COFE reconstructs time labels and predicts rhythmic features with high accuracy

To develop a suitable approach to assign time labels to samples, we relied on our observation that high-dimensional rhythmic time series data (with time labels) projected on a suitable plane produces a closed curve, where the samples are ordered in time (within one cycle) [5, 7, 13, 23]. We formulate this problem as a search for a low-dimensional elliptical approximation of the data that can be solved efficiently using a simple and fast iterative algorithm (Materials and Methods – Theory, S1 Text). Sparse Cyclic PCA (scPCA) finds this circular projection in data without time labels, while also selecting rhythmic features to aid in the circular projection. This procedure can thus be equally cast as dimensionality-reduction or manifold-learning. scPCA combined with a novel cross-validation (CV) scheme for unsupervised learning (Materials & Methods – Implementation) automatically learns the optimal sparsity parameter from the data (S1A-B Fig), reconstructs the circular projection (S1C Fig), assigns time labels to samples (S1D Fig) and identifies rhythmic features in the data (S1E Fig). We implemented scPCA in a Python package named COFE (Fig 1, S2 Text, https://github.com/bharathananth/COFE, https://doi.org/10.5281/zenodo.15167158), and benchmarked it on synthetic and biological (transcriptomic) data.

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Fig 1. Workflow of Cyclic Ordering with Feature Extraction (COFE).

Data without time labels are first optionally pre-filtered and then standardized. A novel cross-validation approach finds a value for the unknown sparsity parameter that guarantees good temporal ordering. Finally, COFE both assigns time labels to the samples and produces a list of rhythmic features in the data.

https://doi.org/10.1371/journal.pbio.3003196.g001

On synthetic data, COFE performed consistently well and its performance improved with increasing signal-to-noise ratios (SNRs), larger number of samples and higher fraction of rhythmic features in the data (S1F-H Fig). We quantified both COFE’s ability to predict time labels of samples and its ability to accurately (precision) and exhaustively (recall) identify the true rhythmic features in synthetic data containing both rhythmic and arrhythmic features with additive noise (S3 Text). COFE predicted the time labels of samples in the data with high accuracy that improved with lower noise, higher number of samples and rhythmic features (S1F Fig). COFE achieved perfect precision except at the smallest sample size (S1G Fig) and approached perfect recall with “better" and “larger" data (like the ordering performance) (S1H Fig).

The CV scheme serves to find the optimal sparsity parameter value as well as quantify the consistency and quality of the temporal ordering. Successful ordering is signalled by COFE with a clear minimum in the CV output (S1B Fig). The minimum imputation error achieved during CV is smaller at higher SNRs, larger sample sizes and larger fractions of rhythmic features (S2A Fig), where better ordering performance was achieved. The optimal sparsity grows with larger and more rhythmic datasets, roughly as the square-root of the number of rhythmic features in the data (S2B-C Fig). When there are no rhythmic features in the data, the output of CV increases monotonically and contains no non-trivial minimum (S2D Fig). Finally, sample times restricted to a fraction of the cycle (as might occur with samples only collected during a part of the circadian cycle), are similarly flagged by a CV output with no non-trivial minimum (S2E-H Fig).

COFE provides high resolution temporal ordering and high precision in finding rhythmic features across a wide range of data settings compared to state-of-the-art methods (S3 Fig). We compared COFE against autoencoder-based (CYCLOPS [15]), graph-based (PAGA [24]) and spectral analysis-based (scPrisma [21]) methods on synthetic data (S3 Text). However, unlike COFE, none of the methods can both assign time labels and identify rhythmic features; the first two do not extract rhythmic features, while the last one does not assign time labels. While COFE and CYCLOPS provided highly accurate time labels across a range of SNRs, scPrisma and PAGA had inferior performance at low and high SNRs, respectively (S3A Fig). COFE excelled in identifying rhythmic features without errors (perfect precision), which was unmatched by the other methods (S3B Fig). The precision of CYCLOPS and PAGA was limited by Type I errors in the cosinor rhythm detection used for rhythmic feature identification [25], whereas scPrisma had very poor performance at low SNRs. PAGA and CYCLOPS exhaustively recovered rhythmic features (along with false positives), which COFE and scPrisma achieved at sufficiently large SNR, number of samples or fraction of rhythmic features.

