# A mixed integer linear program to compress transition probability matrices in Markov chain bootstrapping

Journal article

Cerqueti, R., Falbo, P., Pelizzari, C., Ricca, F. and Scozzari, A. (2017). A mixed integer linear program to compress transition probability matrices in Markov chain bootstrapping. Annals of Operations Research. 248 (1-2), pp. 163-187. https://doi.org/10.1007/s10479-016-2181-9
Authors Cerqueti, R., Falbo, P., Pelizzari, C., Ricca, F. and Scozzari, A. Bootstrapping time series is one of the most acknowledged tools to study the statistical properties of an evolutive phenomenon. An important class of bootstrapping methods is based on the assumption that the sampled phenomenon evolves according to a Markov chain. This assumption does not apply when the process takes values in a continuous set, as it frequently happens with time series related to economic and financial phenomena. In this paper we apply the Markov chain theory for bootstrapping continuous-valued processes, starting from a suitable discretization of the support that provides the state space of a Markov chain of order k≥1. Even for small k, the number of rows of the transition probability matrix is generally too large and, in many practical cases, it may incorporate much more information than it is really required to replicate the phenomenon satisfactorily. The paper aims to study the problem of compressing the transition probability matrix while preserving the “law” characterising the process that generates the observed time series, in order to obtain bootstrapped series that maintain the typical features of the observed time series. For this purpose, we formulate a partitioning problem of the set of rows of such a matrix and propose a mixed integer linear program specifically tailored for this particular problem. We also provide an empirical analysis by applying our model to the time series of Spanish and German electricity prices, and we show that, in these medium size real-life instances, bootstrapped time series reproduce the typical features of the ones under observation.This is a post-peer-review, pre-copyedit version of an article published in Annals of Operations Research volume. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10479-016-2181-9 2017 Annals of Operations Research 248 (1-2), pp. 163-187 Springer 0254-5330 https://doi.org/10.1007/s10479-016-2181-9 https://link.springer.com/article/10.1007%2Fs10479-016-2181-9 Jan 2017 18 Apr 2016 18 Mar 2016 09 Mar 2020 MILP for LSBU.pdfLicenseCC BY 4.0File Access LevelOpen

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Cerqueti, R., Falbo, P. and Pelizzari, C. (2017). Relevant states and memory in Markov chain bootstrapping and simulation. European Journal of Operational Research. 256 (1), pp. 163-177. https://doi.org/10.1016/j.ejor.2016.06.006
##### Long-range properties and data validity for hydrogeological time series: The case of the Paglia river
Ausloos, M., Cerqueti, R. and Lupi, C. (2017). Long-range properties and data validity for hydrogeological time series: The case of the Paglia river. Physica A: Statistical Mechanics and its Applications. 470, pp. 39-50. https://doi.org/10.1016/j.physa.2016.11.137
##### Assessing the relationship between toxicity and economic cost of oncological target agents: A systematic review of clinical trials
Tartari, F., Conti, A. and Cerqueti, R. (2017). Assessing the relationship between toxicity and economic cost of oncological target agents: A systematic review of clinical trials. PLoS ONE. 12 (8), pp. e0183639-e0183639. https://doi.org/10.1371/journal.pone.0183639
##### Data science for assessing possible tax income manipulation: The case of Italy
Ausloos, M., Cerqueti, R. and Mir, T.A. (2017). Data science for assessing possible tax income manipulation: The case of Italy. Chaos, Solitons and Fractals. 104, pp. 238-256. https://doi.org/10.1016/j.chaos.2017.08.012
##### Corruption, evasion and environmental policy: a game theory approach
Cerqueti, R. and Coppier, R. (2016). Corruption, evasion and environmental policy: a game theory approach. IMA Journal of Management Mathematics. 27 (2), pp. 235-253. https://doi.org/10.1093/imaman/dpu019
