Initial commit

_simulations/chain_simulation.py

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+import numpy as np
+import pandas as pd
+from tqdm import tqdm
+
+from _utils.termcolors import termcolors as tc
+
+from _simulations.state import State
+from _simulations.distributions import Distribution
+from _simulations.event import Event
+
+from _stats.schmidt_test import schmidt_test, generalised_schmidt_test as gst
+
+class Chain:
+    """
+    Chain class generates a 1D array of State objects, with randomly chosen paths based on the probabilities given by available branches.
+
+    Arguments:
+        initial_state   The initial state from which to start the chain
+
+    Attributes:
+        chain           The simulated chain of decays
+    """
+
+
+    def __init__(self, initial_state):
+        chain_data, decay_energies = self.generate_random_chain(initial_state)
+        self.chain = [x[0] for x in chain_data]
+        self.decay_energies = decay_energies
+
+        state_id = lambda x: x.id if isinstance(x, State) else ''
+        chain_string = [state_id(x[0]) + f' ==({x[1]})==> ' for x in chain_data]
+        self.id = "".join(chain_string)
+
+    def generate_random_chain(self, initial_state):
+        """Generate a random decay path according to branching probabilities."""
+        chain = [initial_state]
+        decay_energies = []
+        state = initial_state
+
+        while not state.half_life == None: # run until a "stable" (half_life == None) state is found or SF is encountered
+            # Randomly sample a decay branch based on the relative probabilities of each decay
+            b = np.random.choice(state.branches.index.values, p=state.branches['probability'])
+
+            chain[-1] = [chain[-1], b]
+
+            if 'sf' in str(b):
+                decay_energies.append(None)
+                break
+
+            else:
+                excitation_energy = int(state.branches.loc[b]['excitation energy [keV]']) # fixing weird thingy where pandas turns int into float
+                decay_energy = int(state.branches.loc[b]['energy [keV]']) # fixing weird thingy where pandas turns int into float
+
+                decay_energies.append(decay_energy)
+
+                if 'alpha' in str(b): # if alpha decay, find state A-4, Z-2 with corresponding excitation energy
+                    state = State(state.A-4, state.Z-2, excitation_energy)
+
+                elif 'gamma' in str(b):
+                    state = State(state.A, state.Z, excitation_energy)
+
+            chain.append(state)
+
+        return chain, decay_energies
+
+class ChainSimulation:
+    """
+        ChainSimulation simulates N number of decays from a given initial state. Simulations are done using Monte Carlo techniques.
+        For each iteration, a new random path is determined based on branching ratios defined by the Chain object. From this,
+        a cumulative distribution function (CDF) for each step is generated, and a random event time is generated, simulating
+        a random radioactive decay that follows the original exponential distribution of any given state. The result is saved
+        into a pandas DataFrame for further use.
+
+        Arguments:
+            initial_state               The initial state from which to simulate decay chains
+
+        Attributes:
+            run_simulation()            Function to start the simulation.
+                                            int N (default 1000) - The number of decay chains to simulate
+            results                     2D array of the results
+            results_df                  The same results but formatted to a DataFrame
+    """
+    def __init__(self, initial_state):
+        self.initial_state = initial_state
+        self.all_states = initial_state.get_all_states()
+        self.true_half_lives = initial_state.get_true_half_lives()
+        self.result = None
+        self.result_dfs = None
+
+    def run_simulation(self, N=10_000, dist_time_range_factor=5):
+        """
+            Starts the Monte Carlo simulations and updates result attributes.
+
+            Keyword arguments:
+                int N (default 10_000)    The number of decay chains to simulate
+        """
+        chain_simulations = {}
+        temp_dist_dict = {} # temporary dictionary to store CDF and half life, in order to avoid recalculations in the simulation for loop.
+
+        for n in tqdm(range(N)):         
+            chain_obj = Chain(self.initial_state)
+            chain = chain_obj.chain
+            chain_id = chain_obj.id
+