On biological time series (labelled) data, COFE’s performance was only a little worse than the corresponding supervised approach (S4 Fig). In these data (S1 Table), we only evaluated COFE’s ability to predict time labels, since the ground truth regarding the true rhythmic features is unavailable. COFE was able to find the two-dimensional projection for mouse liver time series data (S4A-B Fig) and predicted time labels to within the sampling resolution of the data (S4C Fig). Time labels of longitudinal human time series data from blood monocytes [23](S4D Fig) and skin [13](S4E Fig) could also be predicted almost as accurately as the corresponding supervised approach (using Zeitzeiger) [5] with accuracies of 0.75 h and 1.2 h, respectively. Finally, COFE was able to predict both the time labels and the underlying period of the clock on non-mammalian time series data, such as from different strains of the malaria parasite [26] (S4F Fig).

Therefore, we are confident in COFE’s ability to reconstruct underlying rhythms and identify rhythmic features in high-dimensional biological datasets, when present, and to flag degenerate situations (no rhythms, poor sampling over a cycle) by means of the CV output.

Widespread in vivo population rhythms in human adenocarcinomas

To investigate human rhythms under heretofore inaccessible conditions, we leveraged the high-quality transcriptomic data from human cancer biopsies assembled in The Cancer Genome Atlas (TCGA) database [27]. These data are a collection of patient-derived single samples from a variety of human cancers. We restricted our attention to adenocarcinomas (ACs), which are cancers of epithelial origin that develop in organs and glands and account for 80 to 90 percent of all cancers [28]. We further limited our study to ACs with sufficient large number of samples (>250) within a single histological type (S2 Table), which ensured homogeneity of the samples and comparability of the different ACs (S1 Text). This resulted in a collection of 11 different ACs – ductal breast (BRCA), clear cell (KIRC) and papillary (KIRP) kidney, prostate (PRAD), ovarian (OV), uterine (UCEC), lung (LUAD), colon (COAD), thyroid (THCA), bladder (BLCA) and liver (LIHC) cancers. Samples from each AC were evenly balanced between sexes, except for the four sex-specific ACs (BRCA, PRAD, OV, UCEC) (Fig 2A).

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Fig 2. Circadian population rhythms in human adenocarcinomas (ACs).

(A) Demographics of patients, whose biosamples provided data for training COFE. (B) Number of rhythmic genes and rhythmic core clock genes identified in each AC. (C) Cumulative distribution of the rhythmic gene amplitudes in different ACs. (D) Consensus amplitude and peak times of 15 commonly expressed core clock genes across the ACs (left). The dispersion of peak times of clock genes from the consensus arrangement across the ACs (middle). The relative strength of core clock network in different ACs (right). (E) Intersection of rhythmic gene sets across 11, 10 and 9 different ACs. (F) Reactome-based gene-set enrichment analysis of the common rhythmic genes. (G) Variability of peak times of genes within a pathway compared against the variability of peak times with respect to the core clock in different ACs. The data underlying Figure panels can be found in S1 Data.

