def compute_boltzmann(energies, debug=False):
# create Boltzmann distribution
# https://en.wikipedia.org/wiki/Boltzmann_distribution
# p_i = exp( -E_i / kT ) / ∑ exp( -E_j / kT )
# where:
# p_i is the probability of state i occuring
# E_i is the energy of state i
# E_j is the energy of state j, where the ∑ in the denominator
# iterates over all states j
# first we calculate exp( -E_i / kT ) for all states
if len(energies) == 0:
return []
if debug:
print("Boltzmann Distribution")
divisor = bigfloat.BigFloat.exact(0.0)
for energy in energies:
ep = bigfloat.exp(-energy)
if debug:
print("energy, ep = %g, %g" % (energy, ep))
# divisor = ∑ exp( -E_j / kT )
divisor += ep
probabilities = []
for energy in energies:
# p_i = exp( -E_i / kT ) / divisor
numerator = bigfloat.exp(-energy)
probability = numerator / divisor
# save probability to dictionary
probabilities.append(float(probability))
if debug:
print("%s / %s = %s" % (numerator, divisor, probability))
return probabilities
def compute_boltzmann(energies, debug):
# create Boltzmann distribution
# https://en.wikipedia.org/wiki/Boltzmann_distribution
# p_i = exp( -E_i / kT ) / ∑ exp( -E_j / kT )
# where:
# p_i is the probability of state i occuring
# E_i is the energy of state i
# E_j is the energy of state j, where the ∑ in the denominator
# iterates over all states j
# first we calculate exp( -E_i / kT ) for all states
if len(energies) == 0:
return []
if debug:
print("Boltzmann Distribution")
divisor = BigFloat.exact(0.0)
for energy in energies:
ep = bigfloat.exp(-energy)
if debug:
print("energy, ep = %g, %g" % (energy, ep))
# divisor = ∑ exp( -E_j / kT )
divisor += ep
probabilities = []
for energy in energies:
# p_i = exp( -E_i / kT ) / divisor
numerator = bigfloat.exp(-energy)
probability = numerator / divisor
# save probability to dictionary
probabilities.append(float(probability))
if debug:
print("%s / %s = %s" % (numerator, divisor, probability))
return probabilities
def GauGo_eq(temp, E, E_D, T_pk, theta):
"""Gaussian - Gompertz model, used for estimating trait values at
a given temperature."""
global max_trait
function = max_trait * exp(1) ** (-E * (temp - T_pk) * (temp \
- T_pk) - exp(1) ** (E_D * (temp - T_pk) - theta))
return numpy.array(map(log, function), dtype=numpy.float64)
def JoLe_eq(temp, B0, E, E_D, T_pk):
"""Johnson - Lewin model, used for estimating trait values at
a given temperature."""
global K
function = B0 * exp(1) ** (-E / (K * temp)) / (1 + exp(1) ** (1 / K
* (E_D / T_pk + K * log(E / (E_D - E)) - (E_D / temp))))
return numpy.array(map(log, function), dtype=numpy.float64)
def interval_exp(interval, context_down, context_up):
lower, upper = interval
out_lower, out_upper = [
bf.exp(lower, context_down),
bf.exp(upper, context_up)
]
out_interval = [out_lower, out_upper]
derivs = [out_lower, 0], [0, out_upper]
return out_interval, derivs
def sharpe_schoolf_eq(temp, B0, E, E_D, T_pk):
"""A function to estimate the trait value at a given temperature according
to the Sharpe-Schoolfield model."""
global K
function = B0 * exp(1)**(-E * (
(1 / (K * temp)) -
(1 / (K * 273.15)))) / (1 +
(E /
(E_D - E)) * exp(1)**(E_D / K *
(1 / T_pk - 1 / temp)))
return numpy.array(map(log, function), dtype=numpy.float64)
def test_exp(self):
self.assertEqual(bigfloat.Number("1"),
bigfloat.exp(bigfloat.Number("0")))
self.assertEqual(
bigfloat.Number("1") / bigfloat.exp(bigfloat.Number("1")),
bigfloat.exp(bigfloat.Number("-1")))
self.assertEqual(
bigfloat.Number("1") / bigfloat.exp(bigfloat.Number("123")),
bigfloat.exp(bigfloat.Number("-123")))
self.assertEqual(
bigfloat.Number("1") / bigfloat.exp(bigfloat.Number("0.3")),
bigfloat.exp(bigfloat.Number("-0.3")))
def test_almost_equal(a, b, bound):
err = a - b
self.assertGreaterEqual(bigfloat.Number(bound), err)
self.assertLessEqual(bigfloat.Number("-" + bound), err)
test_almost_equal(bigfloat.exp(bigfloat.Number("1")),
bigfloat.Number("2.71828 18284"), "0.00001")
def test_exp_helper(a1, a2, a3, b, bound):
a = bigfloat.exp(bigfloat.Number(a1)) * bigfloat.exp(
bigfloat.Number(a2)) * bigfloat.exp(bigfloat.Number(a3))
b = bigfloat.exp(bigfloat.Number(b))
test_almost_equal(a, b, bound)
test_exp_helper("1", "1", "1", "3", "0.0000001")
test_exp_helper("2", "3", "7", "12", "0.0000001")
test_exp_helper("0.2", "0.3", "0.9", "1.4", "0.0000001")
def fn_logprior(d_t,R_t,logalpha_pmf,logbeta_pmf,maxlhs,beta_Z,nruleslen,lhs_len):
#The prior will be _proportional_ to this -> we drop the normalization for alpha
#beta_Z is the normalization for beta, except the terms that need to be dropped due to running out of rules.