##### Religion-based urbanization process in Italy: statistical evidence from demographic and economic data
Ausloos, M. and Cerqueti, R. (2016). Religion-based urbanization process in Italy: statistical evidence from demographic and economic data. Quality & Quantity. 50 (4), pp. 1539-1565. https://doi.org/10.1007/s11135-015-0220-2
##### Optimal Investment in Research and Development Under Uncertainty
Cerqueti, R, Marazzina, D and Ventura, M (2016). Optimal Investment in Research and Development Under Uncertainty. Journal of Optimization Theory and Applications. 168 (1), pp. 296-309. https://doi.org/10.1007/s10957-015-0751-7
##### Studies on Regional Wealth Inequalities: The Case of Italy
Ausloos, M and Cerqueti, R (2016). Studies on Regional Wealth Inequalities: The Case of Italy. Acta Physica Polonica A. 129 (5), pp. 959-964. https://doi.org/10.12693/aphyspola.129.959
##### Regularities and discrepancies of credit default swaps: a data science approach through Benford's law
Ausloos, M., Castellano, R. and Cerqueti, R. (2016). Regularities and discrepancies of credit default swaps: a data science approach through Benford's law. Chaos, Solitons and Fractals. 90, pp. 8-17. https://doi.org/10.1016/j.chaos.2016.03.002
##### Bayesian estimation and entropy for economic dynamic stochastic models: An exploration of overconsumption
Argentiero, A., Bovi, M. and Cerqueti, R. (2016). Bayesian estimation and entropy for economic dynamic stochastic models: An exploration of overconsumption. Chaos, Solitons and Fractals. 88, pp. 143-157. https://doi.org/10.1016/j.chaos.2016.03.003
##### Non-exchangeable copulas and multivariate total positivity
Cerqueti, R. and Lupi, C. (2016). Non-exchangeable copulas and multivariate total positivity. Information Sciences. 360, pp. 163-169. https://doi.org/10.1016/j.ins.2016.04.026
##### Sustainable management of fossil fuels: A dynamic stochastic optimization approach with jump-diffusion
Castellano, R., Cerqueti, R. and Spinesi, L. (2016). Sustainable management of fossil fuels: A dynamic stochastic optimization approach with jump-diffusion. European Journal of Operational Research. 255 (1), pp. 288-297. https://doi.org/10.1016/j.ejor.2016.04.052
##### Risk measures on networks and expected utility
Cerqueti, R. and Lupi, C. (2016). Risk measures on networks and expected utility. Reliability Engineering & System Safety. 155, pp. 1-8. https://doi.org/10.1016/j.ress.2016.04.017
##### Innovation, imitation and policy inaction
Cerqueti, R., Quaranta, A.G. and Ventura, M. (2016). Innovation, imitation and policy inaction. Technological Forecasting and Social Change. 111, pp. 22-30. https://doi.org/10.1016/j.techfore.2016.06.001
##### Forecasting macroeconomic fundamentals in economic crises
Bovi, M. and Cerqueti, R. (2016). Forecasting macroeconomic fundamentals in economic crises. Annals of Operations Research. 247 (2), pp. 451-469. https://doi.org/10.1007/s10479-015-1879-4
##### A Universal Rank-Size Law
Ausloos, M. and Cerqueti, R. (2016). A Universal Rank-Size Law. PLoS ONE. 11 (11). https://doi.org/10.1371/journal.pone.0166011
##### A theory of misperception in a stochastic dominance framework and its application to structured financial products
Castellano, R. and Cerqueti, R. (2016). A theory of misperception in a stochastic dominance framework and its application to structured financial products. IMA Journal of Management Mathematics. 29 (1), pp. 23-37. https://doi.org/10.1093/imaman/dpw007
##### A game for exploring political and bureaucratic corruption
Cerqueti, R. and Coppier, R. (2016). A game for exploring political and bureaucratic corruption. IMA Journal of Management Mathematics. 29 (2), pp. 151-173. https://doi.org/10.1093/imaman/dpw017
##### Signatures of systems with non-exchangeable lifetimes: some implications in the analysis of financial risk
Cerqueti, R and Spizzichino, F. (2015). Signatures of systems with non-exchangeable lifetimes: some implications in the analysis of financial risk. in: Kitzos, C (ed.) Springer Proceedingsin Mathematics & Statistics Springer.
##### Approximating Markov Chains for Bootstrapping and Simulation
Cerqueti, R., Falbo, P., Guastaroba, G. and Pelizzari, C. (2015). Approximating Markov Chains for Bootstrapping and Simulation. in: Steland, A (ed.) Stochastic Models, Statistics and Their Applications,Springer Proceedings in Mathematics & Statistics Switzerland Springer.
