Explanation of Simulated Data Table, data

user_data_sim.py

@(@\newcommand{\B}[1]{ {\bf #1} } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\W}[1]{ \; #1 \; }@)@This is dismod_at-20221105 documentation: Here is a link to its current documentation . Explanation of Simulated Data Table, data_sim
See Also
Purpose
Random Effects
Priors
Iota
Other Rates
Covariate Multiplier
Data
Data Subset
meas_noise_effect
Notation Before Simulation
     y
     Capital Delta
     sigma
     E
     delta
Simulation Notation
     z
Source Code

See Also
user_sim_log.py

Purpose
This example explains the data_sim_table by showing that the Adjusted standard deviation for the simulated data is the same as for the original data.

Random Effects
There are no random effects in this example.

Priors
The priors do not matter for this example except for the fact that the truth_var_table values for the model_variables must satisfy the lower and upper limits in the corresponding priors.

Iota
The value iota_true is the simulated true rate for iota. There is only one grid point (one model_variable ) corresponding to iota , hence it is constant in age and time.

Other Rates
For this example the other rates are all zero. This is specified by setting the parent_smooth_id and child_smooth_id to null for the other rates.

Covariate Multiplier
There is one covariate multiplier on the covariate column one and the rate iota. This is a measurement noise covariate multiplier gamma . The true value for this multiplier, used to simulate data, is returned by gamma_true(meas_noise_effect) . There is only one grid point in the covariate multiplier, hence it is constant in age and time. It follows that average noise effect @(@ E_i ( \theta ) @)@ is constant and equal to gamma_true .

Data
There are n_data measurements of Sincidence and each has a standard deviation meas_std (before adding the covariate effect). The meas_value do not affect (do affect) the values in data_sim_table when the density is linear (log scaled ).

Data Subset
Data is only simulated for data_id values that appear in the data_subset table. For this case, this includes all the data_id values in the data table.

meas_noise_effect
see meas_noise_effect .

Notation Before Simulation
The following values do not depend on the simulated data:

y
This is the measured value; see y .

Capital Delta
This is the minimum cv standard deviation corresponding to @(@ y @)@; see Delta .

sigma
This is the transformed standard deviation corresponding to @(@ y @)@; see sigma .

E
This is the average noise effect corresponding to @(@ y @)@; see E .

delta
This is the adjusted standard deviation corresponding to @(@ y @)@; see delta .

Simulation Notation

z
This is the simulated measurement value, before censoring, in the data_sim table; see z .

