SW1PerS: Sliding windows and 1persistence scoring; discovering periodicity in gene expression time series data
 Jose A. Perea^{1, 2}Email author,
 Anastasia Deckard^{3},
 Steve B. Haase^{4, 5} and
 John Harer^{1, 4, 6}
Received: 21 December 2014
Accepted: 10 June 2015
Published: 16 August 2015
Abstract
Background
Identifying periodically expressed genes across different processes (e.g. the cell and metabolic cycles, circadian rhythms, etc) is a central problem in computational biology. Biological time series may contain (multiple) unknown signal shapes of systemic relevance, imperfections like noise, damping, and trending, or limited sampling density. While there exist methods for detecting periodicity, their design biases (e.g. toward a specific signal shape) can limit their applicability in one or more of these situations.
Methods
We present in this paper a novel method, SW1PerS, for quantifying periodicity in time series in a shapeagnostic manner and with resistance to damping. The measurement is performed directly, without presupposing a particular pattern, by evaluating the circularity of a highdimensional representation of the signal. SW1PerS is compared to other algorithms using synthetic data and performance is quantified under varying noise models, noise levels, sampling densities, and signal shapes. Results on biological data are also analyzed and compared.
Results
On the task of periodic/notperiodic classification, using synthetic data, SW1PerS outperforms all other algorithms in the lownoise regime. SW1PerS is shown to be the most shapeagnostic of the evaluated methods, and the only one to consistently classify damped signals as highly periodic. On biological data, and for several experiments, the lists of top 10% genes ranked with SW1PerS recover up to 67% of those generated with other popular algorithms. Moreover, the list of genes from data on the Yeast metabolic cycle which are highlyranked only by SW1PerS, contains evidently noncosine patterns (e.g. ECM33, CDC9, SAM1,2 and MSH6) with highly periodic expression profiles. In data from the Yeast cell cycle SW1PerS identifies genes not preferred by other algorithms, hence not previously reported as periodic, but found in other experiments such as the universal growth rate response of Slavov. These genes are BOP3, CDC10, YIL108W, YER034W, MLP1, PAC2 and RTT101.
Conclusions
In biological systems with low noise, i.e. where periodic signals with interesting shapes are more likely to occur, SW1PerS can be used as a powerful tool in exploratory analyses. Indeed, by having an initial set of periodic genes with a rich variety of signal types, pattern/shape information can be included in the study of systems and the generation of hypotheses regarding the structure of gene regulatory networks.
Keywords
Periodicity Gene expression Time series Sliding windows Persistent homologyBackground
Previous Work
Many methods are available for detecting periodicity in time series data [1, 2], and many have been successfully applied in the task of identifying periodic gene expression. Most of these algorithms can be classified into three broad classes, based on how/if they use reference patterns. In particular: approaches which use sinusoidal curves as a base for comparison, userdefined shape templates, and those that do not use a reference pattern. We provide a brief description below.
Methods in the first class determine the period and measure the strength of periodicity by comparing the input time series to sinusoidal curves with different periods. This includes algorithms which transform a time series into the frequency domain, as with the discrete Fourier transform, and those that fit sinusoidal curves to the target signal. The method introduced in [3] uses a Fourierbased approach and a measure of amplitude (as an indicator of regulation strength) to generate a score, as well as a permutation test to asses significance. COSOPT [4] compares a signal to cosine curves with different phases and periods to measure their correspondence, and then uses empirical resampling to compute significance. LombScargle [5, 6] uses a variation of the discrete Fourier transform to handle unevenly sampled data, and returns a significance score.
Other methods compare the signal to reference curves that are specified by the user. The method of Luan and Li [7], for example, generates a spline function to represent the pattern of known periodic genes, and then uses this shape model to score other signals. JTK_CYCLE [8] determines increasing or decreasing patterns of the observations in both a reference curve and the signal, and then measures the statistical significance of correlation between them.
Other methods, by way of contrast, do not use a set pattern to identify signals of interest, but instead attempt to discover patterns that exist in the data. Address Reduction [9] measures the algorithmic compressibility of the signal; a signal that is more compressible indicates there is a pattern and it might be of biological interest. It is worth noting that noncompressibility does not imply periodicity. An instance of Persistent Homology [10] pairs, in a subtle way, minima and maxima of a time series. This can be used to measure periodicity: if there is only one minimum and maximum pair, it is considered to be a perfect oscillation. Additional oscillations in the time series will create more minimummaximum pairs, indicating a less perfect curve.
A comparative study of the LombScargle, Persistent Homology, JTK_CYCLE and de Lichtenberg methods was undertaken in [1]. One of their main conclusions is that curve shape has considerable impact on the scoring of biological signals; this is specially relevant in exploratory settings where the shapes of interest produced by a particular periodic process are not known.
Our Contribution
SW1PerS, the algorithm introduced here, was designed to help overcome the limitations posed by: Signalshape biases in the rankings of algorithms which use predetermined templates, the effects of damping in periodicity estimation, and the difficulty of interpreting scores derived from pvalues. In a nutshell, SW1PerS transforms the input time series into a highdimensional set of points (also referred to as a point cloud) and interprets periodicity of the original signal as “circularity" of this set. When constructing this point cloud one uses a local normalization process geared toward diminishing the effects of damping. A more in depth description will be presented in the Methods section.
