Go to the MLX, M, PDF, or HTML version of this file. Go back to fan’s MEconTools Toolbox (bookdown), Matlab Code Examples Repository (bookdown), or Math for Econ with Matlab Repository (bookdown).
Examples](https://fanwangecon.github.io/M4Econ/), or** Dynamic Asset This is the example vignette for function: ff_vfi_az_loop from the MEconTools Package. This function solves the dynamic programming problem for a (a,z) model. Households can save a, and face AR(1) shock z. The problem is solved over the infinite horizon.
This is the looped code, it is slow for larger state-space problems. The code uses common grid, with the same state space and choice space grids.
Links to Other Code:
Core Savings/Borrowing Dynamic Programming Solution Functions that are functions in the MEconTools Package. :
Common Choice and States Grid Loop: ff_vfi_az_loop
Common Choice and States Grid Vectorized: ff_vfi_az_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC, Loop:ff_vfi_az_bisec_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, rely on FOC Vectorized: ff_vfi_az_bisec_vec
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Loop:ff_vfi_az_mzoom_loop
States Grid + Continuous Exact Savings as Share of Cash-on-Hand, VALUE comparison, Vectorized: ff_vfi_az_mzoom_vec
The sample codes are written for the standard dynamic savings problem. The code can be adapted for multiple assets, savings and borrowing, discrete and continuous choice, etc. A large proportion of dynamic economic models are based on the underlying structure of solving a model with endogenous states and exogenous shocks, and that is what the (a,z) model does. In general, one can write looped code first to make sure the economics is correct, then vectorized code can be adopted to increase speed.
Call the function with defaults. By default, shows the asset policy function summary. Model parameters can be changed by the mp_params.
%mp_params
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('fl_crra') = 1.5;
mp_params('fl_beta') = 0.94;
% call function
ff_vfi_az_loop(mp_params);
Elapsed time is 2.378952 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ ___
ap 1 1 2 700 100 7 9855.1 14.079 14.408 1.0234 0 50
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
______ ______ ______ ________ _______ _______ ______
r1 0 0 0 0.045213 0.25576 0.61095 1.0362
r2 0 0 0 0.045213 0.25576 0.61095 1.0362
r3 0 0 0 0.045213 0.25576 0.61095 1.0362
r4 0 0 0 0.06647 0.25576 0.61095 1.0362
r5 0 0 0 0.06647 0.25576 0.61095 1.164
r96 43.924 43.924 43.924 43.924 43.924 45.102 45.102
r97 45.102 45.102 45.102 45.102 45.102 46.298 46.298
r98 46.298 46.298 46.298 46.298 46.298 47.513 47.513
r99 47.513 47.513 47.513 47.513 47.513 48.747 48.747
r100 48.747 48.747 48.747 48.747 48.747 50 50
Call the function with different a and z grid size, print out speed:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('ls_ffcmd') = {};
% A grid 50, shock grid 5:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.715890 seconds.
% A grid 750, shock grid 15:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 750;
mp_params('it_z_n') = 15;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 300.576571 seconds.
% A grid 600, shock grid 45:
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 600;
mp_params('it_z_n') = 45;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 910.111661 seconds.
Run the function first without any outputs, but only the timer.
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 50;
mp_params('it_z_n') = 5;
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_timer') = true;
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {};
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.400105 seconds.
