2016:Discovery of Repeated Themes & Sections Results

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Introduction

The task: algorithms take a piece of music as input, and output a list of patterns repeated within that piece. A pattern is defined as a set of ontime-pitch pairs that occurs at least twice (i.e., is repeated at least once) in a piece of music. The second, third, etc. occurrences of the pattern will likely be shifted in time and/or transposed, relative to the first occurrence. Ideally an algorithm will be able to discover all exact and inexact occurrences of a pattern within a piece, so in evaluating this task we are interested in both:

• (1) to what extent an algorithm can discover one occurrence, up to time shift and transposition, and;
• (2) to what extent it can find all occurrences.

The metrics establishment recall, establishment precision and establishment F1 address (1), and the metrics occurrence recall, occurrence precision, and occurrence F1 address (2).

Contribution

Existing approaches to music structure analysis in MIR tend to focus on segmentation (e.g., Weiss & Bello, 2010). The contribution of this task is to afford access to the note content itself (please see the example in Fig. 1A), requiring algorithms to do more than label time windows (e.g., the segmentations in Figs. 1B-D). For instance, a discovery algorithm applied to the piece in Fig. 1A should return a pattern corresponding to the note content of $P_{1}$ and $P_{2}$, as well as a pattern corresponding to the note content of $Q_{1}$. This is because $Q_{1}$ occurs again independently of the accompaniment in bars 19-22 (not shown here). The ground truth also contains nested patterns, such as $P_{1}$ in Fig. 1A being a subset of the sectional repetition $S_{1}$, reflecting the often-hierarchical nature of musical repetition. While we recognise the appealing simplicity of linear segmentation, in the Discovery of Repeated Themes & Sections task we are demanding analysis at a greater level of detail, and have built a ground truth that contains overlapping and nested patterns (Collins et al., 2014).

Figure 1. Pattern discovery v segmentation. (A) Bars 1-12 of Mozart’s Piano Sonata in E-flat major K282 mvt.2, showing some ground-truth themes and repeated sections; (B-D) Three linear segmentations. Numbers below the staff in Fig. 1A and below the segmentation in Fig. 1D indicate crotchet beats, from zero for bar 1 beat 1.

For a more detailed introduction to the task, please see 2015:Discovery_of_Repeated_Themes_&_Sections.

Ground Truth and Algorithms

The ground truth, called the Johannes Kepler University Patterns Test Database (JKUPTD-Aug2013), is based on motifs and themes in Barlow and Morgenstern (1953), Schoenberg (1967), and Bruhn (1993). Repeated sections are based on those marked by the composer. These annotations are supplemented with some of our own where necessary. A Development Database (JKUPDD-Aug2013) enabled participants to try out their algorithms. For each piece in the Development and Test Databases, symbolic and synthesised audio versions are crossed with monophonic and polyphonic versions, giving four versions of the task in total: symPoly, symMono, audPoly, and audMono. There were no submissions to the symPoly category this year, so three versions of the task ran. Submitted algorithms are shown in Table 1.

Sub code	Submission name	Abstract	Contributors
Task Version	symMono
PLM1	SYMCHM	PDF	Matevž Pesek, Urša Medvešek, Aleš Leonardis, Matija Marolt
OL1'14	PatMinr	PDF	Olivier Lartillot
VM2'14	VM2	PDF	Gissel Velarde, David Meredith
Task Version	audMono
WHD1	VMO Motif Discovery	PDF	Cheng-i Wang, Jennifer Hsu, Shlomo Dubnov
WDH1	VMO Motif Discovery FML	PDF	Cheng-i Wang, Jennifer Hsu, Shlomo Dubnov
NF1'14	MotivesExtractor	PDF	Oriol Nieto, Morwaread Farbood
Task Version	audPoly
WHD1	VMO Motif Discovery	PDF	Cheng-i Wang, Jennifer Hsu, Shlomo Dubnov
WDH1	VMO Motif Discovery FML	PDF	Cheng-i Wang, Jennifer Hsu, Shlomo Dubnov
NF1'14	MotivesExtractor	PDF	Oriol Nieto, Morwaread Farbood

Table 1. Algorithms submitted to DRTS. Strong-performing algorithms from 2014 (submission codes ending '14) are included for the sake of comparisons.

Results

(For mathematical definitions of the various metrics, please see 2015:Discovery_of_Repeated_Themes_&_Sections#Evaluation_Procedure.)

symMono

We welcomed a new participant (Ren, 2016) to the symMono version of the task. All other researchers participated in previous years, but some (Meredith, 2016; Pesek, Leonardis, & Marolt, 2016) submitted new versions of algorithms.

To recap some information from previous years, the metrics that can be calculated on a per-pattern basis (establishment recall and occurrence recall) present opportunities to test for significant differences in performance between algorithms. For the remaining metrics, which can only be calculated on a per-piece basis, we do not test for significant differences because the sample size is too small. Still, however, trends in results are evident in the figures and tables below. For instance, in the symMono task version, VM1's (Velarde & Meredith, 2016) establishment F1 score outperforms all other algorithms apart from for one piece (Fig. 18). It is also the stand-out performer on establishment recall (Fig. 14). IR1 does particularly well for some of the smaller/shorter patterns in piece 5 (Fig. 14), which other algorithms seem to miss.

