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How mathematicians predict the future?

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Project for the European Consortium for Mathematics in Industry, ESSIM Summer School 2011 (Milan)

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How mathematicians predict the future?

1. 1. The Problem Univariate Analysis Multivariate Analysis Conclusion How mathematicians predict the future? Instructor: Agnieszka Wyloma´ska n Costanza Catalano, Angela Ciliberti, Gonçalo S. Matos, Allan S. Nielsen, Olga Polikarpova, Mattia Zanella European Summer School in Industrial Mathematics Modelling Week July 30, 2011How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
2. 2. The Problem Univariate Analysis Multivariate Analysis ConclusionSupplied Data for AnalysisHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
3. 3. The Problem Univariate Analysis Multivariate Analysis ConclusionSupplied Data for AnalysisHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
4. 4. The Problem Univariate Analysis Multivariate Analysis ConclusionOur Approach Univariate Analysis Orstein-Unlenbeck Model Autoregressive Model Multivariate Analysis Linear RegressionHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
5. 5. The Problem Univariate Analysis Multivariate Analysis ConclusionOrstein-Uhlenbeck processOrstein-Uhlenbeck process Orstein-Uhlenbeck process The Orstein-Uhlenbeck process (or mean-reverting process) is deﬁned by the following equation: dXt = θ(µ − Xt )dt + σdWt Where Wt is a Wiener process, t ∈ T ⊆ R+ represents time and θ > 0, µ and σ > 0 are time independent constants. Here Xt = log(St ) is the logarithm of the implied/nominal/real inﬂation St .How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
6. 6. The Problem Univariate Analysis Multivariate Analysis ConclusionOrstein-Uhlenbeck processEuler Maruyama method Euler Maruyama method The Euler Maruyama method is a method for the approximate numerical solution of a stochastic diﬀerential equation. In our case, for a partition of [t, t + 1] in n equal subintervals: Xn+1 = Xn + θ(µ − Xn )δ + σ∆Wn Where δ = 1/N is the length of the subintervals, and ∆Wn are independent identically distributed random varibles with expected value of 0 and a variance of δ.How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
7. 7. The Problem Univariate Analysis Multivariate Analysis ConclusionOrstein-Uhlenbeck processEmpirical Distribution for 1 Step PredictionHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
8. 8. The Problem Univariate Analysis Multivariate Analysis ConclusionAR(p)Autoregressive model Autoregressive model The autoregressive model of order p, AR(p), is deﬁned as: p Yt = a0 + ai Yt−i + εt i=1 Where a0 , a1 , . . . , ap are the parameters of the model and εt is independent identically distributed random variables. Here Yt = St − St−1 is the backward diﬀerence of the implied/nominal/real inﬂation St .How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
9. 9. The Problem Univariate Analysis Multivariate Analysis ConclusionAR(p)Autocorrelation Function of the Implied InﬂationHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
10. 10. The Problem Univariate Analysis Multivariate Analysis ConclusionAR(p)ForecastHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
11. 11. The Problem Univariate Analysis Multivariate Analysis ConclusionAR(p)Evolution of Probability DistributionsHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
12. 12. The Problem Univariate Analysis Multivariate Analysis ConclusionConﬁdence BandsConﬁdence Band (close up)How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
13. 13. The Problem Univariate Analysis Multivariate Analysis ConclusionConﬁdence BandsConﬁdence Band (all view)How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
14. 14. The Problem Univariate Analysis Multivariate Analysis ConclusionConﬁdence BandsConﬁdence Band (2 Years Data)How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
15. 15. The Problem Univariate Analysis Multivariate Analysis ConclusionLinear RegressionCorrelation between Time SeriesHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
16. 16. The Problem Univariate Analysis Multivariate Analysis ConclusionLinear Regression Linear Regression The multivariate regression model is: Y = XT β + ε E(Y) = XT β ΣY = σ 2 1 Where Y are the response variables and X are the explanatory variables.How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
17. 17. The Problem Univariate Analysis Multivariate Analysis ConclusionLinear RegressionLinear Regression PredictionHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
18. 18. The Problem Univariate Analysis Multivariate Analysis ConclusionLinear RegressionError in the PredictionHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
19. 19. The Problem Univariate Analysis Multivariate Analysis ConclusionLinear RegressionConﬁdence BandHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
20. 20. The Problem Univariate Analysis Multivariate Analysis ConclusionConclusionFinal Remarks • Summary: Conﬁdence Band and Spread control Implied inﬂation, Real and Nominal seem to be correlatedHow mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry