how mathematicians predict the future?

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


Supplied Data for Analysis



Our Approach

Univariate AnalysisOrstein-Unlenbeck ModelAutoregressive Model

Multivariate AnalysisLinear Regression



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 .




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.




Empirical Distribution for 1 Step Prediction



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 .



AR(p)

Autocorrelation Function of the Implied Inflation



AR(p)

Forecast



AR(p)

Evolution of Probability Distributions



Confidence Bands

Confidence Band (close up)



Confidence Bands

Confidence Band (all view)



Confidence Bands

Confidence Band (2 Years Data)



Linear Regression

Correlation between Time Series



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.



Linear Regression

Linear Regression Prediction



Linear Regression

Error in the Prediction



Linear Regression

Confidence Band



Conclusion

Final Remarks

• Summary:

? Confidence Band and Spread control? Implied inflation, Real and Nominal seem to be correlated


how mathematicians predict the future?

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