visualization + design

Statistics for aneuploidy level `h` = 1 2 3 4 5 6 7 8 9 10

Haploid Genome Coverage Tables

Given a location `x` defined in the context of `h` chromosomes, the probability that position `x` is covered at least `\phi` times is `P_{h,\phi}` and given by $$ P_{h,\phi} = \left( 1 - \sum \frac{1}{k!} \left( \frac{\rho}{h}^k \right) e^{-\rho/h} \right)^h \tag{1} $$

For more details, see Wendl, M.C. and R.K. Wilson. 2008. Aspects of coverage in medical DNA sequencing. BMC Bioinformatics 9: 239.

For a given sequencing redundancy `\rho` (e.g. `\rho`-fold, as determined by the length of the haploid genome) of a haploid genome, the fraction of the haploid genome represented by at least `\phi` reads is reported by `P_{h,\phi}`. Coverage by fewer than `\phi` reads is reported as `1-P_{h,\phi}`. Coverage by exactly `\phi` reads is `P_{h,\phi} - P_{h,\phi+1}`. Entries for which fractional coverage is `\lt 10^{-5}` are not shown.

A rudimentary Monte Carlo simulation of genome coverage is also available, and is a useful supplement to the exact probabilities shown here.

CUSTOM DEPTH AND PLOIDY To create a table with a specific ploidy (e.g. 12) and haploid-equivalent (see below) depth (e.g. `200 \times`), use

http://mkweb.bcgsc.ca/coverage/?aneuploidy=12&depth=200

EXAMPLE 1

Suppose you carried out 3-fold redundant (`\rho=3`) sequencing of a haploid genome (`h=1`). 95.02% of the genome will be covered by at least one read (`P_{1,1}`) while 22.40% will be covered by exactly 3 reads (`P_{1,3} - P_{1,4}`).

EXAMPLE 2

You are sequencing a sample with a tumor content of 25% and you're interested in the depth of sequencing required to detect heterozygous mutations in the tumor. This scenario is equivalent to an aneuploidy = 8 genome—any given allele is present 8 times. If you sequence at (`\rho=200`), then 95% of the bases will be covered at a depth of at least `\phi = 14` (`P_{8,14} = 0.9494`). If you're satisfied with `\phi = 5` then you only need `\rho = 100` since now `P_{8,5} = 0.9580`.

ANALYTICAL vs STOCHASTIC

View plot that compares analytical vs stochastic results.

HAPLOID vs DIPLOID

View plot that compares 100x and 200x coverage of haploid and diploid genomes.

CODE

Download Perl scripts for analytical (to produce the tables below for any `\rho`) and stochastic coverage calculations.

sequencing redundancy for a haploid genome

View table for sequencing redundancy `\rho` = 1 2 3 4 5 6 7 8 9 10 20 25 50 75 100 of a haploid genome.

IMPORTANT The redundancy is always calculated using the size of the haploid genome. For example, if we collect 600 Gb of reads, our sequencing redundancy is `600 / 3 = 200 \times`. We've used the length of the haploid genome (3 Gb) in the calculation. If we now apply this `200 \times` sequencing to a diploid genome, our average coverage will not be `200 \times` but slightly less than `100 \times`.

sequencing redundancy 1-fold (`\rho / h = 1.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.3679	0.0000	1.0000
1	0.3679	0.3679	0.6321
2	0.1839	0.7358	0.2642
3	0.0613	0.9197	0.0803
4	0.0153	0.9810	0.0190
5	0.0031	0.9963	0.0037

sequencing redundancy 2-fold (`\rho / h = 2.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.1353	0.0000	1.0000
1	0.2707	0.1353	0.8647
2	0.2707	0.4060	0.5940
3	0.1804	0.6767	0.3233
4	0.0902	0.8571	0.1429
5	0.0361	0.9473	0.0527
6	0.0120	0.9834	0.0166
7	0.0034	0.9955	0.0045
8	0.0009	0.9989	0.0011
9	0.0002	0.9998	0.0002
10	0.0000	1.0000	0.0000

