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SurrogateCorrAdmissions.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Surrogate functions in admissions markets\n",
"\n",
"Max Kapur | [maxkapur.com](maxkapur.com)\n",
"\n",
"[Surrogates.jl](http://surrogates.sciml.ai/stable/) is an excellent Julia package that allows you to build surrogate models for black box functions. This approach interested me because many of the demand functions I use to model admissions markets are computationally very expensive.\n",
"\n",
"For example, the model considered here assumes that schools admit students based on a convex combination of their characteristics—a [linear assessment function]( hahttps://github.com/maxkapur/SchoolAssessFnRevOpt/blob/main/_ReverseOptimizationSchoolAssessment.ipynb) of the form\n",
"$$u_c(x) = \\sum_{i=1}^n A_{ci} x_i$$\n",
"for each school $c$. The $A_{ci} \\geq 0$, with $\\sum A_{c.} = 1$, are the weights the school gives to the traits $x_i$. The traits are uniformly, independently distributed on $[0,1]$. \n",
"\n",
"The matrix $A$ determines the probability of a student having a given consideration set $C^\\# \\in C!$, which is the $|C|$-dimensional volume of the following polyhedron:\n",
"\n",
"$$\\begin{align}\n",
"A_{c.} x \\geq p_c, \\quad \\forall c \\in C^\\# \\\\\n",
"A_{c.} x < p_c, \\quad \\forall c \\notin C^\\# \\\\\n",
"0 \\leq x \\leq 1\n",
"\\end{align}\n",
"$$\n",
"\n",
"From $C^\\#$, students choose thair favorite school according to a [multinomial logit](https://en.wikipedia.org/wiki/Discrete_choice#Multinomial_choice_without_correlation_among_alternatives) choice model. Letting $V\\left(C^\\#\\right)$ denote the volume of the polyhedron, the demand for school $c$ is\n",
"$$D_c(p) = \n",
"\\sum_{\\substack{C^\\# \\in 2^{C}:\\\\ c \\in C^\\#}}\n",
"\\frac{\\exp{\\delta_c}}{\\sum_{d\\in C^\\#} \\exp{\\delta_d}}\n",
"V\\left(C^\\#\\right)$$\n",
"\n",
"Under these assumptions, the market equilibrium exists and is unique, but computing it is quite difficult in general because there are $2^{|C|}$ possible consideration sets (many with probability zero); hence the demand function is exponential in the number of schools.\n",
"\n",
"## Task at hand\n",
"\n",
"The question here is if we can \"learn\" the form of $D$ from a small number of sample points. If so, and if the approximation is accurate enough, then the surrogate $D$ can be used in place of $D$ itself to search for the equilibrium by a gradient procedure.\n",
"\n",
"In this notebook, I test this idea out and determine that the surrogate optimization approach is **not worth it** for the task at hand. Using the same number of calls to $D$ needed to form a workable approximation, we can run t&acirc;tonnement on $D$ itself and find a more accurate estimate of the equilibrium."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Combinatorics\n",
"using Plots, Plots.PlotMeasures\n",
"using Polyhedra\n",
"using LinearAlgebra\n",
"using Surrogates"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First I define some basic functions."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"normexcessdemand (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\"\"\"\n",
"Contains static information about a school-choice market.\n",
"\"\"\"\n",
"struct Market\n",
" qualities::Array{<:AbstractFloat, 1}\n",
" blends::Array{<:AbstractFloat, 2}\n",
" capacities::Array{<:AbstractFloat, 1}\n",
"end\n",
"\n",
"\n",
"\"\"\"\n",