https://doi.org/10.1371/journal.pbio.3003196.g002

COFE predicted time labels and identified rhythmic features (here genes) for the samples from each AC independently and reproducibly (S1 Text). Surprisingly, COFE reconstructed rhythmic gene expression profiles of 1000s of genes in each AC with the peak time of expression distributed throughout the day (S5 Fig, S6A Fig, S3 Table). The CV outputs with clear minima confirmed that rhythms indeed were present in each AC (S6D Fig). The number of rhythmic genes identified by COFE varied greatly between the different ACs, ranging from about 2000 in uterine AC to over 6000 in ovarian AC out of about 13000 expressed genes in each AC (Fig 2B). When restricted only to the rhythmic genes with at least a two-fold peak-to-trough amplitude, the variability in the number of rhythmic genes across ACs was greatly reduced (S6B Fig). The number of rhythmic genes were independent of the number of samples in the AC (r_s(9) = 0.04,p = 0.92). In general, ACs with more rhythmic genes not only had larger median amplitudes, but also achieved higher amplitudes of gene expression (Fig 2C). The exception to that was BRCA, where the number of high-amplitude rhythmic genes (>4-fold peak-to-trough) was comparable to the ACs with numerous rhythmic genes.

Thus, COFE revealed widespread and heterogeneous in vivo rhythms in human cancers.

Core clock is rhythmic in human adenocarcinomas

Due to the absence of an inherent timescale in the biopsy-derived transcriptomic data, COFE alone cannot ascribe a period to the reconstructed rhythms. To infer this period, we investigated a set of 20 core clock genes that constitute the fundamental rhythm generating mechanism of the cell autonomous clock. Many of the 15 expressed core clock genes were also identified as rhythmic in ACs (Fig 2B, S6C Fig, S3 Table), although the expressed core clock genes were not statistically more rhythmic than other output genes. This is indicative of a rhythmic core clock network in several ACs.

COFE’s time label assignments are consistent within each AC, but time labels from different ACs do not share a common time reference. We therefore constructed the consensus relationships between the peak times of the 15 commonly expressed core clock genes across 11 ACs in the form of an ‘eigen-core clock’ using complex SVD [12]. We used this consensus core clock arrangement to define a time reference for each AC. The arrangement of peak times was reasonably well conserved across ACs (and explained 64% of the inter-AC variation) (Fig 2D). This de novo analysis arranged peak times of most core clock genes in the expected order based on the response elements in their promoters (E-box E-box/D-box D-box RRE) and the peak times spanned half a cycle like in a healthy functional clock [7, 12]. The core clock genes with atypical (disrupted) peak times (circadian phase) were E-box targets NR1D2, TEF, BHLHE40 and E-box/D-box target PER2, which all peaked later in the day than expected (phase delayed).

Simultaneously, the analysis also revealed the strength (amplitude) of distinct clock genes across the different ACs. The D-box targets RORA and PER3 had the largest amplitudes (more than threefold peak-to-trough), while RRE targets NPAS2, NFIL3 and CRY1 had the smallest amplitudes (approximately twofold peak-to-trough) (Fig 2D). The amplitudes in the consensus core clock combine the amplitudes in each AC and degree of peak time consistency of the genes across ACs. The median amplitude of each clock gene across ACs correlated strongly with consensus amplitudes (). NR1D2 not only peaked aberrantly about 8h late, but had extremely consistent peak times in the ACs. Core clock genes with higher amplitudes deviated less from the peak time in the consensus arrangement (Fig 2D, ). Thus, clock genes with larger consensus amplitudes both had higher amplitudes in each individual AC and generally adhered to the consensus peak time across ACs.

We finally evaluated the relative strength of this consensus core clock arrangement in the different ACs. The core clock strength ranged from the strongest in the liver AC and weakest in the breast cancer AC, which was only half as strong. Interestingly, the ordering based on core clock network strength correlated only weakly with an ordering of ACs based on the number of rhythmic genes (r_s(13) = 0.58,p = 0.066).

In summary, there is a functional yet disrupted core clock that is conserved across different cancers, but the similarity of the conserved peak times of the core clock in the cancers to healthy tissue lets us conclude that these rhythms are indeed circadian.