sequencing redundancy 3-fold (`\rho / h = 3.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0498	0.0000	1.0000
1	0.1494	0.0498	0.9502
2	0.2240	0.1991	0.8009
3	0.2240	0.4232	0.5768
4	0.1680	0.6472	0.3528
5	0.1008	0.8153	0.1847
6	0.0504	0.9161	0.0839
7	0.0216	0.9665	0.0335
8	0.0081	0.9881	0.0119
9	0.0027	0.9962	0.0038
10	0.0008	0.9989	0.0011
11	0.0002	0.9997	0.0003
12	0.0001	0.9999	0.0001
13	0.0000	1.0000	0.0000

sequencing redundancy 4-fold (`\rho / h = 4.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0183	0.0000	1.0000
1	0.0733	0.0183	0.9817
2	0.1465	0.0916	0.9084
3	0.1954	0.2381	0.7619
4	0.1954	0.4335	0.5665
5	0.1563	0.6288	0.3712
6	0.1042	0.7851	0.2149
7	0.0595	0.8893	0.1107
8	0.0298	0.9489	0.0511
9	0.0132	0.9786	0.0214
10	0.0053	0.9919	0.0081
11	0.0019	0.9972	0.0028
12	0.0006	0.9991	0.0009
13	0.0002	0.9997	0.0003
14	0.0001	0.9999	0.0001
15	0.0000	1.0000	0.0000

sequencing redundancy 5-fold (`\rho / h = 5.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0067	0.0000	1.0000
1	0.0337	0.0067	0.9933
2	0.0842	0.0404	0.9596
3	0.1404	0.1247	0.8753
4	0.1755	0.2650	0.7350
5	0.1755	0.4405	0.5595
6	0.1462	0.6160	0.3840
7	0.1044	0.7622	0.2378
8	0.0653	0.8666	0.1334
9	0.0363	0.9319	0.0681
10	0.0181	0.9682	0.0318
11	0.0082	0.9863	0.0137
12	0.0034	0.9945	0.0055
13	0.0013	0.9980	0.0020
14	0.0005	0.9993	0.0007
15	0.0002	0.9998	0.0002
16	0.0000	0.9999	0.0001
17	0.0000	1.0000	0.0000

sequencing redundancy 6-fold (`\rho / h = 6.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0025	0.0000	1.0000
1	0.0149	0.0025	0.9975
2	0.0446	0.0174	0.9826
3	0.0892	0.0620	0.9380
4	0.1339	0.1512	0.8488
5	0.1606	0.2851	0.7149
6	0.1606	0.4457	0.5543
7	0.1377	0.6063	0.3937
8	0.1033	0.7440	0.2560
9	0.0688	0.8472	0.1528
10	0.0413	0.9161	0.0839
11	0.0225	0.9574	0.0426
12	0.0113	0.9799	0.0201
13	0.0052	0.9912	0.0088
14	0.0022	0.9964	0.0036
15	0.0009	0.9986	0.0014
16	0.0003	0.9995	0.0005
17	0.0001	0.9998	0.0002
18	0.0000	0.9999	0.0001
19	0.0000	1.0000	0.0000

sequencing redundancy 7-fold (`\rho / h = 7.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0009	0.0000	1.0000
1	0.0064	0.0009	0.9991
2	0.0223	0.0073	0.9927
3	0.0521	0.0296	0.9704
4	0.0912	0.0818	0.9182
5	0.1277	0.1730	0.8270
6	0.1490	0.3007	0.6993
7	0.1490	0.4497	0.5503
8	0.1304	0.5987	0.4013
9	0.1014	0.7291	0.2709
10	0.0710	0.8305	0.1695
11	0.0452	0.9015	0.0985
12	0.0263	0.9467	0.0533
13	0.0142	0.9730	0.0270
14	0.0071	0.9872	0.0128
15	0.0033	0.9943	0.0057
16	0.0014	0.9976	0.0024
17	0.0006	0.9990	0.0010
18	0.0002	0.9996	0.0004
19	0.0001	0.9999	0.0001
20	0.0000	1.0000	0.0000
21	0.0000	1.0000	0.0000

