Advanced Use of gpCAM

The advanced use of gpCAM is about communicating domain knowledge in the form of kernel, acquisition and mean functions, and optimization constraints.

Tailored Mean Functions to Communicate Trends

Often times an overall trend of the model is known in absolute terms or in parametric form. In that case, the user may define their own prior mean function following the example below.

def himmel_blau(gp_obj,x,hyperparameters):

return (x[:,0] ** 2 + x[:,1] - 11.0) ** 2 + (x[:,0] + x[:,1] ** 2 - 7.0) ** 2

Tailored Acquisition Functions for Feature Finding

The acquisition function uses the output of a Gaussian process to steer the experiment or simulation to high-value regions of the search space. You can find an example below.

def upper_confidence_bounds(x,obj):

a = 3.0 #3.0 for 95 percent confidence interval

mean = obj.posterior_mean(x)["f(x)"]

cov = obj.posterior_covariance(x)["v(x)"]

return mean + a * cov ##which is 1-d numpy array

Tailored Kernel Functions for Hard Constraints on the Posterior Mean

Kernel functions are a tremendously powerful tool to communicate hard constraints to the Gaussian process. Examples include the order of differentiability, periodicity, and symmetry of the model function. The kernel can be defined in the way presented below.

def kernel_l2_single_task(x1,x2,hyperparameters,obj):

hps = hyperparameters

distance_matrix = np.zeros((len(x1),len(x2)))

for i in range(len(x1[0])-1):

distance_matrix += abs(np.subtract.outer(x1[:,i],x2[:,i])/hps[1+i])**2

distance_matrix = np.sqrt(distance_matrix)

return hps[0] * obj.matern_kernel_diff1(distance_matrix,1)

Tailored Cost Functions for Optimizing Data Acquisition when Costs are Present

Cost functions are very useful when the main effort of exploration does not come from the data acquisition itself but from the motion through the search space. gpCAM can use cost and cost update functions. You can find examples for both below. If costs are recorded during data acquisition, gpCAM can use them to update the cost function repeatedly.

def l2_cost(origin,x,arguments = None):

offset = arguments["offset"]

slope = arguments["slope"]

return slope*np.linalg.norm(np.abs(np.subtract(origin,x)), axis = 1)+offset

def update_l2_cost_function(costs, bounds, parameters):

print("Cost adjustment in progress...")

print("old cost parameters: ",parameters)

###remove outliers:

origins = []

points = []

motions = []

c = []

cost_per_motion = []

for i in range(len(costs)):

origins.append(costs[i]["origin"])

points.append(costs[i]["point"])

motions.append(abs(costs[i]["origin"] - costs[i]["point"]))

c.append(costs[i]["cost"])

cost_per_motion.append(costs[i]["cost"]/l2_cost(costs[i]["origin"],costs[i]["point"], parameters))

mean_costs_per_distance = np.mean(np.asarray(cost_per_motion))

sd = np.std(np.asarray(cost_per_motion))

for element in cost_per_motion:

if (

element >= mean_costs_per_distance - 2.0 * sd

and element <= mean_costs_per_distance + 2.0 * sd

):

continue

else:

motions.pop(cost_per_motion.index(element))

c.pop(cost_per_motion.index(element))

origins.pop(cost_per_motion.index(element))

points.pop(cost_per_motion.index(element))

cost_per_motion.pop(cost_per_motion.index(element))

res = devo(compute_l2_cost_misfit, bounds, args = (origins, points,c), tol=1e-6, disp=True, maxiter=300,popsize=20,polish=False)

arguments = {"offset": res["x"][0],"slope": res["x"][1]}

print("New cost parameters: ", arguments)

return arguments

Constrained Optimization

This feature is currently in development.