{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"originalKey": "e23719d9-8a24-4208-8439-34e7b8270c79"
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"source": [
"# Visualizations\n",
"\n",
"This tutorial illustrates the core visualization utilities available in Ax."
]
},
{
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""
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"text": [
"[INFO 11-10 20:32:10] ax.utils.notebook.plotting: Injecting Plotly library into cell. Do not overwrite or delete cell.\n"
]
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"data": {
"text/html": [
" \n",
" "
]
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"source": [
"import numpy as np\n",
"from ax.service.ax_client import AxClient\n",
"\n",
"from ax.modelbridge.cross_validation import cross_validate\n",
"from ax.plot.contour import interact_contour\n",
"from ax.plot.diagnostic import interact_cross_validation\n",
"from ax.plot.scatter import(\n",
" interact_fitted,\n",
" plot_objective_vs_constraints,\n",
" tile_fitted,\n",
")\n",
"from ax.plot.slice import plot_slice\n",
"from ax.utils.measurement.synthetic_functions import hartmann6\n",
"from ax.utils.notebook.plotting import render, init_notebook_plotting\n",
"\n",
"init_notebook_plotting()"
]
},
{
"cell_type": "markdown",
"metadata": {
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"originalKey": "8449378f-890e-4e76-8d73-ce2aa4120a69",
"showInput": true
},
"source": [
"## 1. Create experiment and run optimization\n",
"\n",
"The vizualizations require an experiment object and a model fit on the evaluated data. The routine below is a copy of the Service API tutorial, so the explanation here is omitted. Retrieving the experiment and model objects for each API paradigm is shown in the respective tutorials"
]
},
{
"cell_type": "markdown",
"metadata": {
"originalKey": "f7544e06-6c6a-4841-b659-3be6a198a948"
},
"source": [
"#### 1a. Define search space and evaluation function"
]
},
{
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"execution_count": 2,
"metadata": {
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"collapsed": false,
"execution": {
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"executionStartTime": 1627652824829,
"executionStopTime": 1627652824877,
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"outputs": [],
"source": [
"noise_sd = 0.1\n",
"param_names = [f\"x{i+1}\" for i in range(6)] # x1, x2, ..., x6\n",
"\n",
"def noisy_hartmann_evaluation_function(parameterization):\n",
" x = np.array([parameterization.get(p_name) for p_name in param_names])\n",
" noise1, noise2 = np.random.normal(0, noise_sd, 2)\n",
"\n",
" return {\n",
" \"hartmann6\": (hartmann6(x) + noise1, noise_sd),\n",
" \"l2norm\": (np.sqrt((x ** 2).sum()) + noise2, noise_sd)\n",
" }\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"originalKey": "17a51543-298e-47d4-bcd9-33459fe1169e"
},
"source": [
"#### 1b. Create Experiment"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"code_folding": [],
"collapsed": false,
"execution": {
"iopub.execute_input": "2022-11-10T20:32:10.312096Z",
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"shell.execute_reply": "2022-11-10T20:32:10.323206Z"
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"executionStartTime": 1627654956712,
"executionStopTime": 1627654956823,
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"requestMsgId": "905eff52-e649-4bd5-abf0-ff69c1549852"
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"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Starting optimization with verbose logging. To disable logging, set the `verbose_logging` argument to `False`. Note that float values in the logs are rounded to 6 decimal points.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x1. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x2. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x3. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x4. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x5. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x6. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.utils.instantiation: Created search space: SearchSpace(parameters=[RangeParameter(name='x1', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x2', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x3', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x4', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x5', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x6', parameter_type=FLOAT, range=[0.0, 1.0])], parameter_constraints=[]).\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.modelbridge.dispatch_utils: Using Bayesian optimization since there are more ordered parameters than there are categories for the unordered categorical parameters.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.modelbridge.dispatch_utils: Using Bayesian Optimization generation strategy: GenerationStrategy(name='Sobol+GPEI', steps=[Sobol for 12 trials, GPEI for subsequent trials]). Iterations after 12 will take longer to generate due to model-fitting.\n"