We compare SW1PerS (SW) to existing algorithms, specifically: LombScargle (LS), de Lichtenberg (DL), JTK_CYCLE (JTK), and Persistent Homology (PH). The first test evaluates their performance on separating periodic from nonperiodic signals in a synthetic data set. Their biases for different signal shapes is also analyzed. We then examine how the algorithms behave when applied to real data from different periodic processes and species: the cell cycle in yeast, the metabolic cycle in yeast, and circadian rhythms in mouse.
Results
Synthetic data description
The periods and amplitudes were fixed, but the phase shifts were allowed to vary from 0 to the length of the period. The period length was 100 (time units) and the signals covered 200 units of time, so each signal spans two cycles. One thousand signals were generated for each signal shape.
Four noise models were applied to the set of signals, each at five different levels: Gaussian Additive with standard deviation SD equal to 0,12,25,37 and 50, Laplacian Additive with spread b at 0, 8.49, 17.68, 26.16, and 35.36, Gaussian Multiplicative with SD equal to 0, 0.12, 0.25, 0.37 and 0.5, and Laplacian Multiplicative with b={0,0.08,0.18,0.26,0.35}. The standard deviation SD for additive (resp. multiplicative) Gaussian noise and the spread b for additive (resp. multiplicative) Laplacian noise were matched (\(SD = \sqrt {2}b\)) so the distributions would have the same variance. Given the shapes of the distributions, this results in the Laplacian noise model producing signals with more accentuated outliers, as compared to the less extreme behavior of the Gaussian noise. The additive and multiplicative variances were not matched to each other.
Synthetic Data Analysis
In what follows we present our results on the synthetic data. The first analysis of performance is how well an algorithm can distinguish between periodic and nonperiodic signals for several noise models, levels of noise and temporal sampling density. The second explores signal shape bias for each method. For this study JTK, LS, DL, PH and SW1PerS were set to scan for periodicity at a periodlength equal to the true period.
The first thing to notice (see Figs. 2 and 3) is that at the low sampling (17 time points) and low noise regime (SD = 0 to 12 in the additive Gaussian model, SD = 0 to 0.12 in the multiplicative, b=0 to 8.49 in the additive Laplacian and b=0 to 0.08 in the multiplicative), SW has the best performance among the evaluated algorithms in the task of identifying periodic and nonperiodic signals. Moreover, as the number of samples increases and the noise level is kept constant (SD = 0), SW continues to be at the top even as the other algorithms improve their scores. This is due to signals like the contracting cosine and the exponential trend, for Fourierbased methods; e.g. LombScargle and de Lichtenberg. Indeed, for these types of signals the spectral density will not be as concentrated at a single frequency. This, even when there is a clear repeating pattern, which methods like SW and JTK correctly identify.
Classification results deteriorate across the board as noise increases, with DL being the most resilient – specially in highsampling conditions, and SW performing on par with the others. It is worth noting the similarity in spacing and ordering (with respect to signal shape) of the AUC scores between algorithms. This can be interpreted as follows: for all the evaluated methods classification is more accurate for simpler signals (e.g. cosines and square waves) but as shape patterns become more intricate (e.g. contracting cosine and double peaked) correct classification in the presence of noise is more difficult. Indeed, periodicity (interpreted as the repetition of patterns) is more severely affected in complicated signal shapes when random additive noise increases.
If we now turn our attention to Fig. 3, we see a very similar picture to what we have described so far. That is, even with Laplacian noise, which tends to add more accentuated outliers, the relative performance of the algorithms tends to be similar. This can be interpreted as follows: the algorithms presented here are stable, for the most part, for the noise models under consideration. The exception is PH, as can be seen from the figures.
In summary: For the noise models considered here, SW1PerS is the best performer in the nonoise/allsamplings and smallnoise/lowsampling regimes. de Lichtenberg is the most successful in the medium to high noise regime. What we will show next is that SW1PerS has better ranking properties, in that it has a greater richness of signal types at the top of its score distributions.
Methods such as JTK and LS base their score on pvalues. This has a subtle drawback: increasing the number of samples on a periodic curve causes the pvalues to become more significant, muddling comparisons across experiments with different numbers of time points. Since SW1PerS ignores the number of samples in its measure of periodicity, it is more amenable to interexperiment queries.
Significance Analysis
Using synthetic data we have shown that SW1PerS is a powerful method for quantifying periodicity in time series data. And though the score it produces does not have the subtle drawbacks of methods based on pvalues, it is still important to assess its statistical significance.
In what follows we will present a permutation analysis of the SW1PerS score, in order to quantify the probability that observed good scores are due to chance alone. In particular, we compute the empirical probability that a permuted version of a signal gets a better score than the original one. The setup is described below.
For permutation testing we use signals with 25 time points and Gaussian additive noise of 12. One signal was selected for each shape. This set of one signal per shape was then subjected to permutation testing. For permutation testing, each original signal was permuted using python’s ~random.shuffle~ method to create a sample, of size N, of permuted versions. This process was repeated R times. Each one of the permuted signals, along with the original ones, were then run through SW1PerS. For each sample of size N, the pvalue was computed as the proportion of permuted signals with SW1PerS score better than or equal to that of the original version.
The number N of permuted signals was tested at increasing orders of magnitude: 1000, 10,000, 100,000. The number of repetitions R was set to 5. The convergence of the pvalues for 5 (=R) repetitions and 100,000 (=N) permutations was sufficient for analysis. In particular, the standard deviation of the computed pvalues for 5 repetitions, across all shapes, was less than 0.0023.
Computed mean pvalues and standard deviations, across 5 repetitions, for each signal type
Type  Shape  Mean pvalue  Std 

