用分类映射的办法分类两条夹角为0.3度的直线

分类两条直线y=0，和y=x*tanθ，

这个角θ最小可以是多少？

用多次收敛取平均值的办法去测量θ的极小值。收敛标准有16个，每个收敛标准收敛199次，共尝试了θ从9到0.04共24个不同的值，一共收敛了24*16*199次。

得到的数据

θ	10	9	8	7	6	5	4	3	2	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0.09	0.08	0.07	0.06	0.05	0.04
δ	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n	迭代次数n
0.5	195.3317	225.1256	230.8241	194.0603	260.1055	266.7538	256.6181	229.6633	267.9447	289.0704	266.7387	248.5327	280.6683	260.4322	234.9598	243.9296	269.809	261.4724	251.1106	270.0402	229.0402	256.1709	278.3668	291.3618	242.0503
0.4	13914.29	15650.57	17880.11	20209.6	24517.05	29278.5	36627.75	50957.44	77998.87	146102.9	173566.7	176639.8	189926.5	268967	323366.5	369113.9	420655.1	506942.3	2405873	2719082	2048413	2618378	4197245	5604986	4793757
0.3	15479.9	17332.98	19411.22	22473.04	27185.66	31413.58	41070.88	56023.05	83704.67	167101.1	184933.1	196349.9	215094.5	302524.9	354299.6	425524	478569.6	627099	2659858	2958164	2350955	2951109	4596583	6216997	5870306
0.2	16439.69	18223.97	20808.33	24174.77	28656.78	34997.63	43321.73	58718.62	89682.01	176946.5	196522.5	210620.7	236574.3	320900.7	401429.3	466824.3	531129.9	708706.7	2947430	3181111	2600654	3217961	4930519	6764221	6495452
0.1	17919.78	20062.16	22127.07	26230.42	30405.3	36263.06	46928.16	64122.4	96614.49	188302.3	215432.9	232926.9	259838.2	348587.1	417222	507597	572595	775833.8	3044634	3461076	2901890	3572008	5428360	7544252	7407801
0.01	22466.49	25275.75	27902.76	31928.63	37886.42	46846.11	57167.14	77672.2	115687.6	227748.1	263480.7	282903.4	318950.8	411941.7	508047.8	596253.1	643902.1	900253.2	3453155	3998794	3561226	4625189	6999017	9549422	1.10E+07
0.001	29045.94	32983.77	36326.36	40520.33	47857.63	57341.84	70818.73	94979.9	138107.1	274102.4	310804.6	335353.8	371230.4	473280.1	591728.4	654498.1	666656.9	949862.1	3706353	4318767	4046588	5247215	7728852	1.06E+07	1.22E+07
9.00E-04	29906.22	33366.44	36870.04	41179.77	47855.69	58431.72	71149.74	95579.46	139451.3	273893.8	314116.3	335751.5	379128.4	473682	587273.2	659923.1	666305.3	947896.2	3770298	4399940	4049285	5258015	7771985	1.06E+07	1.23E+07
8.00E-04	30505.62	33537.86	37364.13	42257.77	48099.17	58790.35	71160.06	97090.92	139448.4	281822.7	317464.8	340301.6	379438	482171	596464	669215.8	668917.3	951669.8	3743586	4333233	4065487	5250128	7753600	1.07E+07	1.24E+07
7.00E-04	30878.16	34642.3	37722.57	42657.75	49542.59	59781.34	73130.08	98491.56	141650.7	278769.2	320831.5	342315.5	379415.5	488048.1	598162.9	670672.5	672734.9	955540.8	3751633	4316054	4088694	5266705	7786100	1.07E+07	1.24E+07
6.00E-04	31735.77	35227.77	39393.62	43468.07	50670.44	60220.19	74846.17	98957.79	145869.6	280527.3	327816.6	346098.4	388231.7	491714.9	600249.3	669617.7	674191.6	957925.8	3773483	4346545	4109088	5288656	7937419	1.07E+07	1.25E+07
5.00E-04	32248.65	35900.43	39572.38	44431.86	51824.84	61375.43	75278.8	100319.9	146817.3	285198.8	327349.8	350935.4	388693.3	490760.5	608293.4	675595.2	709016.2	964057.8	3793037	4328763	4118079	5286847	7942442	1.08E+07	1.26E+07
4.00E-04	33464.13	37038.02	40763.43	45555.65	52685.73	63106.96	78705.99	101974	151146.5	290272.2	336662.4	356670.8	402343.9	509597.2	614457.4	692573.3	843940.2	966219	3836588	4353527	4136816	5314274	7984687	1.08E+07	1.27E+07
3.00E-04	34532.6	38848.82	42820.11	47207.83	55192.5	65049.65	80288.21	106443.1	152591.6	297656.1	345275.3	363816.7	410187.4	517031.4	623548.3	693778.8	893810.7	970565.6	3794636	4378692	4176950	5431542	8054188	1.08E+07	1.28E+07
2.00E-04	37157.44	41191.23	44549.33	49365.92	58211.46	68042.21	83957.31	110683.5	159184.3	305492.2	355278.5	375129.8	420971.9	533858.9	653484.1	719698.1	915560.8	981738.1	3855110	4505024	4219496	5479412	8086637	1.08E+07	1.30E+07
1.00E-04	40897.64	44706.82	49418.51	53947.74	62068.7	74072.9	91026.79	118315	171038.4	322420.1	372499.6	393522.3	444940.8	565152.9	660316.6	771687.4	955946.7	997062.7	3887811	4585959	4277563	5558074	8128276	1.10E+07	1.33E+07