Source Code


# You can changed the values below and rerun this program
iota_true          = 0.01
meas_std           = 0.001
n_data             = 2000
# You can changed the values above and rerun this program
# ----------------------------------------------------------------------------
import math
import sys
import os
import copy
import numpy
test_program = 'example/user/data_sim.py'
if sys.argv[0] != test_program  or len(sys.argv) != 1 :
   usage  = 'python3 ' + test_program + '\n'
   usage += 'where python3 is the python 3 program on your system\n'
   usage += 'and working directory is the dismod_at distribution directory\n'
   sys.exit(usage)
print(test_program)
#
# import dismod_at
local_dir = os.getcwd() + '/python'
if( os.path.isdir( local_dir + '/dismod_at' ) ) :
   sys.path.insert(0, local_dir)
import dismod_at
#
# change into the build/example/user directory
if not os.path.exists('build/example/user') :
   os.makedirs('build/example/user')
os.chdir('build/example/user')
# ---------------------------------------------------------------------------
# log_density
def log_density(density) :
   assert not density.startswith('cen_')
   return density.startswith('log_')
# ---------------------------------------------------------------------------
# gamma_true
def gamma_true(meas_noise_effect) :
   if meas_noise_effect.startswith('add_std_') :
      result = meas_std
   elif meas_noise_effect.startswith('add_var_') :
      result = meas_std * meas_std
   else :
      assert False
   return result
# ---------------------------------------------------------------------------
# delta_effect
def delta_effect(meas_noise_effect, sigma, E) :
   # add_std
   if meas_noise_effect == 'add_std_scale_all' :
      delta = sigma * (1.0 + E)
   elif meas_noise_effect == 'add_std_scale_none' :
      delta = sigma + E
   elif meas_noise_effect == 'add_std_scale_log' :
      if log_density(density) :
         delta = sigma * (1.0 + E)
      else :
         delta = sigma + E
   # add var
   elif meas_noise_effect == 'add_var_scale_all' :
      delta = sigma * math.sqrt(1.0 + E)
   elif meas_noise_effect == 'add_var_scale_none' :
      delta = math.sqrt( sigma * sigma + E )
   else :
      assert meas_noise_effect == 'add_var_scale_log'
      if log_density(density) :
         delta = sigma * math.sqrt(1.0 + E)
      else :
         delta = math.sqrt( sigma * sigma + E )
   return delta
# ------------------------------------------------------------------------
# Note that the a, t values are not used for this example
def example_db (file_name) :
   # note that the a, t values are not used for this case
   def fun_iota(a, t) :
      return ('prior_iota', None, None)
   def fun_gamma(a, t) :
      return ('prior_gamma', None, None)
   # ----------------------------------------------------------------------
   # age table:
   age_list    = [ 0.0, 100.0 ]
   #
   # time table:
   time_list   = [ 1990.0, 2010.0 ]
   #
   # integrand table:
   integrand_table = [
       { 'name': 'Sincidence' }
   ]
   #
   # node table:
   node_table = [ { 'name':'world', 'parent':'' } ]
   #
   # weight table:
   weight_table = list()
   #
   # covariate table:
   covariate_table = [
      {'name':'one', 'reference':0.0}
   ]
   #
   # mulcov table:
   mulcov_table = [
      {
         'covariate': 'one',
         'type':      'meas_noise',
         'effected':  'Sincidence',
         'group':     'world',
         'smooth':    'smooth_gamma'
      }
   ]
   #
   # avgint table: empty
   avgint_table = list()
   #
   # nslist_table:
   nslist_table = dict()
   # ----------------------------------------------------------------------
   # data table:
   data_table = list()
   # values that are the same for all data rows
   row = {
      'weight':      '',
      'hold_out':     False,
      'node':        'world',
      'subgroup':    'world',
      'one':          1.0   ,
      'age_lower':    50.0,
      'age_upper':    50.0,
      'time_lower':   2000.,
      'time_upper':   2000.,
      'integrand':   'Sincidence',
      'meas_std':     meas_std,
      'eta':          meas_std,
      'nu':           10
   }
   # The censored densities are not included becasue one cannot recover
   # sigma when censoring occurs.
   density_list = [
      'gaussian', 'log_gaussian',
      'laplace',  'log_laplace',
      'students', 'log_students',
   ]
   # values that change between rows:
   for data_id in range( n_data ) :
      if data_id % 2 == 0 :
         row['meas_value'] = 0.9 * iota_true
      else :
         row['meas_value'] = 1.1 * iota_true
      density = density_list[ data_id % len(density_list) ]
      row['density'] = density
      data_table.append( copy.copy(row) )
   #
   # ----------------------------------------------------------------------
   # prior_table
   prior_table = [
      { # prior_iota
         'name':     'prior_iota',
         'density':  'uniform',
         'lower':    iota_true / 100.0,
         'upper':    1.0,
         'mean':     iota_true / 10.0
      },{ # prior_gamma
         'name':     'prior_gamma',
         'density':  'uniform',
         'lower':    0.0,
         'upper':    10.0,
         'mean':     0.01
      }
   ]
   # ----------------------------------------------------------------------
   # smooth table
   name           = 'smooth_iota'
   fun            = fun_iota
   age_id         = 0
   time_id        = 0
   smooth_table = [
      {'name':name, 'age_id':[age_id], 'time_id':[time_id], 'fun':fun }
   ]
   name = 'smooth_iota'
   #
   name = 'smooth_gamma'
   fun  = fun_gamma
   smooth_table.append(
      {'name':name, 'age_id':[age_id], 'time_id':[time_id], 'fun':fun }
   )
   # ----------------------------------------------------------------------
   # rate table:
   rate_table = [
      {  'name':          'iota',
         'parent_smooth': 'smooth_iota',
         'child_smooth':  None
      }
   ]
   # ----------------------------------------------------------------------
   # option_table
   option_table = [
      { 'name':'rate_case',              'value':'iota_pos_rho_zero' },
      { 'name':'parent_node_name',       'value':'world'             },
      { 'name':'random_seed',            'value':'0'                 },