]
}
],
"source": [
"ax_client = AxClient()\n",
"ax_client.create_experiment(\n",
" name=\"test_visualizations\",\n",
" parameters=[\n",
" {\n",
" \"name\": p_name,\n",
" \"type\": \"range\",\n",
" \"bounds\": [0.0, 1.0],\n",
" }\n",
" for p_name in param_names\n",
" ],\n",
" objective_name=\"hartmann6\",\n",
" minimize=True,\n",
" outcome_constraints=[\"l2norm <= 1.25\"]\n",
")"
]
},
{
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"originalKey": "ab892f7c-4830-4c1d-b476-ec1078ec3faf",
"showInput": false
},
"source": [
"#### 1c. Run the optimization and fit a GP on all data"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"code_folding": [],
"collapsed": false,
"execution": {
"iopub.execute_input": "2022-11-10T20:32:10.327168Z",
"iopub.status.busy": "2022-11-10T20:32:10.326688Z",
"iopub.status.idle": "2022-11-10T20:35:04.999970Z",
"shell.execute_reply": "2022-11-10T20:35:04.999389Z"
},
"executionStartTime": 1627654642967,
"executionStopTime": 1627654862819,
"hidden_ranges": [],
"originalKey": "7269a5ba-45c8-4acf-ac83-a5ea8a52d6c1",
"requestMsgId": "c7a4dea8-fd6d-4e1a-84de-ad973ede0cd7"
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/home/runner/work/Ax/Ax/ax/core/observation.py:274: FutureWarning:\n",
"\n",
"In a future version of pandas, a length 1 tuple will be returned when iterating over a groupby with a grouper equal to a list of length 1. Don't supply a list with a single grouper to avoid this warning.\n",
"\n",
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 0 with parameters {'x1': 0.941206, 'x2': 0.128345, 'x3': 0.193663, 'x4': 0.59304, 'x5': 0.67774, 'x6': 0.632615}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 0 with data: {'hartmann6': (-0.013391, 0.1), 'l2norm': (1.666149, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 1 with parameters {'x1': 0.350923, 'x2': 0.817723, 'x3': 0.212011, 'x4': 0.210169, 'x5': 0.206005, 'x6': 0.954653}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 1 with data: {'hartmann6': (-0.295311, 0.1), 'l2norm': (1.445695, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 2 with parameters {'x1': 0.109802, 'x2': 0.062789, 'x3': 0.490375, 'x4': 0.975718, 'x5': 0.572008, 'x6': 0.285686}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 2 with data: {'hartmann6': (-0.17209, 0.1), 'l2norm': (1.164104, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 3 with parameters {'x1': 0.231465, 'x2': 0.964593, 'x3': 0.155377, 'x4': 0.409109, 'x5': 0.756466, 'x6': 0.710501}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 3 with data: {'hartmann6': (0.168145, 0.1), 'l2norm': (1.646876, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 4 with parameters {'x1': 0.00778, 'x2': 0.237625, 'x3': 0.791061, 'x4': 0.711802, 'x5': 0.241862, 'x6': 0.661306}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 4 with data: {'hartmann6': (-0.558998, 0.1), 'l2norm': (1.260731, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 5 with parameters {'x1': 0.7766, 'x2': 0.34743, 'x3': 0.823272, 'x4': 0.807713, 'x5': 0.912812, 'x6': 0.628868}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 5 with data: {'hartmann6': (0.092051, 0.1), 'l2norm': (1.774278, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 6 with parameters {'x1': 0.610301, 'x2': 0.818059, 'x3': 0.683816, 'x4': 0.793503, 'x5': 0.971427, 'x6': 0.025934}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 6 with data: {'hartmann6': (-0.900724, 0.1), 'l2norm': (1.598032, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 7 with parameters {'x1': 0.612836, 'x2': 0.173494, 'x3': 0.39489, 'x4': 0.838601, 'x5': 0.485488, 'x6': 0.826801}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 7 with data: {'hartmann6': (0.166554, 0.1), 'l2norm': (1.424632, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 8 with parameters {'x1': 0.616064, 'x2': 0.886116, 'x3': 0.251193, 'x4': 0.667437, 'x5': 0.838638, 'x6': 0.668467}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 8 with data: {'hartmann6': (-0.066591, 0.1), 'l2norm': (1.673057, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 