Application of Friedman's test to establishment recall per pattern results revealed a significant main effect of algorithm ($\chi ^{2}(3)=54.5062,\ p<.001$). Bonferroni-corrected, pairwise tests suggested the ordering VM1 > DM1 ~ IR1 > PLM1, where > denotes a significant difference and ~ denotes no significant difference. Application of Friedman's test to occurrence recall per pattern results revealed a significant main effect of algorithm ($\chi ^{2}(3)=54.5062,\ p<.001$). Bonferroni-corrected, pairwise tests suggested the ordering VM1 > DM1 ~ PLM1 ~ IR1. One of these tests was only borderline-significant (that between VM1 and PLM1), but this is probably due to averaging results for PLM1 on piece 5 (see below or Fig. 15).

With regards runtimes (Fig. 25), it should be noted that those for PLM1 and IR1 are somewhat harsh, because the submissions had to be run on slower machines than the Linux cluster on which the other submissions were ran. This is a function of programming language/operating system used by the researchers. After running for one week on piece 5 (the longest piece), PLM1 did not produce output, so was assigned mean values over the remaining pieces (Figs. 14 and 15). It was assigned the maximum runtime over the remaining pieces (Fig. 25).

PLM1, DM2, VM1, and VM2 output far fewer patterns than other algorithms (see the n_Q column in Table 4). One potential application of pattern discovery algorithms is for playback accompanied by a visualization of repetitive structure. The relatively few patterns output by these algorithms make them more feasible candidates for this application.

symPoly

Meredith (2016) submitted three algorithms to the symPoly task version, selected according to performance on three-layer F1, precision, and recall metrics, respectively. DM3 outperforms DM2 on establishment recall ($\chi ^{2}(1)=11.9189,\ p<.05$, Bonferroni corrected), which is perhaps to be expected because DM3 was submitted by Meredith (2016) on the basis of a recall metric, whereas DM2 was submitted on the basis of a precision metric. Similarly, DM1 outperforms DM2 on establishment recall ($\chi ^{2}(1)=23.5161,\ p<.005$, Bonferroni corrected). Again, this is not particularly surprising because DM1 was submitted on the basis of an F1 performance metric (which combines recall and precision), whereas DM2 was submitted on the basis of a precision metric. DM3 is generally higher than DM2 on three-layer recall, and DM2 is mostly higher than DM3 on three-layer precision. Both of these results make sense, given the basis on upon which these algorithms were submitted.

Discussion

Since this task began in 2013, it has been observed that the discovery of repeated sections tends to be addressed well by submissions, and the discovery of shorter themes/motifs less well. This in itself is not a particularly surprising observation, since it mirrors the way music analysts might agree on the large-scale repetitive structure of a song/piece, but differ in interpreting its themes and/or motifs. It was exciting to see, however, that WDH1 and WHD1 (Wang et al., 2015a, 2015b) had high establishment recall compared with previous years for seven- and nine-note patterns in piece 1, a five-note pattern in piece 2, and nine- and thirteen-note patterns in piece 4 (see Fig. 14 for details). Whereas WHD1's establishment recall dropped off for six- and nine-note patterns in piece 5, establishment recall remained comparatively high for WDH1. The difference between these submissions was a fixed minimum length (FML) criterion that applied to WDH1 but not to WHD1. The criterion seems to have helped WDH1 to perform well on this aspect of the task.

The PLM1 submission (Pesek et al., 2015) is of interest because it (1) is the first deep architecture to be submitted to the task and (2) has been applied to other MIR tasks such as chord and multi-F0 estimation (Pesek et al., 2014). It outperformed previous algorithms on piece 1 (see Figs. 4-6), demonstrating its potential, but had lower establishment recall for pieces 2-5 (see Fig. 4). For those who have called for multipurpose, integrated MIR solutions, this approach represents an encouraging development.

It is good to see submissions from two new groups this year, and that the emphasis the task places on retrieving and evaluating hierarchical structure is being acknowledged and championed elsewhere (McFee, Nieto, & Bello, 2015). Next year we may switch to new training and test databases that focus not directly on the discovery task itself, but on an application of pattern discovery that can be used as a proxy to evaluate the extent to which algorithms have retrieved relevant repetitive material. If you are interested in helping out with the preparation of a new version of the task, you are welcome to get in touch.

Tom Collins, Leicester, 2015

Results in Detail

symPoly

Figure 2. Establishment recall on a per-pattern basis. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 3. Occurrence recall on a per-pattern basis. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 4. Establishment recall averaged over each piece/movement. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 5. Establishment precision averaged over each piece/movement. Establishment precision answers the following question. On average, how similar is the most similar ground-truth pattern prototype to an algorithm-output pattern?

Figure 6. Establishment F1 averaged over each piece/movement. Establishment F1 is an average of establishment precision and establishment recall.