   ]
   # ----------------------------------------------------------------------
   # subgroup_table
   subgroup_table = [ { 'subgroup':'world', 'group':'world' } ]
   # ----------------------------------------------------------------------
   # create database
   dismod_at.create_database(
      file_name,
      age_list,
      time_list,
      integrand_table,
      node_table,
      subgroup_table,
      weight_table,
      covariate_table,
      avgint_table,
      data_table,
      prior_table,
      smooth_table,
      nslist_table,
      rate_table,
      mulcov_table,
      option_table
   )
   # ----------------------------------------------------------------------
   return
# ===========================================================================
# Run the init command to create the var table
file_name = 'example.db'
example_db(file_name)
program = '../../devel/dismod_at'
#
dismod_at.system_command_prc([ program, file_name, 'init' ])
# -----------------------------------------------------------------------
# read database
new             = False
connection      = dismod_at.create_connection(file_name, new)
var_table       = dismod_at.get_table_dict(connection, 'var')
rate_table      = dismod_at.get_table_dict(connection, 'rate')
integrand_table = dismod_at.get_table_dict(connection, 'integrand')
covariate_table = dismod_at.get_table_dict(connection, 'covariate')
node_table      = dismod_at.get_table_dict(connection, 'node')
# -----------------------------------------------------------------------
# truth table:
# -----------------------------------------------------------------------
meas_noise_effect_list = [
   'add_std_scale_all', 'add_std_scale_none', 'add_std_scale_log',
   'add_var_scale_all', 'add_var_scale_none', 'add_var_scale_log',
]
for meas_noise_effect in meas_noise_effect_list :
   dismod_at.system_command_prc([ program, file_name,
      'set', 'option', 'meas_noise_effect', meas_noise_effect
   ])
   # ------------------------------------------------------------------------
   # truth_var table
   tbl_name     = 'truth_var'
   col_name     = [ 'truth_var_value' ]
   col_type     = [ 'real' ]
   row_list     = list()
   var_id2true  = list()
   for var_id in range( len(var_table) ) :
      var_info        = var_table[var_id]
      truth_var_value = None
      var_type        = var_info['var_type']
      if var_type == 'mulcov_meas_noise' :
         integrand_id  = var_info['integrand_id']
         integrand_name = integrand_table[integrand_id]['integrand_name']
         assert integrand_name == 'Sincidence'
         #
         covariate_id   = var_info['covariate_id']
         covariate_name = covariate_table[covariate_id]['covariate_name' ]
         assert( covariate_name == 'one' )
         #
         truth_var_value = gamma_true(meas_noise_effect)
      else :
         assert( var_type == 'rate' )
         rate_id   = var_info['rate_id']
         rate_name = rate_table[rate_id]['rate_name']
         node_id   = var_info['node_id']
         node_name = node_table[node_id]['node_name']
         assert node_name == 'world'
         #
         truth_var_value = iota_true
      #
      var_id2true.append( truth_var_value )
      row_list.append( [ truth_var_value ] )
   dismod_at.sql_command(connection, 'DROP TABLE IF EXISTS truth_var')
   dismod_at.create_table(connection, tbl_name, col_name, col_type, row_list)
   # -------------------------------------------------------------------------
   # create and check the data_sim table
   dismod_at.system_command_prc([ program, file_name, 'simulate', '1' ])
   #
   # check results in data_sim table
   new               = False
   connection        = dismod_at.create_connection(file_name, new)
   density_table     = dismod_at.get_table_dict(connection, 'density')
   data_table        = dismod_at.get_table_dict(connection, 'data')
   data_subset_table = dismod_at.get_table_dict(connection, 'data_subset')
   data_sim_table    = dismod_at.get_table_dict(connection, 'data_sim')
   #
   # check that all the data_id appear in the data_subset table
   for data_subset_id in range(len(data_subset_table)) :
      data_id = data_subset_table[data_subset_id]['data_id']
      assert data_id == data_subset_id
   #
   # check the values in the data_sim table
   eps99 = 99.0 * sys.float_info.epsilon
   residual_list = list()
   for data_sim_id in range( len(data_sim_table) ) :
      #
      # data_sim table valeus
      row = data_sim_table[data_sim_id]
      simulate_index = row['simulate_index']
      data_subset_id = row['data_subset_id']
      data_sim_value = row['data_sim_value']
      #
      # only one set of data is simulated
      assert simulate_index == 0
      assert data_subset_id == data_sim_id
      #
      # data table values
      data_id        = data_subset_table[data_subset_id]['data_id']
      meas_value     = data_table[data_id]['meas_value']
      Delta          = data_table[data_id]['meas_std']
      eta            = data_table[data_id]['eta']
      density_id     = data_table[data_id]['density_id']
      density        = density_table[density_id]['density_name']
      #
      # Values that do not depend on simulated data
      y         = meas_value
      E         = gamma_true(meas_noise_effect)
      if log_density(density) :
         sigma  = math.log(y + eta + Delta) - math.log(y + eta)
      else :
         sigma  = Delta
      delta     = delta_effect(meas_noise_effect, sigma, E)
      #
      # residual
      z  = data_sim_value  # simulated data withoout censoring
      mu = iota_true       # mean of the simulated data
      if log_density(density) :
         residual = (math.log(z + eta) - math.log(mu + eta)) / delta
      else :
         residual = (z - mu) / delta
      residual_list.append(residual)
   residual_array  = numpy.array( residual_list )
   residual_mean   = residual_array.mean()
   residual_std    = residual_array.std(ddof=1)
   #
   # check that the mean of the residuals is within 2.5 standard deviations
   assert abs(residual_mean) <=  2.5 / numpy.sqrt(n_data)
   # check that the standard deviation of the residuals is near one
   assert abs(residual_std - 1.0) <= 2.5 / numpy.sqrt(n_data)
   #
connection.close()
# -----------------------------------------------------------------------------
print('data_sim.py: OK')
# -----------------------------------------------------------------------------

Input File: example/user/data_sim.py