9 with parameters {'x1': 0.419323, 'x2': 0.084307, 'x3': 0.412628, 'x4': 0.870432, 'x5': 0.671433, 'x6': 0.937125}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 9 with data: {'hartmann6': (-0.00288, 0.1), 'l2norm': (1.549686, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 10 with parameters {'x1': 0.321102, 'x2': 0.720067, 'x3': 0.578888, 'x4': 0.226596, 'x5': 0.488705, 'x6': 0.936341}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 10 with data: {'hartmann6': (-0.336436, 0.1), 'l2norm': (1.371364, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Generated new trial 11 with parameters {'x1': 0.865786, 'x2': 0.308924, 'x3': 0.419366, 'x4': 0.472768, 'x5': 0.062104, 'x6': 0.08089}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:10] ax.service.ax_client: Completed trial 11 with data: {'hartmann6': (-0.038426, 0.1), 'l2norm': (1.095405, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:26] ax.service.ax_client: Generated new trial 12 with parameters {'x1': 0.0, 'x2': 0.195257, 'x3': 0.743973, 'x4': 0.765187, 'x5': 0.316779, 'x6': 0.508665}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:26] ax.service.ax_client: Completed trial 12 with data: {'hartmann6': (-0.225562, 0.1), 'l2norm': (1.271639, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:40] ax.service.ax_client: Generated new trial 13 with parameters {'x1': 0.0, 'x2': 0.269234, 'x3': 0.790091, 'x4': 0.667303, 'x5': 0.06273, 'x6': 0.706032}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:40] ax.service.ax_client: Completed trial 13 with data: {'hartmann6': (-0.508339, 0.1), 'l2norm': (1.427037, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:54] ax.service.ax_client: Generated new trial 14 with parameters {'x1': 0.0, 'x2': 0.217392, 'x3': 0.727826, 'x4': 0.625424, 'x5': 0.429193, 'x6': 0.709425}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:32:54] ax.service.ax_client: Completed trial 14 with data: {'hartmann6': (-0.650362, 0.1), 'l2norm': (1.183348, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:33:09] ax.service.ax_client: Generated new trial 15 with parameters {'x1': 0.0, 'x2': 0.249143, 'x3': 0.860397, 'x4': 0.61219, 'x5': 0.37464, 'x6': 0.76641}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:33:09] ax.service.ax_client: Completed trial 15 with data: {'hartmann6': (-0.844568, 0.1), 'l2norm': (1.481249, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:33:37] ax.service.ax_client: Generated new trial 16 with parameters {'x1': 0.0, 'x2': 0.240716, 'x3': 0.654132, 'x4': 0.593032, 'x5': 0.339848, 'x6': 0.790009}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:33:37] ax.service.ax_client: Completed trial 16 with data: {'hartmann6': (-1.005465, 0.1), 'l2norm': (1.290837, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:34:03] ax.service.ax_client: Generated new trial 17 with parameters {'x1': 0.0, 'x2': 0.224115, 'x3': 0.534221, 'x4': 0.620932, 'x5': 0.363411, 'x6': 0.791665}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:34:03] ax.service.ax_client: Completed trial 17 with data: {'hartmann6': (-0.774214, 0.1), 'l2norm': (1.35333, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:34:27] ax.service.ax_client: Generated new trial 18 with parameters {'x1': 0.0, 'x2': 0.168341, 'x3': 0.611198, 'x4': 0.470563, 'x5': 0.361918, 'x6': 0.739598}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:34:27] ax.service.ax_client: Completed trial 18 with data: {'hartmann6': (-1.672589, 0.1), 'l2norm': (1.278163, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:35:04] ax.service.ax_client: Generated new trial 19 with parameters {'x1': 0.02098, 'x2': 0.181006, 'x3': 0.620755, 'x4': 0.461433, 'x5': 0.349005, 'x6': 0.591645}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 11-10 20:35:04] ax.service.ax_client: Completed trial 19 with data: {'hartmann6': (-2.021935, 0.1), 'l2norm': (1.056877, 0.1)}.\n"
]
}
],
"source": [
"for i in range(20):\n",
" parameters, trial_index = ax_client.get_next_trial()\n",
" # Local evaluation here can be replaced with deployment to external system.\n",
" ax_client.complete_trial(trial_index=trial_index, raw_data=noisy_hartmann_evaluation_function(parameters))"
]
},
{
"cell_type": "markdown",
"metadata": {
"originalKey": "72f4d3e7-fa04-43d0-8451-ded292e705df"
},
"source": [
"## 2. Contour plots\n",
"\n",
"The plot below shows the response surface for `hartmann6` metric as a function of the `x1`, `x2` parameters.\n",
"\n",
"The other parameters are fixed in the middle of their respective ranges, which in this example is 0.5 for all of them."
