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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 03-01 16:37:19] ax.utils.notebook.plotting: Injecting Plotly library into cell. Do not overwrite or delete cell.\n"
]
},
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"text": [
"[INFO 03-01 16:37:19] ax.utils.notebook.plotting: Please see\n",
" (https://ax.dev/tutorials/visualizations.html#Fix-for-plots-that-are-not-rendering)\n",
" if visualizations are not rendering.\n"
]
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" \n",
" "
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"source": [
"import numpy as np\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 interact_fitted, plot_objective_vs_constraints, tile_fitted\n",
"from ax.plot.slice import plot_slice\n",
"from ax.service.ax_client import AxClient, ObjectiveProperties\n",
"from ax.utils.measurement.synthetic_functions import hartmann6\n",
"from ax.utils.notebook.plotting import init_notebook_plotting, render\n",
"\n",
"init_notebook_plotting()"
]
},
{
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"id": "14d19564",
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"showInput": true,
"tags": []
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"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",
"id": "c60a0d62",
"metadata": {
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"tags": []
},
"source": [
"#### 1a. Define search space and evaluation function"
]
},
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"source": [
"noise_sd = 0.1\n",
"param_names = [f\"x{i+1}\" for i in range(6)] # x1, x2, ..., x6\n",
"\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",
" }"
]
},
{
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"id": "aa294115",
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"tags": []
},
"source": [
"#### 1b. Create Experiment"
]
},
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"text": [
"[INFO 03-01 16:37:20] 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 03-01 16:37:20] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x1. If that is not the expected value type, you can explicitly specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x2. If that is not the expected value type, you can explicitly specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x3. If that is not the expected value type, you can explicitly specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x4. If that is not the expected value type, you can explicitly specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x5. If that is not the expected value type, you can explicitly specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x6. If that is not the expected value type, you can explicitly specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] 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 03-01 16:37:20] ax.modelbridge.dispatch_utils: Using Models.BOTORCH_MODULAR since there is at least one ordered parameter and there are no unordered categorical parameters.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.modelbridge.dispatch_utils: Calculating the number of remaining initialization trials based on num_initialization_trials=None max_initialization_trials=None num_tunable_parameters=6 num_trials=None use_batch_trials=False\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.modelbridge.dispatch_utils: calculated num_initialization_trials=12\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.modelbridge.dispatch_utils: num_completed_initialization_trials=0 num_remaining_initialization_trials=12\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.modelbridge.dispatch_utils: `verbose`, `disable_progbar`, and `jit_compile` are not yet supported when using `choose_generation_strategy` with ModularBoTorchModel, dropping these arguments.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.modelbridge.dispatch_utils: Using Bayesian Optimization generation strategy: GenerationStrategy(name='Sobol+BoTorch', steps=[Sobol for 12 trials, BoTorch 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",
" objectives={\"hartmann6\": ObjectiveProperties(minimize=True)},\n",
" outcome_constraints=[\"l2norm <= 1.25\"],\n",
")"
]
},
{
"cell_type": "markdown",
"id": "84f5e23d",
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"showInput": false,
"tags": []
},
"source": [
"#### 1c. Run the optimization and fit a GP on all data"
]
},
{
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"execution_count": 4,
"id": "22b4c26e",
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"requestMsgId": "c7a4dea8-fd6d-4e1a-84de-ad973ede0cd7",
"tags": []
