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synthetic_phenology_max_variance.py
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# -*- coding: utf-8 -*-
######### PROGRAM SYNTHETIC PHENOLOGY MAX VARIANCE ###########
# This programs generates synthetic phenological configurations under certain restrictions
# We solve a multivariate system of inequations to determine the phase space of phenological configurations compatible with the matrix of interactions
# We solve the system using linear programming and a objective function to minimize/maximize
# In this case we maximize the variance associated to the middle dates
import sys
import numpy as np
from scipy.optimize import minimize, Bounds, LinearConstraint, differential_evolution, shgo
import matplotlib.pyplot as plt
from scipy import stats
###############################################
############### FUNCTIONS #####################
###############################################
########### VARIANCE of the MIDDLE DATES ##################
## Calculates the negative variance of the middle dates
def variance(x, n_total, periods):
#Calculating the vector of middle dates
tm = np.empty(shape=(n_total))
for i in range(n_total):
tmiddle = x[i] + (float(periods[i])-1)/2.0
tm[i] = tmiddle
#Calculating the average (mean)
mean = 0
for i in range(n_total):
mean = mean + tm[i]/float(n_total)
#Calculating the variance
suma = 0
for i in range(n_total):
suma = suma + (tm[i] - mean)*(tm[i] - mean)
#We add a negative sign in order to ensure that the variance is maximized
var = -suma/float(n_total)
#Return the value of the variance
return var
###############################################
######## INITIALIZATION OF VARIABLES ##########
###############################################
#reading input argument: index of the network to study. 2 for the Illinois datase, 6 for the Kantsa dataset 1st year and 7 for the Kantsa dataset 2nd year
idoc = sys.argv[1]
#opening files
doc_dimension = open ("general"+idoc+".dat","r")
doc_matrix = open ("matrix"+idoc+".dat","r")
#reading dimension (number of rows, number of columns)
n_rows = int(doc_dimension.readline())
n_cols = int(doc_dimension.readline())
n_total = n_rows + n_cols
doc_dimension.close()
#reading interaction matrix
matrix = []
matrix = [[int(num) for num in line.split('\t')] for line in doc_matrix]
doc_matrix.close()
#calculating the number of links L
L=0
for i in range(n_rows):
for j in range(n_cols):
L = L + matrix[i][j]
###############################################
###### GENERATION OF SYNTHETIC PERIODS ########
###############################################
#Reading periods of activity from file
doc_periods = open ("periods"+idoc+".dat","r")
periods = []
periods_rows = []
for i in range(n_rows):
p = int(doc_periods.readline())
periods_rows.append(p)
periods.append(p)
periods_cols = []
for j in range(n_cols):
p = int(doc_periods.readline())
periods_cols.append(p)
periods.append(p)
###############################################
####### GENERATION OF STARTING DATES ##########
###############################################
###### Formating the constraints using the LienarConstrains object
#### Inequalities must be written in the standard form A*x <= b. We have L inequalities, one per link
A = np.zeros(shape=(L,n_total)) #2-D array of coefficients for x. Size: LxN
lb = np.zeros(shape=(L)) #vector of lower bounds, size L
ub = np.zeros(shape=(L)) #vector of upper bounds, size L
#### Construction of the inequalities
iiter = 0
for i in range(n_rows):
for j in range(n_cols):
#If there's a mutualistic link, construct the two corresponding inequalities
if (matrix[i][j] == 1):
#Coefficients for the starting dates
A[iiter][i] = 1
A[iiter][j+n_rows] = -1
#Lower bound
lb[iiter] = -periods_rows[i] + 1
#Upper bound
ub[iiter] = periods_cols[j] - 1
iiter = iiter + 1
####Formating the constraints using a dictionary
cons_dict = [] #sequence of dictionaries
for i in range(n_rows):
for j in range(n_cols):
#If there's a mutualistic link, construct the two corresponding inequalities
if (matrix[i][j] == 1):
#Lower bound
cons_dict.append({'type': 'ineq', 'fun': lambda x : -x[j+n_rows]+periods_rows[i]-1+x[i]})
#Upper bound
cons_dict.append({'type': 'ineq', 'fun': lambda x : x[j+n_rows]+periods_cols[j]-1-x[i]})
#### Setting the lower and upper bounds (limits of the season). We constrain the starting dates to coincide within one month.