Key cellular processes are enriched for rhythmic genes

To identify the processes enriched for rhythmic genes in different ACs, we performed gene set enrichment analysis [29] using the Reactome database (reactome.db-v1.88.0)[30] with genes scored based on the number of ACs in which the gene was rhythmic. Although only 7 rhythmic genes were shared between all the ACs, the ACs excluding BRCA shared 250 rhythmic genes (Fig 2E). Moreover, the overlap across different 9 out of 11 ACs again excluding BRCA was also 200–300 genes, which were all highly significant (Fisher Exact Test, p < 0.001). The genes with higher scores (i.e., rhythmic in more ACs) were highly enriched in Reactome pathways that fell into four Reactome pathway clusters – mitochondrial translation, respiratory electron transport, mitotic cell cycle, and adaptive immune system (Fig 2F). Respiratory electron transport and mitochondrial translation were the most highly enriched categories () making cellular metabolism the most rhythmic process in ACs.

We next quantified the consistency in peak times of rhythmic genes within the enriched Reactome pathways across different ACs. We computed two metrics that measured the coherence of gene expression peak times within a pathway and the consistency of the peak arrangement across ACs (Fig 2G). This resulted in three types of pathways – pathways with highly synchronized rhythmic genes peaking at the same time with respect to the core clock network (e.g., translation), pathways with synchronized rhythmic genes but the gene expression peaks in each AC at different times with respect to the core clock (e.g., programmed cell death) and pathways with neither synchronized rhythmic gene expression nor fixed times with respect to the core clock in different ACs. The peak activity of genes in respiratory electron transport (with respect to the core clock) varied by utmost 6h across ACs and the genes within this category were synchronized except for a few subunits of respiratory complex I and complex IV.

The identified rhythmic processes are therefore fundamental and highly conserved across the cancers.

Clocks in tumors are uncoupled from clocks in healthy tissue

We thus far presented the results using the core clock in each AC as a reference to define the peak time of rhythmic gene expression. Since the core clock in different tissues in the circadian network typically are not phase-aligned, we would like to determine gene expression with respect to the network (at the organismal level). We therefore first considered the distribution of predicted time labels in each AC over one circadian cycle. If clocks in the ACs are phase-locked to the circadian system, which in turn is entrained to the light-dark cycle, and assuming these tissue biopsies were gained from procedures conducted during the day (standard in most countries), we expect the predicted time labels to be restricted to a fraction of the circadian cycle. Surprisingly, predicted time labels lay throughout the circadian cycle in all ACs (Fig 3A). This is also consistent with the clear minima observed in the CV output for all ACs (S6D Fig). Although the biosamples in TCGA were obtained from several countries, the predicted time labels were only weakly associated with the origin of the samples (MANOVA effect size , S7A Fig), except for a strong effect (MANOVA effect size ) in COAD, and medium effect in UCEC and LIHC (MANOVA effect size ). Simultaneously, we also verified that the sex of the subjects was not associated with time labels in the seven non-sex-specific ACs (MANOVA effect size ) and the staging of the cancers in these patients also only had an effect on the time labels in BRCA, UCEC and KIRP, where the effect was not strong ().

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Fig 3. Coupling of the circadian clock to the cell cycle and the proteome.

(A) Distribution of predicted time labels for the samples in each AC. (B) Difference in time labels predicted by COFE between cancer and patient-matched healthy samples, where both are available. (C) Scores for cell cycle phases (S and G2/M) in the different ACs. The raw and LOESS-smoothed scores per sample are shown. (D) The number of rhythmic and total measured proteins in each AC. (E) The amplitude of rhythmic protein expression (color coded as in (A)) for proteins including modified forms that are rhythmic in at least 10 of the 11 ACs. (F) The raw data and LOESS-smoothed means for the rhythmic protein with the highest amplitude in each AC. (G) Distribution of the difference in peak times of expression of rhythmic proteins and their corresponding rhythmic gene. The data underlying Figure panels can be found in S1 Data.

https://doi.org/10.1371/journal.pbio.3003196.g003

Next, we compared the predicted time labels for tumor tissue samples with the patient-matched healthy tissue samples that were available for 10 of the 11 ACs for a small fraction of the original patients (Fig 3B). The difference between the predicted time labels for healthy and tumor tissue was distributed throughout the circadian cycle in most ACs suggesting that the time differences are almost random. As a negative control, we compared the difference between predicted time labels for patient-matched tumor tissue not used to train COFE (S7B Fig). As expected, these differences clustered around zero.