]
},
{
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"text": [
"[INFO 11-10 20:35:05] ax.service.ax_client: Retrieving contour plot with parameter 'x1' on X-axis and 'x2' on Y-axis, for metric 'hartmann6'. Remaining parameters are affixed to the middle of their range.\n"
]
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"Arm 0_0
hartmann6: -0.0133909 (SEM: 0.1)
x1: 0.941206
x2: 0.128345
x3: 0.193663
x4: 0.59304
x5: 0.67774
x6: 0.632615",
"Arm 10_0
hartmann6: -0.336436 (SEM: 0.1)
x1: 0.321102
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x3: 0.578888
x4: 0.226596
x5: 0.488705
x6: 0.936341",
"Arm 11_0
hartmann6: -0.0384264 (SEM: 0.1)
x1: 0.865786
x2: 0.308924
x3: 0.419366
x4: 0.472768
x5: 0.0621042
x6: 0.0808901",
"Arm 12_0
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"Arm 0_0
hartmann6: -0.0133909 (SEM: 0.1)
x1: 0.941206
x2: 0.128345
x3: 0.193663
x4: 0.59304
x5: 0.67774
x6: 0.632615",
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hartmann6: -0.336436 (SEM: 0.1)
x1: 0.321102
x2: 0.720067
x3: 0.578888
x4: 0.226596
x5: 0.488705
x6: 0.936341",
"Arm 11_0
hartmann6: -0.0384264 (SEM: 0.1)
x1: 0.865786
x2: 0.308924
x3: 0.419366
x4: 0.472768
x5: 0.0621042
x6: 0.0808901",
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hartmann6: -0.225562 (SEM: 0.1)
x1: 0
x2: 0.195257
x3: 0.743973
x4: 0.765187
x5: 0.316779
x6: 0.508665",
"Arm 13_0
hartmann6: -0.508339 (SEM: 0.1)
x1: 0
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x4: 0.667303
x5: 0.0627305
x6: 0.706032",
"Arm 14_0
hartmann6: -0.650362 (SEM: 0.1)
x1: 0
x2: 0.217392
x3: 0.727826
x4: 0.625424
x5: 0.429193
x6: 0.709425",
"Arm 15_0
hartmann6: -0.844568 (SEM: 0.1)
x1: 0
x2: 0.249143
x3: 0.860397
x4: 0.61219
x5: 0.37464
x6: 0.76641",
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hartmann6: -1.00546 (SEM: 0.1)
x1: 0
x2: 0.240716
x3: 0.654132
x4: 0.593032
x5: 0.339848
x6: 0.790009",
"Arm 17_0
hartmann6: -0.774214 (SEM: 0.1)
x1: 0
x2: 0.224115
x3: 0.534221
x4: 0.620932
x5: 0.363411
x6: 0.791665",
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hartmann6: -1.67259 (SEM: 0.1)
x1: 0
x2: 0.168341
x3: 0.611198
x4: 0.470563
x5: 0.361918
x6: 0.739598",
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hartmann6: -0.295311 (SEM: 0.1)
x1: 0.350923
x2: 0.817723
x3: 0.212011
x4: 0.210169
x5: 0.206005
x6: 0.954653",
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hartmann6: -0.17209 (SEM: 0.1)
x1: 0.109802
x2: 0.0627886
x3: 0.490375
x4: 0.975718
x5: 0.572008
x6: 0.285686",
"Arm 3_0
hartmann6: 0.168145 (SEM: 0.1)
x1: 0.231465
x2: 0.964593
x3: 0.155377
x4: 0.409109
x5: 0.756466
x6: 0.710501",
"Arm 4_0
hartmann6: -0.558998 (SEM: 0.1)
x1: 0.00777978
x2: 0.237625
x3: 0.791061
x4: 0.711802
x5: 0.241862
x6: 0.661306",
"Arm 5_0
hartmann6: 0.0920512 (SEM: 0.1)
x1: 0.7766
x2: 0.34743
x3: 0.823272
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