+            event_times = [] # initialise empty list for generated event times
+
+            for i in range(len(chain[:-1])):
+                step = chain[i]
+                last_step = chain[i-1]
+
+                if step.half_life in temp_dist_dict.keys():
+                    dist = temp_dist_dict[step.half_life] # If dist was already generated, load it from temporary dict
+                else:
+                    print(f'CDF for t₁/₂ = {step.half_life}s not found in temporary dictionary. Generating a new one...')
+                    dist = Distribution(step.half_life, dist_time_range_factor*step.half_life) # generate new distribution for given half-life
+                    temp_dist_dict[step.half_life] = dist # add newly generated distribution to dict
+
+                event = Event(dist, parent=last_step, daughter=step)
+                event_times.append(event.event_time)
+
+            if chain_id in chain_simulations.keys():
+                chain_simulations[chain_id].append(event_times)
+            else:
+                chain_simulations[chain_id] = [event_times]
+
+            event_times.append('SF')
+        result_dfs = {}
+
+        for chain_sim in chain_simulations:
+            col_names = chain_sim.split()
+            col_names = [x for x in col_names if not '=' in x] # get column names for all decays (not including the final "stable" state)
+            df = pd.DataFrame(chain_simulations[chain_sim], columns=col_names)
+            result_dfs[chain_sim] = df
+
+        self.result = chain_simulations
+        self.result_dfs = result_dfs
+
+        lifetimes = {}
+
+        for (chain_id, df) in self.result_dfs.items():
+            for column in df.columns[:-1]:
+                try:
+                    lifetimes[column] += df[column].to_numpy()
+                except:
+                    lifetimes[column] = df[column].to_numpy()
+
+        mean_lifetimes = {k:np.mean(v) for (k, v) in lifetimes.items()}
+
+        self.mean_lifetimes = pd.DataFrame(data={
+                                          'Mean Lifetime [s]': mean_lifetimes.values(),
+                                          '"True" Half-life [s]': self.true_half_lives[:-1]}, index=mean_lifetimes.keys())
+
+    # getters
+
+    def get_mean_lifetime(self, A, Z, E=0):
+        """Returns a specific mean lifetime"""
+        try:
+            ret = self.mean_lifetimes.loc[f'{A}.{Z}.{E}']['Mean Lifetime [s]']
+            return ret
+        except:
+            raise KeyError("State not found!")
+
+    # printing functions
+
+    def print_results(self):
+        for i, k in enumerate(self.result_dfs.keys()):
+            print(tc.BOLD+ f"Branch {i+1}: " + '\n' + tc.OKBLUE + k + tc.ENDC, '\n')
+            print(self.result_dfs[k], '\n')
+
+    def print_mean_lifetimes(self):
+        print(tc.OKBLUE + tc.BOLD + "Mean Lifetime of states" + tc.ENDC)
+        print(self.mean_lifetimes, '\n')
+
+    def print_schmidt_test(self):
+        for state in self.all_states[:-1]:
+            print(tc.BOLD + tc.OKBLUE + f"Schmidt Test for {state}" + tc.ENDC)
+
+            ls = []
+            for df in self.result_dfs.values():
+                try: lifetimes = df[state].to_list()
+                except: lifetimes = []
+
+                if lifetimes.__contains__('SF'):
+                    pass
+                else:
+                    ls.append(lifetimes)
+
+            arr = np.concatenate(ls)
+            sigma_theta_exp, conf_int = schmidt_test(arr)
+            lo = conf_int[0]
+            hi = conf_int[1]
+
+            if lo <= sigma_theta_exp <= hi:
+                color = tc.OKGREEN
+            else:
+                color = tc.FAIL
+
+            print('σ_θ: ' + color + str(round(sigma_theta_exp, 3)) + tc.ENDC,
+                    f'[{round(lo, 3)}, {round(hi, 3)}]',
+                    f'({arr.shape[0]} lifetimes)')
+            print()
+        
+    def generalised_schmidt_test(self):
+        print(tc.BOLD + tc.OKBLUE + "Generalised Schmidt Test" + tc.ENDC)
+        for key in self.result_dfs.keys():
+            df = self.result_dfs[key]
+            print(key)
+            sigma_theta_exp, conf_int = gst(df)
+            lo = conf_int[0]
+            hi = conf_int[1]
+
+            if lo <= sigma_theta_exp <= hi:
+                color = tc.OKGREEN
+            else:
+                color = tc.FAIL
+
+            print('σ_θ: ' + color + str(round(sigma_theta_exp, 3)) + tc.ENDC,
+                  f'[{round(lo, 3)}, {round(hi, 3)}]',
+                  f'({df.shape[0]} chains)')
+            print()