Periodic  Cos  0.00005  0.000012 
Cos 2  0.003354  0.000313  
Peak  0.010792  0.000363  
Trend Lin  0.009752  0.00035  
Trend Exp  0.161562  0.001052  
Damp  0.006814  0.000177  
Saw  0.00027  0.000035  
Square  0.00001  0.00001  
Contract  0.262642  0.002222  
Nonperiodic  Flat  0.54663  0.002278 
Line  0.935736  0.001094  
Exp Decay  0.897834  0.000586  
Sigmoid  1  0 
Biological Data Sets
We examined the results of the algorithms on data sets from three microarray experiments (Additional file 2). These experiments were designed to measure periodic gene expression of different processes in different organisms which, as we will show, feature signal shapes which deviate from the usual cosinelike curves.
The wildtype data (WT) from [11] shows periodic gene expression during the cell division cycle (CDC) in budding yeast, S. cerevisiae. A population of wildtype cells were synchronized and samples were taken at 16 minute intervals. The period for the cell cycle in this experiment is estimated to be approximately 95 minutes, and the data sets cover a recovery period and roughly two cell cycles. This data set contains 15 samples, but only the last 13 were used in order to omit a stress response. There are two replicates, WT1 and WT2.
The yeast metabolic cycle (YMC) data of [12] are from S. cerevisiae that were grown to a high density, briefly starved and then given low concentrations of glucose. Samples were taken at variable intervals of 2325 minutes. We evened the sample intervals by changing the times to every 24 minutes. The yeast metabolic cycle is estimated to be approximately 300 minutes; this data set covers approximately three cycles and contains 36 samples.
The mammal circadian rhythm data from [13] is from wildtype mice that were synchronized by entraining them to an environment with 12 h light and 12 h dark for one week. They were then placed into total darkness. Samples were taken from the liver every hour. The period of the circadian rhythm is approximately 24 hours, and this data set covers two circadian cycles and contains 48 samples.
For the yeast cell cycle, the data has a low sampling density of 13 samples for two periods (6.5 samples per cycle). Additionally, the data is damped. The yeast metabolic cycle data has a higher sampling density of 36 samples for three periods (12 samples per cycle). For the circadian rhythm, the data has a higher sampling density of 48 samples for two periods (24 samples per cycle) and the data appear noisier than the yeast cell cycle data.
Biological Data Analysis
Each data set was run through the LS, JTK, DL, and SW algorithms (Parameters in Table S3). We omitted PH from further analysis, as it did not perform as well as the others on the synthetic data. Comparing these algorithms is challenging; unlike in the synthetic data there is no ground truth; the algorithms return pvalues or scores that can be difficult to compare directly, and their score distributions are difficult to interpret (Figures S39S41). We evaluate the performance of SW on biological data, relative to the other algorithms, based on its ability to: find periodic shapes which the other algorithms also identify; find uncommon signals that have nonstandard periodic shapes; and to recover signals of genes that are believed to be part of a given periodic process. In addition, we report sets of genes from overlapping periodic processes found with SW1PerS. We present next the results of these analyses.
Finding common periodic signals.
One of the goals in developing SW was that it would be more shape agnostic, and therefore able to detect a larger range of periodic shapes in the data. SW should, however, recover results from the top of the other algorithm’s lists, which have been shown to detect periodic signals.
Percentage of overlap from the top 10 % and 20 % of probes as ranked by the algorithms
Data  Cell Cycle  Met. Cycle  Circ. Rhy.  