sequencing redundancy 8-fold (`\rho / h = 8.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0003	0.0000	1.0000
1	0.0027	0.0003	0.9997
2	0.0107	0.0030	0.9970
3	0.0286	0.0138	0.9862
4	0.0573	0.0424	0.9576
5	0.0916	0.0996	0.9004
6	0.1221	0.1912	0.8088
7	0.1396	0.3134	0.6866
8	0.1396	0.4530	0.5470
9	0.1241	0.5925	0.4075
10	0.0993	0.7166	0.2834
11	0.0722	0.8159	0.1841
12	0.0481	0.8881	0.1119
13	0.0296	0.9362	0.0638
14	0.0169	0.9658	0.0342
15	0.0090	0.9827	0.0173
16	0.0045	0.9918	0.0082
17	0.0021	0.9963	0.0037
18	0.0009	0.9984	0.0016
19	0.0004	0.9993	0.0007
20	0.0002	0.9997	0.0003
21	0.0001	0.9999	0.0001
22	0.0000	1.0000	0.0000
23	0.0000	1.0000	0.0000

sequencing redundancy 9-fold (`\rho / h = 9.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0001	0.0000	1.0000
1	0.0011	0.0001	0.9999
2	0.0050	0.0012	0.9988
3	0.0150	0.0062	0.9938
4	0.0337	0.0212	0.9788
5	0.0607	0.0550	0.9450
6	0.0911	0.1157	0.8843
7	0.1171	0.2068	0.7932
8	0.1318	0.3239	0.6761
9	0.1318	0.4557	0.5443
10	0.1186	0.5874	0.4126
11	0.0970	0.7060	0.2940
12	0.0728	0.8030	0.1970
13	0.0504	0.8758	0.1242
14	0.0324	0.9261	0.0739
15	0.0194	0.9585	0.0415
16	0.0109	0.9780	0.0220
17	0.0058	0.9889	0.0111
18	0.0029	0.9947	0.0053
19	0.0014	0.9976	0.0024
20	0.0006	0.9989	0.0011
21	0.0003	0.9996	0.0004
22	0.0001	0.9998	0.0002
23	0.0000	0.9999	0.0001
24	0.0000	1.0000	0.0000

sequencing redundancy 10-fold (`\rho / h = 10.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0000	0.0000	1.0000
1	0.0005	0.0000	1.0000
2	0.0023	0.0005	0.9995
3	0.0076	0.0028	0.9972
4	0.0189	0.0103	0.9897
5	0.0378	0.0293	0.9707
6	0.0631	0.0671	0.9329
7	0.0901	0.1301	0.8699
8	0.1126	0.2202	0.7798
9	0.1251	0.3328	0.6672
10	0.1251	0.4579	0.5421
11	0.1137	0.5830	0.4170
12	0.0948	0.6968	0.3032
13	0.0729	0.7916	0.2084
14	0.0521	0.8645	0.1355
15	0.0347	0.9165	0.0835
16	0.0217	0.9513	0.0487
17	0.0128	0.9730	0.0270
18	0.0071	0.9857	0.0143
19	0.0037	0.9928	0.0072
20	0.0019	0.9965	0.0035
21	0.0009	0.9984	0.0016
22	0.0004	0.9993	0.0007
23	0.0002	0.9997	0.0003
24	0.0001	0.9999	0.0001
25	0.0000	1.0000	0.0000
26	0.0000	1.0000	0.0000