" demand(market, cutoffs)\n",
"\n",
"Return demand for each school given a set of cutoffs and ignoring capacity, using\n",
"multinomial logit choice model with one student profile and `t` test scores,\n",
"which represent orthogonal dimensions of student quality. Each school uses\n",
"a convex combination of the tests scores to evaluate students; the rows of `blends`\n",
"give these combinations. `qualities` are the MNL preferability parameters.\n",
"\n",
"A simplification of `demands_pMNL_ttests()` from `DeferredAcceptance` package.\n",
"\"\"\"\n",
"function demand(market ::Market,\n",
" cutoffs ::AbstractArray{<:AbstractFloat, 1};\n",
" )::AbstractArray{<:AbstractFloat, 1}\n",
" qualities = market.qualities\n",
" blends = market.blends\n",
" \n",
" (m, t) = size(blends)\n",
" demands = zeros(m)\n",
" γ = exp.(qualities)\n",
" bounds = vcat([HalfSpace(-col, 0.) for col in eachcol(I(t))],\n",
" [HalfSpace(1col, 1.) for col in eachcol(I(t))])\n",
"\n",
" for C♯ in powerset(1:m) # Possible choice sets\n",
"\n",
" if !isempty(C♯)\n",
" hspaces = HalfSpace{Float64,Array{Float64,1}}[]\n",
" \n",
" # Probability of having this choice set is the volume of\n",
" # this m-dimensional polyhedron. \n",
" for c in C♯\n",
" if cutoffs[c] ≥ 1\n",
" @goto volume_is_zero\n",
" elseif cutoffs[c] > 0\n",
" push!(hspaces, HalfSpace(-blends[c, :], -cutoffs[c]))\n",
" end\n",
" end\n",
" \n",
" for c in setdiff(1:m, C♯)\n",
" if cutoffs[c] ≤ 0\n",
" @goto volume_is_zero\n",
" elseif cutoffs[c] < 1\n",
" push!(hspaces, HalfSpace( blends[c, :], cutoffs[c]))\n",
" end\n",
" end\n",
"\n",
" poly = polyhedron(hrep(vcat(bounds, hspaces)))\n",
" vol = volume(poly)\n",
"# @show vol, poly\n",
"\n",
" mult = vol / sum(γ[C♯])\n",
"\n",
" for c in C♯\n",
" demands[c] += mult * γ[c]\n",
" end\n",
" \n",
" @label volume_is_zero\n",
" end\n",
" end\n",
"\n",
" return demands\n",
"end\n",
"\n",
"\n",
"\"\"\"\n",
" quickeq(market, demand)\n",
"\n",
"Very low-tech tatonnement procedure for finding equibilibrium.\n",
"\"\"\"\n",
"function quickeq(market::Market, demand::Function; maxit::Int=100)\n",
" p = rand(length(market.qualities))\n",
" \n",
" for i in 1:maxit\n",
" p = max.(0, p + .5 * (demand(market, p) - market.capacities) / i^0.001)\n",
"# @show p\n",
" end\n",
" \n",
" return p\n",
"end\n",
"\n",
"normexcessdemand(market, cutoffs) = norm(demand(market, cutoffs) - market.capacities)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here is a sample market with two schools and two dimensions of student quality. Test that the demand function is working with some random cutoffs. Looks like school 1 is getting too few students, and school 2 is getting too many&mdash;disequilibrium."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" -0.1494295854369378\n",
" 0.4660962521036042"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"const savannah = Market([1.1, 0.4] , # Preferability of each school\n",
" [.2 .8; # School assessment functions\n",
" .6 .4] ,\n",
" [.4, .2] # Capacity of each school\n",
" )\n",
" \n",
"demand(savannah, [.6, .2]) - savannah.capacities"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here are the \"real\" market-clearing cutoffs. We can run 500 iterations of t&acirc;tonnement because this market is very small; this will not be the case for a larger market. Observe that the demand matches the capacity of the schools."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"p̂ = [0.4935991474928613, 0.5495075370019941]\n"
]