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"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 0 with parameters {'x1': 0.405293, 'x2': 0.74935, 'x3': 0.328107, 'x4': 0.67813, 'x5': 0.509819, 'x6': 0.051542} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 0 with data: {'hartmann6': (-2.554969, 0.1), 'l2norm': (1.260087, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 1 with parameters {'x1': 0.953876, 'x2': 0.27131, 'x3': 0.817712, 'x4': 0.002659, 'x5': 0.169502, 'x6': 0.035105} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 1 with data: {'hartmann6': (0.118751, 0.1), 'l2norm': (1.262412, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 2 with parameters {'x1': 0.012144, 'x2': 0.067864, 'x3': 0.526769, 'x4': 0.133143, 'x5': 0.005376, 'x6': 0.773202} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 2 with data: {'hartmann6': (-0.767316, 0.1), 'l2norm': (0.755229, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 3 with parameters {'x1': 0.194068, 'x2': 0.344042, 'x3': 0.949107, 'x4': 0.490742, 'x5': 0.933593, 'x6': 0.249598} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 3 with data: {'hartmann6': (0.06035, 0.1), 'l2norm': (1.489811, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 4 with parameters {'x1': 0.890174, 'x2': 0.38554, 'x3': 0.493718, 'x4': 0.901018, 'x5': 0.499325, 'x6': 0.801591} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 4 with data: {'hartmann6': (-0.020413, 0.1), 'l2norm': (1.710817, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 5 with parameters {'x1': 0.678472, 'x2': 0.083581, 'x3': 0.990709, 'x4': 0.833655, 'x5': 0.662268, 'x6': 0.232551} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 5 with data: {'hartmann6': (-0.068442, 0.1), 'l2norm': (1.535639, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 6 with parameters {'x1': 0.978577, 'x2': 0.96843, 'x3': 0.884392, 'x4': 0.899749, 'x5': 0.139095, 'x6': 0.034444} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 6 with data: {'hartmann6': (0.017722, 0.1), 'l2norm': (1.80642, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 7 with parameters {'x1': 0.370112, 'x2': 0.632159, 'x3': 0.832172, 'x4': 0.230053, 'x5': 0.931054, 'x6': 0.781986} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 7 with data: {'hartmann6': (-0.176489, 0.1), 'l2norm': (1.82969, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 8 with parameters {'x1': 0.183521, 'x2': 0.71696, 'x3': 0.632497, 'x4': 0.284422, 'x5': 0.319276, 'x6': 0.375443} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 8 with data: {'hartmann6': (-0.739327, 0.1), 'l2norm': (1.307653, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 9 with parameters {'x1': 0.055582, 'x2': 0.026989, 'x3': 0.642712, 'x4': 0.631488, 'x5': 0.022241, 'x6': 0.920679} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 9 with data: {'hartmann6': (-0.269545, 0.1), 'l2norm': (1.216927, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 10 with parameters {'x1': 0.304531, 'x2': 0.89638, 'x3': 0.936778, 'x4': 0.11165, 'x5': 0.071646, 'x6': 0.007822} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 10 with data: {'hartmann6': (-0.242808, 0.1), 'l2norm': (1.347503, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Generated new trial 11 with parameters {'x1': 0.778506, 'x2': 0.02427, 'x3': 0.603086, 'x4': 0.65902, 'x5': 0.299814, 'x6': 0.659678} using model Sobol.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:20] ax.service.ax_client: Completed trial 11 with data: {'hartmann6': (-0.261256, 0.1), 'l2norm': (1.526738, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:26] ax.service.ax_client: Generated new trial 12 with parameters {'x1': 0.304189, 'x2': 0.743975, 'x3': 0.320002, 'x4': 0.609169, 'x5': 0.428523, 'x6': 0.06511} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:26] ax.service.ax_client: Completed trial 12 with data: {'hartmann6': (-2.202275, 0.1), 'l2norm': (1.193276, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:36] ax.service.ax_client: Generated new trial 13 with parameters {'x1': 0.287856, 'x2': 0.757063, 'x3': 0.286837, 'x4': 0.671959, 'x5': 0.561321, 'x6': 0.0} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:36] ax.service.ax_client: Completed trial 13 with data: {'hartmann6': (-1.859694, 0.1), 'l2norm': (1.279292, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:44] ax.service.ax_client: Generated new trial 14 with parameters {'x1': 0.424433, 'x2': 0.739343, 'x3': 0.3309, 'x4': 0.524169, 'x5': 0.463361, 'x6': 0.066434} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:44] ax.service.ax_client: Completed trial 14 with data: {'hartmann6': (-2.456896, 0.1), 'l2norm': (1.101448, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:51] ax.service.ax_client: Generated new trial 15 with parameters {'x1': 0.443945, 'x2': 0.753923, 'x3': 0.338643, 'x4': 0.619732, 'x5': 0.457005, 'x6': 0.087171} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:51] ax.service.ax_client: Completed trial 