sequencing redundancy 20-fold (`\rho / h = 20.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
4	0.0000	0.0000	1.0000
5	0.0001	0.0000	1.0000
6	0.0002	0.0001	0.9999
7	0.0005	0.0003	0.9997
8	0.0013	0.0008	0.9992
9	0.0029	0.0021	0.9979
10	0.0058	0.0050	0.9950
11	0.0106	0.0108	0.9892
12	0.0176	0.0214	0.9786
13	0.0271	0.0390	0.9610
14	0.0387	0.0661	0.9339
15	0.0516	0.1049	0.8951
16	0.0646	0.1565	0.8435
17	0.0760	0.2211	0.7789
18	0.0844	0.2970	0.7030
19	0.0888	0.3814	0.6186
20	0.0888	0.4703	0.5297
21	0.0846	0.5591	0.4409
22	0.0769	0.6437	0.3563
23	0.0669	0.7206	0.2794
24	0.0557	0.7875	0.2125
25	0.0446	0.8432	0.1568
26	0.0343	0.8878	0.1122
27	0.0254	0.9221	0.0779
28	0.0181	0.9475	0.0525
29	0.0125	0.9657	0.0343
30	0.0083	0.9782	0.0218
31	0.0054	0.9865	0.0135
32	0.0034	0.9919	0.0081
33	0.0020	0.9953	0.0047
34	0.0012	0.9973	0.0027
35	0.0007	0.9985	0.0015
36	0.0004	0.9992	0.0008
37	0.0002	0.9996	0.0004
38	0.0001	0.9998	0.0002
39	0.0001	0.9999	0.0001
40	0.0000	0.9999	0.0001
41	0.0000	1.0000	0.0000
42	0.0000	1.0000	0.0000

sequencing redundancy 25-fold (`\rho / h = 25.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
7	0.0000	0.0000	1.0000
8	0.0001	0.0000	1.0000
9	0.0001	0.0001	0.9999
10	0.0004	0.0002	0.9998
11	0.0008	0.0006	0.9994
12	0.0017	0.0014	0.9986
13	0.0033	0.0031	0.9969
14	0.0059	0.0065	0.9935
15	0.0099	0.0124	0.9876
16	0.0155	0.0223	0.9777
17	0.0227	0.0377	0.9623
18	0.0316	0.0605	0.9395
19	0.0415	0.0920	0.9080
20	0.0519	0.1336	0.8664
21	0.0618	0.1855	0.8145
22	0.0702	0.2473	0.7527
23	0.0763	0.3175	0.6825
24	0.0795	0.3939	0.6061
25	0.0795	0.4734	0.5266
26	0.0765	0.5529	0.4471
27	0.0708	0.6294	0.3706
28	0.0632	0.7002	0.2998
29	0.0545	0.7634	0.2366
30	0.0454	0.8179	0.1821
31	0.0366	0.8633	0.1367
32	0.0286	0.8999	0.1001
33	0.0217	0.9285	0.0715
34	0.0159	0.9502	0.0498
35	0.0114	0.9662	0.0338
36	0.0079	0.9775	0.0225
37	0.0053	0.9854	0.0146
38	0.0035	0.9908	0.0092
39	0.0023	0.9943	0.0057
40	0.0014	0.9966	0.0034
41	0.0009	0.9980	0.0020
42	0.0005	0.9988	0.0012
43	0.0003	0.9993	0.0007
44	0.0002	0.9996	0.0004
45	0.0001	0.9998	0.0002
46	0.0001	0.9999	0.0001
47	0.0000	0.9999	0.0001
48	0.0000	1.0000	0.0000
49	0.0000	1.0000	0.0000