Based on both these observations, we conclude that the clocks in tumors do not have a fixed timing relationship with clocks in healthy tissue.

Circadian clock is coupled with the cell cycle in several ACs

Our previous analysis suggested that tumor clocks are not synchronized with the healthy clocks within the same tissue. We wondered whether the tumor circadian clock might be instead coupled to other rhythmic processes in the cell, such as the cell cycle that was enriched in our Reactome analysis (Fig 2F). To test this, we computed gene expression-based scores for cell cycle phases (S and G2M) (Fig 3C)[31]. Both the scores oscillated over the circadian cycle in all ACs (cosinor, p < 0.003) except KIRC (p = 0.39). The largest amplitude rhythms in the scores were in BRCA, UCEC and LUAD. Surprisingly, the two least rhythmic ACs (BRCA and UCEC) both showed robustly rhythmic scores. These marker scores can also predict the predominant cell cycle phase of cells in these (bulk) tissue samples. The ACs with rhythmic scores indeed show distinct time-of-day dependent cell cycle phases of the constituent cells, with G1 and G2/M phases separated in time (S7C Fig). So, the two important cyclic processes in the cell are synchronized in several human cancers.

Protein expression is rhythmic, even in ACs with few rhythmic genes

Since COFE assigns time labels to samples, we can also assign time labels to measurements from other assays run on the same samples. The Cancer Proteome Atlas (TCPA) database contains the protein expression of a subset of the proteome (258 proteins and modified forms) important in cancer measured using Reverse Phase Protein Array on the same tumor samples in TCGA. We used a cosinor approach [25] to find the rhythmic proteins, including modified forms, in the different ACs (S4 Table). 237 proteins were rhythmic in at least one AC. Moreover, 13 proteins were rhythmic in only one AC, while three were rhythmic in all. Unlike the transcriptome, at least half the measured proteins were rhythmic in all ACs (Fig 3D). Unexpectedly, BRCA had the most rhythmic proteins despite BRCA having among the fewest rhythmic genes and also the least overlap of rhythmic genes with other ACs. 13 proteins were rhythmic in 10 out of 11 ACs (Fig 3E). The nuclear receptor AR, phosphorylated tyrosine kinase SRC and phosphorylated Tuberin were rhythmic with robust amplitudes in all ACs.

We associated the rhythmic proteins from TCPA to the associated pathways defined in [32] (S4 Table). Members of multiple key cancer pathways including activated forms were rhythmic in at least 9 of the 11 ACs: apoptosis (PEA1S, TRANSGLUTAMINASE), cell cycle (ARID1A, BAP1C4, CHK2), epithelial–mesenchymal transition (CLAUDIN7, ECADHERIN), MAPK signaling (MAPK_PT202Y204, MEK1, P38_PT180Y182), metabolism (ACC1, FASN, G6PD), PI3K/AKT signaling (AKT_PS473, PI3KP85) and receptor tyrosine kinase signaling (EGFR_PY1068, HER2_PY1248, HER3_PY1289, SRC_PY416). Nonetheless, the protein with the highest amplitude in each AC varied and sometimes also included proteins that were rhythmic in only a few of the ACs, e.g., P53 and HER2 (Fig 3F).

We next related the rhythmicity of the proteins to the rhythmicity of their underlying genes. The genes of rhythmic proteins were significantly more likely to be rhythmic than arrhythmic proteins (p < 0.001 for each AC, Fisher Exact test) except in the two most rhythmic ACs, OV and PRAD. Similarly, proteins of rhythmic genes were more likely to be rhythmic than proteins of arrhythmic genes (p < 0.02 for each AC, Fisher Exact test). We also compared the peak expression times of rhythmic proteins and their corresponding rhythmic genes (Fig 3G). The peak of rhythmic protein expression was generally synchronized with the peak of rhythmic gene expression for most ACs (average difference h). The notable exception here was LIHC, where the rhythmic proteins peaked on average 6h after their respective rhythmic gene.

