# Pattern Recognition of Gene Expression with Singular Spectrum Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Singular Spectrum Analysis (SSA)

#### 2.1. LS and MV Estimators

#### 2.2. LS Estimate of $\mathbf{S}$

#### 2.3. MV Estimate of $\mathbf{S}$

#### 2.4. Weight Matrix **W**

## 3. Empirical Results

#### 3.1. Simulated Series

Parameter | Original Value | Average | S.D | Ratio | ||
---|---|---|---|---|---|---|

before | after | before | after | |||

A | 200 | 204.2 | 202.6 | 4.19 | 2.64 | $\frac{2.64}{4.19}=0.63$ |

λ | 20 | 19.57 | 19.76 | 0.41 | 0.26 | $\frac{0.26}{0.41}=0.62$ |

**Figure 2.**Distribution of the estimated parameters of A and λ for noisy Bcd and noise-reduced Bcd (thick line).

**X**, 50 eigentriples were obtained, ordered by their contribution (share) in the decomposition stage. We shall say that the series ${Y}_{N}$ is not complex if ${Y}_{N}$ is well approximated by a series with small rank d. For example, series ${y}_{i}={e}^{\alpha i}$ (i = 1, ... , N) has rank 1. For all $2\le L\le N-1$, ${y}_{i}=b{y}_{i-1}$ where $b={e}^{\alpha}$. It should be noted that the number of eigentriples selected as corresponding to the series ${S}_{N}$ has to be at least d. For example, if ${y}_{i}={e}^{\alpha}+{\u03f5}_{i}$ then the window length L should be at least 2 and the first eigentriple is enough for reconstructing the original series if $\left|\right|{S}_{N}\left|\right|\gg \left|\right|{E}_{N}\left|\right|$. Accordingly, the first eigentriple is selected for filtering andtrend extraction.

Simulation Run | RMSE | RRMSE ($\frac{\mathit{SSA}}{\mathit{SDD}}$) | |
---|---|---|---|

SSA | SDD | ||

1 | 2.72 | 4.87 | 0.55 |

2 | 2.71 | 4.83 | 0.56 |

3 | 2.69 | 3.98 | 0.54 |

4 | 2.72 | 4.68 | 0.58 |

5 | 2.73 | 4.96 | 0.54 |

#### 3.2. The Bicode Data

#### 3.2.1. Data Description

#### 3.2.2. The Results

Time Class | w-Correlation | Time Class | w-Correlation |
---|---|---|---|

10 | 0.008 | 14(3) | 0.001 |

11 | 0.005 | 14(4) | 0.002 |

12 | 0.007 | 14(5) | 0.001 |

13 | 0.003 | 14(6) | 0.001 |

14(1) | 0.001 | 14(7) | 0.001 |

14(2) | 0.001 | 14(8) | 0.001 |

**Figure 3.**Temporal dynamics of Bcd expression, cleavage cycles 10 to 13. Red and green lines show the trend extracted by SSA

_{MV}and SDD, respectively.

**Figure 4.**Temporal dynamics of Bcd expression, cleavage cycles 14. Red and green lines show the trend extracted by SSA

_{MV}and SDD, respectively.