lbound = 100
ubound = 200
veclb = np.empty(shape=(n_total))
vecub = np.empty(shape=(n_total))
for i in range(n_rows):
veclb[i] = lbound
vecub[i] = ubound - periods_rows[i]+1
for j in range(n_cols):
veclb[j+n_rows] = lbound
vecub[j+n_rows] = ubound - periods_cols[j]+1
season_bounds = Bounds(veclb,vecub) #Valid for the methods 'minimize' and 'differential_evolution'
seq_bounds = np.column_stack((np.array(veclb),np.array(vecub)))
seq_bounds = seq_bounds.tolist()
#### We solve the system of inequalities by linear programming
########Local+global minimization using the functions 'minimize' and 'differential_evolution'
Nsample = 15
#We start constructing a population by solving iteratively the problem with a local minimizer, then use each of these solutions as a initializer
init_pop = []
for i in range(Nsample):
#Draw an initial point
t0 = np.random.uniform(lbound,ubound,size=(n_total))
#Maximize the entropy with the function 'minimize
res = minimize(variance, t0, args = (n_total,periods), method='trust-constr', bounds=season_bounds, constraints=(LinearConstraint(A, lb, ub, keep_feasible=False)), tol=0.1,callback=None, options={'disp':False})
#Save the initial times
ti = res.x
init_pop.append(ti)
init_pop = np.reshape(init_pop,(Nsample,n_total))
print(init_pop)
#Maximize the entropy using the global optimizer 'differential_evolution' and the local solutions as input
res=differential_evolution(variance, bounds=season_bounds, args=(n_total, periods), strategy='best1exp', maxiter=1000000000000, tol=0.1,disp=False, polish=True, init=init_pop,constraints=(LinearConstraint(A, lb, ub,keep_feasible=False)))
#Print results
ti = res.x
print ("Solution of starting dates=", ti)
print("Results=",res)
#Initial times for rows
ti_rows = []
for i in range(n_rows):
ti_rows.append(int(round(ti[i])))
#Initial times for columns
ti_cols = []
for i in range(n_cols):
ti_cols.append(int(round(ti[i+n_rows])))
###############################################
###### RESULTS: Plots and writing files########
###############################################
####Construct matrices of presence/absence
#Minimum initial date
min_ti_rows = min(ti_rows)
min_ti_cols = min(ti_cols)
min_ti = min(min_ti_rows, min_ti_cols)
#Ending date
tf_rows = []
for i in range(n_rows):
tf = ti_rows[i] + periods_rows[i]
tf_rows.append(tf)
tf_cols = []
for i in range(n_cols):
tf = ti_cols[i] + periods_cols[i]
tf_cols.append(tf)
#Maximum final date
max_tf_rows = max(tf_rows)
max_tf_cols = max(tf_cols)
max_tf = max(max_tf_rows, max_tf_cols)
#Matrix of presence for rows
matrix_rows = []
for i in range(n_rows):
matrix_rows.append([])
ti = ti_rows[i]
tf = tf_rows[i]
for t in range (min_ti, max_tf):
#within the period of activity, append a 1
if (ti <= t < tf):
matrix_rows[i].append(1)
#outside the period of activity, append a 0
else:
matrix_rows[i].append(0)
#Matrix of presence for columns
matrix_cols = []
for i in range(n_cols):
matrix_cols.append([])
ti = ti_cols[i]
tf = tf_cols[i]
for t in range (min_ti, max_tf):
#within the period of activity, append a 1
if (ti <= t < tf):
matrix_cols[i].append(1)
#outside the period of activity, append a 0
else:
matrix_cols[i].append(0)
######## Writing the periods and starting dates on a file
#Opening files
doc_periods = open ("maxvar_periods"+idoc+".dat","w+")
doc_ti = open ("maxvar_ti"+idoc+".dat","w+")
#Writing periods
for i in range(n_total):
doc_periods.write(str(periods[i])+"\n")
#Writing initial times
for i in range(n_rows):
doc_ti.write(str(ti_rows[i])+"\n")
for i in range(n_cols):
doc_ti.write(str(ti_cols[i])+"\n")
#####Drawing figure: we plot the timeline of periods, distinguishing between plants (rows) and pollinators (columns)
fig, axs = plt.subplots(2)
axs[0].imshow(matrix_rows,cmap='Greens', aspect='auto')
axs[0].set(ylabel='Plant species')
axs[0].xaxis.set_ticks_position('none')
axs[0].set_xticks([])
plt.xlabel('Day of the year')
plt.ylabel('Pollinator species')
axs[1].imshow(matrix_cols,cmap='BuPu', aspect='auto')
plt.draw()
plt.show()
fig.savefig('maxvar_periods'+idoc+'.png')
plt.clf()