Figure 7. Occurrence recall ($c=.75$) averaged over each piece/movement. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 8. Occurrence precision ($c=.75$) averaged over each piece/movement. Occurrence precision answers the following question. On average, how similar is the most similar discovered ground-truth occurrence set to a set of algorithm-output pattern occurrences?

Figure 9. Occurrence F1 ($c=.75$) averaged over each piece/movement. Occurrence F1 is an average of occurrence precision and occurrence recall.

Figure 10. Three-layer recall averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment recall), three-layer recall uses $2|P\cap Q|/(|P|+|Q|)$, which is a kind of F1 measure.

Figure 11. Three-layer precision averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment precision), three-layer precision uses $2|P\cap Q|/(|P|+|Q|)$, which is a kind of F1 measure.

Figure 12. Three-layer F1 (TLF) averaged over each piece/movement. TLF is an average of three-layer precision and three-layer recall.

Figure 13. Log runtime of the algorithm for each piece/movement.

symMono

(Submission PLM1 did not complete on piece 5. The task captain took the decision to assign the mean of the evaluation metrics for PLM1 calculated across the remaining pieces. Apart from runtime, in which case the maximum across the remaining pieces was assigned.)

Figure 14. Establishment recall on a per-pattern basis. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 15. Occurrence recall on a per-pattern basis. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 16. Establishment recall averaged over each piece/movement. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 17. Establishment precision averaged over each piece/movement. Establishment precision answers the following question. On average, how similar is the most similar ground-truth pattern prototype to an algorithm-output pattern?

Figure 18. Establishment F1 averaged over each piece/movement. Establishment F1 is an average of establishment precision and establishment recall.

Figure 19. Occurrence recall ($c=.75$) averaged over each piece/movement. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 20. Occurrence precision ($c=.75$) averaged over each piece/movement. Occurrence precision answers the following question. On average, how similar is the most similar discovered ground-truth occurrence set to a set of algorithm-output pattern occurrences?

Figure 21. Occurrence F1 ($c=.75$) averaged over each piece/movement. Occurrence F1 is an average of occurrence precision and occurrence recall.

Figure 22. Three-layer recall averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment recall), three-layer recall uses $2|P\cap Q|/(|P|+|Q|)$, which is a kind of F1 measure.

Figure 23. Three-layer precision averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment precision), three-layer precision uses $2|P\cap Q|/(|P|+|Q|)$, which is a kind of F1 measure.

Figure 24. Three-layer F1 (TLF) averaged over each piece/movement. TLF is an average of three-layer precision and three-layer recall.

Figure 25. Log runtime of the algorithm for each piece/movement.

Tabular Versions of Plots

symPoly

AlgIdx	AlgStub	Piece	n_P	n_Q	P_est	R_est	F1_est	P_occ(c=.75)	R_occ(c=.75)	F_1occ(c=.75)	P_3	R_3	TLF_1	runtime	FRT	FFTP_est	FFP	P_occ(c=.5)	R_occ(c=.5)	F_1occ(c=.5)	P	R	F_1
1	DM1	piece1	5	30.000	0.328	0.431	0.372	0.000	0.000	0.000	0.215	0.326	0.259	74.000	0.000	0.366	0.330	0.284	0.430	0.342	0.000	0.000	0.000
1	DM1	piece2	5	63.000	0.565	0.661	0.609	0.508	0.936	0.658	0.438	0.468	0.452	727.000	0.000	0.285	0.357	0.485	0.944	0.641	0.000	0.000	0.000
1	DM1	piece3	10.000	28.000	0.703	0.685	0.694	0.520	0.784	0.625	0.620	0.536	0.575	35.000	0.000	0.447	0.688	0.496	0.749	0.597	0.000	0.000	0.000
1	DM1	piece4	5	19.000	0.410	0.495	0.448	0.000	0.000	0.000	0.327	0.340	0.333	2.000	0.000	0.416	0.271	0.255	0.564	0.351	0.000	0.000	0.000
1	DM1	piece5	13.000	56.000	0.766	0.612	0.680	0.543	0.941	0.688	0.634	0.467	0.538	1894.000	0.000	0.391	0.686	0.513	0.890	0.651	0.000	0.000	0.000
2	DM2	piece1	5	10.000	0.353	0.291	0.319	0.000	0.000	0.000	0.242	0.174	0.202	77.000	0.000	0.291	0.268	0.424	0.156	0.228	0.000	0.000	0.000
2	DM2	piece2	5	10.000	0.320	0.306	0.313	0.000	0.000	0.000	0.295	0.253	0.273	710.000	0.000	0.306	0.449	0.388	0.692	0.497	0.000	0.000	0.000
2	DM2	piece3	10.000	10.000	0.777	0.497	0.606	0.710	0.802	0.753	0.737	0.450	0.559	33.000	0.000	0.435	0.711	0.651	0.775	0.708	0.000	0.000	0.000
2	DM2	piece4	5	10.000	0.519	0.476	0.496	0.565	0.952	0.709	0.450	0.293	0.355	1.000	0.000	0.280	0.626	0.441	0.839	0.578	0.000	0.000	0.000
2	DM2	piece5	13.000	10.000	0.857	0.397	0.543	0.614	0.913	0.734	0.717	0.358	0.478	1734.000	0.000	0.391	0.686	0.597	0.857	0.704	0.000	0.000	0.000
3	DM3	piece1	5	41.000	0.357	0.528	0.426	0.406	0.439	0.422	0.257	0.422	0.319	509.000	0.000	0.498	0.439	0.369	0.516	0.430	0.000	0.000	0.000
3	DM3	piece2	5	71.000	0.621	0.691	0.654	0.605	0.945	0.738	0.559	0.599	0.578	2109.000	0.000	0.290	0.305	0.533	0.908	0.672	0.000	0.000	0.000
3	DM3	piece3	10.000	25.000	0.594	0.589	0.592	0.528	0.777	0.629	0.585	0.522	0.551	89.000	0.000	0.302	0.687	0.481	0.728	0.580	0.000	0.000	0.000
3	DM3	piece4	5	24.000	0.465	0.488	0.476	0.355	0.765	0.485	0.352	0.337	0.344	2.000	0.000	0.415	0.583	0.279	0.617	0.384	0.000	0.000	0.000
3	DM3	piece5	13.000	75.000	0.697	0.470	0.561	0.644	0.948	0.767	0.657	0.443	0.529	4765.000	0.000	0.205	0.739	0.598	0.881	0.713	0.000	0.000	0.000