Top #  590  1180  933  1866  4510  9020 
Top %  10 %  20 %  10 %  20 %  10 %  20 % 
SW ∩DL  51 %  59 %  36 %  56 %  64 %  68 % 
SW ∩LS  52 %  60 %  67 %  78 %  67 %  59 % 
SW ∩JTK  51 %  59 %  60 %  73 %  67 %  66 % 
All  26 %  42 %  23 %  42 %  53 %  55 % 
Complete Venn diagrams (Figures S42S44) and tables of percent overlap (Tables S4S6) can be found in the supplements.
Number and percentage of probes in the top 10 % of rankings from each algorithm that are in a consensus set. That is, those which appear in the top 10 % of rankings for at least three algorithms
Data Set  Alg  #Consensus  %Consensus 

Yeast Cell Cycle  sw  316  0.90 
Consensus: 353  dl  289  0.82 
ls  298  0.84  
jtk  311  0.88  
Yeast Met. Cycle  sw  553  0.93 
Consensus: 596  dl  345  0.58 
ls  563  0.94  
jtk  541  0.91  
Mammal Circadian  sw  3090  0.82 
Consensus: 3767  dl  3330  0.88 
ls  3640  0.97  
jtk  3636  0.97 
As shown, SW has the highest percentage of probes (90 %) in the consensus for the yeast cell cycle. SW has second highest percentage (93 % compared to 94 % for LS) in the yeast metabolic cycle. In the mammal circadian set, SW has 82 % in the consensus set, while the other algorithms have higher percentages (8897 %). These analyses suggest that SW1PerS is able to identify a large portion of genes labeled as highly periodic, even when the labelling process has been done with very different algorithms.
Finding uncommon periodic signals.
All signals in this figure are listed in the 3,656 probes (39 % of all probes on the array) identified as periodic in [12]. They use an autocorrelation function with a period determined by LombScargle. These signals are ranked very highly by SW, are not necessarily highly periodic according to the other algorithms under consideration, and have shapes which are very unusual. Notice that a repetition across three periods makes it highly unlikely for these shapes to be artifacts.
Finding signals that are part of a periodic process.
To determine if the algorithms recover genes associated with periodic processes, we examine their rankings of genes associated with the yeast cell cycle and the circadian rhythm. The lists of genes were created from previous studies that locate the binding sites of genes known to be part of the given periodic process.
For the yeast cell cycle, the ChIPchip data of [14] includes nine known cell cycle transcription factors: Mbp1, Swi4, Swi6, Mcm1, Fkh1, Fkh2, Ndd1, Swi5, and Ace2. From this data set, we selected a list of 141 genes as targets of these transcription factors. For the mouse circadian rhythm, the ChipSeq data of [15] includes seven known circadian transcription factors: BMAL1, CLOCK, NPAS2, PER1, PER2, CRY1, and CRY2. From this data set, we selected 361 genes as targets of these transcription factors. See Methods for our inclusion criteria.
Overlap between algorithm rankings and binding data for Yeast Cell Cycle data. The percent of probes in the top X % of rankings for each algorithm that are in the set of bindings targets that we compiled from the ChIPchip data of Simon, et al, 2001
Alg  5 %  10 %  15 %  20 % 