#log p(d_star) = log \alpha(m|lbda) + sum_{i=1...m} log beta(l_i | eta) + log gamma(r_i | l_i)
#The length of the list (m) is R_t
#Get logalpha (length of list) (overloaded notation in this code, unrelated to the prior hyperparameter alpha)
logprior = 0.
logalpha = logalpha_pmf[R_t] #this is proportional to logalpha - we have dropped the normalization for truncating based on total number of rules
logprior += logalpha
empty_rulelens = []
nlens = zeros(maxlhs+1)
for i in range(R_t):
l_i = lhs_len[d_t[i]]
logbeta = logbeta_pmf[l_i] - log(beta_Z - sum([bigfloat.exp(logbeta_pmf[l_j]) for l_j in empty_rulelens])) #The correction for exhausted rule lengths
#Finally loggamma
loggamma = -log(nruleslen[l_i] - nlens[l_i])
#And now check if we have exhausted all rules of a certain size
nlens[l_i] += 1
if nlens[l_i] == nruleslen[l_i]:
empty_rulelens.append(l_i)
elif nlens[l_i] > nruleslen[l_i]:
raise Exception
#Add 'em in
logprior += logbeta
logprior += loggamma
#All done
return logprior
def TanSig(x):
# n=len(x)
# if(n==1):
# print x
y = 2 / (1 + bigfloat.exp((numpy.multiply(-2, x)))) - 1
# print y
return y
def partition_functions_for_each_configuration(self):
"""
Inversely solve the free energy to get partition function for :math:`j` th configuration by
.. math::
Z_{j}(T, V) = \exp \\bigg( -\\frac{ F_{j}(T, V) }{ k_B T } \\bigg).
:return: A matrix, the partition function of each configuration of each volume.
"""
try:
import bigfloat
except ImportError:
raise ImportError(
"Install ``bigfloat`` package to use {0} object!".format(
self.__class__.__name__))
with bigfloat.precision(self.precision):
return np.array([
bigfloat.exp(d)
for d in # shape = (# of volumes for each configuration, 1)
logsumexp(
-self.aligned_free_energies_for_each_configuration.T /
(K * self.temperature),
axis=1,
b=self.degeneracies)
])
def sigmoidx(x, derivative):
# the function
s = 1 / (1 + bfloat.exp(-x / 20, bfloat.precision(30)))
# if derivative is true, the output is the derivative
if derivative == True:
return (s + s * (1 - s))
else:
return s
def sigmoid(s):
y = []
for k in s:
z=[]
for l in k:
z.append( bigfloat.div( 1. , bigfloat.add(1,bigfloat.exp(-l,bigfloat.precision(precision)))))
y.append(z)
return np.array(y)
def schoolf_eq(temp, B0, E, E_D, T_pk):
"""Schoolfield model, used for estimating trait values at
a given temperature."""
global K
function = B0 * exp(1) ** (-E * ((1/(K*temp)) - (1/(K*283.15)))) / (1 + (E/(E_D - E)) * exp(1) ** (E_D / K
* (1 / T_pk - 1 / temp)))
return numpy.array(map(log, function), dtype=numpy.float64)
def predict_dpi(x, s):
num = 0
den = 0
for i in range(len(s)):
y_i = s[i, len(x)]
x_i = s[i, :len(x)]
exp = bg.exp(-0.25 * norm(x - x_i)**2)
num += y_i * exp
den += exp
return num / den
def predict_dpi(x, s):
num = 0
den = 0
for i in range(len(s)):
y_i = s[i, len(x)]
x_i = s[i, :len(x)]
ex = bg.exp(-0.25 * (math.pow(euclidean_distance(x, x_i), 2)))
num = bg.add(num, bg.mul(y_i, ex))
den = bg.add(den, ex)
return bg.div(num, den)
def GauGo(params, temp, data):
"""Gaussian - Gompertz model, used by the optimizer."""