15 with data: {'hartmann6': (-2.457146, 0.1), 'l2norm': (1.112872, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:56] ax.service.ax_client: Generated new trial 16 with parameters {'x1': 0.417616, 'x2': 0.69061, 'x3': 0.36041, 'x4': 0.606034, 'x5': 0.458493, 'x6': 0.016772} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:37:56] ax.service.ax_client: Completed trial 16 with data: {'hartmann6': (-2.228838, 0.1), 'l2norm': (1.071268, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:38:03] ax.service.ax_client: Generated new trial 17 with parameters {'x1': 0.425905, 'x2': 0.729397, 'x3': 0.286408, 'x4': 0.619346, 'x5': 0.496688, 'x6': 0.112599} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:38:03] ax.service.ax_client: Completed trial 17 with data: {'hartmann6': (-2.432175, 0.1), 'l2norm': (1.125472, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:38:08] ax.service.ax_client: Generated new trial 18 with parameters {'x1': 0.431538, 'x2': 0.818916, 'x3': 0.298256, 'x4': 0.624864, 'x5': 0.464648, 'x6': 0.057468} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:38:08] ax.service.ax_client: Completed trial 18 with data: {'hartmann6': (-2.773538, 0.1), 'l2norm': (1.206975, 0.1)}.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:38:12] ax.service.ax_client: Generated new trial 19 with parameters {'x1': 0.417722, 'x2': 0.880657, 'x3': 0.323639, 'x4': 0.564667, 'x5': 0.497552, 'x6': 0.071858} using model BoTorch.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:38:12] ax.service.ax_client: Completed trial 19 with data: {'hartmann6': (-3.115826, 0.1), 'l2norm': (1.391717, 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(\n",
" trial_index=trial_index, raw_data=noisy_hartmann_evaluation_function(parameters)\n",
" )"
]
},
{
"cell_type": "markdown",
"id": "20f84393",
"metadata": {
"originalKey": "72f4d3e7-fa04-43d0-8451-ded292e705df",
"papermill": {
"duration": 0.047457,
"end_time": "2024-03-01T16:38:12.758607",
"exception": false,
"start_time": "2024-03-01T16:38:12.711150",
"status": "completed"
},
"tags": []
},
"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."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "64f0f02b",
"metadata": {
"code_folding": [],
"execution": {
"iopub.execute_input": "2024-03-01T16:38:12.836199Z",
"iopub.status.busy": "2024-03-01T16:38:12.835695Z",
"iopub.status.idle": "2024-03-01T16:38:13.649159Z",
"shell.execute_reply": "2024-03-01T16:38:13.648440Z"
},
"executionStartTime": 1627654870209,
"executionStopTime": 1627654871972,
"hidden_ranges": [],
"originalKey": "843df85c-965d-4a83-9fe1-696225d81c0f",
"papermill": {
"duration": 0.857307,
"end_time": "2024-03-01T16:38:13.653857",
"exception": false,
"start_time": "2024-03-01T16:38:12.796550",
"status": "completed"
},
"requestMsgId": "4a643541-867c-46b6-868d-64337920c2a3",
"tags": []
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 03-01 16:38:12] 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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"# this could alternately be done with `ax.plot.contour.plot_contour`\n",
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hartmann6: -2.55497 (SEM: 0.1)
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x6: 0.0515415",
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hartmann6: 0.118751 (SEM: 0.1)
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hartmann6: -0.0684416 (SEM: 0.1)
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hartmann6: -2.43217 (SEM: 0.1)
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"Arm 0_0
hartmann6: -2.55497 (SEM: 0.1)
x1: 0.405293
x2: 0.74935
x3: 0.328107
x4: 0.67813
x5: 0.509819
x6: 0.0515415",
"Arm 1_0
hartmann6: 0.118751 (SEM: 0.1)
x1: 0.953876
x2: 0.27131
x3: 0.817712
x4: 0.00265851
x5: 0.169502
x6: 0.0351052",
"Arm 2_0
hartmann6: -0.767316 (SEM: 0.1)
x1: 0.0121443
x2: 0.0678637
x3: 0.526769
x4: 0.133143
x5: 0.00537562
x6: 0.773202",
"Arm 3_0
hartmann6: 0.0603503 (SEM: 0.1)
x1: 0.194068
x2: 0.344042
x3: 0.949107
x4: 0.490742
x5: 0.933593
x6: 0.249598",
"Arm 4_0
hartmann6: -0.0204126 (SEM: 0.1)
x1: 0.890174
x2: 0.38554
x3: 0.493718
x4: 0.901018
x5: 0.499325
x6: 0.801591",
"Arm 5_0
hartmann6: -0.0684416 (SEM: 0.1)
x1: 0.678472
x2: 0.0835807
x3: 0.990709
x4: 0.833655
x5: 0.662268
x6: 0.232551",
"Arm 6_0
hartmann6: 0.0177218 (SEM: 0.1)
x1: 0.978577
x2: 0.96843
x3: 0.884392
x4: 0.899749
x5: 0.139095
x6: 0.0344437",
"Arm 7_0
hartmann6: -0.176489 (SEM: 0.1)
x1: 0.370112
x2: 0.632159
x3: 0.832172
x4: 0.230053
x5: 0.931054
x6: 0.781986",
"Arm 8_0
hartmann6: -0.739327 (SEM: 0.1)
x1: 0.183521
x2: 0.71696
x3: 0.632497
x4: 0.284422
x5: 0.319276
x6: 0.375443",
"Arm 9_0
hartmann6: -0.269545 (SEM: 0.1)
x1: 0.0555822
x2: 0.0269895
x3: 0.642712
x4: 0.631488
x5: 0.0222407
x6: 0.920679",
"Arm 10_0
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