sequencing redundancy 50-fold (`\rho / h = 50.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
24	0.0000	0.0000	1.0000
25	0.0000	0.0000	1.0000
26	0.0001	0.0001	0.9999
27	0.0001	0.0001	0.9999
28	0.0002	0.0003	0.9997
29	0.0004	0.0005	0.9995
30	0.0007	0.0009	0.9991
31	0.0011	0.0016	0.9984
32	0.0017	0.0027	0.9973
33	0.0026	0.0044	0.9956
34	0.0038	0.0070	0.9930
35	0.0054	0.0108	0.9892
36	0.0075	0.0162	0.9838
37	0.0102	0.0238	0.9762
38	0.0134	0.0340	0.9660
39	0.0172	0.0474	0.9526
40	0.0215	0.0646	0.9354
41	0.0262	0.0861	0.9139
42	0.0312	0.1123	0.8877
43	0.0363	0.1435	0.8565
44	0.0412	0.1798	0.8202
45	0.0458	0.2210	0.7790
46	0.0498	0.2669	0.7331
47	0.0530	0.3167	0.6833
48	0.0552	0.3697	0.6303
49	0.0563	0.4249	0.5751
50	0.0563	0.4812	0.5188
51	0.0552	0.5375	0.4625
52	0.0531	0.5927	0.4073
53	0.0501	0.6458	0.3542
54	0.0464	0.6959	0.3041
55	0.0422	0.7423	0.2577
56	0.0376	0.7845	0.2155
57	0.0330	0.8221	0.1779
58	0.0285	0.8551	0.1449
59	0.0241	0.8836	0.1164
60	0.0201	0.9077	0.0923
61	0.0165	0.9278	0.0722
62	0.0133	0.9443	0.0557
63	0.0105	0.9576	0.0424
64	0.0082	0.9682	0.0318
65	0.0063	0.9764	0.0236
66	0.0048	0.9827	0.0173
67	0.0036	0.9875	0.0125
68	0.0026	0.9911	0.0089
69	0.0019	0.9938	0.0062
70	0.0014	0.9957	0.0043
71	0.0010	0.9970	0.0030
72	0.0007	0.9980	0.0020
73	0.0005	0.9987	0.0013
74	0.0003	0.9991	0.0009
75	0.0002	0.9994	0.0006
76	0.0001	0.9996	0.0004
77	0.0001	0.9998	0.0002
78	0.0001	0.9999	0.0001
79	0.0000	0.9999	0.0001
80	0.0000	0.9999	0.0001
81	0.0000	1.0000	0.0000
82	0.0000	1.0000	0.0000
83	0.0000	1.0000	0.0000

sequencing redundancy 75-fold (`\rho / h = 75.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
42	0.0000	0.0000	1.0000
43	0.0000	0.0000	1.0000
44	0.0000	0.0000	1.0000
45	0.0001	0.0001	0.9999
46	0.0001	0.0001	0.9999
47	0.0001	0.0002	0.9998
48	0.0002	0.0004	0.9996
49	0.0003	0.0006	0.9994
50	0.0005	0.0009	0.9991
51	0.0007	0.0014	0.9986
52	0.0011	0.0021	0.9979
53	0.0015	0.0032	0.9968
54	0.0021	0.0047	0.9953
55	0.0028	0.0068	0.9932
56	0.0038	0.0096	0.9904
57	0.0050	0.0134	0.9866
58	0.0065	0.0184	0.9816
59	0.0082	0.0249	0.9751
60	0.0103	0.0331	0.9669
61	0.0126	0.0433	0.9567
62	0.0153	0.0560	0.9440
63	0.0182	0.0712	0.9288
64	0.0213	0.0894	0.9106
65	0.0246	0.1107	0.8893
66	0.0279	0.1353	0.8647
67	0.0313	0.1632	0.8368
68	0.0345	0.1945	0.8055
69	0.0375	0.2290	0.7710
70	0.0402	0.2665	0.7335
71	0.0424	0.3066	0.6934
72	0.0442	0.3490	0.6510
73	0.0454	0.3932	0.6068
74	0.0460	0.4386	0.5614
75	0.0460	0.4846	0.5154
76	0.0454	0.5307	0.4693
77	0.0442	0.5761	0.4239
78	0.0425	0.6203	0.3797
79	0.0404	0.6628	0.3372
80	0.0379	0.7032	0.2968
81	0.0350	0.7411	0.2589
82	0.0321	0.7761	0.2239
83	0.0290	0.8082	0.1918
84	0.0259	0.8371	0.1629
85	0.0228	0.8630	0.1370
86	0.0199	0.8858	0.1142
87	0.0172	0.9057	0.0943
88	0.0146	0.9229	0.0771
89	0.0123	0.9375	0.0625
90	0.0103	0.9498	0.0502
91	0.0085	0.9601	0.0399
92	0.0069	0.9685	0.0315
93	0.0056	0.9754	0.0246
94	0.0044	0.9810	0.0190
95	0.0035	0.9854	0.0146
96	0.0027	0.9889	0.0111
97	0.0021	0.9917	0.0083
98	0.0016	0.9938	0.0062
99	0.0012	0.9954	0.0046
100	0.0009	0.9966	0.0034
101	0.0007	0.9976	0.0024
102	0.0005	0.9983	0.0017
103	0.0004	0.9988	0.0012
104	0.0003	0.9991	0.0009
105	0.0002	0.9994	0.0006
106	0.0001	0.9996	0.0004
107	0.0001	0.9997	0.0003
108	0.0001	0.9998	0.0002
109	0.0000	0.9999	0.0001
110	0.0000	0.9999	0.0001
111	0.0000	0.9999	0.0001
112	0.0000	1.0000	0.0000
113	0.0000	1.0000	0.0000
114	0.0000	1.0000	0.0000
115	0.0000	1.0000	0.0000