global max_trait
# Get the parameters.
E = params['E'].value
E_D = params['E_D'].value
T_pk = params['T_pk'].value
theta = params['theta'].value
# Prevent E being higher than E_D, by punishing the optimizer with huge
# numbers when that is the case.
if E >= E_D:
return 1e10
# Otherwise, fit the model!
function = max_trait * exp(1) ** (-E * (temp - T_pk) * (temp \
- T_pk) - exp(1) ** (E_D * (temp - T_pk) - theta))
return numpy.array(map(log, function) - data, dtype=numpy.float64)
def JoLe(params, temp, data):
"""Johnson - Lewin model, used by the optimizer."""
global K
# Get the parameters.
E = params['E'].value
E_D = params['E_D'].value
T_pk = params['T_pk'].value
B0 = params['B0'].value
# Prevent E being higher than E_D, by punishing the optimizer with huge
# numbers when that is the case.
if E >= E_D:
return 1e10
# Otherwise, fit the model!
function = B0 * exp(1) ** (-E / (K * temp)) / (1 + exp(1) ** (1 / K
* (E_D / T_pk + K * log(E / (E_D - E)) - (E_D / temp))))
return numpy.array(map(log, function) - data, dtype=numpy.float64)
def degdist_bigfloat(self, maxdeg, n, mindeg=0):
"""Returns the degree distribution from 0 to maxdeg degree.
It should be use with the (iterated) link probability measure.
Parameters:
maxdeg: the maximal degree for we calculate the degree distribution
n: the size of the network (the number of vertices)
Returns:
rho: the degree distribution as a list with length of maxdeg+1.
The index k gives the probability of having degree k.
"""
assert isinstance(n, int) and isinstance(maxdeg, int) and n > maxdeg
import bigfloat
context = bigfloat.Context(precision=10)
divs = self.divs
n_intervals = len(divs)
lengths = [divs[i] - divs[i - 1] for i in xrange(1, n_intervals)]
lengths.insert(0, divs[0])
log_lengths = numpy.log(lengths)
# Eq. 5, where ...
avgdeg = [
bigfloat.BigFloat(n * sum(
[self.probs[i][j] * lengths[j] for j in xrange(n_intervals)]),
context=context) for i in xrange(n_intervals)
]
#log_factorial = [ 0.5*bigfloat.log(2*math.pi, context=context) + (d+.5)*bigfloat.log(d, context=context) - d
# for d in xrange(1,maxdeg+1) ]
log_factorial = [
bigfloat.log(bigfloat.factorial(k), context=context)
for k in xrange(1, maxdeg + 1)
]
log_factorial.insert(0, 0)
rho = [bigfloat.BigFloat(0, context=context)] * (maxdeg + 1)
# Eq. 4
for i in xrange(n_intervals):
# Eq. 5
log_rho_i = [
(bigfloat.mul(k, bigfloat.log(avgdeg[i]), context=context) -
log_factorial[k] - avgdeg[i])
for k in xrange(mindeg, maxdeg + 1)
]
log_rho_i_length = [
log_rho_i[k] + log_lengths[i]
for k in xrange(mindeg, maxdeg + 1)
]
for k in xrange(mindeg, maxdeg + 1):
rho[k] += bigfloat.exp(log_rho_i_length[k], context=context)
return rho
def sharpe_schoolf(params, temp, data):
"""A function to be used by the optimizer to fit the
Sharpe-Schoolfield model."""
global K
E = params['E'].value
E_D = params['E_D'].value
T_pk = params['T_pk'].value
B0 = params['B0'].value
# Penalize the optimizer when E becomes much higher than E_D.
if E >= E_D:
return 1e10
function = B0 * exp(1)**(-E * (
(1 / (K * temp)) -
(1 / (K * 273.15)))) / (1 +
(E /
(E_D - E)) * exp(1)**(E_D / K *
(1 / T_pk - 1 / temp)))
return numpy.array(map(log, function) - data, dtype=numpy.float64)
def relative_likelihood_result_calculator(population):
"""
Given a population, this function calculates the relative likelihood result (it is a way to make
the likelihood result bigger) to every chromosome.
Args:
population : LIST[Chromosome(), Chromosome(), ...]
A list filled with 'Chromosome' objects
"""
with bf.quadruple_precision:
total = sum_likelihood_result(population)
for i in range(0, len(population)):
log_likelihood_result = bf.exp(bf.BigFloat(str(population[i].get_log_likelihood_result())))
population[i].set_relative_likelihood_result(float(bf.div(log_likelihood_result, total)))
def relative_likelihood_result_calculator(population):
"""
Given a population, this function calculates the relative likelihood result (it is a way to make
the likelihood result bigger) to every chromosome.