##### The impact of innovation on companies’ performance: an entropy-based analysis of the STAR market segment of the Italian Stock Exchange
Bartolacci, F., Castellano, N.G. and Cerqueti, R. (2015). The impact of innovation on companies’ performance: an entropy-based analysis of the STAR market segment of the Italian Stock Exchange. Technology Analysis & Strategic Management. 27 (1), pp. 102-123. https://doi.org/10.1080/09537325.2014.952624
##### Patent Valuation under Spatial Point Processes with Delayed and Decreasing Jump Intensity
Cerqueti, R. and Ventura, M. (2015). Patent Valuation under Spatial Point Processes with Delayed and Decreasing Jump Intensity. The B.E. Journal of Theoretical Economics. 15 (2). https://doi.org/10.1515/bejte-2013-0104
##### Risk and uncertainty in the patent race: a probabilistic model
Cerqueti, R and Ventura, M (2015). Risk and uncertainty in the patent race: a probabilistic model. IMA Journal of Management Mathematics. 26 (1), pp. 39-62. https://doi.org/10.1093/imaman/dpt020
##### On the coexistence of innovators and imitators
Cerqueti, R., Tramontana, F. and Ventura, M. (2015). On the coexistence of innovators and imitators. Technological Forecasting and Social Change. 90 (B), pp. 487-496. https://doi.org/10.1016/j.techfore.2014.03.011
##### A review of aggregation techniques for agent-based models: understanding the presence of long-term memory
Cerqueti, R. and Rotundo, G. (2015). A review of aggregation techniques for agent-based models: understanding the presence of long-term memory. Quality & Quantity. 49 (4), pp. 1693-1717. https://doi.org/10.1007/s11135-014-9995-9
##### Evidence of economic regularities and disparities of Italian regions from aggregated tax income size data
Cerqueti, R. and Ausloos, M. (2015). Evidence of economic regularities and disparities of Italian regions from aggregated tax income size data. Physica A: Statistical Mechanics and its Applications. 421, pp. 187-207. https://doi.org/10.1016/j.physa.2014.11.027
##### Corruptibility and tax evasion
Cerqueti, R. and Coppier, R. (2015). Corruptibility and tax evasion. European Journal of Law and Economics. 39 (2), pp. 355-373. https://doi.org/10.1007/s10657-013-9406-z
##### Approximating multivariate Markov chains for bootstrapping through contiguous partitions
Cerqueti, R., Falbo, P., Guastaroba, G. and Pelizzari, C. (2015). Approximating multivariate Markov chains for bootstrapping through contiguous partitions. OR Spectrum. 37 (3), pp. 803-841. https://doi.org/10.1007/s00291-015-0397-8
##### Socio-economical analysis of Italy: The case of hagiotoponym cities
Cerqueti, R. and Ausloos, M. (2015). Socio-economical analysis of Italy: The case of hagiotoponym cities. The Social Science Journal. 52 (4), pp. 561-564. https://doi.org/10.1016/j.soscij.2015.03.004
##### Cross ranking of cities and regions: population versus income
Cerqueti, R. and Ausloos, M. (2015). Cross ranking of cities and regions: population versus income. Journal of Statistical Mechanics: Theory and Experiment. 2015 (7), pp. P07002-P07002. https://doi.org/10.1088/1742-5468/2015/07/p07002
##### A disutility-based drift control for exchange rates
Castellano, R, Cerqueti, R and D'Ecclesia, R. L (2014). A disutility-based drift control for exchange rates. Optimization. 63 (2), pp. 255-269. https://doi.org/10.1080/02331934.2011.641016
##### A quantitative view on policymakers’ goal, institutions and tax evasion
Bovi, M and Cerqueti, R (2014). A quantitative view on policymakers’ goal, institutions and tax evasion. Quality & Quantity. 48 (3), pp. 1493-1510. https://doi.org/10.1007/s11135-013-9849-x
##### Mean–Variance portfolio selection in presence of infrequently traded stocks
Castellano, R and Cerqueti, R (2014). Mean–Variance portfolio selection in presence of infrequently traded stocks. European Journal of Operational Research. 234 (2), pp. 442-449. https://doi.org/10.1016/j.ejor.2013.04.024
##### Exhaustion of resources: a marked temporal process framework
Cerqueti, R (2014). Exhaustion of resources: a marked temporal process framework. Stochastic Environmental Research and Risk Assessment. 28 (4), pp. 1023-1033. https://doi.org/10.1007/s00477-013-0798-7
##### Symmetries of systems with spatially bounded domains
Cerqueti, R and Sarnari, F (2014). Symmetries of systems with spatially bounded domains. Journal of Advanced Research in Dynamical and Control Systems. 6 (1), pp. 11-25.