sequencing redundancy 100-fold (`\rho / h = 100.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
61	0.0000	0.0000	1.0000
62	0.0000	0.0000	1.0000
63	0.0000	0.0000	1.0000
64	0.0000	0.0000	1.0000
65	0.0000	0.0001	0.9999
66	0.0001	0.0001	0.9999
67	0.0001	0.0002	0.9998
68	0.0002	0.0003	0.9997
69	0.0002	0.0004	0.9996
70	0.0003	0.0007	0.9993
71	0.0004	0.0010	0.9990
72	0.0006	0.0014	0.9986
73	0.0008	0.0020	0.9980
74	0.0011	0.0028	0.9972
75	0.0015	0.0040	0.9960
76	0.0020	0.0055	0.9945
77	0.0026	0.0074	0.9926
78	0.0033	0.0100	0.9900
79	0.0042	0.0133	0.9867
80	0.0052	0.0175	0.9825
81	0.0064	0.0226	0.9774
82	0.0078	0.0291	0.9709
83	0.0094	0.0369	0.9631
84	0.0112	0.0463	0.9537
85	0.0132	0.0575	0.9425
86	0.0154	0.0708	0.9292
87	0.0176	0.0861	0.9139
88	0.0201	0.1038	0.8962
89	0.0225	0.1238	0.8762
90	0.0250	0.1463	0.8537
91	0.0275	0.1714	0.8286
92	0.0299	0.1989	0.8011
93	0.0322	0.2288	0.7712
94	0.0342	0.2610	0.7390
95	0.0360	0.2952	0.7048
96	0.0375	0.3312	0.6688
97	0.0387	0.3687	0.6313
98	0.0395	0.4074	0.5926
99	0.0399	0.4468	0.5532
100	0.0399	0.4867	0.5133
101	0.0395	0.5266	0.4734
102	0.0387	0.5660	0.4340
103	0.0376	0.6047	0.3953
104	0.0361	0.6423	0.3577
105	0.0344	0.6784	0.3216
106	0.0325	0.7128	0.2872
107	0.0303	0.7453	0.2547
108	0.0281	0.7756	0.2244
109	0.0258	0.8037	0.1963
110	0.0234	0.8294	0.1706
111	0.0211	0.8529	0.1471
112	0.0188	0.8740	0.1260
113	0.0167	0.8928	0.1072
114	0.0146	0.9095	0.0905
115	0.0127	0.9241	0.0759
116	0.0110	0.9368	0.0632
117	0.0094	0.9478	0.0522
118	0.0079	0.9572	0.0428
119	0.0067	0.9651	0.0349
120	0.0056	0.9718	0.0282
121	0.0046	0.9773	0.0227
122	0.0038	0.9819	0.0181
123	0.0031	0.9857	0.0143
124	0.0025	0.9888	0.0112
125	0.0020	0.9912	0.0088
126	0.0016	0.9932	0.0068
127	0.0012	0.9948	0.0052
128	0.0010	0.9960	0.0040
129	0.0007	0.9970	0.0030
130	0.0006	0.9977	0.0023
131	0.0004	0.9983	0.0017
132	0.0003	0.9987	0.0013
133	0.0003	0.9991	0.0009
134	0.0002	0.9993	0.0007
135	0.0001	0.9995	0.0005
136	0.0001	0.9996	0.0004
137	0.0001	0.9997	0.0003
138	0.0001	0.9998	0.0002
139	0.0000	0.9999	0.0001
140	0.0000	0.9999	0.0001
141	0.0000	0.9999	0.0001
142	0.0000	1.0000	0.0000
143	0.0000	1.0000	0.0000
144	0.0000	1.0000	0.0000
145	0.0000	1.0000	0.0000