In summary, there was broad agreement between the independent analyses that identified transcript and protein rhythms.

A large, diverse array of small molecule drugs target rhythmic genes in tumor tissue

If the rhythmic genes include targets of approved and experimental cancer drugs, time-of-day changes in the target could be potentially leveraged to improve drug efficacy (chronotherapy)[33]. We identified rhythmic gene targets of a list of 481 FDA-approved, clinical candidates and putative small molecules that target key cancer pathways or processes from the Cancer Therapeutics Response Portal [34, 35] (S5 Table). 51 of the 53 FDA-approved drugs in the database targeted a rhythmic gene on average in around five different ACs and included several that had rhythmic targets in most ACs (Fig 4). The targets of two widely-used chemotherapeutic drugs, Teniposide and Gemcitabine, were rhythmic in all but one AC. Half the rhythmic FDA approved drug targets had better than 3-fold peak-to-trough amplitude. In 30% of cases, FDA approved drugs had more than one rhythmic target in an AC. Moreover, the distribution of FDA approved drug target amplitudes were similar across the different ACs with a few exceptions. Interestingly, targets of tamoxifen and fulvestrant in BRCA had the largest amplitudes, despite BRCA being one of the least rhythmic ACs. The peak time of the FDA approved drug target expression was in general similar across ACs in which they were rhythmic except for tacrolimus, microsporic, thalidomide and dasatinib that differed by up to 12h between ACs (Fig 4B).

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Fig 4. Rhythmic targets of FDA-approved and clinical candidate cancer drugs.

The -fold amplitude (A) and peak time relative to the core clock (B) of the largest rhythmic target gene in each AC of FDA approved and clinical candidate drugs that target cancer pathways or processes. All FDA-approved drugs and clinical candidates with rhythmic targets in at least 7 ACs are included. Drugs with multiple rhythmic gene targets in an AC are boxed in black. The data underlying Figure panels can be found in S1 Data.

https://doi.org/10.1371/journal.pbio.3003196.g004

91 of the 95 clinical candidates targeted rhythmic genes in some AC. Although clinical candidates also typically had rhythmic targets in about five ACs like the FDA approved drugs, five candidate drugs (PHA-793887, SNS-032, XL765, dinaciclib, rigosertib) had rhythmic targets in all ACs. Amplitudes of clinical candidate drug targets were generally smaller than the FDA approved drugs (Fig 4A). Similar to the FDA approved drugs, candidate drug targets in BRCA and UCEC (the least rhythmic ACs) had the highest median amplitude of 2-fold peak-to-trough. Clinical candidate drugs too could have targets with greatly different peak times (e.g., AZD4547, tosedostat), but the candidate drugs with rhythmic targets in all tissues had consistent phase across ACs.

Almost all putative cancer drugs in the database (248/267) targeted a rhythmic gene (S8 Fig). These putative drugs were largely similar to the clinical candidates in the average number of ACs with rhythmic targets and number of drugs with rhythmic targets in all ACs. Combinations of drugs involving selumetinib targeted rhythmic genes most broadly (S8A Fig). Putative drug targets in BRCA again and LIHC had the highest-median amplitude of 3-fold peak-to-trough. Compared to the other two drug classes, peak times of putative drugs were consistent across ACs, although they included some drugs with diversely peaking targets (selumetinib+navitoclax) (S8B Fig).

Theory

The p features measured at N (unknown) points in time are assembled in a data matrix . We approximate the matrix along two principal components (PCs) and with corresponding loading vectors and . To achieve a variance interpretation for the projection, the data matrix is column-centered, i.e., for each j. As such, this is a standard rank-2 reduction of the matrix , like PCA:

(1)

We refer to the objective function in the first line of Eq 1 as the matrix approximation error.