Run the function and show policy function for savings choice. For ls_ffcmd, ls_ffsna, ls_ffgrh, can include these: ‘v’, ‘ap’, ‘c’, ‘y’, ‘coh’, ‘savefraccoh’. These are value, aprime savings choice, consumption, income, cash on hand, and savings fraction as cash-on-hand.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
% ls_ffcmd: summary print which outcomes
mp_support('ls_ffcmd') = {};
% ls_ffsna: detail print which outcomes
mp_support('ls_ffsna') = {'savefraccoh'};
% ls_ffgrh: graphical print which outcomes
mp_support('ls_ffgrh') = {'savefraccoh'};
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.410866 seconds.
xxx ff_vfi_az_vec, outcome=savefraccoh xxxxxxxxxxxxxxxxxxxxxxxxxxx
group a mean_z_0_4858 mean_z_0_67798 mean_z_0_9462 mean_z_1_3205 mean_z_1_8429
_____ ________ _____________ ______________ _____________ _____________ _____________
1 0 0 0 0.071865 0.20862 0.36462
2 0.002975 0 0 0.071698 0.20827 0.36418
3 0.016829 0 0 0.070928 0.20666 0.36216
4 0.046375 0 0.0029827 0.069341 0.20331 0.35793
5 0.095198 0.0038183 0.044243 0.11681 0.27649 0.35114
6 0.1663 0.054362 0.084837 0.17517 0.26637 0.34171
7 0.26234 0.099899 0.13609 0.16422 0.25383 0.41847
8 0.38568 0.15381 0.19428 0.22348 0.32132 0.40047
9 0.53852 0.21153 0.25554 0.28573 0.39055 0.47258
10 0.72291 0.26934 0.31659 0.34814 0.36175 0.44538
11 0.94076 0.3247 0.37504 0.40848 0.42229 0.50941
12 1.1939 0.37617 0.42941 0.46521 0.4802 0.57087
13 1.484 0.53695 0.47898 0.51743 0.5344 0.5291
14 1.8128 0.57847 0.52356 0.56473 0.58429 0.58056
15 2.1817 0.61468 0.56329 0.6071 0.62958 0.62823
16 2.5924 0.6462 0.5985 0.64475 0.67028 0.67186
17 3.0463 0.67365 0.62963 0.67804 0.60721 0.71141
18 3.5449 0.69762 0.65713 0.70737 0.6404 0.65255
19 4.0894 0.71859 0.68142 0.73318 0.67021 0.68509
20 4.6813 0.73701 0.70293 0.75587 0.6969 0.71446
21 5.3218 0.75325 0.722 0.77584 0.72078 0.74089
22 6.0121 0.76763 0.73895 0.79344 0.74211 0.76461
23 6.7536 0.7804 0.75407 0.80897 0.76119 0.78587
24 7.5474 0.7918 0.76759 0.8227 0.77824 0.80491
25 8.3948 0.80201 0.77972 0.83486 0.79351 0.82194
26 9.2967 0.81119 0.79063 0.84567 0.80719 0.83719
27 10.254 0.81947 0.80049 0.8553 0.81948 0.85083
28 11.269 0.82697 0.80941 0.86389 0.83053 0.86306
29 12.342 0.83379 0.81752 0.87159 0.84048 0.87401
30 13.473 0.84001 0.8249 0.87849 0.84946 0.88384
31 14.665 0.84569 0.83165 0.8847 0.85759 0.82241
32 15.918 0.8509 0.83782 0.8903 0.86495 0.83188
33 17.233 0.8557 0.8435 0.89536 0.87163 0.84053
34 18.611 0.86012 0.84872 0.89995 0.8777 0.84844
35 20.053 0.86421 0.85354 0.90411 0.88324 0.85568
36 21.56 0.86799 0.858 0.9079 0.8883 0.86231
37 23.133 0.87151 0.86214 0.91136 0.89292 0.86841
38 24.773 0.87479 0.86598 0.91452 0.89716 0.87401
39 26.481 0.87784 0.86955 0.91741 0.90105 0.87917
40 28.258 0.8807 0.87289 0.92007 0.90463 0.88393
41 30.104 0.88337 0.87601 0.92251 0.90793 0.88833
42 32.021 0.88588 0.87893 0.92475 0.91097 0.8924
43 34.01 0.88824 0.88166 0.92683 0.91378 0.89617
44 36.07 0.89046 0.88423 0.92874 0.91638 0.89966
45 38.204 0.89256 0.88665 0.93052 0.91879 0.90291
46 40.412 0.89453 0.9403 0.93216 0.92102 0.90592
47 42.695 0.8964 0.94141 0.93368 0.9231 0.90873
48 45.053 0.89817 0.94245 0.9351 0.92504 0.91135
49 47.488 0.89985 0.94341 0.93642 0.92684 0.9138