0.5	0.675724	0.778792	0.798505	0.671326	0.8998	0.922799	0.887736	0.794489	0.926919	1	0.922747	0.859765	0.970934	0.90093	0.812812	0.843842	0.933368	0.904528	0.868683	0.934168	0.792334	0.886189	0.962973	1.007927	0.83734
0.4	0.095236	0.10712	0.12238	0.138324	0.167807	0.200396	0.250698	0.348778	0.533862	1	1.187976	1.209009	1.29995	1.840942	2.213279	2.526396	2.879169	3.469761	16.46698	18.61073	14.02034	17.92146	28.728	38.36326	32.81082
0.3	0.092638	0.103727	0.116165	0.134488	0.16269	0.187991	0.245785	0.335264	0.500922	1	1.106714	1.175036	1.287212	1.81043	2.120271	2.546506	2.863952	3.752812	15.91766	17.70284	14.06906	17.66062	27.50779	37.205	35.13026
0.2	0.092908	0.102991	0.117597	0.136622	0.161952	0.197787	0.24483	0.331844	0.506831	1	1.110633	1.190307	1.336983	1.813547	2.268648	2.638224	3.001642	4.005204	16.65719	17.97781	14.69741	18.18607	27.86447	38.2275	36.70857
0.1	0.095165	0.106542	0.117508	0.1393	0.161471	0.192579	0.249217	0.340529	0.513082	1	1.14408	1.236984	1.3799	1.85121	2.215704	2.69565	3.040829	4.120151	16.16886	18.38043	15.41081	18.96954	28.82791	40.06458	39.33994
0.01	0.098646	0.110981	0.122516	0.140193	0.166352	0.205693	0.25101	0.341044	0.507963	1	1.156895	1.242177	1.400454	1.80876	2.230744	2.618038	2.827256	3.952846	15.16217	17.55797	15.63669	20.30835	30.73139	41.92975	48.18644
0.001	0.105968	0.120334	0.132528	0.147829	0.174598	0.209199	0.258366	0.346513	0.503852	1	1.1339	1.223462	1.354349	1.726655	2.158786	2.387787	2.432146	3.465355	13.52179	15.75604	14.76305	19.14327	28.19695	38.73537	44.68858
9.00E-04	0.109189	0.121823	0.134614	0.150349	0.174724	0.213337	0.259771	0.348965	0.509144	1	1.146854	1.225845	1.384217	1.729436	2.144164	2.409412	2.432714	3.460816	13.76555	16.0644	14.78414	19.19727	28.37591	38.5603	44.86755
8.00E-04	0.108244	0.119003	0.13258	0.149945	0.170672	0.208608	0.252499	0.344511	0.494809	1	1.12647	1.207502	1.346371	1.710902	2.116451	2.374599	2.373539	3.376839	13.28348	15.37574	14.42569	18.62919	27.51233	38.08589	43.86378
7.00E-04	0.110766	0.124269	0.135318	0.153022	0.177719	0.214447	0.262332	0.353309	0.508129	1	1.150886	1.227953	1.361038	1.750724	2.145728	2.405834	2.413232	3.427713	13.45784	15.48253	14.66695	18.8927	27.93027	38.50134	44.61283
6.00E-04	0.113129	0.125577	0.140427	0.154951	0.180626	0.214668	0.266805	0.352756	0.519984	1	1.168573	1.233742	1.383936	1.752824	2.139718	2.386996	2.403301	3.414733	13.45139	15.49419	14.64773	18.85255	28.29464	38.31549	44.58581
5.00E-04	0.113074	0.125879	0.138754	0.155793	0.181715	0.215202	0.263952	0.351754	0.514789	1	1.147795	1.230494	1.362885	1.720766	2.132875	2.368857	2.486042	3.380301	13.29963	15.17805	14.43933	18.53741	27.84879	37.74883	44.17618
4.00E-04	0.115285	0.127598	0.140432	0.156941	0.181505	0.217406	0.271146	0.351305	0.520706	1	1.159816	1.228746	1.386092	1.755584	2.116832	2.385945	2.90741	3.328666	13.21721	14.99809	14.25151	18.3079	27.50759	37.1753	43.77628
3.00E-04	0.116015	0.130516	0.143858	0.158599	0.185424	0.21854	0.269735	0.357604	0.512644	1	1.159981	1.222272	1.378058	1.737009	2.094861	2.330806	3.00283	3.260694	12.74839	14.71057	14.0328	18.24771	27.0587	36.24465	43.16978
2.00E-04	0.121631	0.134836	0.145828	0.161595	0.19055	0.22273	0.274826	0.362312	0.521075	1	1.162971	1.227952	1.378012	1.747537	2.139119	2.355864	2.997002	3.213627	12.61934	14.74677	13.81212	17.93634	26.47085	35.30276	42.54276
1.00E-04	0.126846	0.13866	0.153274	0.167321	0.192509	0.22974	0.282324	0.366959	0.530483	1	1.155324	1.220526	1.380003	1.752846	2.048001	2.393422	2.96491	3.092433	12.05821	14.22355	13.26705	17.23861	25.2102	34.05152	41.16793