SW rank  14  24  31  35 
DL rank  39  53  56  59 
LS rank  9  22  35  38 
JTK rank  10  19  31  40 
Overlap between algorithm rankings and binding data for Mammal Circadian data. The percent of probes in the top X % of rankings for each algorithm that are in the set of bindings targets that we compiled from the ChIPSeq data of Koike, et al, 2012. Note that the array for the circadian data set has multiple probes for some genes and duplicates were not removed
Alg  5 %  10 %  15 %  20 % 

SW rank  32  50  63  74 
DL rank  41  54  69  70 
LS rank  37  53  65  71 
JTK rank  35  55  70  78 
In contrast with SW, LS and JTK have pushed a larger portions of these genes to the lowest rank. See supplement for a comparison of rankings for yeast cell cycle (Figure S45) and mammal circadian rhythm (Figure S46) for a selected set of known genes.
Discovery of signals from multiple processes.
To determine if SW finds genes involved in multiple processes in budding yeast, we compared cell cycle genes preferred by SW with results from other experiments in yeast. To create the list of genes that SW prefers, we selected the top 10 % of ranked results from SW that were not in the top 10 % of ranked results for the yeast cell cycle (WT1) on either DL, JTK or LS. This results in 151 probes with 148 unique systematic names. To filter out probes that are potentially more noise than signal, we compared the replicates WT1 and WT2 using a combined score from SW and JTK (see Supplements, section 8). A cutoff of 0.05 yielded a list of 77 probes.
Number of probes from the top 10 % of SW, not in the top 10 % of other algorithms, filtered for noise using the replicates, that overlap with other data sets. We also show the numbers of these probes not identified in Orlando 2008, Spellman 1998, and not in either of these data sets
Dataset  Overlap  ¬Orlando  ¬Spellman  ¬Either 

YMC  36 (47 %)  21 (27 %)  30 (39 %)  18 (23 %) 
GRR Pos  3 (4 %)  2 (3 %)  3 (4 %)  2 (3 %) 
GRR Neg  13 (17 %)  8 (10 %)  12 (16 %)  8 (10 %) 
Discussion
The results from the synthetic data show that SW is comparable to other popular algorithms for most signal shapes, noise levels, and sampling densities. Additionally, SW outperforms DL, LS, JTK, and PH on the low noise regime and across all sampling densities. This analysis has shown that SW1PerS performs well on data that has shapes which occur in biological systems from different organisms, and that it is well behaved under sampling densities and noise levels found in microarray data sets. SW1PerS shows less bias against damped signals, which occur frequently for instance in the yeast cell cycle data.
The analysis of the biological data shows that SW1PerS is able to recover many of the signals other algorithms find, and can additionally discover noncosine shapes that other algorithms might exclude. We believe that finding signals with a greater diversity of shapes well outweighs the cost of giving higher ranks to signals that might appear to be noise. SW also appears to detect different types of biological processes than the other algorithms based on GO enrichment (see Supplemental table S7S14).

SW1PerS, in the low noise range, has been shown to be the most shapeagnostic algorithm out of the methods studied here.

SW1PerS is able to effectively estimate periodicity even as the period length changes from oscillation to oscillation. We saw this, for instance, with the contracting cosine in the synthetic data.

The score that SW1PerS returns has a geometric interpretation, and can be compared across different data sets.

SW1PerS can be used on data with low temporal resolution and uneven time spacing.

While the algorithm requires the selection of certain parameters (e.g. window size, embedding dimension, etc), the theory behind the method suggests reasonable values.

Even though the innerworkings of the SW1PerS algorithm are quite different from the other methods studied here, it is able to recover – to a large extent – what other algorithms find.

The implementation we have of SW1PerS has been clocked at between 0.5sec and 1.0sec per signal, on a laptop computer. Hence, runningtime can be an issue. We expect that as better algorithms for computing 1persistent homology and more computational resources become available, this problem can be mitigated.

The probability distribution for the SW1PerS score, even for the additive Gaussian noise model, has not been described as of yet. Hence, we lack a principled way of producing pvalues. And though studying this distribution is out of the scope of the present article, we have used synthetic data – where the ground truth is known – to assess the performance of SW1PerS relative to other algorithms. In addition, permutation tests were also performed to evaluate significance and positive results were obtained as shown in Table 1.

As we have observed with the synthetic data, SW1PerS tends to degrade as noise increases, and it does so at a faster rate than some of the other methods studied here. Signal processing, however, is a rich field with highly successful denoising algorithms that can be brought to bear in this problem.