50 50 0.90144 0.9443 0.93765 0.92853 0.91608
Run the function and show summaries for savings and fraction of coh saved:
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 9;
mp_support('ls_ffcmd') = {'ap', 'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_support('bl_vfi_store_all') = true; % store c(a,z), y(a,z)
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 3.281815 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
ap 1 1 2 900 100 9 12904 14.338 14.524 1.013 0 50
savefraccoh 2 2 2 900 100 9 619.51 0.68834 0.26953 0.39157 0 0.93151
xxx TABLE:ap xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
______ ______ ______ __________ ________ _______ _______ ______ ______
r1 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r2 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r3 0 0 0 0 0.092813 0.25576 0.61095 1.0362 1.6023
r4 0 0 0 0.00051272 0.092813 0.25576 0.61095 1.0362 1.6023
r5 0 0 0 0.0029004 0.092813 0.25576 0.61095 1.0362 1.6023
r96 43.924 43.924 43.924 43.924 43.924 45.102 45.102 45.102 46.298
r97 45.102 45.102 45.102 45.102 45.102 46.298 46.298 46.298 47.513
r98 46.298 46.298 46.298 46.298 46.298 47.513 47.513 47.513 48.747
r99 47.513 47.513 47.513 47.513 47.513 48.747 48.747 48.747 50
r100 48.747 48.747 48.747 48.747 48.747 50 50 50 50
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7 c8 c9
_______ _______ _______ __________ ________ _______ _______ _______ _______
r1 0 0 0 0 0.070073 0.15255 0.28789 0.38573 0.47121
r2 0 0 0 0 0.070045 0.1525 0.28781 0.38565 0.47114
r3 0 0 0 0 0.069914 0.15228 0.28748 0.3853 0.4708
r4 0 0 0 0.00048613 0.069636 0.1518 0.28676 0.38454 0.47007
r5 0 0 0 0.0027273 0.069182 0.15101 0.28559 0.38329 0.46886
r96 0.92625 0.92358 0.92022 0.916 0.91072 0.92836 0.91992 0.90945 0.92033
r97 0.92676 0.92416 0.92088 0.91677 0.91162 0.92918 0.92095 0.91073 0.92169
r98 0.92727 0.92473 0.92153 0.91752 0.91249 0.92998 0.92194 0.91196 0.923
r99 0.92776 0.92528 0.92216 0.91824 0.91333 0.93076 0.92291 0.91315 0.92426
r100 0.92823 0.92581 0.92277 0.91895 0.91416 0.93151 0.92384 0.91431 0.90252
Show only save fraction of cash on hand:
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';
Solve the model with several different interest rates and discount factor:
% Lower Savings Incentives
mp_params('fl_beta') = 0.80;
mp_params('fl_r') = 0.01;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 0.825240 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ ______ ______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 357.49 0.5107 0.2755 0.53945 0 0.80531
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _______ _______ __________ ________
r1 0 0 0 0 0 0.0002246 0.041573
r2 0 0 0 0 0 0.00022455 0.041566
r3 0 0 0 0 0 0.0012689 0.041533
r4 0 0 0 0 0 0.001266 0.041462
r5 0 0 0 0 0 0.0034759 0.041345
r96 0.78455 0.78145 0.79995 0.79456 0.7876 0.77865 0.76719
r97 0.78669 0.78366 0.77972 0.79679 0.78998 0.78122 0.77001
r98 0.78878 0.78582 0.78197 0.79897 0.79231 0.78374 0.77276
r99 0.79084 0.78794 0.78417 0.77927 0.79459 0.7862 0.77545
r100 0.79285 0.79001 0.78633 0.78154 0.79682 0.7886 0.77808
% Higher Savings Incentives
mp_params('fl_beta') = 0.95;
mp_params('fl_r') = 0.04;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 2.386791 seconds.
