Example of simple GP fit, adapted from Stan's example-models repository.
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
sns.set()
x = np.array([-5, -4.9, -4.8, -4.7, -4.6, -4.5, -4.4, -4.3, -4.2, -4.1, -4,
-3.9, -3.8, -3.7, -3.6, -3.5, -3.4, -3.3, -3.2, -3.1, -3, -2.9,
-2.8, -2.7, -2.6, -2.5, -2.4, -2.3, -2.2, -2.1, -2, -1.9, -1.8,
-1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1, -0.9, -0.8, -0.7,
-0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0, 0.1, 0.2, 0.3, 0.4, 0.5,
0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8,
1.9, 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3, 3.1,
3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4, 4.1, 4.2, 4.3, 4.4,
4.5, 4.6, 4.7, 4.8, 4.9, 5])
y = np.array([1.04442478194401, 0.948306088493654, 0.357037759697332, 0.492336514646604,
0.520651364364746, 0.112629866592809, 0.470995468454158, -0.168442254267804,
0.0720344402575861, -0.188108980535916, -0.0160163306512027,
-0.0388792158617705, -0.0600673630622568, 0.113568725264636,
0.447160403837629, 0.664421188556779, -0.139510743820276, 0.458823971660986,
0.141214654640904, -0.286957663528091, -0.466537724021695, -0.308185884317105,
-1.57664872694079, -1.44463024170082, -1.51206214603847, -1.49393593601901,
-2.02292464164487, -1.57047488853653, -1.22973445533419, -1.51502367058357,
-1.41493587255224, -1.10140254663611, -0.591866485375275, -1.08781838696462,
-0.800375653733931, -1.00764767602679, -0.0471028950122742, -0.536820626879737,
-0.151688056391446, -0.176771681318393, -0.240094952335518, -1.16827876746502,
-0.493597351974992, -0.831683011472805, -0.152347043914137, 0.0190364158178343,
-1.09355955218051, -0.328157917911376, -0.585575679802941, -0.472837120425201,
-0.503633622750049, -0.0124446353828312, -0.465529814250314,
-0.101621725887347, -0.26988462590405, 0.398726664193302, 0.113805181040188,
0.331353802465398, 0.383592361618461, 0.431647298655434, 0.580036473774238,
0.830404669466897, 1.17919105883462, 0.871037583886711, 1.12290553424174,
0.752564860804382, 0.76897960270623, 1.14738839410786, 0.773151715269892,
0.700611498974798, 0.0412951045437818, 0.303526087747629, -0.139399513324585,
-0.862987735433697, -1.23399179134008, -1.58924289116396, -1.35105117911049,
-0.990144529089174, -1.91175364127672, -1.31836236129543, -1.65955735224704,
-1.83516148300526, -2.03817062501248, -1.66764011409214, -0.552154350554687,
-0.547807883952654, -0.905389222477036, -0.737156477425302, -0.40211249920415,
0.129669958952991, 0.271142753510592, 0.176311762529962, 0.283580281859344,
0.635808289696458, 1.69976647982837, 1.10748978734239, 0.365412229181044,
0.788821368082444, 0.879731888124867, 1.02180766619069, 0.551526067300283])
N = len(y)
Original Stan model:
// Fit a Gaussian process's hyperparameters
// for squared exponential prior
data {
int<lower=1> N;
vector[N] x;
vector[N] y;
}
transformed data {
vector[N] mu;
for (i in 1:N)
mu[i] <- 0;
}
parameters {
real<lower=0> eta_sq;
real<lower=0> rho_sq;
real<lower=0> sigma_sq;
}
model {
matrix[N,N] Sigma;
// off-diagonal elements
for (i in 1:(N-1)) {
for (j in (i+1):N) {
Sigma[i,j] <- eta_sq * exp(-rho_sq * pow(x[i] - x[j],2));
Sigma[j,i] <- Sigma[i,j];
}
}
// diagonal elements
for (k in 1:N)
Sigma[k,k] <- eta_sq + sigma_sq; // + jitter
eta_sq ~ cauchy(0,5);
rho_sq ~ cauchy(0,5);
sigma_sq ~ cauchy(0,5);
y ~ multi_normal(mu,Sigma);
}
from pymc3 import Model, MvNormal, HalfCauchy, sample, traceplot
import theano.tensor as T
with Model() as gp_fit:
mu = np.zeros(N)
η_sq = HalfCauchy('η_sq', 5)
ρ_sq = HalfCauchy('ρ_sq', 5)
σ_sq = HalfCauchy('σ_sq', 5)
Sigma = T.stack([[η_sq * np.exp(-ρ_sq * (x[i] - x[j])**2) * (σ_sq*int(i==j) or 1)
for i in range(N)]
for j in range(N)])
obs = MvNormal('obs', mu, Sigma, observed=y)
Applied log-transform to η_sq and added transformed η_sq_log to model. Applied log-transform to ρ_sq and added transformed ρ_sq_log to model. Applied log-transform to σ_sq and added transformed σ_sq_log to model.
