092732667922974
296 1.9074759483337402
297 1.905681848526001
298 1.903890609741211
299 1.9021025896072388
300 1.900317907333374
301 1.8985365629196167
302 1.8967583179473877
303 1.8949832916259766
304 1.8932111263275146
305 1.891442060470581
306 1.8896763324737549
307 1.887913703918457
308 1.8861541748046875
309 1.8843978643417358
310 1.882644772529602
311 1.8808945417404175
312 1.8791474103927612
313 1.8774032592773438
314 1.8756628036499023
315 1.873924970626831
316 1.8721903562545776
317 1.8704588413238525
318 1.868730068206787
319 1.8670048713684082
320 1.8652822971343994
321 1.863563060760498
322 1.861846685409546
323 1.8601336479187012
324 1.8584232330322266
325 1.8567159175872803
326 1.8550119400024414
327 1.8533108234405518
328 1.8516125679016113
329 1.8499172925949097
330 1.8482252359390259
331 1.8465361595153809
332 1.8448498249053955
333 1.8431668281555176
334 1.8414862155914307
335 1.8398091793060303
336 1.8381346464157104
337 1.836463212966919
338 1.8347947597503662
339 1.8331294059753418
340 1.8314666748046875
341 1.829807162284851
342 1.8281506299972534
343 1.8264967203140259
344 1.8248461484909058
345 1.8231980800628662
346 1.8215528726577759
347 1.8199106454849243
348 1.8182711601257324
349 1.8166348934173584
350 1.8150012493133545
351 1.813370704650879
352 1.8117430210113525
353 1.8101179599761963
354 1.8084959983825684
355 1.8068767786026
356 1.805260419845581
357 1.8036469221115112
358 1.802036166191101
359 1.8004279136657715
360 1.7988229990005493
361 1.7972208261489868
362 1.7956211566925049
363 1.7940247058868408
364 1.7924308776855469
365 1.790839672088623
366 1.7892513275146484
367 1.787665843963623
368 1.7860831022262573
369 1.7845028638839722
370 1.7829258441925049
371 1.7813513278961182
372 1.7797796726226807
373 1.7782106399536133
374 1.7766443490982056
375 1.775080919265747
376 1.7735202312469482
377 1.7719619274139404
378 1.770406723022461
379 1.768853783607483
380 1.7673041820526123
381 1.7657568454742432
382 1.7642122507095337
383 1.7626705169677734
384 1.7611315250396729
385 1.7595946788787842
386 1.7580609321594238
387 1.7565298080444336
388 1.7550013065338135
389 1.7534751892089844
390 1.7519521713256836
391 1.7504315376281738
392 1.7489136457443237
393 1.7473981380462646
394 1.7458856105804443
395 1.7443755865097046
396 1.7428679466247559
397 1.741363286972046
398 1.739861011505127
399 1.7383613586425781
400 1.7368645668029785
401 1.7353700399398804
402 1.7338781356811523
403 1.732388973236084
404 1.7309024333953857
405 1.7294180393218994
406 1.7279365062713623
407 1.7264575958251953
408 1.7249810695648193
409 1.7235071659088135
410 1.7220356464385986
411 1.720566987991333
412 1.7191007137298584
413 1.717637062072754
414 1.7161755561828613
415 1.714716911315918
416 1.713260531425476
417 1.7118067741394043
418 1.710355520248413
419 1.7089070081710815
420 1.707460641860962
421 1.7060167789459229
422 1.704575538635254
423 1.703136682510376
424 1.7017005681991577
425 1.7002664804458618
426 1.6988348960876465
427 1.6974060535430908
428 1.6959795951843262
429 1.6945555210113525
430 1.6931337118148804
431 1.6917147636413574
432 1.6902978420257568
433 1.6888833045959473
434 1.687471628189087
435 1.6860618591308594
436 1.684654712677002
437 1.6832499504089355
438 1.6818475723266602
439 1.6804475784301758
440 1.679050087928772
441 1.6776549816131592
442 1.6762621402740479
443 1.6748714447021484
444 1.6734834909439087
445 1.6720978021621704
446 1.6707143783569336
447 1.6693333387374878
448 1.6679548025131226
449 1.6665784120559692
450 1.6652045249938965
451 1.663832664489746
452 1.6624635457992554
453 1.6610963344573975
454 1.6597316265106201
455 1.6583693027496338
456 1.6570093631744385
457 1.655651330947876
458 1.6542961597442627
459 1.6529430150985718
460 1.6515917778015137
461 1.6502432823181152
462 1.6488969326019287
463 1.6475532054901123
464 1.6462111473083496
465 1.6448715925216675
466 1.643534541130066
467 1.6421995162963867
468 1.6408666372299194
469 1.639535903930664
470 1.6382079124450684
471 1.6368818283081055