Args:
population : LIST[Chromosome(), Chromosome(), ...]
A list filled with 'Chromosome' objects
"""
with bf.quadruple_precision:
total = sum_likelihood_result(population)
for i in range(0, len(population)):
log_likelihood_result = bf.exp(bf.BigFloat(str(population[i].get_log_likelihood_result())))
population[i].set_relative_likelihood_result(float(bf.div(log_likelihood_result, total)))
def schoolf(params, temp, data):
"""Schoolfield model, used by the optimizer."""
global K
E = params['E'].value
E_D = params['E_D'].value
T_pk = params['T_pk'].value
B0 = params['B0'].value
if E >= E_D:
return 1e10
function = B0 * exp(1) ** (-E * ((1/(K*temp)) - (1/(K*283.15)))) / (1 + (E/(E_D - E)) * exp(1) ** (E_D / K
* (1 / T_pk - 1 / temp)))
return numpy.array(map(log, function) - data, dtype=numpy.float64)
def get_boltzmann_distribution(energy_by_arm):
R = 8.3144621 # gas constant
T = 293.15 # room temperature
factor = 4184.0 # joules_per_kcal
boltzmann_distribution = []
for dG in energy_by_arm:
ps = []
total = bigfloat.BigFloat(0)
for energy in dG:
p = bigfloat.exp((-energy*factor)/(R*T), bigfloat.precision(1000))
ps.append(p)
total = bigfloat.add(total, p)
normal_ps = []
for p in ps:
normal_ps.append(float(bigfloat.div(p,total)))
boltzmann_distribution.append(numpy.array(normal_ps))
return boltzmann_distribution
def predict_dpi(x, s):
"""Predict the average price change deltap_i, 1 <= i <= 3.
Args:
x: A numpy array of floats representing previous 180, 360, or 720 prices.
s: A 2-dimensional numpy array generated by choose_effective_centers().
Returns:
A big float representing average price change deltap_i.
"""
num = 0
den = 0
for i in range(len(s)):
y_i = s[i, len(x)]
x_i = s[i, :len(x)]
exp = bg.exp(-0.25 * norm(x - x_i)**2)
num += y_i * exp
den += exp
return num / den
def sum_likelihood_result(population):
"""
Sums all the likelihood results values on a population.
Args:
population : LIST[Chromosome(), Chromosome(), ...]
A list filled with 'Chromosome' objects
Returns:
INT
The sum of all likelihood results on a population
"""
with bf.quadruple_precision:
total = bf.BigFloat("0.0")
for i in range(0, len(population)):
log_likelihood_result = bf.exp(bf.BigFloat(str(population[i].get_log_likelihood_result())))
total = bf.add(total, log_likelihood_result)
return total
def sum_likelihood_result(population):
"""
Sums all the likelihood results values on a population.
Args:
population : LIST[Chromosome(), Chromosome(), ...]
A list filled with 'Chromosome' objects
Returns:
INT
The sum of all likelihood results on a population
"""
with bf.quadruple_precision:
total = bf.BigFloat("0.0")
for i in range(0, len(population)):
log_likelihood_result = bf.exp(bf.BigFloat(str(population[i].get_log_likelihood_result())))
total = bf.add(total, log_likelihood_result)
return total
def bayesdl_mcmc(numiters,thinning,alpha,lbda,eta,X,Y,nruleslen,lhs_len,maxlhs,permsdic,burnin,rseed,d_init):
#initialize
perms = []
if rseed:
random.seed(rseed)
#Do some pre-computation for the prior
beta_Z,logalpha_pmf,logbeta_pmf = prior_calculations(lbda,len(X),eta,maxlhs)
if d_init: #If we want to begin our chain at a specific place (e.g. to continue a chain)
d_t = Pickle.loads(d_init)
d_t.extend([i for i in range(len(X)) if i not in d_t])
R_t = d_t.index(0)
N_t = compute_rule_usage(d_t,R_t,X,Y)
else:
d_t,R_t,N_t = initialize_d(X,Y,lbda,eta,lhs_len,maxlhs,nruleslen) #Otherwise sample the initial value from the prior
#Add to dictionary which will store the sampling results
a_t = Pickle.dumps(d_t[:R_t+1]) #The antecedent list in string form
if a_t not in permsdic:
permsdic[a_t][0] = fn_logposterior(d_t,R_t,N_t,alpha,logalpha_pmf,logbeta_pmf,maxlhs,beta_Z,nruleslen,lhs_len) #Compute its logposterior
if burnin == 0:
permsdic[a_t][1] += 1 #store the initialization sample
#iterate!