},
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 0.4\n",
" 0.1999999999999999"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p̂ = quickeq(savannah, demand; maxit=500)\n",
"@show p̂\n",
"demand(savannah, p̂)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To show visually that the equilibrium is unique, plot the norm of the excess demand vector in cutoff space. Since the total capacity is less than one, both schools will be completely full at equilibrium, and the equilibrium coincides with the minimum of this function, when the norm is zero.\n",
"\n",
"(The \"ridges\" in the graph are the artifact of a [known bug](https://github.com/JuliaPolyhedra/Polyhedra.jl/issues/249) in Polyhedra.jl, which is used to compute the volume of the polyhedra described above. Since these ridges do not occur at the equilibrium, they do not affect our computations.)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
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" 1048.37,1253.68 1031.61,1265.04 1030.66,1265.7 1012.95,1277.61 1006.9,1281.55 995.238,1289.37 981.878,1298.05 977.528,1300.97 959.817,1312.4 956.319,1314.56 \n",
" 942.106,1323.65 930.047,1331.07 924.395,1334.67 906.684,1345.45 902.994,1347.57 888.973,1355.96 874.721,1364.08 871.263,1366.14 853.552,1375.93 844.54,1380.59 \n",
" 835.841,1385.29 818.13,1394.13 811.522,1397.09 800.419,1402.34 782.708,1409.8 772.229,1413.6 764.998,1416.37 747.287,1421.8 729.576,1425.87 711.865,1428.18 \n",
" 694.154,1428.15 676.443,1424.88 658.733,1416.96 654.218,1413.6 641.022,1401.02 638.18,1397.09 628.96,1380.59 623.311,1365.96 622.733,1364.08 619.002,1347.57 \n",
" 616.693,1331.07 615.544,1314.56 615.367,1298.05 616.016,1281.55 617.385,1265.04 619.394,1248.53 621.98,1232.03 623.311,1225.16 625.166,1215.52 628.893,1199.01 \n",
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" 676.443,1059.54 680.174,1050.45 687.191,1033.94 694.154,1018 694.4,1017.44 701.847,1000.93 709.486,984.422 711.865,979.452 717.35,967.915 725.429,951.409 \n",
" 729.576,943.184 733.733,934.902 742.27,918.395 747.287,908.967 751.047,901.888 760.076,885.381 764.998,876.631 769.367,868.874 778.937,852.367 782.708,846.037 \n",
" 788.803,835.86 798.978,819.353 800.419,815.23 804.56,802.847 818.13,788.445 820.249,786.34 831.105,769.833 835.841,762.606 842.019,753.326 852.941,736.819 \n",
" 853.552,735.882 863.897,720.312 871.263,709.066 874.784,703.805 885.607,687.298 888.973,682.034 896.343,670.791 906.684,654.583 906.88,654.285 917.363,637.778 \n",
" 924.395,626.387 927.648,621.271 937.816,604.764 942.106,597.568 947.838,588.257 957.651,571.75 959.817,567.969 967.368,555.243 976.786,538.736 977.528,537.382 \n",
" 986.15,522.229 995.137,505.723 995.238,505.528 1004.09,489.216 1012.61,472.709 1012.95,472.02 1021.08,456.202 1029.1,439.695 1030.66,436.304 1037,423.188 \n",
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" 251.383,142.571 233.673,142.571 233.673,159.078 233.673,175.585 233.673,192.092 233.673,208.599 233.673,225.106 \n",
" 233.673,241.612 233.673,258.119 233.673,274.626 233.673,291.133 233.673,307.64 233.673,324.147 233.673,340.654 233.673,357.161 233.673,373.668 233.673,390.174 \n",
" 233.673,406.681 233.673,423.188 233.673,439.695 233.673,456.202 233.673,472.709 233.673,489.216 233.673,505.723 233.673,522.229 233.673,538.736 233.673,555.243 \n",
" 233.673,571.75 233.673,588.257 233.673,604.764 233.673,621.271 233.673,637.778 233.673,654.285 233.673,670.791 233.673,687.298 233.673,703.805 233.673,720.312 \n",