download these results as csv

Table 2. Tabular version of Figures 4-13.

AlgIdx	AlgStub	Piece	n_P	R_est	R_occ(c=.75)	R_occ(c=.5)
1	DM1	piece1	5
0.882	0.900	0.500	0.450	0.867
0.866	0.613	0.000	0.000	0.778
0.866	0.613	0.000	0.000	0.778
1	DM1	piece2	5
0.444	0.875	0.500	0.908	0.927
0.00000	0.802	0.000	0.908	0.923
0.00000	0.802	0.000	0.908	0.923
1	DM1	piece3	10.000
0.455	0.605	0.767	0.364	0.500	0.250	0.045	0.562	0.417	0.333
0.00000	0.000	0.678	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.678	0.000	0.000	0.000	0.000	0.000	0.000	0.000
1	DM1	piece4	8
1.00000	0.857	0.385	0.444	0.867	0.350	0.250	0.846
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
1	DM1	piece5	13.000
0.560	0.120	0.967	0.984	0.857	0.172	0.130	0.333	0.122	0.675	0.636	0.938	0.752
0.00000	0.000	0.967	0.949	0.711	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.752
0.00000	0.000	0.967	0.949	0.711	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.752
2	DM2	piece1	5
1.00000	1.000	0.125	0.000	0.682
0.866	0.613	0.000	0.000	0.000
0.866	0.613	0.000	0.000	0.000
2	DM2	piece2	5
0.056	0.348	0.021	0.876	0.419
0.00000	0.000	0.000	0.875	0.000
0.00000	0.000	0.000	0.875	0.000
2	DM2	piece3	10.000
0.349	0.612	0.619	0.163	0.500	0.250	0.045	0.562	0.146	0.083
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2	DM2	piece4	8
0.667	0.538	0.625	0.571	0.469	0.643	0.450	0.591
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2	DM2	piece5	13.000
0.094	0.092	0.350	0.273	0.254	0.122	0.051	0.069	0.084	0.462	0.243	0.938	0.515
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.000
3	DM3	piece1	5
0.882	0.857	0.333	0.273	0.867
0.866	0.613	0.000	0.000	0.778
0.866	0.613	0.000	0.000	0.778
3	DM3	piece2	5
0.800	0.438	0.333	0.672	0.920
0.739	0.000	0.000	0.000	0.913
0.739	0.000	0.000	0.000	0.913
3	DM3	piece3	10.000
0.600	0.674	0.722	1.000	0.579	0.355	0.065	0.795	1.000	0.667
0.00000	0.000	0.000	0.190	0.000	0.000	0.000	0.751	0.667	0.000
0.00000	0.000	0.000	0.190	0.000	0.000	0.000	0.751	0.667	0.000
3	DM3	piece4	8
1.00000	0.857	0.385	0.364	0.875	0.619	0.600	0.857
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
3	DM3	piece5	13.000
0.591	0.500	0.846	0.980	0.281	0.145	0.067	0.075	0.083	0.835	0.286	0.936	0.742
0.00000	0.000	0.846	0.947	0.000	0.000	0.000	0.000	0.000	0.504	0.000	0.924	0.000
0.00000	0.000	0.846	0.947	0.000	0.000	0.000	0.000	0.000	0.504	0.000	0.924	0.000
4	VM1	piece1	5
0.944	1.000	1.000	0.364	0.938
0.775	0.771	0.784	0.000	0.782
0.775	0.771	0.784	0.000	0.782
4	VM1	piece2	5
0.800	0.812	0.600	0.936	0.934
0.614	0.734	0.000	0.936	0.934
0.614	0.734	0.000	0.936	0.934
4	VM1	piece3	10.000
0.647	0.752	0.846	1.000	0.579	0.314	0.500	0.742	1.000	1.000
0.00000	0.752	0.724	0.831	0.000	0.000	0.000	0.000	0.905	1.000
0.00000	0.752	0.724	0.831	0.000	0.000	0.000	0.000	0.905	1.000
4	VM1	piece4	8
1.00000	0.857	0.600	1.000	1.000	1.000	0.900	1.000
0.860	0.429	0.000	1.000	1.000	0.912	0.750	1.000
0.860	0.429	0.000	1.000	1.000	0.912	0.750	1.000
4	VM1	piece5	13.000
0.857	0.714	0.989	0.993	0.968	0.400	0.125	0.529	0.500	0.771	0.800	0.668	0.573
0.857	0.000	0.989	0.993	0.968	0.000	0.000	0.000	0.000	0.366	0.746	0.000	0.000
0.857	0.000	0.989	0.993	0.968	0.000	0.000	0.000	0.000	0.366	0.746	0.000	0.000
5	VM2	piece1	5
0.727	0.719	0.625	0.111	0.667
0.00000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000
5	VM2	piece2	5
0.538	0.812	0.600	0.920	0.934
0.00000	0.734	0.000	0.920	0.934
0.00000	0.734	0.000	0.920	0.934
5	VM2	piece3	10.000
0.786	0.744	0.962	0.278	0.395	0.360	0.250	0.677	0.571	0.190
0.384	0.000	0.498	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.384	0.000	0.498	0.000	0.000	0.000	0.000	0.000	0.000	0.000
5	VM2	piece4	8
0.667	0.571	0.600	1.000	1.000	0.714	0.800	0.769
0.00000	0.000	0.000	0.911	1.000	0.000	0.733	0.769
0.00000	0.000	0.000	0.911	1.000	0.000	0.733	0.769
5	VM2	piece5	13.000
0.609	0.087	0.974	0.982	0.633	0.600	0.333	0.111	0.084	0.800	0.667	0.652	0.488
0.00000	0.000	0.974	0.982	0.000	0.000	0.000	0.000	0.000	0.388	0.000	0.000	0.000
0.00000	0.000	0.974	0.982	0.000	0.000	0.000	0.000	0.000	0.388	0.000	0.000	0.000
6	IR1	piece1	5
0.765	0.684	0.714	0.750	0.600
0.074	0.000	0.000	0.375	0.000
0.074	0.000	0.000	0.375	0.000
6	IR1	piece2	5
0.727	0.812	0.600	0.024	0.154
0.00000	0.109	0.000	0.000	0.000
0.00000	0.109	0.000	0.000	0.000
6	IR1	piece3	10.000
0.647	0.062	0.742	0.833	0.237	0.474	0.250	0.839	0.857	0.750
0.00000	0.000	0.000	0.144	0.000	0.000	0.000	0.468	0.238	0.188
0.00000	0.000	0.000	0.144	0.000	0.000	0.000	0.468	0.238	0.188
6	IR1	piece4	8
0.600	0.429	0.000	0.800	0.133	0.143	0.778	0.692
0.00000	0.000	0.000	0.177	0.000	0.000	0.389	0.000
0.00000	0.000	0.000	0.177	0.000	0.000	0.389	0.000
6	IR1	piece5	13.000
0.500	0.625	0.015	0.081	0.333	0.364	0.700	0.889	0.800	0.359	0.652	0.134	0.181
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.136	0.198	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.136	0.198	0.000	0.000	0.000	0.000
7	PLM1	piece1	5
0.941	0.842	0.077	0.000	0.000
0.309	0.265	0.000	0.000	0.000
0.309	0.265	0.000	0.000	0.000
7	PLM1	piece2	5
0.333	0.552	0.125	0.116	0.053
0.00000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000
7	PLM1	piece3	10.000
0.312	0.357	0.553	0.146	0.474	0.344	0.062	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
7	PLM1	piece4	8
0.250	0.500	0.000	0.286	0.000	1.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	1.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	1.000	0.000	0.000
7	PLM1	piece5	13.000
0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263
0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054
0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054

download these results as csv

Table 3. Tabular version of Figures 2 and 3.