VIEW ALL

news + thoughts

Nasa to send our human genome discs to the Moon

Sat 23-03-2024

We'd like to say a ‘cosmic hello’: mathematics, culture, palaeontology, art and science, and ... human genomes.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca

▲ SANCTUARY PROJECT | A cosmic hello of art, science, and genomes. (details)

▲ SANCTUARY PROJECT | Benoit Faiveley, founder of the Sanctuary project gives the Sanctuary disc a visual check at CEA LeQ Grenoble (image: Vincent Thomas). (details)

▲ SANCTUARY PROJECT | Sanctuary team examines the Life disc at INRIA Paris Saclay (image: Benedict Redgrove) (details)

Comparing classifier performance with baselines

Sat 23-03-2024

All animals are equal, but some animals are more equal than others. —George Orwell

This month, we will illustrate the importance of establishing a baseline performance level.

Baselines are typically generated independently for each dataset using very simple models. Their role is to set the minimum level of acceptable performance and help with comparing relative improvements in performance of other models.

▲ Nature Methods Points of Significance column: Comparing classifier performance with baselines. (read)

Unfortunately, baselines are often overlooked and, in the presence of a class imbalance5, must be established with care.

Megahed, F.M, Chen, Y-J., Jones-Farmer, A., Rigdon, S.E., Krzywinski, M. & Altman, N. (2024) Points of significance: Comparing classifier performance with baselines. Nat. Methods 20.

Happy 2024 π Day—
sunflowers ho!

Sat 09-03-2024

Celebrate π Day (March 14th) and dig into the digit garden. Let's grow something.

▲ 2024 π DAY | A garden of 1,000 digits of π. (details)

How Analyzing Cosmic Nothing Might Explain Everything

Thu 18-01-2024

Huge empty areas of the universe called voids could help solve the greatest mysteries in the cosmos.

My graphic accompanying How Analyzing Cosmic Nothing Might Explain Everything in the January 2024 issue of Scientific American depicts the entire Universe in a two-page spread — full of nothing.

▲ How Analyzing Cosmic Nothing Might Explain Everything. Text by Michael Lemonick (editor), art direction by Jen Christiansen (Senior Graphics Editor), source: SDSS

The graphic uses the latest data from SDSS 12 and is an update to my Superclusters and Voids poster.

Michael Lemonick (editor) explains on the graphic:

“Regions of relatively empty space called cosmic voids are everywhere in the universe, and scientists believe studying their size, shape and spread across the cosmos could help them understand dark matter, dark energy and other big mysteries.

To use voids in this way, astronomers must map these regions in detail—a project that is just beginning.

Shown here are voids discovered by the Sloan Digital Sky Survey (SDSS), along with a selection of 16 previously named voids. Scientists expect voids to be evenly distributed throughout space—the lack of voids in some regions on the globe simply reﬂects SDSS’s sky coverage.”

voids

Sofia Contarini, Alice Pisani, Nico Hamaus, Federico Marulli Lauro Moscardini & Marco Baldi (2023) Cosmological Constraints from the BOSS DR12 Void Size Function Astrophysical Journal 953:46.