## 4. Conclusions

## Conflicts of Interest

## References

- How Instructors Request a Complimentary Sample. Available online: http://highered.mcgraw-hill.com (accessed on 30 June 2014).
- Surkova, S.; Kosman, D.; Kozlov, K.; Manu; Myasnikova, E.; Samsonova, A.A.; Spirov, A.; Vanario-Alonso, C.E.; Samsonova, M.; Reinitz, J. Characterization of the Drosophila segment determination morphome. Dev. Biol.
**2007**, 313, 844–862. [Google Scholar] [CrossRef] [PubMed] - Wolpert, L. Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol.
**1969**, 27, 1–47. [Google Scholar] [CrossRef] - Nusslein-Volhard, C.; Wieschaus, E. Mutations affecting segment number and polarity in Drosophila. Nature
**1980**, 287, 795–801. [Google Scholar] [CrossRef] [PubMed] - Liu, W.; Niranjan, M. Gaussian process modelling for bicoid mRNA regulation in spatio-temporal Bicoid profile. Bioinformatics
**2011**, 28, 366–372. [Google Scholar] [CrossRef] [PubMed] - Holloway, D.M.; Harrison, L.G.; Kosman, D.; Vanario-Alonso, C.E.; Spirov, A.V. Analysis of pattern precision shows that Drosophila segmentation develops substantial independence from gradients of maternal gene products. Dev. Dyn.
**2006**, 235, 2949–2960. [Google Scholar] [CrossRef] [PubMed] - Nusslein-Volhard, C.N. The bicoid morphogen papers (I): Account from CNV. Cell
**2004**, 116, 1–5. [Google Scholar] [CrossRef] - Grimm, O.; Coppey, M.; Wieschaus, E. Modelling the bicoid gradient. Development
**2010**, 137, 2253–2264. [Google Scholar] [CrossRef] [PubMed] - Liu, J.; He, F.; Ma, J. Morphogen gradient formation and action. Fly
**2011**, 5, 242–246. [Google Scholar] [CrossRef] [PubMed] - Houchmandzadeh, B.; Wieschaus, E.; Leibler, S. Establishment of developmental precision and proportions in the early Drosophila embryo. Nature
**2002**, 415, 798–802. [Google Scholar] [CrossRef] [PubMed] - Gregor, T.; Tank, D.W.; Wieschaus, E.F.; Bialek, W. Stability and Nuclear Dynamics of the Bicoid Morphogen Gradient. Cell
**2007**, 153, 153–164. [Google Scholar] [CrossRef] [PubMed] - Hilfinger, H.; Paulsson, J. Separating intrinsic from extrinsic fluctuations in dynamic biological systems. Proc. Natl. Acad. Sci. USA
**2011**, 108, 12167–12172. [Google Scholar] [CrossRef] [PubMed] - Hassani, H.; Zokaei, M.; von Rosenc, D.; Amiri, S.; Ghodsi, M. Does noise reduction matter for curve fitting in growth curve models? Comput. Methods Programs Biomed.
**2009**, 96, 173–181. [Google Scholar] [CrossRef] [PubMed] - Hassani, H.; Dionisio, A.; Ghodsi, M. The effect of noise reduction in measuring the linear and nonlinear dependency of financial markets. Nonlin. Anal.: Real World Appl.
**2010**, 11, 492–502. [Google Scholar] [CrossRef] - Du, L.P.; Wu, S.H.; Liew, A.W.C.; Smith, D.K.; Yan, H. Spectral Analysis of Microarray Gene Expression Time Series Data of Plasmodium Falciparum. Int. J. Bioinf. Res. Appl.
**2008**, 4, 337–349. [Google Scholar] [CrossRef] [PubMed] - Hassani, H. Singular spectrum analysis: Methodology and comparison. J. Data Sci.
**2007**, 5, 239–257. [Google Scholar] - Hassani, H.; Thomakos, D. A review on singular spectrum analysis for economic and financial time series. Stat. Interface
**2010**, 3, 377–397. [Google Scholar] [CrossRef] - Hassani, H.; Soofi, S.; Zhigljavsky, A. Predicting Inflation Dynamics with Singular Spectrum Analysis. J. R. Stat. Soc. Ser. A (Stat. Soc.)
**2012**, 176, 743–760. [Google Scholar] [CrossRef] - Golyandina, N.E.; Holloway, D.M.; Lopesc, F.J.P.; Spirov, A.V.; Spirova, E.N.; Usevich, K.D. Measuring gene expression noise in early Drosophila embryos: nucleus-to-nucleus variability. Int. Conf. Comput. Sci.