symMono

AlgIdx	AlgStub	Piece	n_P	n_Q	P_est	R_est	F1_est	P_occ(c=.75)	R_occ(c=.75)	F_1occ(c=.75)	P_3	R_3	TLF_1	runtime	FRT	FFTP_est	FFP	P_occ(c=.5)	R_occ(c=.5)	F_1occ(c=.5)	P	R	F_1
1	DM1	piece1	5	44.000	0.410	0.720	0.523	0.470	0.689	0.559	0.237	0.537	0.328	152.000	0.000	0.518	0.606	0.388	0.695	0.498	0.000	0.000	0.000
1	DM1	piece2	5	47.000	0.535	0.731	0.618	0.518	0.904	0.659	0.453	0.616	0.522	90.000	0.000	0.208	0.344	0.434	0.883	0.582	0.000	0.000	0.000
1	DM1	piece3	10.000	13.000	0.484	0.430	0.455	0.678	0.678	0.678	0.481	0.386	0.429	2.000	0.000	0.359	0.542	0.402	0.605	0.483	0.000	0.000	0.000
1	DM1	piece4	8	24.000	0.331	0.625	0.432	0.524	0.771	0.624	0.274	0.425	0.333	1.000	0.000	0.519	0.497	0.445	0.802	0.572	0.042	0.125	0.062
1	DM1	piece5	13.000	72.000	0.740	0.558	0.636	0.538	0.930	0.682	0.640	0.445	0.525	688.000	0.000	0.263	0.650	0.517	0.869	0.648	0.000	0.000	0.000
2	DM2	piece1	5	10.000	0.709	0.561	0.627	0.687	0.697	0.692	0.448	0.435	0.441	139.000	0.000	0.436	0.534	0.541	0.699	0.610	0.100	0.200	0.133
2	DM2	piece2	5	10.000	0.395	0.344	0.368	0.524	0.875	0.656	0.375	0.277	0.319	64.000	0.000	0.208	0.344	0.350	0.875	0.500	0.000	0.000	0.000
2	DM2	piece3	10.000	10.000	0.520	0.333	0.406	0.000	0.000	0.000	0.490	0.342	0.403	2.000	0.000	0.333	0.534	0.370	0.536	0.438	0.000	0.000	0.000
2	DM2	piece4	8	10.000	0.290	0.569	0.384	0.000	0.000	0.000	0.205	0.237	0.220	1.000	0.000	0.197	0.225	0.166	0.405	0.236	0.000	0.000	0.000
2	DM2	piece5	13.000	10.000	0.821	0.273	0.409	0.564	0.926	0.701	0.683	0.234	0.349	575.000	0.000	0.263	0.650	0.540	0.865	0.665	0.000	0.000	0.000
3	DM3	piece1	5	44.000	0.364	0.642	0.465	0.518	0.717	0.602	0.224	0.520	0.313	156.000	0.000	0.518	0.606	0.404	0.736	0.521	0.000	0.000	0.000
3	DM3	piece2	5	47.000	0.544	0.633	0.585	0.673	0.886	0.765	0.508	0.571	0.538	234.000	0.000	0.572	0.578	0.545	0.824	0.656	0.021	0.200	0.038
3	DM3	piece3	10.000	14.000	0.633	0.646	0.639	0.700	0.536	0.607	0.534	0.523	0.528	4.000	0.000	0.294	0.588	0.502	0.565	0.531	0.071	0.100	0.083
3	DM3	piece4	8	26.000	0.376	0.695	0.488	0.398	0.804	0.532	0.260	0.404	0.316	2.000	0.000	0.671	0.505	0.287	0.709	0.409	0.038	0.125	0.059
3	DM3	piece5	13.000	60.000	0.618	0.490	0.547	0.570	0.866	0.688	0.589	0.414	0.486	848.000	0.000	0.311	0.705	0.528	0.806	0.638	0.000	0.000	0.000
4	VM1	piece1	5	7	0.557	0.849	0.673	0.360	0.777	0.492	0.344	0.362	0.353	61.569	0.000	0.849	0.482	0.372	0.777	0.503	0.000	0.000	0.000