average(0.4:1e-4)	0.107649	0.11999	0.132919	0.149685	0.175354	0.209888	0.26022	0.348897	0.513218	1	1.147924	1.220134	1.361297	1.767278	2.152345	2.454956	2.735065	3.514797	14.11971	16.15065	14.46165	18.53527	27.87105	37.90077	41.97517
1/θ	0.1	0.111111	0.125	0.142857	0.166667	0.2	0.25	0.333333	0.5	1	1.111111	1.25	1.428571	1.666667	2	2.5	3.333333	5	10	11.11111	12.5	14.28571	16.66667	20	25

迭代次数随着角度θ的减小而逐渐增加

用分类映射的办法分类两条夹角为0.3度的直线

假设1：完全相同的两个对象无法被分成两类，与之对应的分类迭代次数为无穷大，分类准确率是50%，50%。相等收敛标准下迭代次数越大表明二者差异越小。

这一现象可以用假设1解释，因为θ减小将导致两条直线趋近重合，并导致差异减小，从而迭代次数增加。

另外发现迭代次数在0.4<=θ<=10内有一个非常明显的规律

用分类映射的办法分类两条夹角为0.3度的直线

角θ的迭代次数与当θ=1时的迭代次数的比约为1/θ。

比如当θ=0.6，收敛标准=1e-4时，n0.6的迭代次数实测为565152，用公式估算=322420/0.6=537366误差约为5%。

再观察分类准确率的数据

	10	9	8	7	6	5	4	3	2	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0.09	0.08	0.07	0.06	0.05	0.04
δ	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave	平均准确率p-ave
0.5	0.501347	0.500349	0.5	0.500465	0.500181	0.500008	0.500186	0.499902	0.500088	0.500058	0.500417	0.500093	0.499915	0.500043	0.500362	0.499809	0.500028	0.499759	0.500166	0.500101	0.499957	0.500158	0.499889	0.50005	0.500043
0.4	0.965457	0.967872	0.977319	0.965643	0.962161	0.924279	0.849533	0.654043	0.620018	0.529583	0.534296	0.53143	0.515324	0.510666	0.500274	0.500503	0.51903	0.506133	0.495249	0.499899	0.49605	0.486945	0.5	0.497138	0.505809
0.3	0.979269	0.979417	0.984598	0.974613	0.981173	0.97992	0.976807	0.931113	0.772146	0.547513	0.513965	0.530126	0.516329	0.515837	0.502889	0.506289	0.519357	0.5105	0.499106	0.5	0.498055	0.489545	0.504015	0.5	0.503229
0.2	0.98297	0.986837	0.989935	0.979264	0.98855	0.985882	0.977925	0.982726	0.97196	0.645862	0.530299	0.559523	0.512407	0.563915	0.50843	0.513314	0.527407	0.525181	0.514977	0.501131	0.497	0.4995	0.5095	0.497	0.504304
0.1	0.986819	0.991279	0.99349	0.983701	0.991153	0.990673	0.986299	0.986138	0.979013	0.921229	0.929317	0.889746	0.790874	0.971769	0.520312	0.793568	0.5	0.940548	0.4985	0.4995	0.503852	0.5	0.511005	0.504497	0.516148
0.01	0.994842	0.995472	0.996307	0.99146	0.995342	0.995211	0.994942	0.992905	0.989357	0.994261	0.99459	0.992812	0.992616	0.987146	0.997487	0.997111	0.511955	0.541201	0.5	0.5	0.500658	0.874975	0.508	0.5	0.5