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 479.94 0.68563 0.27152 0.39602 0 0.93121
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ __________ ________ _______ _______ _______
r1 0 0 0 0.07007 0.17967 0.30874 0.43404
r2 0 0 0 0.070042 0.17961 0.30866 0.43396
r3 0 0 0 0.069911 0.17935 0.30833 0.4336
r4 0 0 0 0.069633 0.17881 0.30762 0.43284
r5 0 0 0.00049972 0.069179 0.17792 0.30645 0.43158
r96 0.92489 0.92134 0.91672 0.91072 0.92717 0.91691 0.92776
r97 0.92544 0.92198 0.91747 0.91162 0.92802 0.91801 0.92895
r98 0.92598 0.9226 0.9182 0.91249 0.92885 0.91908 0.9301
r99 0.9265 0.9232 0.91891 0.91333 0.92965 0.92011 0.93121
r100 0.927 0.92379 0.9196 0.91416 0.93042 0.9211 0.90914
Here, again, show fraction of coh saved in summary tabular form, but also show it graphically.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {'savefraccoh'};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 100;
mp_params('it_z_n') = 7;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';
Solve the model with different risk aversion levels, higher preferences for risk:
% Lower Risk Aversion
mp_params('fl_crra') = 0.5;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 1.327261 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ ______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 450.35 0.64336 0.2803 0.43568 0 0.90711
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ _______ _________ ________ _______ _______
r1 0 0 0 0.0060341 0.093241 0.19572 0.30604
r2 0 0 0 0.0060316 0.093213 0.19567 0.30599
r3 0 0 0 0.0060204 0.09308 0.19546 0.30574
r4 0 0 0 0.0059964 0.092798 0.19501 0.3052
r5 0 0 0 0.012229 0.092335 0.19427 0.30431
r96 0.90049 0.89703 0.89253 0.88669 0.90296 0.89297 0.90379
r97 0.90128 0.89791 0.89351 0.88781 0.90404 0.89429 0.88181
r98 0.90205 0.89876 0.89447 0.88891 0.9051 0.89557 0.88337
r99 0.9028 0.89959 0.89541 0.88998 0.90612 0.89681 0.88489
r100 0.90354 0.9004 0.89632 0.89101 0.90711 0.89802 0.88636
When risk aversion increases, at every state-space point, the household wants to save more.
% Higher Risk Aversion
mp_params('fl_crra') = 5;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 2.680109 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 700 100 7 500.59 0.71513 0.25488 0.35641 0 0.94324
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c6 c7
_______ _______ ________ _______ _______ _______ _______
r1 0 0 0.044811 0.15534 0.25694 0.40177 0.48276
r2 0 0 0.044787 0.15528 0.25686 0.40168 0.48268
r3 0 0 0.044678 0.15499 0.2565 0.40124 0.48228
r4 0 0 0.044445 0.15437 0.25572 0.40032 0.48143
r5 0 0 0.064784 0.15337 0.25445 0.39879 0.48003
r96 0.92489 0.92134 0.94129 0.93513 0.92717 0.91691 0.92776
r97 0.92544 0.92198 0.9418 0.9358 0.92802 0.91801 0.92895
r98 0.92598 0.9226 0.9423 0.93644 0.92885 0.91908 0.9301
r99 0.9265 0.9232 0.94278 0.93706 0.92965 0.92011 0.93121
r100 0.927 0.92379 0.94324 0.93765 0.93042 0.9211 0.90914
Increase the standard deviation of the Shock.