for itr in range(numiters):
#Sample from proposal distribution
d_star,Jratio,R_star,step = proposal(d_t,R_t,X,Y,alpha)
#Compute the new posterior value, if necessary
a_star = Pickle.dumps(d_star[:R_star+1])
if a_star not in permsdic:
N_star = compute_rule_usage(d_star,R_star,X,Y)
permsdic[a_star][0] = fn_logposterior(d_star,R_star,N_star,alpha,logalpha_pmf,logbeta_pmf,maxlhs,beta_Z,nruleslen,lhs_len)
#Compute the metropolis acceptance probability
q = bigfloat.exp(permsdic[a_star][0] - permsdic[a_t][0] + Jratio)
u = random.random()
if u < q:
#then we accept the move
d_t = list(d_star)
R_t = int(R_star)
a_t = str(a_star)
#else: pass
if itr > burnin and itr % thinning == 0:
##store
permsdic[a_t][1] += 1
perms.append(a_t)
return permsdic,perms
def predict_dpi(x, s):
"""Predict the average price change Δp_i, 1 <= i <= 3.
Args:
x: A numpy array of floats representing previous 180, 360, or 720 prices.
s: A 2-dimensional numpy array generated by choose_effective_centers().
Returns:
A big float representing average price change Δp_i.
"""
num = 0
den = 0
for i in range(len(s)):
y_i = s[i, len(x)]
x_i = s[i, :len(x)]
exp = bg.exp(-0.25 * norm(x - x_i) ** 2)
num += y_i * exp
den += exp
return num / den
def degdist_bigfloat(self, maxdeg, n, mindeg=0):
"""Returns the degree distribution from 0 to maxdeg degree.
It should be use with the (iterated) link probability measure.
Parameters:
maxdeg: the maximal degree for we calculate the degree distribution
n: the size of the network (the number of vertices)
Returns:
rho: the degree distribution as a list with length of maxdeg+1.
The index k gives the probability of having degree k.
"""
assert isinstance(n, int) and isinstance(maxdeg, int) and n > maxdeg
import bigfloat
context = bigfloat.Context(precision=10)
divs = self.divs
n_intervals = len(divs)
lengths = [divs[i] - divs[i-1] for i in xrange(1, n_intervals)]
lengths.insert(0, divs[0])
log_lengths = numpy.log(lengths)
# Eq. 5, where ...
avgdeg = [bigfloat.BigFloat(n*sum([self.probs[i][j]*lengths[j]
for j in xrange(n_intervals)]), context=context) for i in xrange(n_intervals)]
#log_factorial = [ 0.5*bigfloat.log(2*math.pi, context=context) + (d+.5)*bigfloat.log(d, context=context) - d
# for d in xrange(1,maxdeg+1) ]
log_factorial = [bigfloat.log(bigfloat.factorial(k), context=context)
for k in xrange(1, maxdeg+1)]
log_factorial.insert(0, 0)
rho = [bigfloat.BigFloat(0, context=context)] * (maxdeg+1)
# Eq. 4
for i in xrange(n_intervals):
# Eq. 5
log_rho_i = [(bigfloat.mul(k, bigfloat.log(avgdeg[i]), context=context) - log_factorial[k] - avgdeg[i])
for k in xrange(mindeg, maxdeg+1)]
log_rho_i_length = [log_rho_i[k] + log_lengths[i]
for k in xrange(mindeg, maxdeg+1)]
for k in xrange(mindeg, maxdeg+1):
rho[k] += bigfloat.exp(log_rho_i_length[k], context=context)
return rho
def calculate_operational_niche_width(fitted_params, B_pk, T_ref):
"""Calculate the operational niche width for this fit, i.e.
the difference in temperatures between the thermal optimum (T_pk)
and the temperature at which performance is half of the
maximum (B_pk/2)."""
dist_to_half_B_pk = 999999
current_temperature = -9999
for temp in numpy.arange(263.15, fitted_params['T_pk'].value + 0.01, 0.01):
trait_val_at_temp = exp(1)**sharpe_schoolf_eq(
numpy.array([temp]), fitted_params['B0'].value,
fitted_params['E'].value, fitted_params['E_D'].value,
fitted_params['T_pk'].value)
current_dist = abs((B_pk / 2.0) - trait_val_at_temp)
if current_dist < dist_to_half_B_pk:
dist_to_half_B_pk = current_dist
current_temperature = temp
return fitted_params['T_pk'].value - current_temperature
def _static_part(self) -> Vector:
"""
Calculate the static contribution to the partition function.
:return: The static contribution on the temperature-volume grid.