_simulations/distributions.py

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+import numpy as np
+from matplotlib import pyplot as plt
+
+from _utils.termcolors import termcolors as tc
+
+class Distribution:
+    """
+    Generates a PDF and CDF for any exponential decay using the half-life.
+
+    Arguments:
+        half_life   Half life of the isotope
+        time_range  User-defined range for the distribution
+        dt          Time interval for the time axis
+        A           Amplitude in exponential decay equation
+    
+    Attributes:
+        half_life   Half life of the isotope
+        dt          Time interval for the decay graph
+        time_range  User-defined range for the distribution
+        exponential The exponential decay distribution
+        pdf         The (normalised) Probability Density Function (PDF) of the exponential distribution
+        cdf         The cumulative sum of the PDF
+    """
+
+    def __init__(self, half_life, time_range, time_start=0, A=1):
+        self.half_life = half_life
+        self.dt = 5*half_life/10_000
+        self.time_range = np.arange(time_start, time_range, self.dt)
+
+        self.exponential = A * np.exp(- np.log(2) * self.time_range/self.half_life) # exponential decay formula
+        self.pdf = self.calculate_pdf(self.exponential, self.time_range)
+        self.cdf = self.calculate_cdf(self.pdf)
+
+    def calculate_pdf(self, exp, time_range):
+        pdf = exp / np.trapz(exp, x=time_range) # normalize the distribution
+        return pdf
+
+    def calculate_cdf(self, pdf):
+        cdf = np.cumsum(pdf) / np.sum(pdf) # cumulative sum
+        return cdf
+
+    def plot(self, function='all'):
+        """
+        Plot the exponential distribution ('exp'), Probability Density Function ('pdf'), Cumulative Density Function ('cdf'), or all functions.
+
+        Keyword arguments:
+        function    Choose between 'all', 'exp', 'pdf', 'cdf' (default 'all')
+        """
+        dists = {'exp': self.exponential, 'pdf': self.pdf, 'cdf': self.cdf}
+        if function == 'all':
+            for d in dists.keys():
+                y = dists[d]
+                plt.plot(self.time_range, y, label=d)
+        elif function in dists.keys():
+            y = dists[function]
+            plt.plot(self.time_range, y, label=function)
+        else:
+            raise ValueError(tc.WARNING + "Invalid function. Leave function kwarg empty to plot all, or choose between 'exp', 'pdf' or 'cdf'." + tc.ENDC)
+
+        plt.legend()
+        plt.grid() 
+        plt.show()

_simulations/event.py

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+import bisect
+import random
+
+class Event:
+    def __init__(self, dist, parent=None, daughter=None):
+        self.parent = parent
+        self.daughter = daughter
+
+        if not parent == daughter == None:
+            self.name = f'{parent.name} => {daughter.name}'
+        
+        self.event_time = self.generate_event_time(dist.cdf, dist.time_range)
+    
+    def generate_event_time(self, cdf, time_range):
+        d_bin = time_range[1] - time_range[0] # define discrete bin
+        r = random.random()
+        i = bisect.bisect_left(cdf, r) # from the left, find the nearest value to r in the CDF
+            
+        if i:
+            pass
+        else:
+            i = 0
+
+        t = time_range[i]
+        t += random.random() * d_bin # this avoids issues with the discretization selection (for continuum quantities)
+
+        return t