Nico Hamaus, Alice Pisani, Jin-Ah Choi, Guilhem Lavaux, Benjamin D. Wandelt & Jochen Weller (2020) Journal of Cosmology and Astroparticle Physics 2020:023.

Sloan Digital Sky Survey Data Release 12

constellation figures

Alan MacRobert (Sky & Telescope), Paulina Rowicka/Martin Krzywinski (revisions & Microscopium)

stars

Hoffleit & Warren Jr. (1991) The Bright Star Catalog, 5th Revised Edition (Preliminary Version).

cosmology

H₀ = 67.4 km/(Mpc·s), Ω_m = 0.315, Ω_v = 0.685. Planck collaboration Planck 2018 results. VI. Cosmological parameters (2018).

Error in predictor variables

Tue 02-01-2024

It is the mark of an educated mind to rest satisfied with the degree of precision that the nature of the subject admits and not to seek exactness where only an approximation is possible. —Aristotle

In regression, the predictors are (typically) assumed to have known values that are measured without error.

Practically, however, predictors are often measured with error. This has a profound (but predictable) effect on the estimates of relationships among variables – the so-called “error in variables” problem.

▲ Nature Methods Points of Significance column: Error in predictor variables. (read)

Error in measuring the predictors is often ignored. In this column, we discuss when ignoring this error is harmless and when it can lead to large bias that can leads us to miss important effects.

Altman, N. & Krzywinski, M. (2024) Points of significance: Error in predictor variables. Nat. Methods 20.

Background reading

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple linear regression. Nat. Methods 12:999–1000.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nat. Methods 13:541–542 (2016).

Das, K., Krzywinski, M. & Altman, N. (2019) Points of significance: Quantile regression. Nat. Methods 16:451–452.

Convolutional neural networks

Tue 02-01-2024

Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. – Richard Feynman

Following up on our Neural network primer column, this month we explore a different kind of network architecture: a convolutional network.

The convolutional network replaces the hidden layer of a fully connected network (FCN) with one or more filters (a kind of neuron that looks at the input within a narrow window).

▲ Nature Methods Points of Significance column: Convolutional neural networks. (read)

Even through convolutional networks have far fewer neurons that an FCN, they can perform substantially better for certain kinds of problems, such as sequence motif detection.

Derry, A., Krzywinski, M & Altman, N. (2023) Points of significance: Convolutional neural networks. Nature Methods 20:1269–1270.

Background reading

Derry, A., Krzywinski, M. & Altman, N. (2023) Points of significance: Neural network primer. Nature Methods 20:165–167.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nature Methods 13:541–542.

visualization + design

Haploid Genome Coverage Tables

sequencing redundancy for a haploid genome

sequencing redundancy 1-fold (`\rho / h = 1.0`)

sequencing redundancy 2-fold (`\rho / h = 2.0`)

sequencing redundancy 3-fold (`\rho / h = 3.0`)

sequencing redundancy 4-fold (`\rho / h = 4.0`)

sequencing redundancy 5-fold (`\rho / h = 5.0`)

sequencing redundancy 6-fold (`\rho / h = 6.0`)

sequencing redundancy 7-fold (`\rho / h = 7.0`)

sequencing redundancy 8-fold (`\rho / h = 8.0`)

sequencing redundancy 9-fold (`\rho / h = 9.0`)

sequencing redundancy 10-fold (`\rho / h = 10.0`)

sequencing redundancy 20-fold (`\rho / h = 20.0`)

sequencing redundancy 25-fold (`\rho / h = 25.0`)

sequencing redundancy 50-fold (`\rho / h = 50.0`)

sequencing redundancy 75-fold (`\rho / h = 75.0`)

sequencing redundancy 100-fold (`\rho / h = 100.0`)

Nasa to send our human genome discs to the Moon

Comparing classifier performance with baselines

Happy 2024 π Day—sunflowers ho!

How Analyzing Cosmic Nothing Might Explain Everything

voids

constellation figures

stars

cosmology

Error in predictor variables

Background reading

Convolutional neural networks

Background reading

Happy 2024 π Day—
sunflowers ho!