In this section, we compare the periodicity detection performance of the Fourier-score-based algorithm 31, M-estimator 29, and the Laplace periodogram using two sets of real data. First, we use the Saccharomyces cerevisiae Alpha (6075 genes), CDC15 (5673 genes), and CDC28 (6214 genes) experiments from 23. The time-course gene expression datasets of these three experiments are obtained from Cyclebase.org http://www.cyclebase.org. The expression datasets obtained have been normalized to a common scale based on the percentage of the cell division cycle where the sampling occurs. Furthermore, the magnitudes of the gene expression have also been normalized to have a standard deviation of one 32. Next, we compare the performance of the Fourier-score-based algorithm and the Laplace periodogram using the circadian oscillation dataset for auxin signaling in Arabidopsis33.

We now compare the detection performances of the Fourier score, Laplace periodogram, and M-estimator on the same Saccharomyces cerevisiae Alpha, CDC15, and CDC28 datasets from 23, but with added impulsive noise to the dataset. For each gene, there is a 0.1 chance of having an impulse noise at a random position of magnitude +7 or -7 in its time-course gene expression, replacing its original value. Note that the range of the peaks of the magnitude in these experiments varies between 3 and 5. We generate for each of the three experiments 50 such generated datasets with randomly placed impulse noises. We then perform periodicity detection with the three methods on these randomly generated datasets and averaged the results for each experiment. However, in our simulations, we observed that the performance for each of the algorithms when ranked using the estimated p-values is no better than if we had randomly ranked the genes. Therefore, instead of ranking the genes by their p-value, here we rank them by using their magnitude instead of using the estimated p-values. This observation can be explained by the presence of the outlier impulsive noise, whose influence on the spectrum is not removed by the random permutations, thus resulting magnitude for a periodically expressed time series and its permutations do not differ by a significant amount. The results of these plots are then given in Figures 5, 6, 7.

From these figures we can see that with the addition of impulse noise, the Laplace periodogram on average gives better detection performance than Fourier score for most of the combinations, and only in the CDC28-B3 combination does the Laplace periodogram achieves worse detection accuracy than Fourier score. However, it should be noted that a lot of the genes in benchmark set B3 are not involved in the transcriptional regulation, thus only a very small amount of genes in B3 are expected to be periodic 31. What is surprising in these simulations is that even when impulse noises are added, the M-estimator-based method, which is designed to be robust in the presence of outliers, performs the worst out of the three methods compared. A closer look at the M-estimator proposed in 29, we see that the M-estimator is based on finding

β ˜ = min ⁡ β ∑ t = 1 N ρ { y t − x i T β σ N } , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGaf8NSdiMbaGaacqGH9aqpdaWfqaqaaiGbc2gaTjabcMgaPjabc6gaUbWcbaGae8NSdigabeaakmaaqahabaGaeqyWdi3aaiWaaKqbagaadaWcaaqaaiabdMha5naaBaaabaGaemiDaqhabeaacqGHsislieWacqGF4baEdaqhaaqaaiabdMgaPbqaaiabdsfaubaacqWFYoGyaeaacqaHdpWCdaWgaaqaaiabd6eaobqabaaaaaGccaGL7bGaayzFaaaaleaacqWG0baDcqGH9aqpcqaIXaqmaeaacqWGobGta0GaeyyeIuoakiabcYcaSaaa@4E00@

where ρ is a symmetric function with the Tukey's biweight function as its derivative, and σ_N a scaling factor. The Tukey's biweight function is given as

