Selection of optimal factorization rank

A crucial step in data analysis with NMF is the determination of the optimal factorization rank, i.e. the number of columns of the matrix W or equivalently the number of rows of the matrix H. No consensus method for an automatic evaluation of the optimal factorization rank has been found to date. Instead, the decomposition is usually performed iteratively over a range of possible factorization ranks and different quality measures are computed for every tested factorization ranks. Many quality measures have been proposed:

• The Frobenius reconstruction error, i.e. the Frobenius norm of the residuals of the decomposition: ||W ⋅ H − V||_F

• Criteria to assess the stability of the decomposition:

  • The cophenetic correlation coefficient
  • An Amari type distance
  • Silhouette values over clusters of patterns extracted iteratively at the same factorization rank

The package ButchR provides a function to visualize all factorization metrics.

Frobenius reconstruction error

The accesor function FrobError returns a data frame with as many columns as the number of decomposed factorization ranks and as many rows as there are iterations per factorization rank.

These residuals are helpful to select the decompostion that is closer to the original matrix.

ButchR will recommend an optimal K value if there is a local minima across the mean Frobenius error of multiple factorization ranks.

FrobError(rna_norm_nmf_exp)

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## [1] "\nNo optimal K found, please select manualy\n"

k2	k3	k4	k5	k6	k7	k8	k9	k10
0.1975693	0.1833415	0.1671235	0.1532516	0.1481093	0.1340936	0.1367746	0.1352234	0.1176797
0.1968017	0.1816921	0.1629583	0.1545430	0.1470479	0.1380185	0.1313077	0.1300730	0.1169848
0.1973312	0.1792726	0.1674623	0.1527080	0.1452187	0.1325158	0.1341041	0.1318661	0.1200745
0.1965124	0.1770754	0.1614196	0.1540932	0.1437228	0.1324107	0.1272746	0.1293386	0.1176948
0.1968222	0.1763261	0.1655435	0.1521472	0.1438162	0.1389070	0.1308415	0.1295302	0.1155311
0.1977374	0.1870041	0.1622449	0.1507951	0.1465456	0.1379959	0.1390313	0.1243427	0.1211344
0.1963016	0.1832476	0.1610669	0.1517311	0.1495082	0.1414465	0.1323888	0.1272976	0.1230526
0.1987224	0.1787391	0.1606174	0.1531967	0.1403434	0.1394498	0.1359764	0.1341932	0.1184785
0.1974589	0.1766975	0.1628189	0.1513562	0.1415927	0.1410059	0.1366363	0.1267888	0.1273292
0.1971803	0.1785069	0.1713303	0.1591989	0.1381672	0.1319049	0.1371715	0.1273660	0.1233168

Factorization quality metrics and optimal K

Besides the Frobenius reconstruction error, ButchR also estimates the cophenetic correlation coefficient, the Amari type distance and Silhouette values.

The values of the computed factorization metrics can be accessed with OptKStats:

OptKStats(rna_norm_nmf_exp)

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rank_id	k	FrobError_min	FrobError_mean	FrobError_sd	FrobError_cv	sumSilWidth	meanSilWidth	copheneticCoeff	meanAmariDist
k2	2	0.1963016	0.1972437	0.0006984	0.0035407	19.77115	0.9885575	0.9773151	0.0027460
k3	3	0.1763261	0.1801903	0.0035052	0.0194528	22.77287	0.7590957	0.8144873	0.0533091
k4	4	0.1606174	0.1642586	0.0034874	0.0212309	31.81031	0.7952578	0.7598790	0.0437001
k5	5	0.1507951	0.1533021	0.0023846	0.0155550	25.97285	0.5194569	0.6734895	0.0851761
k6	6	0.1381672	0.1444072	0.0035920	0.0248740	33.09691	0.5516152	0.6077940	0.0875744
k7	7	0.1319049	0.1367749	0.0036902	0.0269798	38.07320	0.5439029	0.6412591	0.0939794
k8	8	0.1272746	0.1341507	0.0036326	0.0270786	34.12018	0.4265023	0.5583952	0.1325817
k9	9	0.1243427	0.1296020	0.0033991	0.0262273	36.35971	0.4039967	0.6838229	0.1413364
k10	10	0.1155311	0.1201276	0.0036093	0.0300453	43.32247	0.4332247	0.6127411	0.1341529

Based on the results of the factorization quality metrics, an optimal number of signatures (k) must be chosen, following these guidelines:

• Minize the Frobenius error, the coefficient of variation and the mean Amari distance.
• Maximize the sum and mean silhouette width and the cophenic coefficient.

## Plot K stats
gg_plotKStats(rna_norm_nmf_exp)

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