A heuristic for moment-matching scenario generation

Transcription

1 Lukáš Adam / 21

2 Table of contents 1 Introduction 2 Basic algorithm 3 Modified algorithm 4 Results 2 / 21

3 Introduction Consider an optimization stochastic problem. To solve the problem one must 1 Convert the continuous normal distribution into a discrete one 2 Solve the resulting problem With increasing number of random variables the importance of the first part increases. We will be interested purely in the first part. 3 / 21

4 Goal Problem how to represent the random variable Especially with multidimensional distributions Goal: to generate from a joint distribution with specified values of the first four marginal moments and correlations Intention: to generate from one dimensional standard normal distribution and using an iterative procedure to achieve the goal This iterative procedure combines simulation, Cholesky decomposition and various transformations 4 / 21

5 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 higher than 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. 5 / 21

6 Cholesky decomposition Theorem Consider a symmetric real matrix R. Then R is positive definite if and only if there is a regular lower triangular matrix L such that R = LL T. Similarly, R is positive semidefinite if and only if the decomposition still holds true but L may have zeros on its main diagonal. 6 / 21

7 Correlation matrix assumptions 1 Correlation matrix R is possible, hence it is symmetric, positive semidefinite with ones on the main diagonal. Otherwise the Cholesky decomposition fails. If it is not satisfied, either check the input data or find a closest correlation 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 others and hence, the dimension of the problem may be reduced. 7 / 21

8 Transformations 1 Cubic transformation To generate univariate distributions with specific moments. 2 Matrix transformation To transform a multivariate distribution to obtain a given correlation matrix. Destroys higher order moments. 8 / 21

9 Cubic transformation Y i = a + bx i + cxi 2 + dxi 3 EY i = a + bex i + cexi 2 + dexi 3 EYi 2 = a d 2 EXi 6... EYi 4 = a d 4 EXi 12 If all the moments of X and Y are known, the system may be solved for (a, b, c, d). It may happen that the system has no solution, in such a case minimize the distance of the discrepancies. 9 / 21

10 Summary 1 Take some random variable X i with the same number of outcomes as Y i 2 Calculate the first 12 moments of X i 3 Compute the parameters a, b, c, d 4 Compute the outcomes as Y i = a + bx i + cxi 2 + dxi 3 10 / 21

11 Matrix transformation Y = LX with L being a lower triangular matrix. For X i assume zero means and variances equal to one. This implies that Y i has zero means, variances equal to one and Y has correlation matrix R. Higher order moments EY 3 i = EY 4 i 3 = i j=1 i j=1 L 3 ijex 3 j L 4 ij(ex 4 j 3) 11 / 21

12 Inverse transformation EX 4 i EX 3 i = 1 L 3 (EYi 3 ii 3 = 1 L 4 (EYi 4 ii i 1 j=1 L 3 ijex 3 j ) i 1 3 j=1 L 4 ij(ex 4 j 3)) As L ii > 0, the inverse transformation is correctly defined. 12 / 21

13 Input phase Goal: generate a dicrete approximation of Z with moments TARMOM and correlation matrix R. Matrix transformation needs zero means and variances equal to 1. Instead of Z generate Y with moments α = TARMOM β = TARMOM 1 and set Z = αy + β. MOM 3 = TARMOM 3 α 3 MOM 4 = TARMOM 4 α 4 13 / 21

14 Derive moments of independent univariate random variables X i such that 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. 14 / 21

15 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 to obtain the transformed moments for X i. 2 Transform Y = LX to obtain moments MOM and correlations R. 3 Transform Z = αy + β to obtain target moments TARMOM. 15 / 21

16 Problems Sample correlation of X is not equal precisely to I (the scenario number s would have to be high enough). Matrix transformation Y = LX destroys third and fourth moments. 16 / 21

17 Modification 1 1 Generate n univariate random variables with moments TRSFMOM and correlation R 1 close to I. Set p = 1 2 If d(r p, I ) ε x, stop with X p and Xp 1. Otherwise continue to the next step. 3 Do Cholesky decomposition R p = L p L T p and backward transformation Xp = L 1 p X p, which has zero correlations but wrong moments. 4 Do cubic transformation with TRSFMOM to obtain X p+1 with right moments and wrong correlations. 5 Increase p, compute R p and return to step / 21

18 Modification 2 1 Set Y 1 = LX with both moments and correlations incorrect (due to moments different from zero). Set p = 1 and compute R 1 the correlation matrix of Y 1. 2 If d(r p, R) ε y, stop with Y p. Otherwise continue to the next step. 3 Do Cholesky decomposition R p = L p L T p and backward transformation Yp = L 1 p Y p, which has zero correlations but wrong moments. 4 Do forward transform Yp = LYp to obtain Yp with correct correlation but incorrect moments. 5 Do cubic transformation with MOM to obtain Y p+1 with right moments and wrong correlations. 6 Increase p, compute R p and return to step / 21

19 Outputs Two possible outputs Y Y p with correct moments but incorrect correlation with d(r p, R) ε y. Y p with incorrect moments but correct correlation. Perform linear transformation Z = αy + β to reflect the original data. 19 / 21

20 Convergence No convergence proof provided. On the other hand used for more than two years in Gjensidige Nor Asset 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 first modification). 20 / 21

21 Numerical results Very good. The generation of 1000 scenarios with 20 random variables took less than one minute (Pentium III). The running time may decrease with increased number of scenarios (better convergence for more scenarios). 21 / 21