472 1.6355578899383545
473 1.6342363357543945
474 1.6329169273376465
475 1.6315997838974
476 1.6302850246429443
477 1.6289722919464111
478 1.6276617050170898
479 1.6263536214828491
480 1.625047206878662
481 1.6237432956695557
482 1.6224416494369507
483 1.621141791343689
484 1.619844675064087
485 1.6185493469238281
486 1.6172564029693604
487 1.6159652471542358
488 1.6146764755249023
489 1.6133900880813599
490 1.6121056079864502
491 1.6108232736587524
492 1.6095432043075562
493 1.6082650423049927
494 1.6069891452789307
495 1.6057153940200806
496 1.6044435501098633
497 1.603174090385437
498 1.6019065380096436
499 1.6006414890289307
500 1.5993781089782715
501 1.5981168746948242
502 1.5968579053878784
503 1.595600962638855
504 1.594346284866333
505 1.5930932760238647
506 1.5918426513671875
507 1.590593934059143
508 1.5893473625183105
509 1.5881028175354004
510 1.5868604183197021
511 1.5856201648712158
512 1.5843816995620728
513 1.5831451416015625
514 1.5819109678268433
515 1.5806787014007568
516 1.5794485807418823
517 1.5782203674316406
518 1.57699453830719
519 1.575770378112793
520 1.5745482444763184
521 1.5733280181884766
522 1.5721100568771362
523 1.5708940029144287
524 1.569679856300354
525 1.5684678554534912
526 1.5672576427459717
527 1.5660494565963745
528 1.5648434162139893
529 1.5636391639709473
530 1.5624370574951172
531 1.56123685836792
532 1.5600388050079346
533 1.5588421821594238
534 1.557647943496704
535 1.5564558506011963
536 1.555265188217163
537 1.554076910018921
538 1.5528901815414429
539 1.5517053604125977
540 1.5505229234695435
541 1.549342155456543
542 1.5481634140014648
543 1.54698646068573
544 1.5458112955093384
545 1.5446382761001587
546 1.5434671640396118
547 1.5422980785369873
548 1.541130542755127
549 1.5399651527404785
550 1.5388014316558838
551 1.537639856338501
552 1.5364799499511719
553 1.5353221893310547
554 1.5341660976409912
555 1.5330119132995605
556 1.5318593978881836
557 1.5307092666625977
558 1.5295605659484863
559 1.5284137725830078
560 1.5272687673568726
561 1.5261257886886597
562 1.52498459815979
563 1.5238454341888428
564 1.5227077007293701
565 1.5215721130371094
566 1.520438313484192
567 1.5193061828613281
568 1.5181759595870972
569 1.517047643661499
570 1.5159211158752441
571 1.514796257019043
572 1.5136733055114746
573 1.51255202293396
574 1.5114328861236572
575 1.510315179824829
576 1.5091995000839233
577 1.5080854892730713
578 1.506973385810852
579 1.505862832069397
580 1.5047541856765747
581 1.5036473274230957
582 1.5025423765182495
583 1.5014389753341675
584 1.5003373622894287
585 1.4992375373840332
586 1.4981393814086914
587 1.4970431327819824
588 1.4959485530853271
589 1.4948556423187256
590 1.4937645196914673
591 1.4926753044128418
592 1.491587519645691
593 1.4905016422271729
594 1.489417552947998
595 1.4883350133895874
596 1.48725426197052
597 1.486175298690796
598 1.485097885131836
599 1.4840223789215088
600 1.4829485416412354
601 1.4818761348724365
602 1.4808056354522705
603 1.4797366857528687
604 1.4786696434020996
605 1.4776041507720947
606 1.4765405654907227
607 1.4754784107208252
608 1.4744179248809814
609 1.4733591079711914
610 1.4723021984100342
611 1.471246600151062
612 1.4701929092407227
613 1.4691407680511475
614 1.468090295791626
615 1.4670414924621582
616 1.4659945964813232
617 1.4649490118026733
618 1.4639050960540771
619 1.4628626108169556
620 1.4618220329284668
621 1.4607831239700317
622 1.4597456455230713
623 1.4587100744247437
624 1.4576759338378906
625 1.4566435813903809
626 1.4556126594543457
627 1.4545834064483643
628 1.453555703163147
629 1.4525295495986938
630 1.451505184173584
631 1.4504821300506592
632 1.4494609832763672
633 1.4484412670135498
634 1.447422981262207
635 1.446406602859497
636 1.4453916549682617
637 1.444378137588501
638 1.443366289138794
639 1.442355990409851
640 1.441347360610962
641 1.440340280532837
642 1.439334750175476
643 1.4383306503295898
644 1.4373283386230469
645 1.4363274574279785
646 1.4353282451629639
647 1.4343302249908447
648 1.4333341121673584
649 1.4323391914367676