ψ ( x ) = { x ( 1 − x 2 c 2 ) 2 for | x | < c , 0 for | x | > c , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiYdKNaeiikaGIaemiEaGNaeiykaKIaeyypa0ZaaiqaaeaafaqaaeGacaaabaGaemiEaG3aaeWaaeaacqaIXaqmcqGHsisljuaGdaWcaaqaaiabdIha4naaCaaabeqaaiabikdaYaaaaeaacqWGJbWydaahaaqabeaacqaIYaGmaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaaakeaacqqGMbGzcqqGVbWBcqqGYbGCcqGG8baFcqWG4baEcqGG8baFcqGH8aapcqWGJbWycqGGSaalaeaacqaIWaamaeaacqqGMbGzcqqGVbWBcqqGYbGCcqGG8baFcqWG4baEcqGG8baFcqGH+aGpcqWGJbWycqGGSaalaaaacaGL7baaaaa@577B@

where c is the constant that determines the shape of the curve and the cutoff point above which the residual of the sample and the current fit contributes zero weighting, thus completely removing the contribution of possible outliers that exceeds a certain magnitude. However, this approach requires a value for c which may vary from dataset to dataset. The optimal value for c may depend on the magnitude of the peaks and the magnitude of the outliers. When given an unknown dataset without a benchmark set of periodically expressed genes, it is uncertain how c can be tuned to achieve the optimal result, thus demonstrates a disadvantage of the M-estimator-based method. Note that for both simulations with and without the presence of outliers, we used the default value of c which came with the program provided by the authors of 29.

For time series samples y = [y₁, ..., y_N ], the classical periodogram is computed as

G N ( ω ) = 1 N | ∑ t = 1 N y t exp ⁡ ( − j t ω ) | 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4raC0aaSbaaSqaaiabd6eaobqabaGccqGGOaakcqaHjpWDcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabd6eaobaakmaaemaabaWaaabCaeaacqWG5bqEdaWgaaWcbaGaemiDaqhabeaakiGbcwgaLjabcIha4jabcchaWjabcIcaOiabgkHiTiabdQgaQjabdsha0jabeM8a3jabcMcaPaWcbaGaemiDaqNaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaakiaawEa7caGLiWoadaahaaWcbeqaaiabikdaYaaakiabcYcaSaaa@506A@

for the frequency range ω ∈ (0, 2π). For frequencies ω=2πkN MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyYdCNaeyypa0tcfa4aaSaaaeaacqaIYaGmcqaHapaCcqWGRbWAaeaacqWGobGtaaaaaa@3479@, k = 1, 2, ..., the classical periodogram can be alternatively written as

G N ( ω ) = 1 4 N ‖ β ˜ N ( ω ) ‖ 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4raC0aaSbaaSqaaiabd6eaobqabaGccqGGOaakcqaHjpWDcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabisda0aaacqWGobGtdaqbdaqaaGGadiqb=j7aIzaaiaWaaSbaaeaacqWGobGtaeqaaiabcIcaOiabeM8a3jabcMcaPaGaayzcSlaawQa7amaaCaaabeqaaiabikdaYaaakiabcYcaSaaa@4265@

where β˜N MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGaf8NSdiMbaGaadaWgaaWcbaGaemOta4eabeaaaaa@2EDE@ is given by

β ˜ N ≜ arg ⁡ min ⁡ β ∈ ℝ 2 ∑ t = 1 N | y t − x t T ( ω ) β | 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGaf8NSdiMbaGaadaWgaaWcbaGaemOta4eabeaakiablYLiajGbcggaHjabckhaYjabcEgaNnaaxababaGagiyBa0MaeiyAaKMaeiOBa4galeaacqWFYoGycqGHiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab+1risnaaCaaameqabaGaeGOmaidaaaWcbeaakmaaqahabaWaaqWaaeaacqWG5bqEdaWgaaWcbaGaemiDaqhabeaakiabgkHiTGqadiab9Hha4naaDaaaleaacqWG0baDaeaacqWGubavaaGccqGGOaakcqaHjpWDcqGGPaqkcqWFYoGyaiaawEa7caGLiWoadaahaaWcbeqaaiabikdaYaaaaeaacqWG0baDcqGH9aqpcqaIXaqmaeaacqWGobGta0GaeyyeIuoakiabcYcaSaaa@6169@