**2012**, 9, 373–382. [Google Scholar] [CrossRef] [PubMed] - Spirov, A.V.; Golyandina, N.E.; Holloway, D.M.; Alexandrov, T.; Spirova, E.N.; Lopes, F.J.P. Measuring Gene Expression Noise in Early Drosophila Embryos: The Highly Dynamic Compartmentalized Micro-environment of the Blastoderm Is One of the Main Sources of Noise. Evol. Comput. Mach. Learn. Data Mining Bioinf.
**2012**, 7246, 177–188. [Google Scholar] - Holloway, D.M.; Lopes, F.J.P.; da Fontoura Costa, L.; Travencolo, B.A.N.; Golyandina, N.; Usevich, K.; Spirov, A.V. Gene Expression Noise in Spatial Patterning: Hunchback Promoter Structure Affects Noise Amplitude and Distribution in Drosophila Segmentation. PLoS Comput. Biol.
**2011**, 7, e1001069. [Google Scholar] [CrossRef] [PubMed] - Hassani, H. Singular spectrum analysis based on the minimum variance estimator. Nonlin. Anal.: Real World Appl.
**2010**, 11, 2065–2077. [Google Scholar] [CrossRef] - Alexandrov, T.; Golyandina, N.; Sprov, A. Singular spectrum analysis of gene expression profiles of early drosophila embryo: Exponential-in-distance patterns. Res. Lett. Signal Process
**2010**, 2008, 5. [Google Scholar] - Ghodsi, M.; Hassani, H.; Sanei, S.; Hicks, Y. The use of noise information for detection of temporomandibular disorder. Biomed. Signal Process. Control
**2010**, 4, 79–85. [Google Scholar] [CrossRef] - Ghodsi, M.; Hassani, H.; Sanei, S. Extracting fetal heart signal from noisy maternal ECG by multivariate singular spectrum analysis. Stat. Interface
**2010**, 3, 399–411. [Google Scholar] [CrossRef] - Vivian, T.; Tang, Y.; Yan, H. Noise Reduction in Microarray Gene Expression Data Based on Spectral Analysis. Int. J. Mach. Learning Cybern.
**2012**, 3, 51–57. [Google Scholar] [CrossRef] - Mamou, J.; Feleppa, E.J. Singular Spectrum Analysis Applied to Ultrasonic Detection and Imaging of Brachytherapy Seeds. J. Acoust. Soc. Am.
**2007**, 121, 1790–1801. [Google Scholar] [CrossRef] [PubMed] - De Moor, B. The singular value decomposition and long and short spaces on noisy matrices. IEEE Trans. Signal Process.
**1993**, 41, 2826–2838. [Google Scholar] [CrossRef] - Golyandina, N.; Nekrutkin, V.; Zhigljavsky, A. Analysis of Time Series Structure: SSA and related techniques; Chapman & Hall/CRC: Boca Raton, FL, USA, 2001. [Google Scholar]
- Hassani, H.; Heravi, H.; Zhigljavsky, A. Forecasting European Industrial Production with Singular Spectrum Analysis. Int. J. Forec.
**2009**, 25, 103–118. [Google Scholar] [CrossRef] - Van Huffel, S. Enhanced resolution based on minimum variance estimation and exponential data modeling. Signal Process
**1993**, 33, 333–355. [Google Scholar] [CrossRef] - A Database of Segmentation gene Expression in Drosophila. Available online: http://urchin.spbcas.ru/flyex/ (accessed on 30 June 2014).
- Pisarev, A.; Poustelnikova, E.; Samsonova, M.; Reinitz, J. FlyEx, the quantitative atlas on segmentation gene expression at cellular resolution. Nucl. Acids Res.
**2008**. [Google Scholar] [CrossRef] [PubMed] - Poustelnikova, E.; Pisarev, A.; Blagov, M.; Samsonova, M.; Reinitz, J. A database for management of gene expression data in situ. Bioinformatics
**2008**, 20, 2212–2221. [Google Scholar]

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Hassani, H.; Ghodsi, Z. Pattern Recognition of Gene Expression with Singular Spectrum Analysis. *Med. Sci.* **2014**, *2*, 127-139.
https://doi.org/10.3390/medsci2030127

**AMA Style**

Hassani H, Ghodsi Z. Pattern Recognition of Gene Expression with Singular Spectrum Analysis. *Medical Sciences*. 2014; 2(3):127-139.
https://doi.org/10.3390/medsci2030127

**Chicago/Turabian Style**

Hassani, Hossein, and Zara Ghodsi. 2014. "Pattern Recognition of Gene Expression with Singular Spectrum Analysis" *Medical Sciences* 2, no. 3: 127-139.
https://doi.org/10.3390/medsci2030127