Optimal and efficient designs for fMRI experiments via two-level circulant almost orthogonal arrays. (English) Zbl 1460.62124
Summary: In this paper, we investigate a class of optimal circulant \(\{ 0 , 1 \}\)-arrays other than the previously known class of optimal designs for fMRI experiments with a single type of stimulus. We suppose throughout the paper that \(n \equiv 2\pmod 4\) and discuss the asymptotic optimality and the D-efficiency of \(k \times n\) circulant almost orthogonal arrays (CAOAs) with 2 levels (presence/absence of the stimulus), strength 2 and bandwidth 1, denoted by CAOA \(( n , k , 2 , 2 , 1 )\). We show that for \(n \equiv 2\pmod 4\) the largest possible value of \(k\) for statistically optimal CAOA \(( n , k , 2 , 2 , 1 )\) cannot exceed \(n / 2\). We also clarify that CAOA \(( n , k , 2 , 2 , 1 )\) with high D-efficiency and \(k\) greater than \(n / 2\) can be obtained via perfect binary sequences. By applying algebraic constructions for perfect binary sequences and by computer search, lists of such efficient CAOAs and the new class of optimal CAOAs are provided.
MSC:
| 62K05 | Optimal statistical designs |
| 05B15 | Orthogonal arrays, Latin squares, Room squares |
| 05B30 | Other designs, configurations |
Software:
SugarReferences:
| [1] | Arasu, K. T., Sequences and arrays with desirable correlation properties, (Crnković, D.; Tonchev, V., Information Security, Coding Theory and Related Combinatorics. Information Security, Coding Theory and Related Combinatorics, NATO Science for Peace and Security Series, 29 (2011), IOS Press), 136-171 · Zbl 1323.05021 |
| [2] | Arasu, K. T.; Ding, C.; Helleseth, T.; Kumar, P. V.; Martinsen, H. M., Almost difference sets and their sequences with optimal autocorrelation, IEEE Trans. Inform. Theory, 47, 7, 2934-2943 (2001) · Zbl 1008.05027 |
| [3] | Arasu, K. T.; Little, Z., Balanced perfect sequences of period 38 and 50, J. Comb. Inf. Syst. Sci., 35, 1-2, 91-95 (2010) · Zbl 1269.94021 |
| [4] | Arasu, K. T.; Pott, A., Perfect binary sequences of even period, J. Stat. Appl., 4, 2-3, 169-178 (2009) · Zbl 1294.05033 |
| [5] | Boynton, M. G.; Engel, S. S.; Glover, H. G.; Heeger, J. D., Linear systems analysis of functional magnetic resonance imaging in human V1, J. Neurosci., 16, 13, 4207-4221 (1996) |
| [6] | Buračas, T. G.; Boynton, M. G., Efficient design of event-related fMRI experiments using M-sequences, NeuroImage, 16, 3, 801-813 (2002) |
| [7] | Cai, Y.; Ding, C., Binary sequences with optimal autocorrelation, Theoret. Comput. Sci., 410, 24-25, 2316-2322 (2009) · Zbl 1170.94008 |
| [8] | Cheng, C.-S., Optimality of certain asymmetrical experimental designs, Ann. Statist., 6, 1239-1261 (1978) · Zbl 0396.62055 |
| [9] | Cheng, C.-S.; Kao, M.-H., Optimal experimental designs for fMRI via circulant biased weighing designs, Ann. Statist., 43, 6, 2565-2587 (2015) · Zbl 1331.62381 |
| [10] | Cheng, C.-S.; Kao, M.-H.; Phoa, F. K.H., Optimal and efficient designs for functional brain imaging experiments, J. Statist. Plann. Inference, 181, 71-80 (2017) · Zbl 1358.62090 |
| [11] | Ding, C.; Helleseth, T.; Martinsen, H., New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inform. Theory, 47, 1, 428-433 (2001) · Zbl 1019.94009 |
| [12] | Hämmerlin, G.; Hoffmann, K.-H., (Numerical Mathematics. Numerical Mathematics, Undergrad. Texts Math. (1991), Springer-Verlag: Springer-Verlag New York) · Zbl 0709.65001 |
| [13] | Hollon, J. R., Investigation of Communication and Radar System Optimization: New Computational and Theoretical Methods (2018), Wright State University, (Ph.D. thesis) |
| [14] | Kao, M.-H., On the optimality of extended maximal length linear feedback shift register sequences, Statist. Probab. Lett., 83, 6, 1479-1483 (2013) · Zbl 1283.62157 |
| [15] | Kao, M.-H., A new type of experimental designs for event-related fMRI via Hadamard matrices, Statist. Probab. Lett., 84, 108-112 (2014) · Zbl 1400.62164 |
| [16] | Kao, M.-H., Universally optimal fMRI designs for comparing hemodynamic response functions, Statist. Sinica, 25, 2, 499-506 (2015) · Zbl 1534.62110 |
| [17] | Kao, M.-H.; Stufken, J., Optimal design for event-related fMRI studies, (Dean, A.; Morris, M.; Stufken, J.; Bingham, D., Handbook of Design and Analysis of Experiments. Handbook of Design and Analysis of Experiments, Chapman & Hall/CRC Handbooks of Modern Statistical Methods (2015), CRC Press: CRC Press Boca Raton), 895-924 · Zbl 1369.62172 |
| [18] | Kiefer, J., General equivalence theory for optimum designs (approximate theory), Ann. Statist., 2, 5, 849-879 (1974) · Zbl 0291.62093 |
| [19] | Lazar, N. A., (The Statistical Analysis of Functional MRI Data. The Statistical Analysis of Functional MRI Data, Statistics for Biology and Health (2008), Springer-Verlag: Springer-Verlag New York) · Zbl 1312.62004 |
| [20] | Lempel, A.; Cohn, M.; Eastman, W., A class of balanced binary sequences with optimal autocorrelation properties, IEEE Trans. Inform. Theory, IT-23, 1, 38-42 (1977) · Zbl 0359.94018 |
| [21] | Lin, Y.-L.; Phoa, F. K.H.; Kao, M.-H., Circulant partial Hadamard matrices: Construction via general difference sets and its application to fMRI experiments, Statist. Sinica, 27, 4, 1715-1724 (2017) · Zbl 1392.62236 |
| [22] | Lin, Y.-L.; Phoa, F. K.H.; Kao, M.-H., Optimal design of fMRI experiments using circulant (almost-)orthogonal arrays, Ann. Statist., 45, 6, 2483-2510 (2017) · Zbl 1456.05024 |
| [23] | Miezin, M. F.; Maccotta, L.; Ollinger, M. J.; Petersen, E. S.; Buckner, R. L., Characterizing the hemodynamic response: effects of presentation rate, sampling procedure, and the possibility of ordering brain activity based on relative timing, NeuroImage, 11, 6, 735-759 (2000) |
| [24] | Sidelnikov, V. M., Some \(k\)-valued pseudo-random sequences and nearly equidistant codes, Probl. Peredachi Inf., 5, 1, 16-22 (1969) · Zbl 0295.94032 |
| [25] | Tamura, N.; Taga, A.; Kitagawa, S.; Banbara, M., Compiling finite linear CSP into SAT, Constraints, 14, 2, 254-272 (2009) · Zbl 1186.68076 |
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