With xtT(ω)≜[cos⁡(ωt),sin⁡(ωt)] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacbmGae8hEaG3aa0baaSqaaiabdsha0bqaaiabdsfaubaakiabcIcaOiabeM8a3jabcMcaPiablYLiajabcUfaBjGbcogaJjabc+gaVjabcohaZjabcIcaOiabeM8a3jabdsha0jabcMcaPiabcYcaSiGbcohaZjabcMgaPjabc6gaUjabcIcaOiabeM8a3jabdsha0jabcMcaPiabc2faDbaa@4A8F@. In other words, β˜N MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGaf8NSdiMbaGaadaWgaaWcbaGaemOta4eabeaaaaa@2EDE@ is the solution to the least-squares regression of y to the regressor x_t37.

To overcome the weakness of classical and Lomb-Scargle periodograms in dealing with outliers and heavy-tailed noise, it is proposed in 30 that the L₂ norm in (6) be replaced with the L₁ norm, thus replacing the least squares with least absolute deviation (LAD). Thus, the Laplace periodogram can be computed for ω=2πkN MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyYdCNaeyypa0tcfa4aaSaaaeaacqaIYaGmcqaHapaCcqWGRbWAaeaacqWGobGtaaaaaa@3479@, k = 1, 2, ...,

L N ( ω ) = 1 4 N ‖ γ ˜ N ( ω ) ‖ 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaW0aaSbaaSqaaiabd6eaobqabaGccqGGOaakcqaHjpWDcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabisda0aaakiabd6eaonaafmaabaaccmGaf83SdCMbaGaadaWgaaWcbaGaemOta4eabeaakiabcIcaOiabeM8a3jabcMcaPaGaayzcSlaawQa7amaaCaaaleqabaGaeGOmaidaaOGaeiilaWcaaa@429F@

where we replace the least squares coefficient β˜N MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGaf8NSdiMbaGaadaWgaaWcbaGaemOta4eabeaaaaa@2EDE@ with the LAD coefficient,

γ ˜ N ≜ arg ⁡ min ⁡ γ ∈ ℝ 2 ∑ t = 1 N | y t − x t T ( ω ) γ | . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGaf83SdCMbaGaadaWgaaWcbaGaemOta4eabeaakiablYLiajGbcggaHjabckhaYjabcEgaNnaaxababaGagiyBa0MaeiyAaKMaeiOBa4galeaacqWFZoWzcqGHiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab+1risnaaCaaameqabaGaeGOmaidaaaWcbeaakmaaqahabaWaaqWaaeaacqWG5bqEdaWgaaWcbaGaemiDaqhabeaakiabgkHiTGqadiab9Hha4naaDaaaleaacqWG0baDaeaacqWGubavaaGccqGGOaakcqaHjpWDcqGGPaqkcqWFZoWzaiaawEa7caGLiWoaaSqaaiabdsha0jabg2da9iabigdaXaqaaiabd6eaobqdcqGHris5aOGaeiOla4caaa@606B@

Note here that the magnitude at each angular frequency can be computed independent of the other frequencies, meaning that if we know exactly the periodicity that we are looking for, there is no need to compute the LAD coefficients for the entire frequency spectrum. In 29, the proposed robust estimators are used in the Fisher's g-statistics, which has a denominator that is computed with contribution from all the frequency components. Thus the LAD periodogram has an advantage in terms of computational complexity when the periodicity is known, by foregoing the evaluation of the magnitudes of the rest of the frequency components.