"""
try:
import bigfloat
except ImportError:
raise ImportError(
"You need to install ``bigfloat`` package to use {0} object!".
format(self.__class__.__name__))
with bigfloat.precision(self.precision):
return np.array([
bigfloat.exp(d)
for d in # shape = (# of volumes for each configuration, 1)
logsumexp(-self.static_energies / (K * self.temperature),
axis=1,
b=self.degeneracies)
])
def bigfloat_exp(self, hypothesis):
return [bf.exp(i) for i in hypothesis]
# Deal with overflow in exp using numpy
import bigfloat
bigfloat.exp(5000, bigfloat.precision(100))
# -> BigFloat.exact('2.9676283840236670689662968052896e+2171', precision=100)
# Deal with overflow in exp using numpy
import bigfloat
bigfloat.exp(5000,bigfloat.precision(100))
# -> BigFloat.exact('2.9676283840236670689662968052896e+2171', precision=100)
def parallel_ONW_for_bootstrap(arg):
global temporary_dataset
numpy.random.seed(arg)
resampled_indices = numpy.floor(
numpy.random.rand(len(temporary_dataset)) *
len(temporary_dataset)).astype(int)
resampled_dataset = numpy.array(temporary_dataset)[
resampled_indices, :].tolist()
(B0_start, E_start, T_pk_start,
E_D_start) = generate_starting_values(resampled_dataset)
temps = []
trait_vals = []
for row in resampled_dataset:
# Convert temperatures to Kelvin.
temps.append(float(row[4]) + 273.15)
trait_vals.append(log(float(row[5])))
# Convert temps and trait_vals to numpy arrays.
temps = numpy.array(temps, dtype=numpy.float64)
trait_vals = numpy.array(trait_vals, dtype=numpy.float64)
# Prepare the parameters and their bounds.
params = Parameters()
params.add('B0', value=B0_start)
params.add('E', value=E_start, min=0.00001, max=10)
params.add('E_D', value=E_D_start, min=0.00001, max=30)
params.add('T_pk', value=T_pk_start, min=273.15 - 10, max=273.15 + 150)
try:
# Try and fit!
result = minimize(sharpe_schoolf,
params,
args=(temps, trait_vals),
xtol=1e-12,
ftol=1e-12,
maxfev=100000)
except Exception:
# If fitting failed, return None.
result = None
if result is not None:
# In the highly unlikely scenario that E == E_D, add a tiny number to
# E_D to avoid division by zero.
if params['E'].value == params['E_D'].value:
params['E_D'].value += 0.000000000000001
B_pk = exp(1)**sharpe_schoolf_eq(numpy.array([params['T_pk'].value]),
params['B0'].value, params['E'].value,
params['E_D'].value,
params['T_pk'].value)
current_operational_niche_width = calculate_operational_niche_width(
params, B_pk[0], T_ref=273.15)
return (current_operational_niche_width)
else:
return (None)
def fit_sharpe_schoolfield(dataset, B0_start, E_start, T_pk_start, E_D_start):
"""Fit the Sharpe-Schoolfield model."""
global K
# Store temperatures and log-transformed trait values.
temps = []
trait_vals = []
for row in dataset:
# Convert temperatures to Kelvin.
temps.append(float(row[4]) + 273.15)
trait_vals.append(log(float(row[5])))
# Convert temps and trait_vals to numpy arrays.
temps = numpy.array(temps, dtype=numpy.float64)
trait_vals = numpy.array(trait_vals, dtype=numpy.float64)
# Prepare the parameters and their bounds.
params = Parameters()
params.add('B0', value=B0_start)
params.add('E', value=E_start, min=0.00001, max=10)
params.add('E_D', value=E_D_start, min=0.00001, max=30)
params.add('T_pk', value=T_pk_start, min=273.15 - 10, max=273.15 + 150)
try:
# Try and fit!
result = minimize(sharpe_schoolf,
params,
args=(temps, trait_vals),
xtol=1e-12,
ftol=1e-12,
maxfev=100000)
except Exception:
# If fitting failed, return.
return None
# Check if we have a covariance matrix (just in case we don't for any reason).
if result.covar is None:
return None
# Since we're still here, fitting was successful!
# In the highly unlikely scenario that E == E_D, add a tiny number to
# E_D to avoid division by zero.
if params['E'].value == params['E_D'].value:
