Computes a suite of diagnostic values for an MLMC estimation problem.
Usage
mlmc.test(
mlmc_l,
N,
L,
N0,
eps.v,
Lmin,
Lmax,
alpha = NA,
beta = NA,
gamma = NA,
parallel = NA,
silent = FALSE,
...
)Arguments
- mlmc_l
a user supplied function which provides the estimate for level \(l\). It must take at least two arguments, the first is the level number to be simulated and the second the number of paths. Additional arguments can be taken if desired: all additional
...arguments to this function are forwarded to the user definedmlmc_lfunction.The user supplied function should return a named list containing one element named
sumsand second namedcost, where:sumsis a vector of length six \(\left(\sum Y_i, \sum Y_i^2, \sum Y_i^3, \sum Y_i^4, \sum X_i, \sum X_i^2\right)\) where \(Y_i\) are iid simulations with expectation \(E[P_0]\) when \(l=0\) and expectation \(E[P_l-P_{l-1}]\) when \(l>0\), and \(X_i\) are iid simulations with expectation \(E[P_l]\). Note that this differs from the main
mlmc()driver, which only requires the first two of these elements in order to calculate the estimate. The remaining elements are required bymlmc.test()since they are used for convergence tests, kurtosis, and telescoping sum checks.costis a scalar with the total cost of the paths simulated. For example, in the financial options samplers included in this package, this is calculated as \(NM^l\), where \(N\) is the number of paths requested in the call to the user function
mlmc_l, \(M\) is the refinement cost factor (\(M=2\) formcqmc06_l()and \(M=4\) foropre_l()), and \(l\) is the level being sampled.
See the function (and source code of)
opre_l()andmcqmc06_l()in this package for an example of user supplied level samplers.- N
number of samples to use in convergence tests, kurtosis, telescoping sum check.
- L
number of levels to use in convergence tests, kurtosis, telescoping sum check.
- N0
initial number of samples which are used for the first 3 levels and for any subsequent levels which are automatically added in the complexity tests. Must be \(> 0\).
- eps.v
a vector of one or more target accuracies for the complexity tests. Must all be \(> 0\).
- Lmin
the minimum level of refinement for complexity tests. Must be \(\ge 2\).
- Lmax
the maximum level of refinement for complexity tests. Must be \(\ge\)
Lmin.- alpha
the weak error, \(O(2^{-\alpha l})\). Must be \(> 0\) if specified. If
NAthenalphawill be estimated.- beta
the variance, \(O(2^{-\beta l})\). Must be \(> 0\) if specified. If
NAthenbetawill be estimated.- gamma
the sample cost, \(O(2^{\gamma l})\). Must be \(> 0\) if specified. If
NAthengammawill be estimated.- parallel
if an integer is supplied, R will fork
parallelparallel processes. This is done for the convergence tests section by splitting theNsamples as evenly as possible across cores when sampling each level. This is also done for the MLMC complexity tests by passing theparallelargument on to themlmc()driver when targeting each accuracy level ineps.- silent
set to TRUE to supress running output (identical output can still be printed by printing the return result)
- ...
additional arguments which are passed on when the user supplied
mlmc_lfunction is called
Value
An mlmc.test object which contains all the computed diagnostic values.
This object can be printed or plotted (see plot.mlmc.test).
Details
See one of the example level sampler functions (e.g. opre_l()) for example usage.
This function is based on GPL-2 'Matlab' code by Mike Giles.