##### Benford’s law predicted digit distribution of aggregated income taxes: the surprising conformity of Italian cities and regions
Mir, T. A, Ausloos, M and Cerqueti, R (2014). Benford’s law predicted digit distribution of aggregated income taxes: the surprising conformity of Italian cities and regions. The European Physical Journal B. 87 (11). https://doi.org/10.1140/epjb/e2014-50525-2
##### New results on the convergence of random matrices
Cerqueti, R and Costantini, M (2013). New results on the convergence of random matrices. Statistics. 47 (4), pp. 663-671. https://doi.org/10.1080/02331888.2011.648639
##### Roots and effects of financial misperception in a stochastic dominance framework
Castellano, R and Cerqueti, R (2013). Roots and effects of financial misperception in a stochastic dominance framework. Quality & Quantity. 47 (6), pp. 3371-3389. https://doi.org/10.1007/s11135-012-9726-z
##### Extension of dependence properties to semi-copulas and applications to the mean–variance model
Cerqueti, R and Spizzichino, F (2013). Extension of dependence properties to semi-copulas and applications to the mean–variance model. Fuzzy Sets and Systems. 220, pp. 99-108. https://doi.org/10.1016/j.fss.2012.08.011
##### A Tabu Search heuristic procedure in Markov chain bootstrapping
Cerqueti, R, Falbo, P, Guastaroba, G and Pelizzari, C (2013). A Tabu Search heuristic procedure in Markov chain bootstrapping. European Journal of Operational Research. 227 (2), pp. 367-384. https://doi.org/10.1016/j.ejor.2012.11.009
##### Economic interactions and social tolerance: A dynamic perspective
Cerqueti, R, Correani, L and Garofalo, G (2013). Economic interactions and social tolerance: A dynamic perspective. Economics Letters. 120 (3), pp. 458-463. https://doi.org/10.1016/j.econlet.2013.05.032
##### Illegal finance and usurers behaviour
Barone, R, Cerqueti, R and Quaranta, A. G. (2012). Illegal finance and usurers behaviour. European Journal of Law and Economics. 34 (2), pp. 265-277. https://doi.org/10.1007/s10657-010-9183-x
##### The perspective of a bank in granting credits: an optimization model
Cerqueti, R and Quaranta, A. G (2012). The perspective of a bank in granting credits: an optimization model. Optimization Letters. 6 (5), pp. 867-882. https://doi.org/10.1007/s11590-011-0310-6
##### Financing policies via stochastic control: a dynamic programming approach
Cerqueti, R (2012). Financing policies via stochastic control: a dynamic programming approach. Journal of Global Optimization. 53 (3), pp. 539-561. https://doi.org/10.1007/s10898-011-9725-y
##### The role of diversity in persistence aggregation
Cerqueti, R and Rotundo, G (2012). The role of diversity in persistence aggregation. International Journal of Intelligent Systems. 27 (2), pp. 176-187. https://doi.org/10.1002/int.21519
##### Corruption, growth and ethnic fractionalization: a theoretical model
Cerqueti, R, Coppier, R and Piga, G (2012). Corruption, growth and ethnic fractionalization: a theoretical model. Journal of Economics. 106 (2), pp. 153-181. https://doi.org/10.1007/s00712-011-0253-5
##### Optimal consumption/investment problem with light stocks: A mixed continuous-discrete time approach
Castellano, R and Cerqueti, R (2012). Optimal consumption/investment problem with light stocks: A mixed continuous-discrete time approach. Applied Mathematics and Computation. 218 (12), pp. 6887-6898. https://doi.org/10.1016/j.amc.2011.12.065
##### Why is the Tax Evasion so Persistent?
Cerqueti, R and Bovi, M (2011). Why is the Tax Evasion so Persistent? in: Tavidze, A (ed.) Progress in Economics Nova Science Publishers, Inc..
Castellano, R and Cerqueti, R (2011). The optimal bid/ask spread in a Specialist System. Economic Modelling. 28 (5), pp. 2247-2253. https://doi.org/10.1016/j.econmod.2011.06.019
##### Economic growth, corruption and tax evasion
Cerqueti, R and Coppier, R (2011). Economic growth, corruption and tax evasion. Economic Modelling. 28 (1-2), pp. 489-500. https://doi.org/10.1016/j.econmod.2010.07.006
##### Testing for rational bubbles in the presence of structural breaks: Evidence from nonstationary panels
Cerqueti, R and Costantini, M (2011). Testing for rational bubbles in the presence of structural breaks: Evidence from nonstationary panels. Journal of Banking & Finance. 35 (10), pp. 2598-2605. https://doi.org/10.1016/j.jbankfin.2011.02.011
##### Memory Property in Heterogeneously Populated Markets
Cerqueti, R and Rotundo, G (2010). Memory Property in Heterogeneously Populated Markets. in: Greco, S, Marques Pereira, RA, Squillante, M, Yager, RR and Kacprzyk, J (ed.) Preferences and Decisions - Studies in Fuzziness and Soft Computing Berlin Springer. pp. 53-69
##### Firms clustering in presence of technological renewal processes
Cerqueti, R and Rotundo, G (2010). Firms clustering in presence of technological renewal processes. in: Puu, T and Panchuk, A (ed.) Nonlinear economic dynamics Nova Science Publishers, Inc..