" 233.673,736.819 233.673,753.326 233.673,769.833 233.673,786.34 233.673,802.847 233.673,819.353 233.673,835.86 233.673,852.367 233.673,868.874 233.673,885.381 \n",
" 233.673,901.888 233.673,918.395 233.673,934.902 233.673,951.409 233.673,967.915 233.673,984.422 233.673,1000.93 233.673,1017.44 233.673,1033.94 233.673,1050.45 \n",
" 233.673,1066.96 233.673,1083.46 233.673,1099.97 233.673,1116.48 233.673,1132.98 233.673,1149.49 233.673,1166 233.673,1182.5 233.673,1199.01 233.673,1215.52 \n",
" 233.673,1232.03 233.673,1248.53 233.673,1265.04 233.673,1281.55 233.673,1298.05 233.673,1314.56 233.673,1331.07 233.673,1347.57 233.673,1364.08 233.673,1380.59 \n",
" 233.673,1397.09 233.673,1413.6 233.673,1430.11 233.673,1446.62 233.673,1463.12 233.673,1479.63 233.673,1496.14 233.673,1512.64 233.673,1529.15 233.673,1545.66 \n",
" 233.673,1562.16 233.673,1578.67 233.673,1595.18 233.673,1611.68 233.673,1628.19 233.673,1644.7 233.673,1661.2 233.673,1677.71 233.673,1694.22 233.673,1710.73 \n",
" 233.673,1727.23 233.673,1743.74 233.673,1760.25 233.673,1776.75 233.673,1793.26 \n",
" \"/>\n",
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" 393.07,1793.26 410.781,1793.26 428.492,1793.26 446.203,1793.26 463.913,1793.26 481.624,1793.26 499.335,1793.26 517.046,1793.26 534.757,1793.26 552.468,1793.26 \n",
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" 2004.76,703.805 2004.76,687.298 2004.76,670.791 2004.76,654.285 2004.76,637.778 2004.76,621.271 2004.76,604.764 2004.76,588.257 2004.76,571.75 2004.76,555.243 \n",
" 2004.76,538.736 2004.76,522.229 2004.76,505.723 2004.76,489.216 2004.76,472.709 2004.76,456.202 2004.76,439.695 2004.76,423.188 2004.76,406.681 2004.76,390.174 \n",
" 2004.76,373.668 2004.76,357.161 2004.76,340.654 2004.76,324.147 2004.76,307.64 2004.76,291.133 2004.76,274.626 2004.76,258.119 2004.76,241.612 2004.76,225.106 \n",
" 2004.76,208.599 2004.76,192.092 2004.76,175.585 2004.76,159.078 2004.76,142.571 1987.05,142.571 1969.33,142.571 \n",
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" 1566.25,522.229 1570.67,538.736 1575.22,555.243 1579.7,571.336 1579.81,571.75 1584.3,588.257 1588.63,604.764 1592.72,621.271 1596.46,637.778 1597.41,642.474 \n",
" 1599.74,654.285 1602.46,670.791 1604.55,687.298 1605.9,703.805 1606.44,720.312 1606.08,736.819 1604.73,753.326 1602.34,769.833 1598.84,786.34 1597.41,786.491 \n",
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" 906.684,1458.84 901.391,1463.12 888.973,1473.53 881.492,1479.63 871.263,1488.27 861.679,1496.14 853.552,1503.1 841.978,1512.64 835.841,1518.01 822.464,1529.15 \n",
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" 443.941,1397.09 446.203,1389.82 448.905,1380.59 454.116,1364.08 459.625,1347.57 463.913,1335.48 465.394,1331.07 471.333,1314.56 477.551,1298.05 481.624,1287.82 \n",
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" 649.767,934.902 658.62,918.395 658.733,918.189 667.623,901.888 676.443,885.956 676.762,885.381 686.067,868.874 694.154,854.762 695.529,852.367 705.177,835.86 \n",
" 711.865,824.614 715.007,819.353 723.384,802.847 729.576,794.058 735.196,786.34 745.337,769.833 747.287,766.63 755.459,753.326 764.998,737.638 765.501,736.819 \n",
" 775.485,720.312 782.708,708.163 785.331,703.805 795.056,687.298 800.419,677.991 804.624,670.791 814.037,654.285 818.13,646.92 823.287,637.778 832.353,621.271 \n",
" 835.841,614.733 841.246,604.764 849.931,588.257 853.552,581.15 858.426,571.75 866.699,555.243 871.263,545.807 874.748,538.736 882.579,522.229 888.973,508.209 \n",