0.001	0.995392	0.998774	0.997809	0.992874	0.996284	0.996626	0.996575	0.993472	0.99002	0.997613	0.994103	0.995118	0.995784	0.996	0.99449	0.9985	0.508003	0.5	0.988	0.5	0.501281	0.5	0.5	0.5025	0.5
9.00E-04	0.995389	0.99843	0.99791	0.992389	0.996776	0.996671	0.99693	0.993018	0.990379	0.998018	0.994038	0.995043	0.995731	0.996	0.994515	0.9985	0.507741	0.5	0.988	0.5	0.501274	0.5	0.5	0.5025	0.5
8.00E-04	0.995324	0.99853	0.997563	0.993065	0.997299	0.996771	0.997163	0.993183	0.990204	0.99846	0.994143	0.995349	0.995467	0.996	0.994073	0.9985	0.507603	0.5	0.91201	0.5	0.5015	0.5	0.5	0.5	0.5
7.00E-04	0.99553	0.998462	0.997814	0.993568	0.997216	0.996621	0.997503	0.993161	0.990754	0.999171	0.994422	0.995337	0.995116	0.996	0.994158	0.998646	0.50748	0.5	0.5	0.5	0.5015	0.5	0.5	0.5	0.5
6.00E-04	0.995701	0.998153	0.997477	0.993611	0.997038	0.996734	0.997163	0.993317	0.991601	0.999329	0.994794	0.99507	0.994882	0.996	0.994264	0.999153	0.507442	0.5	0.5	0.5	0.5015	0.5	0.5	0.5	0.5
5.00E-04	0.995807	0.998377	0.997364	0.992163	0.996402	0.996794	0.997319	0.993678	0.991779	0.9995	0.995254	0.995191	0.99497	0.996	0.993719	0.9995	0.581188	0.5	0.5	0.5	0.5015	0.5	0.5	0.5	0.5
4.00E-04	0.996681	0.998332	0.997437	0.99199	0.996595	0.996812	0.997384	0.994161	0.992859	0.9995	0.995364	0.995035	0.995274	0.996	0.993882	0.9995	0.905487	0.5	0.507236	0.5	0.500691	0.5	0.5	0.5	0.5
3.00E-04	0.996693	0.998889	0.99749	0.991543	0.996111	0.996399	0.99751	0.993445	0.992739	0.9995	0.995638	0.994972	0.996121	0.996	0.993704	0.999643	0.9965	0.5	0.961063	0.5	0.5005	0.5	0.5	0.5	0.5
2.00E-04	0.997003	0.999332	0.998131	0.99245	0.996814	0.996565	0.997631	0.993932	0.993583	0.9995	0.995653	0.994324	0.997023	0.996	0.994352	1	0.996573	0.501005	0.981779	0.5	0.5005	0.5	0.5	0.5	0.5
1.00E-04	0.996279	0.998724	0.99858	0.993922	0.997259	0.997435	0.997844	0.994296	0.994786	0.9995	0.995111	0.99446	0.996342	0.9965	0.996186	1	0.997	0.506156	0.9845	0.5	0.5005	0.5	0.5	0.5	0.502025

用分类映射的办法分类两条夹角为0.3度的直线

在0.3<=θ<=10这个区间，可以明显的观察到pave有上升的趋势，分类准确率很高，接近100%。但当θ<0.3以后pave急剧下降，到接近50%。也就是当θ<0.3以后虽然网络仍然可以收敛但已经不能有效分类。

也就是用映射法分类两条直线，θ最小值约为0.3度。

用分类映射的办法分类两条夹角为0.3度的直线

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