params['E_D'].value += 0.000000000000001
# Calculate B(T_ref), B_pk, and the operational niche width.
B_T_ref = exp(1)**sharpe_schoolf_eq(numpy.array([273.15]),
params['B0'].value, params['E'].value,
params['E_D'].value,
params['T_pk'].value)
B_pk = exp(1)**sharpe_schoolf_eq(numpy.array([params['T_pk'].value]),
params['B0'].value, params['E'].value,
params['E_D'].value, params['T_pk'].value)
operational_niche_width = calculate_operational_niche_width(params,
B_pk[0],
T_ref=273.15)
robjects.r("rm(list=ls())")
robjects.r.assign("K", K)
robjects.r.assign("covar_mat", result.covar)
robjects.r.assign(
"Schoolf_coeffs",
robjects.FloatVector((params['B0'].value, params['E'].value,
params['E_D'].value, params['T_pk'].value)))
B_T_ref_stderr = robjects.r(
"deltamethod( ~ x1 / (1 + (x2/(x3 - x2)) * exp(x3 / K * (1 / x4 - 1 / 273.15))), Schoolf_coeffs, covar_mat)^2"
)[0]
B_T_ref_1_4_stderr = robjects.r(
"deltamethod( ~ (x1 / (1 + (x2/(x3 - x2)) * exp(x3 / K * (1 / x4 - 1 / 273.15))))^(1/4), Schoolf_coeffs, covar_mat)^2"
)[0]
log_E_stderr = robjects.r(
"deltamethod(~ log(x2), Schoolf_coeffs, covar_mat)^2")[0]
T_pk_squared_stderr = robjects.r(
"deltamethod(~ x4^2, Schoolf_coeffs, covar_mat)^2")[0]
log_E_D_stderr = robjects.r(
"deltamethod(~ log(x3), Schoolf_coeffs, covar_mat)^2")[0]
B_pk_stderr = robjects.r(
"deltamethod( ~ (x1 * exp((-x2) * (1/(K * x4) - 1/(K * 273.15)))) / (1 + (x2/(x3 - x2))), Schoolf_coeffs, covar_mat)^2"
)[0]
log_B_pk_stderr = robjects.r(
"deltamethod( ~ log((x1 * exp((-x2) * (1/(K * x4) - 1/(K * 273.15)))) / (1 + (x2/(x3 - x2)))), Schoolf_coeffs, covar_mat)^2"
)[0]
[operational_niche_width_stderr, log_operational_niche_width_stderr
] = bootstrap_operational_niche_width(dataset)
# Calculate the fitted trait values.
pred = exp(1)**sharpe_schoolf_eq(temps, params['B0'].value,
params['E'].value, params['E_D'].value,
params['T_pk'].value)
pred = numpy.array(pred, dtype=numpy.float64)
# Collect measured trait values without log transformation.
trait_vals_no_log = []
for row in dataset:
trait_vals_no_log.append(float(row[5]))
# Calculate the residual sum of squares.
residuals = trait_vals_no_log - pred
rss = sum(residuals**2)
# Temperatures ...
temperatures = ""
for element in temps:
temperatures += str(element - 273.15) + ','
temperatures = temperatures[:-1]
# Trait Values ...
trait_values = ""
for element in trait_vals_no_log:
trait_values += str(element) + ','
trait_values = trait_values[:-1]
# If, for whatever reason, the residual sum of squares is
# 'not a number' or infinite, then return.
if numpy.isnan(rss) or numpy.isinf(rss):
return None
# Calculate the total sum of squares.
tss = sum((trait_vals_no_log - numpy.mean(trait_vals_no_log))**2)
# Calculate the R-squared value.
if tss == 0:
fit_goodness = 1
else:
fit_goodness = 1 - (rss / tss)
# Number of data points before and after T_pk ...
count_temps = []
points_before_pk = 0
points_after_pk = 0
for row in dataset:
if float(row[4]) in count_temps:
continue
else:
count_temps.append(float(row[4]))
if float(row[4]) < (params['T_pk'].value - 273.15):
points_before_pk += 1
elif float(row[4]) > (params['T_pk'].value - 273.15):
points_after_pk += 1
result_line = [
dataset[0][1], dataset[0][7], dataset[0][8], temperatures,
trait_values, dataset[0][2],
str(B_T_ref[0]**(1 / 4.0)),
str(B_T_ref_1_4_stderr),
str(log(params['E'].value)),
str(log_E_stderr),
str(log(operational_niche_width)),
str(log_operational_niche_width_stderr),
str((params['T_pk'].value - 273.15)**2),
str(T_pk_squared_stderr),
str(log(B_pk[0])),
str(log_B_pk_stderr),
str(log(params['E_D'].value)),
str(log_E_D_stderr),
str(fit_goodness),
str(points_before_pk),
str(points_after_pk)
]
return result_line
def schoolfield_model(dataset, B0_start, E_start, T_pk_start, E_D_start):
"""Prepare the Schoolfield model."""