650 1.4313459396362305
651 1.430354356765747
652 1.4293643236160278
653 1.4283756017684937
654 1.4273885488510132
655 1.4264030456542969
656 1.4254188537597656
657 1.4244362115859985
658 1.4234552383422852
659 1.422475814819336
660 1.4214977025985718
661 1.4205210208892822
662 1.419546127319336
663 1.4185724258422852
664 1.417600393295288
665 1.4166297912597656
666 1.4156606197357178
667 1.4146931171417236
668 1.413727045059204
669 1.4127624034881592
670 1.4117990732192993
671 1.4108375310897827
672 1.409877061843872
673 1.4089181423187256
674 1.4079610109329224
675 1.4070050716400146
676 1.406050443649292
677 1.405097484588623
678 1.4041460752487183
679 1.403195858001709
680 1.4022471904754639
681 1.4012999534606934
682 1.4003541469573975
683 1.3994098901748657
684 1.398466944694519
685 1.397525429725647
686 1.39658522605896
687 1.3956464529037476
688 1.3947093486785889
689 1.3937735557556152
690 1.392838954925537
691 1.3919060230255127
692 1.390974521636963
693 1.3900442123413086
694 1.3891154527664185
695 1.388188123703003
696 1.3872621059417725
697 1.3863372802734375
698 1.3854141235351562
699 1.3844921588897705
700 1.383571743965149
701 1.3826526403427124
702 1.3817347288131714
703 1.380818486213684
704 1.3799035549163818
705 1.3789899349212646
706 1.378077745437622
707 1.377166748046875
708 1.3762571811676025
709 1.3753490447998047
710 1.3744423389434814
711 1.3735367059707642
712 1.3726327419281006
713 1.371729850769043
714 1.37082839012146
715 1.3699283599853516
716 1.3690295219421387
717 1.3681321144104004
718 1.3672358989715576
719 1.3663411140441895
720 1.365447759628296
721 1.3645555973052979
722 1.3636648654937744
723 1.362775444984436
724 1.3618872165679932
725 1.3610005378723145
726 1.3601148128509521
727 1.359230637550354
728 1.3583476543426514
729 1.3574659824371338
730 1.3565857410430908
731 1.3557066917419434
732 1.354828953742981
733 1.353952407836914
734 1.3530771732330322
735 1.352203369140625
736 1.3513309955596924
737 1.3504595756530762
738 1.3495895862579346
739 1.3487210273742676
740 1.347853183746338
741 1.3469871282577515
742 1.346122145652771
743 1.3452584743499756
744 1.3443961143493652
745 1.3435348272323608
746 1.342674970626831
747 1.3418161869049072
748 1.340958833694458
749 1.3401026725769043
750 1.339247703552246
751 1.3383941650390625
752 1.3375418186187744
753 1.3366906642913818
754 1.3358407020568848
755 1.3349921703338623
756 1.3341443538665771
757 1.3332983255386353
758 1.3324532508850098
759 1.3316094875335693
760 1.3307669162750244
761 1.3299256563186646
762 1.3290854692459106
763 1.3282465934753418
764 1.327408790588379
765 1.326572299003601
766 1.3257369995117188
767 1.3249027729034424
768 1.3240700960159302
769 1.3232386112213135
770 1.322407841682434
771 1.3215787410736084
772 1.3207505941390991
773 1.3199235200881958
774 1.3190979957580566
775 1.3182733058929443
776 1.317449927330017
777 1.316627860069275
778 1.3158068656921387
779 1.3149869441986084
780 1.3141683340072632
781 1.313350796699524
782 1.3125346899032593
783 1.3117194175720215
784 1.3109055757522583
785 1.3100926876068115
786 1.3092811107635498
787 1.308470606803894
788 1.3076610565185547
789 1.3068530559539795
790 1.3060460090637207
791 1.3052400350570679
792 1.3044353723526
793 1.3036316633224487
794 1.302829384803772
795 1.302027940750122
796 1.3012276887893677
797 1.300428867340088
798 1.2996307611465454
799 1.298833966255188
800 1.2980382442474365
801 1.2972438335418701
802 1.2964503765106201
803 1.295657992362976
804 1.294866919517517
805 1.294076919555664
806 1.2932878732681274
807 1.2925001382827759
808 1.2917133569717407
809 1.2909278869628906
810 1.2901431322097778
811 1.2893598079681396
812 1.2885775566101074
813 1.2877963781356812
814 1.2870162725448608
815 1.286237120628357
816 1.2854591608047485
817 1.2846821546554565
818 1.2839064598083496
819 1.2831318378448486
820 1.282358169555664
821 1.281585693359375
822 1.280814290046692
823 1.2800438404083252
824 1.2792747020721436
825 1.2785063982009888
826 1.2777392864227295