To ensure that PCs represent circular processes, we enforce a circular constraint, (), i.e., to constrain the samples in PC space to lie on a circle. We term the resulting and cyclic principal components. To identify rhythmic features, we constrain the loading vectors and to be suitably sparse, so that non-zero loading coefficients correspond to rhythmic features. Sparsity is enforced via an l₁ constraint with a hyperparameter s to keep the optimization tractable – the smaller the value of s, the sparser the solution. The resulting optimization, which we term sparse cyclic PCA (scPCA), also includes a l₂-regularization on the loading vectors () to make the problem well-defined:

(2)

Sparse cyclic PCA can be also considered an approximation of the periodic signals in the data by their Fourier series (at the fundamental frequency) under Gaussian noise or by a sinusoidal (cosinor) signal with additive noise (section A in S1 Text).

The optimization in Eq 2 is hard to solve due to the cross term. The cross term can be eliminated by imposing an additional orthogonality constraint on the loading vectors s or on the cyclic principal components s. But, such an optimization is not tractable. We therefore simply drop the cross term:

(3)

This simplified optimization in Eq 3 is bi-convex (section B in S1 Text) and can be solved to find a local maximum using Alternate Convex Search (alternating maximization) [46].

Alternating maximization.

We can maximize Eq 3 by sequentially and iteratively updating the s, the s, and d until convergence, starting from a random initial choice of vectors. The bi-convexity ensures that the score is monotonically non-decreasing during this update. For a given , we can solve for by means of Lagrange multipliers to incorporate the circular constraints:

resulting in the update

(4)

The solutions to and given are decoupled and need to maximize the projections and independently. For a given sparsity parameter s, the optimal is proportional to but with its elements soft-thresholded and scaled to satisfy both the l₁ and l₂ constraints, similar to sparse PCA [47]. (Given a threshold and input , the soft-thresholding function outputs ). Guillemot et al. [48] provide a simple algorithm to compute the soft-thresholding parameter for any vector given the l₁ constraint s resulting in an update of the form:

(5)

Finally, given , from Eq 3 the update for d can be easily computed as:

(6)

Implementation - Cyclic Ordering with Feature Extraction (COFE)

The scPCA algorithm described above needs to address some additional issues before it can be used for rhythm profiling without a time series.

1. We standardize the features to have zero-mean and unit-variance (z-score) to ensure that they are equally weighted with respect to the l₁ constraint for sparsity.
2. The solution to the optimization, found via alternating maximization, only guarantees a local maximum. We therefore solve scPCA from multiple (restarts number of times) initial conditions for each choice of hyperparameter s, where the s are randomly initialized to lie on the unit circle and the elements s are initialized from sparsity-inducing Laplacian distributions and scaled to satisfy the necessary constraints. We then choose the solution with the smallest matrix approximation error Eq 1 rather than the one with the best simplified objective function Eq 3 to mitigate to an extent the effect of the neglected cross term.
3. How do we choose a level of sparsity s for the regularized problem Eq 3? Since scPCA is an unsupervised learning technique, we do not have access to the true time labels to help us choose s. Hence, we implemented a form of CV for unsupervised learning [51].
   To perform K-fold CV, 1/K elements of the data matrix are deleted in each fold. For the deleted elements to be evenly distributed across rows (samples) and columns (features), random deletion patterns are generated as circular row and column shifts of different pseudo-diagonals of the data matrix as defined in [52].
   Deleted elements are initially imputed by column means (i.e, the feature means). We then alternately run scPCA and update the deleted elements using the scPCA-based approximation until both converge (within user-defined tolerances). The error between the original values of the deleted elements and their final imputed values from the low rank approximation serves as the CV error metric [51], which is a form of an expectation-maximization algorithm.
   Repeated K-fold CV (repeats number of times) is performed for a range of s values chosen by the user. The CV error (average imputation error) shows a sharp phase transition at the optimal s (S1B Fig), when rhythms exist and all the rhythmic features above noise have been detected.