with gp_fit:
gp_trace = sample(1000)
Assigned NUTS to η_sq_log Assigned NUTS to ρ_sq_log Assigned NUTS to σ_sq_log
--------------------------------------------------------------------------- RecursionError Traceback (most recent call last) <ipython-input-16-87dffea1edf3> in <module>() 1 with gp_fit: 2 ----> 3 gp_trace = sample(1000) /Users/fonnescj/Repositories/pymc3/pymc3/sampling.py in sample(draws, step, start, trace, chain, njobs, tune, progressbar, model, random_seed) 122 model = modelcontext(model) 123 --> 124 step = assign_step_methods(model, step) 125 126 if njobs is None: /Users/fonnescj/Repositories/pymc3/pymc3/sampling.py in assign_step_methods(model, step, methods) 67 68 # Instantiate all selected step methods ---> 69 steps += [s(vars=selected_steps[s]) for s in selected_steps if selected_steps[s]] 70 71 if len(steps)==1: /Users/fonnescj/Repositories/pymc3/pymc3/sampling.py in <listcomp>(.0) 67 68 # Instantiate all selected step methods ---> 69 steps += [s(vars=selected_steps[s]) for s in selected_steps if selected_steps[s]] 70 71 if len(steps)==1: /Users/fonnescj/Repositories/pymc3/pymc3/step_methods/nuts.py in __init__(self, vars, scaling, step_scale, is_cov, state, Emax, target_accept, gamma, k, t0, model, profile, **kwargs) 67 68 if isinstance(scaling, dict): ---> 69 scaling = guess_scaling(Point(scaling, model=model), model=model, vars = vars) 70 71 /Users/fonnescj/Repositories/pymc3/pymc3/tuning/scaling.py in guess_scaling(point, vars, model) 107 model = modelcontext(model) 108 try: --> 109 h = find_hessian_diag(point, vars, model=model) 110 except NotImplementedError: 111 h = fixed_hessian(point, vars, model=model) /Users/fonnescj/Repositories/pymc3/pymc3/tuning/scaling.py in find_hessian_diag(point, vars, model) 101 """ 102 model = modelcontext(model) --> 103 H = model.fastfn(hessian_diag(model.logpt, vars)) 104 return H(Point(point, model=model)) 105 /Users/fonnescj/Repositories/pymc3/pymc3/memoize.py in memoizer(*args, **kwargs) 12 13 if key not in cache: ---> 14 cache[key] = obj(*args, **kwargs) 15 16 return cache[key] /Users/fonnescj/Repositories/pymc3/pymc3/theanof.py in hessian_diag(f, vars) 101 102 if vars: --> 103 return -t.concatenate([hessian_diag1(f, v) for v in vars], axis=0) 104 else: 105 return empty_gradient /Users/fonnescj/Repositories/pymc3/pymc3/theanof.py in <listcomp>(.0) 101 102 if vars: --> 103 return -t.concatenate([hessian_diag1(f, v) for v in vars], axis=0) 104 else: 105 return empty_gradient /Users/fonnescj/Repositories/pymc3/pymc3/theanof.py in hessian_diag1(f, v) 92 return gradient1(g[i], v)[i] 93 ---> 94 return theano.map(hess_ii, idx)[0] 95 96 /Users/fonnescj/anaconda3/lib/python3.5/site-packages/theano/scan_module/scan_views.py in map(fn, sequences, non_sequences, truncate_gradient, go_backwards, mode, name) 67 go_backwards=go_backwards, 68 mode=mode, ---> 69 name=name) 70 71 /Users/fonnescj/anaconda3/lib/python3.5/site-packages/theano/scan_module/scan.py in scan(fn, sequences, outputs_info, non_sequences, n_steps, truncate_gradient, go_backwards, mode, name, profile, allow_gc, strict) 743 # and outputs that needs to be separated 744 --> 745 condition, outputs, updates = scan_utils.get_updates_and_outputs(fn(*args)) 746 if condition is not None: 747 as_while = True /Users/fonnescj/Repositories/pymc3/pymc3/theanof.py in hess_ii(i) 90 91 def hess_ii(i): ---> 92 return gradient1(g[i], v)[i] 93 94 return theano.map(hess_ii, idx)[0] /Users/fonnescj/Repositories/pymc3/pymc3/theanof.py in gradient1(f, v) 42 def gradient1(f, v): 43 """flat gradient of f wrt v""" ---> 44 return t.flatten(t.grad(f, v, disconnected_inputs='warn')) 45 46 /Users/fonnescj/anaconda3/lib/python3.5/site-packages/theano/gradient.py in grad(cost, wrt, consider_constant, disconnected_inputs, add_names, known_grads, return_disconnected, null_gradients) 458 459 var_to_app_to_idx = _populate_var_to_app_to_idx( --> 460 outputs, wrt, consider_constant) 461 462 # build a dict mapping var to the gradient of cost with respect to var /Users/fonnescj/anaconda3/lib/python3.5/site-packages/theano/gradient.py in _populate_var_to_app_to_idx(outputs, wrt, consider_constant) 885 # add all variables that are true ancestors of the cost 886 for output in outputs: --> 887 account_for(output) 888 889 # determine which variables have elements of wrt as a true /Users/fonnescj/anaconda3/lib/python3.5/site-packages/theano/gradient.py in account_for(var) 881 if i not in idx: 882 idx.append(i) --> 883 account_for(ipt) 884 885 # add all variables that are true ancestors of the cost ... last 1 frames repeated, from the frame below ... /Users/fonnescj/anaconda3/lib/python3.5/site-packages/theano/gradient.py in account_for(var) 881 if i not in idx: 882 idx.append(i) --> 883 account_for(ipt) 884 885 # add all variables that are true ancestors of the cost RecursionError: maximum recursion depth exceeded
traceplot(gp_trace)