how mathematicians predict the future?
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The Problem Univariate Analysis Multivariate Analysis Conclusion
How mathematicians predict the future?
Instructor: Agnieszka Wy lomanska
Costanza Catalano, Angela Ciliberti, Gonçalo S. Matos, Allan S. Nielsen,Olga Polikarpova, Mattia Zanella
European Summer School in Industrial MathematicsModelling Week
July 30, 2011
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Supplied Data for Analysis
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Supplied Data for Analysis
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Our Approach
Univariate AnalysisOrstein-Unlenbeck ModelAutoregressive Model
Multivariate AnalysisLinear Regression
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Orstein-Uhlenbeck process
Orstein-Uhlenbeck process
Orstein-Uhlenbeck processThe Orstein-Uhlenbeck process (or mean-reverting process) isdefined 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/realinflation St .
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Orstein-Uhlenbeck process
Euler Maruyama method
Euler Maruyama methodThe Euler Maruyama method is a method for the approximatenumerical solution of a stochastic differential equation. In ourcase, 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 areindependent identically distributed random varibles with expectedvalue of 0 and a variance of δ.
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Orstein-Uhlenbeck process
Empirical Distribution for 1 Step Prediction
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Autoregressive model
Autoregressive modelThe autoregressive model of order p, AR(p), is defined as:
Yt = a0 +p∑
i=1aiYt−i + εt
Where a0, a1, . . . , ap are the parameters of the model and εt isindependent identically distributed random variables.
Here Yt = St − St−1 is the backward difference of theimplied/nominal/real inflation St .
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Autocorrelation Function of the Implied Inflation
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Forecast
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Evolution of Probability Distributions
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Confidence Bands
Confidence Band (close up)
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Confidence Bands
Confidence Band (all view)
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Confidence Bands
Confidence Band (2 Years Data)
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Correlation between Time Series
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Linear RegressionThe multivariate regression model is:
Y = XTβ + ε
E(Y) = XTβ
ΣY = σ21
Where Y are the response variables and X are the explanatoryvariables.
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Linear Regression Prediction
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Error in the Prediction
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Confidence Band
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
The Problem Univariate Analysis Multivariate Analysis Conclusion
Conclusion
Final Remarks
• Summary:
? Confidence Band and Spread control? Implied inflation, Real and Nominal seem to be correlated
How mathematicians predict the future? ESSIMEuropean Consortium for Mathematics in Industry
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