4	VM1	piece2	5	7	0.412	0.817	0.548	0.486	0.805	0.606	0.320	0.430	0.367	393.263	0.000	0.472	0.074	0.399	0.764	0.524	0.286	0.400	0.333
4	VM1	piece3	10.000	7	0.773	0.738	0.755	0.442	0.823	0.575	0.627	0.521	0.569	12.921	0.000	0.628	0.557	0.533	0.703	0.606	0.000	0.000	0.000
4	VM1	piece4	8	7	0.932	0.920	0.926	0.644	0.869	0.740	0.749	0.620	0.679	66.093	0.000	0.812	0.735	0.526	0.855	0.651	0.286	0.250	0.267
4	VM1	piece5	13.000	7	0.819	0.684	0.745	0.512	0.755	0.610	0.675	0.400	0.503	51.341	0.000	0.566	0.549	0.414	0.656	0.507	0.286	0.154	0.200
5	VM2	piece1	5	5	0.540	0.570	0.555	0.000	0.000	0.000	0.286	0.207	0.240	18.250	0.000	0.570	0.286	0.291	0.490	0.365	0.000	0.000	0.000
5	VM2	piece2	5	7	0.446	0.761	0.562	0.649	0.863	0.741	0.357	0.488	0.413	32.966	0.000	0.419	0.128	0.427	0.630	0.509	0.143	0.200	0.167
5	VM2	piece3	10.000	7	0.690	0.521	0.594	0.865	0.441	0.584	0.609	0.471	0.531	3.580	0.000	0.393	0.541	0.662	0.561	0.607	0.000	0.000	0.000
5	VM2	piece4	8	6	0.842	0.765	0.802	0.579	0.837	0.684	0.732	0.504	0.597	5.863	0.000	0.721	0.711	0.410	0.732	0.525	0.167	0.125	0.143
5	VM2	piece5	13.000	7	0.739	0.540	0.624	0.910	0.781	0.841	0.677	0.434	0.529	37.876	0.000	0.420	0.554	0.517	0.636	0.570	0.000	0.000	0.000
6	IR1	piece1	5	100.000	0.557	0.703	0.622	0.123	0.188	0.149	0.177	0.256	0.209	388.120	0.000	0.499	0.208	0.175	0.191	0.183	0.000	0.000	0.000
6	IR1	piece2	5	100.000	0.685	0.464	0.553	0.074	0.109	0.088	0.093	0.105	0.099	7384.700	0.000	0.459	0.084	0.070	0.096	0.081	0.000	0.000	0.000
6	IR1	piece3	10.000	100.000	0.820	0.569	0.672	0.212	0.301	0.249	0.274	0.201	0.232	52.520	0.000	0.449	0.279	0.228	0.223	0.225	0.000	0.000	0.000
6	IR1	piece4	8	80.000	0.544	0.447	0.491	0.371	0.283	0.321	0.313	0.252	0.279	1.800	0.000	0.246	0.263	0.358	0.364	0.361	0.000	0.000	0.000
6	IR1	piece5	13.000	100.000	0.807	0.433	0.564	0.028	0.172	0.048	0.113	0.095	0.103	37865.940	0.000	0.433	0.105	0.063	0.135	0.086	0.000	0.000	0.000
7	PLM1	piece1	5	4	0.839	0.372	0.515	0.860	0.295	0.439	0.399	0.190	0.258	14436.000	0.000	0.372	0.399	0.775	0.287	0.419	0.000	0.000	0.000
7	PLM1	piece2	5	6	0.242	0.236	0.239	0.000	0.000	0.000	0.194	0.199	0.196	103741.000	0.000	0.122	0.138	0.552	0.276	0.368	0.000	0.000	0.000
7	PLM1	piece3	10.000	3	0.463	0.225	0.303	0.000	0.000	0.000	0.537	0.275	0.364	129.000	0.000	0.225	0.537	0.549	0.500	0.524	0.000	0.000	0.000
7	PLM1	piece4	8	2	0.750	0.254	0.380	1.000	1.000	1.000	0.734	0.208	0.325	411.000	0.000	0.254	0.734	0.719	0.651	0.683	0.500	0.125	0.200

download these results as csv

Table 4. Tabular version of Figures 16-25.

AlgIdx	AlgStub	Piece	n_P	R_est	R_occ(c=.75)	R_occ(c=.5)
1	DM1	piece1	5
0.882	0.900	0.500	0.450	0.867
0.866	0.613	0.000	0.000	0.778
0.866	0.613	0.000	0.000	0.778
1	DM1	piece2	5
0.444	0.875	0.500	0.908	0.927
0.00000	0.802	0.000	0.908	0.923
0.00000	0.802	0.000	0.908	0.923
1	DM1	piece3	10.000
0.455	0.605	0.767	0.364	0.500	0.250	0.045	0.562	0.417	0.333
0.00000	0.000	0.678	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.678	0.000	0.000	0.000	0.000	0.000	0.000	0.000
1	DM1	piece4	8
1.00000	0.857	0.385	0.444	0.867	0.350	0.250	0.846
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
1	DM1	piece5	13.000
0.560	0.120	0.967	0.984	0.857	0.172	0.130	0.333	0.122	0.675	0.636	0.938	0.752
0.00000	0.000	0.967	0.949	0.711	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.752
0.00000	0.000	0.967	0.949	0.711	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.752
2	DM2	piece1	5
1.00000	1.000	0.125	0.000	0.682
0.866	0.613	0.000	0.000	0.000
0.866	0.613	0.000	0.000	0.000
2	DM2	piece2	5
0.056	0.348	0.021	0.876	0.419
0.00000	0.000	0.000	0.875	0.000
0.00000	0.000	0.000	0.875	0.000
2	DM2	piece3	10.000
0.349	0.612	0.619	0.163	0.500	0.250	0.045	0.562	0.146	0.083
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2	DM2	piece4	8
0.667	0.538	0.625	0.571	0.469	0.643	0.450	0.591
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2	DM2	piece5	13.000
0.094	0.092	0.350	0.273	0.254	0.122	0.051	0.069	0.084	0.462	0.243	0.938	0.515
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.926	0.000
3	DM3	piece1	5
0.882	0.857	0.333	0.273	0.867
0.866	0.613	0.000	0.000	0.778
0.866	0.613	0.000	0.000	0.778
3	DM3	piece2	5
0.800	0.438	0.333	0.672	0.920
0.739	0.000	0.000	0.000	0.913
0.739	0.000	0.000	0.000	0.913
3	DM3	piece3	10.000
0.600	0.674	0.722	1.000	0.579	0.355	0.065	0.795	1.000	0.667
0.00000	0.000	0.000	0.190	0.000	0.000	0.000	0.751	0.667	0.000
0.00000	0.000	0.000	0.190	0.000	0.000	0.000	0.751	0.667	0.000
3	DM3	piece4	8
1.00000	0.857	0.385	0.364	0.875	0.619	0.600	0.857
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
0.750	0.643	0.000	0.000	0.867	0.000	0.000	0.846
3	DM3	piece5	13.000
0.591	0.500	0.846	0.980	0.281	0.145	0.067	0.075	0.083	0.835	0.286	0.936	0.742
0.00000	0.000	0.846	0.947	0.000	0.000	0.000	0.000	0.000	0.504	0.000	0.924	0.000
0.00000	0.000	0.846	0.947	0.000	0.000	0.000	0.000	0.000	0.504	0.000	0.924	0.000
4	VM1	piece1	5
0.944	1.000	1.000	0.364	0.938
0.775	0.771	0.784	0.000	0.782
0.775	0.771	0.784	0.000	0.782
4	VM1	piece2	5
0.800	0.812	0.600	0.936	0.934
0.614	0.734	0.000	0.936	0.934
0.614	0.734	0.000	0.936	0.934
4	VM1	piece3	10.000
0.647	0.752	0.846	1.000	0.579	0.314	0.500	0.742	1.000	1.000
0.00000	0.752	0.724	0.831	0.000	0.000	0.000	0.000	0.905	1.000
0.00000	0.752	0.724	0.831	0.000	0.000	0.000	0.000	0.905	1.000
4	VM1	piece4	8
1.00000	0.857	0.600	1.000	1.000	1.000	0.900	1.000