" 890.131,505.723 897.492,489.216 904.534,472.709 906.684,467.422 911.35,456.202 917.877,439.695 924.065,423.188 924.395,422.255 930.041,406.681 935.671,390.174 \n",
" 940.946,373.668 942.106,369.772 945.961,357.161 950.636,340.654 954.941,324.147 958.874,307.64 959.817,303.267 962.509,291.133 965.787,274.626 968.68,258.119 \n",
" 971.188,241.612 973.31,225.106 975.046,208.599 976.397,192.092 977.362,175.585 977.528,170.852 977.954,159.078 978.153,142.571 977.528,142.571 \n",
" 959.817,142.571 942.106,142.571 924.395,142.571 906.684,142.571 888.973,142.571 871.263,142.571 853.552,142.571 835.841,142.571 818.13,142.571 800.419,142.571 \n",
" 782.708,142.571 764.998,142.571 747.287,142.571 729.576,142.571 711.865,142.571 694.154,142.571 676.443,142.571 658.733,142.571 641.022,142.571 623.311,142.571 \n",
" 605.6,142.571 587.889,142.571 570.178,142.571 552.468,142.571 534.757,142.571 517.046,142.571 499.335,142.571 481.624,142.571 463.913,142.571 446.203,142.571 \n",
" 428.492,142.571 410.781,142.571 393.07,142.571 375.359,142.571 357.648,142.571 339.938,142.571 322.227,142.571 304.516,142.571 286.805,142.571 269.094,142.571 \n",
" 251.383,142.571 233.673,142.571 233.673,159.078 233.673,175.585 233.673,192.092 233.673,208.599 233.673,225.106 \n",
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]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function plot_doer()\n",
" xs = ys = 0:0.01:1\n",
"\n",
" pl = plot(xlim=(0, 1), ylim=(0, 1), camera=(60, 45), size=(570, 500), #background_color=:gray5,\n",
" xlabel=\"Cutoff at 1\", ylabel=\"Cutoff at 2\")\n",
"\n",
" plot!(pl, xs, ys, \n",
" (x, y)->normexcessdemand(savannah, [x, y]),\n",
" st=:contourf, color=:summer, lc=:black, lw=.5, title=\"Norm of excess demand\")\n",
" \n",
" scatter!(pl, [p̂[1]], [p̂[2]], c=:black, label=\"Equilibrium\", marker=:X)\n",
" \n",
" return pl\n",
"end\n",
"\n",
"norm_excess_demand = plot_doer()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Since the excess demand is nonzero everywhere except at the equilibrium, minimizing the norm of the excess demand is equivalent to finding the cutoffs. So, let's try to create a surrogate for the norm of the excess demand function and minimize it using Surrogates.jl."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(::RadialBasis{Surrogates.var\"#1#2\", Int64, Vector{Tuple{Float64, Float64}}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64, Bool}) (generic function with 1 method)"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function surrogatenormexcessdemand(market::Market; n_samples::Int=35)\n",
" m = length(market.qualities)\n",
" \n",
" lb = zeros(m)\n",
" ub = ones(m)\n",
" \n",
" ps = sample(n_samples, lb, ub, SobolSample())\n",
" zs = [normexcessdemand(market, collect(p)) for p in ps]\n",
"\n",
" return RadialBasis(ps, zs, lb, ub)\n",
"end\n",
"\n",
"surr = surrogatenormexcessdemand(savannah)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We built the surrogate using 35 samples, and we will let `surrogate_optimize()` collect 5 more, for a total cost of 40 calls to `demand()`."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 0.49368421723188044\n",
" 0.5495630287863199"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p_surropt40 = surrogate_optimize(p->normexcessdemand(savannah, collect(p)),\n",
" DYCORS(), # Fiddle with this; DYCORS hangs sometimes\n",
" [0, 0],\n",
" [1, 1],\n",