An implementation of the proposed algorithm in MATLAB can be found at http://kuoching.googlepages.com

To solve for the LAD coefficients, we can convert (8) into a set of equations and constraints to be solved using linear programming 38. Let γ = [γ₁γ₂]^T, and x_t = [x_t,1 x_t,2]^T, where x_t,1 = cos(ωt), and x_t,2 = sin(ωt). We first consider the following equation,

y t − ∑ j = 1 2 γ j x i , j = u t − v t , t = 1 , ⋯ , N , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdMha5naaBaaaleaacqWG0baDaeqaaOGaeyOeI0YaaabCaeaacqaHZoWzdaWgaaWcbaGaemOAaOgabeaakiabdIha4naaBaaaleaacqWGPbqAcqGGSaalcqWGQbGAaeqaaOGaeyypa0JaemyDau3aaSbaaSqaaiabdsha0bqabaGccqGHsislcqWG2bGDdaWgaaWcbaGaemiDaqhabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqaIYaGma0GaeyyeIuoakiabcYcaSaqaaiabdsha0jabg2da9iabigdaXiabcYcaSiabl+UimjabcYcaSiabd6eaojabcYcaSaaaaaa@515F@

where u_t and v_t are non-negative variables. By setting γ_j = b_j - c_j, where b_j and c_j are non-negative variables, we can obtain the best L₁ approximation by solving the following linear programming problem:

Minimize ∑ t = 1 N ( u t + v t ) subject to y t = ∑ j = 1 2 ( b j − c j ) x t , j + u t − v t , t = 1 , ⋯ , N , and b j , c j , u t , v t ≥ 0. MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabmGaaaqaaiabb2eanjabbMgaPjabb6gaUjabbMgaPjabb2gaTjabbMgaPjabbQha6jabbwgaLbqaamaaqahabaGaeiikaGIaemyDau3aaSbaaSqaaiabdsha0bqabaGccqGHRaWkcqWG2bGDdaWgaaWcbaGaemiDaqhabeaakiabcMcaPaWcbaGaemiDaqNaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaakeaacqqGZbWCcqqG1bqDcqqGIbGycqqGQbGAcqqGLbqzcqqGJbWycqqG0baDcqqGGaaicqqG0baDcqqGVbWBaeaafaqabeqacaaabaGaemyEaK3aaSbaaSqaaiabdsha0bqabaGccqGH9aqpdaaeWbqaaiabcIcaOiabdkgaInaaBaaaleaacqWGQbGAaeqaaOGaeyOeI0Iaem4yam2aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkcqWG4baEdaWgaaWcbaGaemiDaqNaeiilaWIaemOAaOgabeaakiabgUcaRiabdwha1naaBaaaleaacqWG0baDaeqaaOGaeyOeI0IaemODay3aaSbaaSqaaiabdsha0bqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaeGOmaidaniabggHiLdGccqGGSaalaeaacqWG0baDcqGH9aqpcqaIXaqmcqGGSaalcqWIVlctcqGGSaalcqWGobGtcqGGSaalaaaabaGaeeyyaeMaeeOBa4MaeeizaqgabaGaemOyai2aaSbaaSqaaiabdQgaQbqabaGccqGGSaalcqWGJbWydaWgaaWcbaGaemOAaOgabeaakiabcYcaSiabdwha1naaBaaaleaacqWG0baDaeqaaOGaeiilaWIaemODay3aaSbaaSqaaiabdsha0bqabaGccqGHLjYScqaIWaamcqGGUaGlaaaaaa@9458@

To solve the LAD approximation for non-uniformly spaced samples, we follow the same steps to solve for the LAD coefficient in the following,

γ ˜ N ≜ arg ⁡ min ⁡ γ ∈ ℝ 2 ∑ τ = 1 N | y t r − x t r T ( ω ) γ | . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@640E@

where [t₁, t₂, ..., t_N ] are the N non-uniformly sampled time instants.

In this formulation, the LAD coefficients can be easily solved using standard algorithms for solving linear programming problems. For our implementation in MATLAB, we used the LINPROG function in the Optimization Toolbox. In terms of computational time required to process the data, for experiment Alpha which consists of 6075 genes and 18 samples each, the total time to compute 1000 permutations for the p-value analysis takes approximately 24 hours on a Pentium Core 2 CPU at 2.66 GHz, which is similar to the amount of time taken by the M-estimator-based method, also implemented in MATLAB using the ROBUSTFIT function in the Statistics Toolbox.