mp_support = containers.Map('KeyType','char', 'ValueType','any');
mp_support('bl_print_params') = false;
mp_support('bl_print_iterinfo') = false;
mp_support('ls_ffcmd') = {'savefraccoh'};
mp_support('ls_ffsna') = {};
mp_support('ls_ffgrh') = {};
mp_params = containers.Map('KeyType','char', 'ValueType','any');
mp_params('it_a_n') = 150;
mp_params('it_z_n') = 15;
mp_params('fl_a_max') = 50;
mp_params('st_grid_type') = 'grid_powerspace';
% graph color spectrum
mp_params('cl_colors') = 'copper';
Lower standard deviation of shock:
% Lower Risk Aversion
mp_params('fl_shk_std') = 0.10;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 13.492999 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
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i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1506.3 0.66947 0.28673 0.4283 0 0.93222
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.14061 0.1891 0.24154 0.2699 0.32439
r2 0 0 0 0 0 0.1406 0.18908 0.24152 0.26988 0.32437
r3 0 0 0 0 0 0.14053 0.189 0.24142 0.26977 0.32426
r4 0 0 0 0 0 0.14038 0.18881 0.2412 0.26956 0.32402
r5 0 0 0 0 0 0.14013 0.18851 0.24085 0.2692 0.32362
r146 0.93087 0.92957 0.92815 0.92661 0.92492 0.92712 0.92403 0.92069 0.91706 0.91312
r147 0.93121 0.92994 0.92854 0.92702 0.92537 0.92768 0.92465 0.92135 0.91778 0.91391
r148 0.93156 0.9303 0.92893 0.92743 0.92581 0.92823 0.92525 0.92201 0.91849 0.91467
r149 0.93189 0.93065 0.9293 0.92783 0.92623 0.92878 0.92584 0.92264 0.91918 0.91542
r150 0.93222 0.931 0.92967 0.92823 0.92665 0.9293 0.92641 0.92327 0.91986 0.91616
Higher shock standard deviation: low shock high asset save more, high shock more asset save less, high shock low asset save more:
% Higher Risk Aversion
mp_params('fl_shk_std') = 0.40;
ff_vfi_az_loop(mp_params, mp_support);
Elapsed time is 18.680264 seconds.
----------------------------------------
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CONTAINER NAME: mp_ffcmd ND Array (Matrix etc)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
i idx ndim numel rowN colN sum mean std coefvari min max
_ ___ ____ _____ ____ ____ ______ _______ _______ ________ ___ _______
savefraccoh 1 1 2 2250 150 15 1678.8 0.74614 0.22779 0.30529 0 0.93141
xxx TABLE:savefraccoh xxxxxxxxxxxxxxxxxx
c1 c2 c3 c4 c5 c11 c12 c13 c14 c15
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______
r1 0 0 0 0 0 0.53612 0.59853 0.67884 0.73891 0.77675
r2 0 0 0 0 0 0.53609 0.5985 0.67882 0.73889 0.77674
r3 0 0 0 0 0 0.53594 0.59839 0.67873 0.73883 0.77669
r4 0 0 0 0 0 0.53563 0.59814 0.67853 0.73868 0.77658
r5 0 0 0 0 0 0.53511 0.59774 0.67821 0.73843 0.7764
r146 0.92696 0.9262 0.92513 0.92359 0.92142 0.91653 0.9078 0.88992 0.86057 0.80415
r147 0.92721 0.92647 0.92541 0.9239 0.92176 0.91741 0.90895 0.89144 0.84828 0.79341
r148 0.92746 0.92673 0.92569 0.92421 0.9221 0.91827 0.91007 0.87813 0.83621 0.78284
r149 0.9277 0.92698 0.92596 0.9245 0.92243 0.9191 0.89605 0.86507 0.82436 0.77245
r150 0.92794 0.92724 0.92623 0.9248 0.92276 0.90467 0.88233 0.85227 0.81273 0.76223