827 1.2769731283187866
828 1.2762082815170288
829 1.2754443883895874
830 1.2746813297271729
831 1.2739195823669434
832 1.2731587886810303
833 1.2723991870880127
834 1.2716405391693115
835 1.2708829641342163
836 1.2701263427734375
837 1.2693709135055542
838 1.2686166763305664
839 1.2678629159927368
840 1.2671104669570923
841 1.2663593292236328
842 1.2656089067459106
843 1.2648594379425049
844 1.2641113996505737
845 1.263364315032959
846 1.262618064880371
847 1.2618730068206787
848 1.2611287832260132
849 1.2603857517242432
850 1.2596435546875
851 1.2589024305343628
852 1.258162498474121
853 1.2574235200881958
854 1.256685495376587
855 1.2559484243392944
856 1.255212426185608
857 1.2544775009155273
858 1.253743290901184
859 1.2530105113983154
860 1.252278447151184
861 1.2515474557876587
862 1.2508172988891602
863 1.2500884532928467
864 1.2493603229522705
865 1.2486333847045898
866 1.2479071617126465
867 1.2471821308135986
868 1.2464580535888672
869 1.2457349300384521
870 1.2450129985809326
871 1.2442917823791504
872 1.2435716390609741
873 1.2428523302078247
874 1.2421340942382812
875 1.2414170503616333
876 1.2407007217407227
877 1.239985466003418
878 1.2392712831497192
879 1.2385578155517578
880 1.2378456592559814
881 1.2371342182159424
882 1.2364236116409302
883 1.235714077949524
884 1.235005497932434
885 1.2342979907989502
886 1.2335911989212036
887 1.2328855991363525
888 1.2321809530258179
889 1.2314770221710205
890 1.230774164199829
891 1.230072259902954
892 1.2293713092803955
893 1.2286713123321533
894 1.2279720306396484
895 1.227273941040039
896 1.226576805114746
897 1.2258803844451904
898 1.2251850366592407
899 1.2244905233383179
900 1.2237968444824219
901 1.2231043577194214
902 1.2224128246307373
903 1.221721887588501
904 1.221031904220581
905 1.2203432321548462
906 1.219655156135559
907 1.218968152999878
908 1.218281865119934
909 1.2175967693328857
910 1.2169122695922852
911 1.2162288427352905
912 1.2155463695526123
913 1.2148644924163818
914 1.2141838073730469
915 1.2135040760040283
916 1.212825059890747
917 1.2121469974517822
918 1.2114696502685547
919 1.210793375968933
920 1.2101178169250488
921 1.2094436883926392
922 1.208769679069519
923 1.2080971002578735
924 1.2074252367019653
925 1.206754446029663
926 1.206084132194519
927 1.20541512966156
928 1.2047467231750488
929 1.2040791511535645
930 1.2034125328063965
931 1.2027469873428345
932 1.2020819187164307
933 1.2014179229736328
934 1.200755000114441
935 1.2000926733016968
936 1.1994314193725586
937 1.1987707614898682
938 1.1981110572814941
939 1.1974523067474365
940 1.1967942714691162
941 1.1961373090744019
942 1.1954810619354248
943 1.1948256492614746
944 1.1941710710525513
945 1.1935176849365234
946 1.1928645372390747
947 1.192212700843811
948 1.1915614604949951
949 1.190911054611206
950 1.1902618408203125
951 1.1896131038665771
952 1.1889652013778687
953 1.1883184909820557
954 1.1876723766326904
955 1.1870269775390625
956 1.186382532119751
957 1.1857389211654663
958 1.1850961446762085
959 1.184454321861267
960 1.1838130950927734
961 1.1831727027893066
962 1.1825333833694458
963 1.1818945407867432
964 1.181256651878357
965 1.180619716644287
966 1.179983377456665
967 1.179348111152649
968 1.1787135601043701
969 1.178079605102539
970 1.177446961402893
971 1.1768145561218262
972 1.1761831045150757
973 1.175552487373352
974 1.1749228239059448
975 1.1742939949035645
976 1.1736657619476318
977 1.1730386018753052
978 1.1724120378494263
979 1.1717863082885742
980 1.17116117477417
981 1.170536994934082
982 1.169913649559021
983 1.1692910194396973
984 1.1686692237854004
985 1.1680481433868408
986 1.1674280166625977
987 1.1668084859848022
988 1.1661899089813232
989 1.165571928024292
990 1.1649549007415771
991 1.1643385887145996
992 1.1637232303619385
993 1.1631081104278564
994 1.162494421005249
995 1.1618812084197998
996 1.1612685918807983
997 1.1606569290161133
998 1.160046100616455
999 1.1594359874725342

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001

图像显示如下，我们可以看到处在Pass和 Not Pass之间的是2.5小时