_simulations/state.py

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+import pandas as pd
+import yaml
+
+class State:
+    """
+        Object to describe a unique quantum state using the nucleon count A, proton number Z and the excitation energy E.
+        
+        Arguments:
+            A                   Nucleon count of the isotope
+            Z                   Proton number of the isotope
+            excitation_energy   The energy level occupied, relative to the ground state E=0
+
+        Attributes:
+            id                  The unique identifier of the state
+            A                   Nucleon count of the isotope
+            Z                   Proton number of the isotope
+            E                   Excitation energy relative to ground state
+            half_life           The half-life of the isotope
+            branches            DataFrame object containing the available decay branches for this state
+    """
+
+    # Initialize state database as class-level attribute
+    with open('state_db.yml', 'r') as file:
+        DATABASE = yaml.safe_load(file)
+
+    def __init__(self, A, Z, excitation_energy=0):
+        state_id, db_state = self.find_state(A, Z, excitation_energy) # Find the state in the existing database
+
+        if db_state == None:
+            raise KeyError("State not found in database.")
+        
+        else:
+            self.id = state_id
+            self.A = A
+            self.Z = Z
+            self.E = excitation_energy
+            self.half_life = db_state['half_life']
+            self.name = db_state['name']
+
+            if self.half_life == None: # if there is no half life, we have a "stable" state and don't need branches
+                self.branches = None
+            
+            else:
+                self.branches = self.unpack_branches(db_state)
+
+    def get_all_states(self):
+        return list(self.DATABASE.keys())
+
+    def get_true_half_lives(self):
+        return [self.DATABASE[x]['half_life'] for x in self.DATABASE.keys()]
+
+    def find_state(self, A, Z, excitation_energy):
+        try:
+            state_id = f'{A}.{Z}.{excitation_energy}'
+            return state_id, self.DATABASE[f'{A}.{Z}.{excitation_energy}']
+        except KeyError:
+            return None, None 
+
+    def unpack_branches(self, db_state):
+        """Takes a state from YML database and unpacks into a Pandas DataFrame"""
+
+        cols = ['probability', 'energy [keV]', 'excitation energy [keV]']
+        df = pd.DataFrame.from_dict(db_state['branches'], orient='index', columns=cols)
+        return df

_stats/schmidt_test.py

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+from bisect import bisect_left
+
+import numpy as np
+import scipy as scp
+import pandas as pd
+
+def schmidt_test(decay_times):
+    """
+    Calculates σ_Θ_exp following the original Schmidt test.
+
+    Keyword arguments:
+        decay_times         1D array of decay times to be tested.
+
+    Returns:
+        sigma_theta_exp     The σ_Θ_exp value for the given data.
+        confidence_interval The upper and lower limit to the confidence interval.
+
+    """
+    
+    sigma_theta_exp = np.nanstd(np.log(decay_times))
+
+    if len(decay_times) <= 100: # Schmidt (2000) has tabulated values for n <= 100 events
+        dat = pd.read_pickle("_stats/st_sigma_theta_exp_properties.pkl") # load tabulated values
+        i = bisect_left(dat.loc[:,'n'], len(decay_times)) # find the closest value in tabulated values
+        lim_l = dat.loc[i,'lower limit of σ_Θ_exp']
+        lim_h = dat.loc[i,'upper limit of σ_Θ_exp']
+    else: # if n > 100, calculate limits based on analytical formula
+        lim_l = 1.28-2.15/np.sqrt(len(decay_times))
+        lim_h = 1.28+2.15/np.sqrt(len(decay_times))
+
+    confidence_interval = (lim_l, lim_h)
+    return sigma_theta_exp, confidence_interval
+
+def g_nan_mean(data):
+    """
+    Code written by Anton
+    """
+
+    if len(np.shape(data)) == 1:
+        return data
+    ret = np.empty(np.shape(data)[0])
+    for i in range(np.shape(data)[0]):
+        temp = 1
+        steps = 0
+        for j in range(np.shape(data)[1]):
+            if isinstance(data, pd.DataFrame) and ~np.isnan(data.iloc[i,j]):
+                temp *= data.iloc[i,j]
+            elif not isinstance(data, pd.DataFrame) and ~np.isnan(data[i,j]):
+                temp *= data[i,j]
+            else:
+                break
+            steps += 1
+        ret[i] = temp**(1./steps)
+    return ret
+
+def generalised_schmidt_test(df):
+    """
+    Generalised Schmidt Test
+
+    Keyword arguments:
+        df    DataFrame containing simulated chain event times  
+    """
+
+    if any(df.iloc[:,-1].str.contains('SF')):
+        df = df.iloc[:,:-1]
+
+    theta = np.log(df)
+    theta_var = np.square(theta - np.nanmean(theta, axis=0))
+    gen_Schmidt_temp = g_nan_mean(theta_var)
+    sigma_theta_exp = np.sqrt(np.mean(gen_Schmidt_temp))
+
+    if len(df) <= 100: # Schmidt (2000) has tabulated values for n <= 100 events
+        dat = pd.read_pickle("_stats/st_sigma_theta_exp_properties.pkl") # load tabulated values
+        i = bisect_left(dat.loc[:,'n'], len(df)) # find the closest value in tabulated values
+        lim_l = dat.loc[i,'lower limit of σ_Θ_exp']
+        lim_h = dat.loc[i,'upper limit of σ_Θ_exp']
+    else: # if n > 100, calculate limits based on analytical formula
+        lim_l = 1.28-2.15/np.sqrt(len(df))
+        lim_h = 1.28+2.15/np.sqrt(len(df))
+
+    confidence_interval = (lim_l, lim_h)
+
+    return sigma_theta_exp, confidence_interval

