luk a s adam - univerzita karlovamsekce.karlin.mff.cuni.cz/~vorisek/seminar/1314z/1314z_adam.pdf ·...

Introduction Basic algorithm Modified algorithm Results

A heuristic for moment-matching scenario generation

Lukas Adam

4. 11. 2013

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Table of contents

1 Introduction

2 Basic algorithm

3 Modified algorithm

4 Results

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Introduction

Consider an optimization stochastic problem. To solve the problemone must

1 Convert the continuous normal distribution into a discrete one2 Solve the resulting problem

With increasing number of random variables the importance of thefirst part increases.

We will be interested purely in the first part.

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Goal

Problem how to represent the random variable

Especially with multidimensional distributions

Goal: to generate from a joint distribution with specified values ofthe first four marginal moments and correlations

Intention: to generate from one–dimensional standard normaldistribution and using an iterative procedure to achieve the goal

This iterative procedure combines simulation, Choleskydecomposition and various transformations

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General idea

Generate n discrete univariate random variables from N(0, 1).

Perform the cubic transformation to reach the prescribed moments.

Transform them so that the correlation is satisfied.

This transformation will distort the marginal moments of higherthan second order.

Start with different higher moments and repeat.

Produces exact results only if random variables are independent.Instead of it, a possible outcome error is allowed.

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Cholesky decomposition

Theorem

Consider a symmetric real matrix R. Then R is positive definite ifand only if there is a regular lower triangular matrix L such thatR = LLT .Similarly, R is positive semidefinite if and only if the decompositionstill holds true but L may have zeros on its main diagonal.

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Correlation matrix assumptions

1 Correlation matrix R is possible, hence it is symmetric, positivesemidefinite with ones on the main diagonal.

Otherwise the Cholesky decomposition fails.If it is not satisfied, either check the input data or find a closestcorrelation matrix.

2 R is positive definite.

It is again checked by the Cholesky decomposition.If this is not the case, some variables can be computed from othersand hence, the dimension of the problem may be reduced.

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Transformations

1 Cubic transformation

To generate univariate distributions with specific moments.

2 Matrix transformation

To transform a multivariate distribution to obtain a givencorrelation matrix.Destroys higher order moments.

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Cubic transformation

Yi = a + bXi + cX 2i + dX 3

i

EYi = a + bEXi + cEX 2i + dEX 3

i

EY 2i = a2 + · · · + d2EX 6

i

. . .

EY 4i = a4 + · · · + d4EX 12

i

If all the moments of X and Y are known, the system may besolved for (a, b, c , d).

It may happen that the system has no solution, in such a caseminimize the distance of the discrepancies.

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Summary

1 Take some random variable Xi with the same number of outcomesas Yi

2 Calculate the first 12 moments of Xi

3 Compute the parameters a, b, c, d4 Compute the outcomes as Yi = a + bXi + cX 2

i + dX 3i

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Matrix transformation

Y = LX with L being a lower triangular matrix.

For Xi assume zero means and variances equal to one. This impliesthat Yi has zero means, variances equal to one and Y hascorrelation matrix R.

Higher order moments

EY 3i =

i∑j=1

L3ijEX 3

j

EY 4i − 3 =

i∑j=1

L4ij(EX 4

j − 3)

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Inverse transformation

EX 3i =

1

L3ii

(EY 3i −

i−1∑j=1

L3ijEX 3

j )

EX 4i − 3 =

1

L4ii

(EY 4i − 3 −

i−1∑j=1

L4ij(EX 4

j − 3))

As Lii > 0, the inverse transformation is correctly defined.

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Input phase

Goal: generate a dicrete approximation of Z with momentsTARMOM and correlation matrix R.

Matrix transformation needs zero means and variances equal to 1.

Instead of Z generate Y with moments

α = TARMOM122

β = TARMOM1

MOM3 =TARMOM3

α3

MOM4 =TARMOM4

α4

and set Z = αY + β.

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Derive moments of independent univariate random variables Xi suchthat Y = LX will have the target moments and correlations.

Summary

1 Specify TARMOM and R for Z .2 Find the normalized moments MOM for Y .3 Find the transformed moments TRSFMOM for X .

Fast phase: does not depend on number of scenarios s.

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Output phase

Repeat the procedure from input phase to generate s scenarios.

Summary

1 Generate n times from N(0, 1) and use the cubic transformation toobtain the transformed moments for Xi .

2 Transform Y = LX to obtain moments MOM and correlations R.3 Transform Z = αY + β to obtain target moments TARMOM.

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Problems

Sample correlation of X is not equal precisely to I (the scenarionumber s would have to be high enough).

Matrix transformation Y = LX destroys third and fourth moments.

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Modification 1

1 Generate n univariate random variables with moments TRSFMOMand correlation R1 close to I . Set p = 1

2 If d(Rp, I ) ≤ εx , stop with Xp and X ∗p−1. Otherwise continue to the

next step.

3 Do Cholesky decomposition Rp = LpLTp and backward

transformation X ∗p = L−1

p Xp, which has zero correlations but wrongmoments.

4 Do cubic transformation with TRSFMOM to obtain Xp+1 with rightmoments and wrong correlations.

5 Increase p, compute Rp and return to step 2.

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Modification 2

1 Set Y1 = LX ∗ with both moments and correlations incorrect (dueto moments different from zero). Set p = 1 and compute R1 thecorrelation matrix of Y1.

2 If d(Rp,R) ≤ εy , stop with Yp. Otherwise continue to the next step.

3 Do Cholesky decomposition Rp = LpLTp and backward

transformation Y ∗p = L−1

p Yp, which has zero correlations but wrongmoments.

4 Do forward transform Y ∗∗p = LY ∗

p to obtain Y ∗∗p with correct

correlation but incorrect moments.

5 Do cubic transformation with MOM to obtain Yp+1 with rightmoments and wrong correlations.

6 Increase p, compute Rp and return to step 2.

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Outputs

Two possible outputs Y

Yp with correct moments but incorrect correlation withd(Rp,R) ≤ εy .

Y ∗∗p with incorrect moments but correct correlation.

Perform linear transformation Z = αY + β to reflect the originaldata.

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Convergence

No convergence proof provided.

On the other hand used for more than two years in Gjensidige NorAsset Management.

The results may not exist but they cannot be bad.

Fix to this

Rerun the algorithm.Check data consistency (zero variance, positive skewness).Increase the number of scenarios (improves the quality of the firstmodification).

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Numerical results

Very good.

The generation of 1000 scenarios with 20 random variables took lessthan one minute (Pentium III).

The running time may decrease with increased number of scenarios(better convergence for more scenarios).

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luk a s adam - univerzita karlovamsekce.karlin.mff.cuni.cz/~vorisek/seminar/1314z/1314z_adam.pdf ·...

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