0.860	0.429	0.000	1.000	1.000	0.912	0.750	1.000
0.860	0.429	0.000	1.000	1.000	0.912	0.750	1.000
4	VM1	piece5	13.000
0.857	0.714	0.989	0.993	0.968	0.400	0.125	0.529	0.500	0.771	0.800	0.668	0.573
0.857	0.000	0.989	0.993	0.968	0.000	0.000	0.000	0.000	0.366	0.746	0.000	0.000
0.857	0.000	0.989	0.993	0.968	0.000	0.000	0.000	0.000	0.366	0.746	0.000	0.000
5	VM2	piece1	5
0.727	0.719	0.625	0.111	0.667
0.00000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000
5	VM2	piece2	5
0.538	0.812	0.600	0.920	0.934
0.00000	0.734	0.000	0.920	0.934
0.00000	0.734	0.000	0.920	0.934
5	VM2	piece3	10.000
0.786	0.744	0.962	0.278	0.395	0.360	0.250	0.677	0.571	0.190
0.384	0.000	0.498	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.384	0.000	0.498	0.000	0.000	0.000	0.000	0.000	0.000	0.000
5	VM2	piece4	8
0.667	0.571	0.600	1.000	1.000	0.714	0.800	0.769
0.00000	0.000	0.000	0.911	1.000	0.000	0.733	0.769
0.00000	0.000	0.000	0.911	1.000	0.000	0.733	0.769
5	VM2	piece5	13.000
0.609	0.087	0.974	0.982	0.633	0.600	0.333	0.111	0.084	0.800	0.667	0.652	0.488
0.00000	0.000	0.974	0.982	0.000	0.000	0.000	0.000	0.000	0.388	0.000	0.000	0.000
0.00000	0.000	0.974	0.982	0.000	0.000	0.000	0.000	0.000	0.388	0.000	0.000	0.000
6	IR1	piece1	5
0.765	0.684	0.714	0.750	0.600
0.074	0.000	0.000	0.375	0.000
0.074	0.000	0.000	0.375	0.000
6	IR1	piece2	5
0.727	0.812	0.600	0.024	0.154
0.00000	0.109	0.000	0.000	0.000
0.00000	0.109	0.000	0.000	0.000
6	IR1	piece3	10.000
0.647	0.062	0.742	0.833	0.237	0.474	0.250	0.839	0.857	0.750
0.00000	0.000	0.000	0.144	0.000	0.000	0.000	0.468	0.238	0.188
0.00000	0.000	0.000	0.144	0.000	0.000	0.000	0.468	0.238	0.188
6	IR1	piece4	8
0.600	0.429	0.000	0.800	0.133	0.143	0.778	0.692
0.00000	0.000	0.000	0.177	0.000	0.000	0.389	0.000
0.00000	0.000	0.000	0.177	0.000	0.000	0.389	0.000
6	IR1	piece5	13.000
0.500	0.625	0.015	0.081	0.333	0.364	0.700	0.889	0.800	0.359	0.652	0.134	0.181
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.136	0.198	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.136	0.198	0.000	0.000	0.000	0.000
7	PLM1	piece1	5
0.941	0.842	0.077	0.000	0.000
0.309	0.265	0.000	0.000	0.000
0.309	0.265	0.000	0.000	0.000
7	PLM1	piece2	5
0.333	0.552	0.125	0.116	0.053
0.00000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000
7	PLM1	piece3	10.000
0.312	0.357	0.553	0.146	0.474	0.344	0.062	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
7	PLM1	piece4	8
0.250	0.500	0.000	0.286	0.000	1.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	1.000	0.000	0.000
0.00000	0.000	0.000	0.000	0.000	1.000	0.000	0.000
7	PLM1	piece5	13.000
0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263	0.263
0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054
0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054	0.054

download these results as csv

Table 5. Tabular version of Figures 14 and 15.

References

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