global K
# Store temperatures and log-transformed trait values.
temps = []
trait_vals = []
for row in dataset:
# Convert temperatures to Kelvin.
temps.append(float(row[4]) + 273.15)
trait_vals.append(log(float(row[5])))
# Convert temps and trait_vals to numpy arrays.
temps = numpy.array(temps, dtype=numpy.float64)
trait_vals = numpy.array(trait_vals, dtype=numpy.float64)
# Prepare the parameters and their bounds.
params = Parameters()
params.add('B0', value=B0_start)
params.add('E', value=E_start, min=0.00001, max=30)
params.add('E_D', value=E_D_start, min=0.00001, max=50)
params.add('T_pk', value=T_pk_start, min=273.15, max=273.15 + 150)
try:
# Set the random seed.
numpy.random.seed(1337)
# Try and fit!
minimize(schoolf, params, args=(temps, trait_vals))
except Exception:
# If fitting failed, return.
return None, None, None, None
# Since we're still here, fitting was successful!
points_before_pk = 0
points_after_pk = 0
# Collect measured trait values without log transformation.
trait_vals_no_log = []
for row in dataset:
trait_vals_no_log.append(float(row[5]))
# Get the number of data points before and after T_pk.
if float(row[4]) < (params['T_pk'].value - 273.15):
points_before_pk += 1
elif float(row[4]) > (params['T_pk'].value - 273.15):
points_after_pk += 1
# Calculate the estimated trait values.
pred = exp(
1) ** schoolf_eq(temps,
params['B0'].value,
params['E'].value,
params['E_D'].value,
params['T_pk'].value)
pred = numpy.array(pred, dtype=numpy.float64)
# Calculate the residual sum of squares.
residuals = trait_vals_no_log - pred
rss = sum(residuals ** 2)
# If, for whatever reason, the residual sum of squares is
# 'not a number' or infinite, then return.
if numpy.isnan(rss) or numpy.isinf(rss):
return None, None, None, None
# Calculate the total sum of squares.
tss = sum((trait_vals_no_log - numpy.mean(trait_vals_no_log)) ** 2)
# If the total sum of squares is 0, then we have a perfect fit!
if tss == 0:
fit_goodness = 1
# Otherwise, calculate the R-Squared value.
else:
fit_goodness = 1 - (rss / tss)
# Get the values of the three Information Criteria for this fit.
AIC_schoolfield = AICrss(len(temps), 4, rss)
BIC_schoolfield = BICrss(len(temps), 4, rss)
HQC_schoolfield = HQCrss(len(temps), 4, rss)
# Create a list of Species, Standardised Species, Reference, Trait,
# Latitude, Longitude ...
result_line = [
dataset[0][0], dataset[0][1], dataset[0][2], dataset[0][3],
dataset[0][7], dataset[0][8]
]
# Temperatures ...
temperatures = ""
for element in temps:
temperatures += str(element - 273.15) + ','
temperatures = temperatures[:-1]
result_line.append(temperatures)
# Trait Values ...
trait_values = ""
for element in trait_vals_no_log:
trait_values += str(element) + ','
trait_values = trait_values[:-1]
result_line.append(trait_values)
# B0 and B0_stderr ...
result_line.append(str(params['B0'].value))
result_line.append(str(params['B0'].stderr))
# E and E_stderr ...
result_line.append(str(params['E'].value))
result_line.append(str(params['E'].stderr))
# T_pk and T_pk_stderr ...
result_line.append(str(params['T_pk'].value - 273.15))
result_line.append(str(params['T_pk'].stderr))
# E_D and E_D_stderr ...
result_line.append(str(params['E_D'].value))
result_line.append(str(params['E_D'].stderr))
# No theta and theta_stderr for this model!
result_line.append("NA")
result_line.append("NA")
# R-squared ...
result_line.append(str(fit_goodness))
# Formula ...
result_line.append(
"log(Trait_value) = log(B0 * exp(-E * ((1/(" + str(K) \
+ " * temp)) - (1/(" + str(K) \
+ " * 273.15))))/(1 + (E/(E_D - E)) * exp(E_D/" \
+ str(K) + " * (1/T_pk - 1/temp))))")
# Model name ...
result_line.append("Schoolfield")
# Implementation ...
result_line.append("lmfit.minimize (Python package)")
# Number of data points before and after T_pk ...
result_line.append(str(points_before_pk))
result_line.append(str(points_after_pk))
return AIC_schoolfield, BIC_schoolfield, HQC_schoolfield, result_line
def H(g, j):
return exp(j+g+0.5) / (j+g+0.5)**(j+0.5)
def gamma(z, g, N):
return (z+g-0.5)**(z-0.5) / exp(z+g-0.5) * L(g, z, N)
def H(g, j):
return exp(j+g+0.5) / (j+g+0.5)**(j+0.5)
def integrand(self, x):
return 1.0 - (1.0 - self.lasum(x)
/ bigfloat.exp(x, bigfloat.precision(100)))**self.n