Author
Louis Aslett <louis.aslett@durham.ac.uk>
Mike Giles <Mike.Giles@maths.ox.ac.uk>
Tigran Nagapetyan <nagapetyan@stats.ox.ac.uk>
Examples
# \donttest{
# Example calls with realistic arguments
# Financial options using an Euler-Maruyama discretisation
tst <- mlmc.test(opre_l, N = 2000000,
L = 5, N0 = 1000,
eps.v = c(0.005, 0.01, 0.02, 0.05, 0.1),
Lmin = 2, Lmax = 6,
option = 1)
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 2e+06 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0209e+01 1.0209e+01 1.6139e+02 1.6139e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 2.0971e-01 1.0420e+01 4.4439e+00 2.0102e+02 2.0055e+01 2.9916e-02 4.0000e+00
#> 2 2.9164e-02 1.0443e+01 1.0641e+00 2.1243e+02 1.2387e+01 9.3236e-02 1.6000e+01
#> 3 5.8030e-03 1.0457e+01 2.7256e-01 2.1538e+02 7.5182e+00 1.1818e-01 6.4000e+01
#> 4 1.6226e-03 1.0450e+01 6.8529e-02 2.1643e+02 6.2241e+00 1.3021e-01 2.5600e+02
#> 5 5.6870e-04 1.0456e+01 1.7268e-02 2.1633e+02 5.9348e+00 8.0291e-02 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 2.122090 (exponent for MLMC weak convergence)
#> beta = 1.997195 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.0449e+01 4.625e+07 2.955e+09 63.88 19941583 1650853 405719 103214 25829
#> 0.0100 1.0454e+01 8.433e+06 1.838e+08 21.80 4257398 354804 85692 21643
#> 0.0200 1.0449e+01 2.127e+06 4.595e+07 21.60 1068793 88086 22057 5520
#> 0.0500 1.0440e+01 2.519e+05 1.813e+06 7.20 159399 11284 2959
#> 0.1000 1.0495e+01 6.324e+04 4.532e+05 7.17 35177 3016 1000
#>
tst
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 2e+06 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0209e+01 1.0209e+01 1.6139e+02 1.6139e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 2.0971e-01 1.0420e+01 4.4439e+00 2.0102e+02 2.0055e+01 2.9916e-02 4.0000e+00
#> 2 2.9164e-02 1.0443e+01 1.0641e+00 2.1243e+02 1.2387e+01 9.3236e-02 1.6000e+01
#> 3 5.8030e-03 1.0457e+01 2.7256e-01 2.1538e+02 7.5182e+00 1.1818e-01 6.4000e+01
#> 4 1.6226e-03 1.0450e+01 6.8529e-02 2.1643e+02 6.2241e+00 1.3021e-01 2.5600e+02
#> 5 5.6870e-04 1.0456e+01 1.7268e-02 2.1633e+02 5.9348e+00 8.0291e-02 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 2.122090 (exponent for MLMC weak convergence)
#> beta = 1.997195 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.0449e+01 4.625e+07 2.955e+09 63.88 19941583 1650853 405719 103214 25829
#> 0.0100 1.0454e+01 8.433e+06 1.838e+08 21.80 4257398 354804 85692 21643
#> 0.0200 1.0449e+01 2.127e+06 4.595e+07 21.60 1068793 88086 22057 5520
#> 0.0500 1.0440e+01 2.519e+05 1.813e+06 7.20 159399 11284 2959
#> 0.1000 1.0495e+01 6.324e+04 4.532e+05 7.17 35177 3016 1000
#>
plot(tst)
# Financial options using a Milstein discretisation
tst <- mlmc.test(mcqmc06_l, N = 20000,
L = 8, N0 = 200,
eps.v = c(0.005, 0.01, 0.02, 0.05, 0.1),
Lmin = 2, Lmax = 10,
option = 1)
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 20000 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0133e+01 1.0133e+01 1.9810e+02 1.9810e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 1.8339e-01 1.0238e+01 1.4650e-01 2.0666e+02 4.3668e+01 1.2791e-01 2.0000e+00
#> 2 1.0178e-01 1.0207e+01 4.1696e-02 2.0649e+02 3.1277e+01 2.1643e-01 4.0000e+00
#> 3 5.4679e-02 1.0390e+01 1.2096e-02 2.1192e+02 2.2094e+01 2.0862e-01 8.0000e+00
#> 4 2.7491e-02 1.0312e+01 3.0769e-03 2.1030e+02 1.2999e+01 1.7120e-01 1.6000e+01
#> 5 1.3946e-02 1.0439e+01 8.5509e-04 2.2017e+02 1.3315e+01 1.8132e-01 3.2000e+01
#> 6 6.9500e-03 1.0260e+01 2.1296e-04 2.1398e+02 1.1626e+01 2.9709e-01 6.4000e+01
#> 7 3.5764e-03 1.0542e+01 5.3863e-05 2.1728e+02 9.9385e+00 4.4704e-01 1.2800e+02
#> 8 1.8070e-03 1.0587e+01 1.3390e-05 2.1707e+02 9.4222e+00 6.9319e-02 2.5600e+02
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.960913 (exponent for MLMC weak convergence)
#> beta = 1.919459 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.0447e+01 1.355e+07 2.964e+09 218.65 11903219 229699 88580 33052 12456 4446 1600 569 220
#> 0.0100 1.0438e+01 3.363e+06 3.708e+08 110.27 2957414 56143 25072 8068 2875 1106 426 155