SW1PerS does not recover the phase or period length.
Keeping all this in mind, the analyses presented here have shown the benefits of applying SW1PerS, especially in exploratory situations where signal shapes might not be known and a broad set of candidates is desirable.
Conclusions
We have presented in this paper a new algorithm, SW1PerS, for quantifying periodicity in time series data. The algorithm has been extensively tested and compared to other popular methods in the literature, using both synthetic and biological data. Specifically, with a vast synthetic data set spanning 14 different signal types (10 periodic and 4 nonperiodic), 4 noise models, 5 noise levels and 3 sampling densities, it was shown that SW1PerS outperforms the other algorithms presented here in the lownoise and lowsampling regimes. Moreover, it exhibits at the top of its rankings the most variety in signal types, making it the most shapeagnostic and the only one to identify damped signals as highly periodic. In the biological data SW1PerS recovers, to a large extent, what other algorithms have identified in previous work. Moreover, it was also able to discover signals with interesting shapes, which were overlooked by the other methods.
By using SW1PerS along with other algorithms that complement its strengths and lessen its weaknesses, it can be used as a powerful tool in exploratory analyses. Indeed, in biological systems with low noise, i.e. where periodic signals with interesting shapes are more likely to occur, SW1PerS can be used to identify an initial set of periodic genes with a rich variety of signal types. Patterns and shape information can then be included in the study of systems, as well as in the generation of hypotheses regarding the structure of gene regulatory networks.
Methods
The SW1PerS Algorithm
The way SW1PerS recognizes periodicity is simple: It measures the existence of a distinctive pattern in the graph of the signal, and quantifies the extent to which it repeats. The quantification step, in contrast with other methods, does not involve the usual measures of correlation. Instead we use tools from topological data analysis [24], a new set of techniques that probe/quantify the shape of data, to measure the circularity of a point cloud derived from the time series.
The repetition of a pattern in the graph of g is thus associated with the circular arrangement of the snippets, while its distinctiveness corresponds to the size of the “hole” in the middle of the arrangement. Notice that the term “pattern” applies to any type of snippet; this is what gives SW1PerS its shapeagnostic nature.
We formalize this construction as follows: let M be a positive integer (usually larger than twice the number of time points) and set \(\tau = \frac {w}{M}\) for some window size w∈(0,2π). The theory behind SW1PerS [25] implies that a good window size should be close to \(\frac {2\pi M}{L(M+1)}\), where L is the number of expected periods.
This ameliorates the effects of damping and trending in the original time series, and also makes SW1PerS amplitude blind.
Dealing with Noise
We present two denoising paradigms included in the SW1PerS pipeline; the first operates on time series, and the second focuses on noise at the pointcloud level.
Simple Moving Average
Simple Moving Average often yields satisfactory results given its local nature, and that it can be applied to time series with low time resolution (S≥13). A limitation, however, is that it can remove fine features and peaklike behavior. Thus, we restrict k to values so that g _{ s−k },…,g _{ s },…,g _{ s+k } does not span more than a third of the window size w.
MeanShift
Has appeared numerous times in the statistics literature, and more recently in the work of [27]. It can be seen as a pointcloudlevel version of moving average, in which each point of the cloud is replaced by the average of those close to it. Intuitively, this has a tightening effect. Closeness to a point can be defined as being among its qth nearest neighbors for some integer q, or by being no farther than ε away for some constant ε>0. It is the second option we use in this paper. Since in SW1PerS the sliding window point cloud has been pointwise meancentered and normalized, it follows that it lies on the surface of the unit sphere in \(\mathbb {R}^{M+1}\). Hence we measure distance between two such points x,y via the angle between them and deem them to be close^{2} if \(\measuredangle (\mathbf {x},\mathbf {y}) < \frac {\pi }{16}\). Once each point has been replaced by the average of those no more than \(\frac {\pi }{16}\) away, we proceed to pointwise normalizing the resulting cloud.
Availability and Supporting Data
An implementation of SW1PerS can be found at http://cms.math.duke.edu/harer/?q=downloads.
Endnotes
^{1} In practice we use \(T= \{\frac {j(2\pi w)}{200}  j = 0, 1, \ldots,\)
^{2} This constant was set experimentally based on performance on the synthetic data.
Declarations
Acknowledgements
We would like to thank Dave Orlando for his implementation of DL, Yuriy Mileyko for that of PH and John Hogenesch for his implementation of the JTK algorithm. We thank Sara Bristow, Adam Leman, and Christina Kelliher for informative discussions. J. Perea would like to thank the IMA at the University of Minnesota for its support during portions of this project. This research was supported by the Defense Advanced Research Projects Agency through grant [D12AP00001].
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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