_utils/termcolors.py

@@ -0,0 +1,10 @@
+class termcolors:
+    HEADER = '\033[95m'
+    OKBLUE = '\033[94m'
+    OKCYAN = '\033[96m'
+    OKGREEN = '\033[92m'
+    WARNING = '\033[93m'
+    FAIL = '\033[91m'
+    ENDC = '\033[0m'
+    BOLD = '\033[1m'
+    UNDERLINE = '\033[4m'

example.py

@@ -0,0 +1,33 @@
+from _simulations.chain_simulation import ChainSimulation
+from _simulations.state import State
+
+from _stats.schmidt_test import generalised_schmidt_test as gst
+
+from _utils.termcolors import termcolors as tc
+
+NUMBER_OF_SIMS = 1
+
+# define a state
+initial_state = State(A=288, Z=115)
+
+# initialise the decay chain simulation
+sim = ChainSimulation(initial_state=initial_state)
+
+# run the simulation
+sim.run_simulation(NUMBER_OF_SIMS)
+
+# print all dataframes
+sim.print_results()
+
+# print mean lifetimes
+sim.print_mean_lifetimes()        
+
+# print statistics of individual steps
+sim.print_schmidt_test()
+
+# print statistics of all the chains (INACCURATE CONFIDENCE INTERVALS!!!)
+sim.generalised_schmidt_test()
+
+# get a specific lifetime
+x = sim.get_mean_lifetime(A=288, Z=115)
+print(x)

state_db.yml

@@ -0,0 +1,35 @@
+# Data adapted from U. Forsberg “Element 115.”
+# Lund University, 2016.
+# https://lup.lub.lu.se/record/c5df0fbd-7eb1-46a3-a47e-c5c35887859e
+
+288.115.0:
+  name: '288-Fl'
+  half_life: 0.2
+  branches: {'alpha1': [1.0, 10300, 0]}
+
+284.113.0:
+  name: '284-Nh'
+  half_life: 0.7
+  branches: {'alpha1': [0.9, 10100, 0],
+             'sf': [0.1, null, null]}
+
+280.111.0:
+  name: '280-Rg'
+  half_life: 6
+  branches: {'alpha1': [0.8, 10000, 0],
+             'sf': [0.2, null, null]}
+
+276.109.0:
+  name: '276-Mt'
+  half_life: 0.8
+  branches: {'alpha1': [1.0, 9900, 0]}
+
+272.107.0:
+  name: '272-Bh'
+  half_life: 9
+  branches: {'alpha1': [1.0, 9100, 0]}
+
+268.105.0:
+  name: '268-Db'
+  half_life: 93600
+  branches: {'sf': [1.0, null, null]}