#> 0.0200 1.0468e+01 9.014e+05 4.565e+07 50.64 795790 15288 8075 2046 735 266 96
#> 0.0500 1.0432e+01 1.313e+05 3.758e+06 28.62 116815 2610 840 297 127 46
#> 0.1000 1.0251e+01 3.492e+04 4.486e+05 12.85 31940 613 200 65 27
#>
tst
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 20000 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0133e+01 1.0133e+01 1.9810e+02 1.9810e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 1.8339e-01 1.0238e+01 1.4650e-01 2.0666e+02 4.3668e+01 1.2791e-01 2.0000e+00
#> 2 1.0178e-01 1.0207e+01 4.1696e-02 2.0649e+02 3.1277e+01 2.1643e-01 4.0000e+00
#> 3 5.4679e-02 1.0390e+01 1.2096e-02 2.1192e+02 2.2094e+01 2.0862e-01 8.0000e+00
#> 4 2.7491e-02 1.0312e+01 3.0769e-03 2.1030e+02 1.2999e+01 1.7120e-01 1.6000e+01
#> 5 1.3946e-02 1.0439e+01 8.5509e-04 2.2017e+02 1.3315e+01 1.8132e-01 3.2000e+01
#> 6 6.9500e-03 1.0260e+01 2.1296e-04 2.1398e+02 1.1626e+01 2.9709e-01 6.4000e+01
#> 7 3.5764e-03 1.0542e+01 5.3863e-05 2.1728e+02 9.9385e+00 4.4704e-01 1.2800e+02
#> 8 1.8070e-03 1.0587e+01 1.3390e-05 2.1707e+02 9.4222e+00 6.9319e-02 2.5600e+02
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.960913 (exponent for MLMC weak convergence)
#> beta = 1.919459 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.0447e+01 1.355e+07 2.964e+09 218.65 11903219 229699 88580 33052 12456 4446 1600 569 220
#> 0.0100 1.0438e+01 3.363e+06 3.708e+08 110.27 2957414 56143 25072 8068 2875 1106 426 155
#> 0.0200 1.0468e+01 9.014e+05 4.565e+07 50.64 795790 15288 8075 2046 735 266 96
#> 0.0500 1.0432e+01 1.313e+05 3.758e+06 28.62 116815 2610 840 297 127 46
#> 0.1000 1.0251e+01 3.492e+04 4.486e+05 12.85 31940 613 200 65 27
#>
plot(tst)
# }
# Toy versions for CRAN tests
tst <- mlmc.test(opre_l, N = 10000,
L = 5, N0 = 1000,
eps.v = c(0.025, 0.1),
Lmin = 2, Lmax = 6,
option = 1)
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 10000 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0144e+01 1.0144e+01 1.5933e+02 1.5933e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 2.0461e-01 1.0494e+01 4.3545e+00 2.0017e+02 1.6552e+01 1.6836e-01 4.0000e+00
#> 2 3.4172e-02 1.0188e+01 1.0920e+00 2.0313e+02 1.3486e+01 3.8611e-01 1.6000e+01
#> 3 2.7394e-03 1.0609e+01 2.7241e-01 2.1473e+02 8.9020e+00 4.7445e-01 6.4000e+01
#> 4 4.6225e-03 1.0487e+01 7.0387e-02 2.2204e+02 6.2642e+00 1.4175e-01 2.5600e+02
#> 5 2.3568e-03 1.0375e+01 1.6529e-02 2.1034e+02 5.6902e+00 1.2895e-01 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 1.576591 (exponent for MLMC weak convergence)
#> beta = 2.003820 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0250 1.0442e+01 1.352e+06 2.932e+07 21.68 682155 56440 13888 3468
#> 0.1000 1.0326e+01 6.651e+04 4.333e+05 6.52 38193 3079 1000
#>
tst <- mlmc.test(mcqmc06_l, N = 10000,
L = 8, N0 = 1000,
eps.v = c(0.025, 0.1),
Lmin = 2, Lmax = 10,
option = 1)
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 10000 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 9.7083e+00 9.7083e+00 1.8872e+02 1.8872e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 1.8889e-01 1.0493e+01 1.5689e-01 2.1033e+02 4.2469e+01 6.9418e-01 2.0000e+00
#> 2 1.0293e-01 1.0345e+01 4.1077e-02 2.1021e+02 2.5473e+01 2.8726e-01 4.0000e+00
#> 3 5.2791e-02 1.0138e+01 1.1696e-02 2.1027e+02 2.0845e+01 2.9683e-01 8.0000e+00
#> 4 2.7545e-02 1.0462e+01 3.1582e-03 2.1956e+02 1.3111e+01 3.3625e-01 1.6000e+01
#> 5 1.4108e-02 1.0503e+01 7.9526e-04 2.1093e+02 1.2555e+01 2.9865e-02 3.2000e+01
#> 6 6.8579e-03 1.0170e+01 2.0129e-04 2.1102e+02 9.9169e+00 3.8931e-01 6.4000e+01
#> 7 3.5937e-03 1.0490e+01 5.4733e-05 2.1709e+02 1.2669e+01 3.6073e-01 1.2800e+02
#> 8 1.7843e-03 1.0185e+01 1.3757e-05 2.1383e+02 9.7348e+00 3.4885e-01 2.5600e+02
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.965258 (exponent for MLMC weak convergence)
#> beta = 1.924650 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0250 1.0446e+01 5.278e+05 2.881e+07 54.59 468086 9101 3436 1207 505 183 65
#> 0.1000 1.0334e+01 3.549e+04 4.684e+05 13.20 28562 1000 1000 76 20
#>