##### Light Stocks and Wealth Allocation
Cerqueti, R and Castellano, R (2010). Light Stocks and Wealth Allocation. in: Kasımbeyli, R, Dinçer, C, Özpeynirci, S and Sakalauskas, L (ed.) 24th mini euro conference on continuous optimization and information based technologies in the financial sector - MEC EUROPT 2010: selected papers Vilnius "Technika".
##### Some Nonparametric Asymptotic Results for a Class of Stochastic Processes
Cerqueti, R and Costantini, M (2010). Some Nonparametric Asymptotic Results for a Class of Stochastic Processes. Communications in Statistics - Theory and Methods. 39 (14), pp. 2552-2560. https://doi.org/10.1080/03610920903061021
##### Options with underlying asset driven by a fractional brownian motion: crossing barriers estimates
Rotundo, G and Cerqueti, R (2010). Options with underlying asset driven by a fractional brownian motion: crossing barriers estimates. New Mathematics and Natural Computation. 06 (01), pp. 109-118. https://doi.org/10.1142/s1793005710001633
##### A Dynamic Stochastic Model of Asset Pricing with Heterogeneous Beliefs
Brianzoni, S, Cerqueti, R and Michetti, E (2010). A Dynamic Stochastic Model of Asset Pricing with Heterogeneous Beliefs. Computational Economics. 35 (2), pp. 165-188. https://doi.org/10.1007/s10614-009-9189-z
##### A Characterization of the Dickey-Fuller Distribution With Some Extensions to the Multivariate Case
Cerqueti, R, Costantini, M and Lupi, C (2009). A Characterization of the Dickey-Fuller Distribution With Some Extensions to the Multivariate Case. in: Martinez, WL (ed.) Proceedings of the Joint Statistical Meeting (Business and Economic Statistics Section), 1-6 August 2009, Washington DC, USA American Statistical Association.
##### Dynamic Programming via Measurable Selection
Cerqueti, R (2009). Dynamic Programming via Measurable Selection. Pacific Journal of Optimization. 5 (1), pp. 169-181.
##### Tax revenues, fiscal corruption and “shame” costs
Cerqueti, R and Coppier, R (2009). Tax revenues, fiscal corruption and “shame” costs. Economic Modelling. 26 (6), pp. 1239-1244. https://doi.org/10.1016/j.econmod.2009.06.005
##### A spatial mixed Poisson framework for combination of excess-of-loss and proportional reinsurance contracts
Cerqueti, R, Foschi, R and Spizzichino, F (2009). A spatial mixed Poisson framework for combination of excess-of-loss and proportional reinsurance contracts. Insurance: Mathematics and Economics. 45 (1), pp. 59-64. https://doi.org/10.1016/j.insmatheco.2009.03.001
##### Companies' decisions for profit maximization: a structural model
Cerqueti, R and Rotundo, G. (2009). Companies' decisions for profit maximization: a structural model. Applied Mathematical Sciences. 3 (27), pp. 1327-1340.
##### Asymptotic solutions of a generalized eigenvalue problem
Cerqueti, R. and Costantini, M. (2009). Asymptotic solutions of a generalized eigenvalue problem. Applied Mathematical Sciences. 3 (60), pp. 2985-2999.
##### On the asymptotic behaviour of random matrices in a multivariate statistical model
Cerqueti, R and Costantini, M (2008). On the asymptotic behaviour of random matrices in a multivariate statistical model. Statistics & Probability Letters. 78 (14), pp. 2039-2045. https://doi.org/10.1016/j.spl.2008.01.051
##### Dynamics of financial time series in an inhomogeneous framework
Cerqueti, R and Rotundo, G (2007). Dynamics of financial time series in an inhomogeneous framework. in: Sibillo, M and Perna, C (ed.) Mathematical and statistical methods in insurance and finance Berlin Springer. pp. 67-74
##### Productivity and costs for firms in presence of technology renewal processes
Cerqueti, R and Rotundo, G (2007). Productivity and costs for firms in presence of technology renewal processes. International Transactions in Operational Research. 14 (6), pp. 521-534. https://doi.org/10.1111/j.1475-3995.2007.00611.x
##### Microeconomic modeling of financial time series with long term memory
Cerqueti, R and Rotundo, G (2003). Microeconomic modeling of financial time series with long term memory. 2003 IEEE International Conference on Computational Intelligence for Financial Engineering. Proceedings. Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.1109/cifer.2003.1196260