" surr,\n",
" SobolSample(),\n",
" num_new_samples=5)[1] |> collect"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 0.3999239816657382\n",
" 0.19996856833794543"
]
},
"execution_count": 8,
"metadata": {},
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"source": [
"demand(savannah, p_surropt40)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see how well t&acirc;tonnement does with 40 evaluations:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 0.40000000030304833\n",
" 0.20000000032769302"
]
},
"execution_count": 9,
"metadata": {},
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],
"source": [
"p_tat40 = quickeq(savannah, demand; maxit=40)\n",
"demand(savannah, p_tat40)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The error is several orders of magnitude smaller when using plain t&acirc;tonnement and the same number of demand computations. Hence, we will not pursue this approach further.\n",
"\n",
"Instead, let's try to approximate the demand function directly, which is vector-valued. Here is a plot of each of its components. In this view, it looks like the equilibrium is just at a \"random\" point in the middle, but if you inspect the colorbar, you can see that the demand for each school matches its capacity, meaning the school has no incentive to raise or lower its cutoff."
]
},
{
"cell_type": "code",
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]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function plot_doer()\n",
" xs = ys = 0:0.02:1\n",
"\n",
" pl = plot(xlim=(0, 1), ylim=(0, 1), camera=(60, 45), size=(1200, 500), #background_color=:gray5,\n",
" xlabel=\"Cutoff at 1\", ylabel=\"Cutoff at 2\", layout=2, margin=30px)\n",
"\n",
" plot!(pl, xs, ys, \n",
" (x, y)->demand(savannah, [x, y])[1],\n",
" st=:contourf, color=:viridis, subplot=1, lc=:black, lw=.5, title=\"Demand at 1\", legend=nothing)\n",
"\n",
" plot!(pl, xs, ys,\n",
" (x, y)->demand(savannah, [x, y])[2],\n",
" st=:contourf, color=:viridis, subplot=2, lc=:black, lw=.5, title=\"Demand at 2\")\n",
"\n",
" for i in 1:2 scatter!(pl, [p̂[1]], [p̂[2]], c=:black, label=\"Equilibrium\", marker=:X, subplot=i) end\n",
" \n",
" return pl\n",
"end\n",
"\n",
"savannah_excess_demand = plot_doer()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Build a surrogate. I will use the radial basis form, which is advertised as an efficient choice for vector-valued functions like this one."
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"surrogatedemand (generic function with 1 method)"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function surrogatedemand(market::Market, demand::Function; n_samples::Int=40)\n",
" m = length(market.qualities)\n",
" \n",
" lb = zeros(m)\n",
" ub = ones(m)\n",
" \n",
" ps = sample(n_samples, lb, ub, SobolSample())\n",
" zs = [demand(market, collect(p)) for p in ps]\n",
"\n",
" return RadialBasis(ps, zs, lb, ub)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(::RadialBasis{Surrogates.var\"#1#2\", Int64, Vector{Tuple{Float64, Float64}}, Vector{Vector{Float64}}, Vector{Float64}, Vector{Float64}, Matrix{Float64}, Float64, Bool}) (generic function with 1 method)"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"surr = surrogatedemand(savannah, demand)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compare the surrogate with the actual function plotted above."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
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