Bootstrap particle filter for epidemic forecasting¶

Parameters¶

Particle filter parameters are provided by default_params().

Observation model parameters (and fixed parameters for the process model, if any) should be added to this parameter dictionary, so that all parameters pertaining to the simulation are stored together.

def get_params():
    """The default simulation parameters."""
    # The particle filter parameters.
    params = pypfilt.default_params(Model, px_count=1000)
    # Provide an observation model.
    params['log_llhd_fn'] = log_llhd
    # System model parameters.
    params['sys'] = {'noise_alpha': 5e-3, 'noise_dx': 0.025}
    # Observation model parameters.
    params['obs'] = {'sdev': 0.05}
    # Set a fixed PRNG seed.
    params['resample']['prng_seed'] = 42
    return params

Observations¶

Observations are represented as dictionaries that have the following keys:

{'date': datetime.datetime(...),  # When the observation was made
 'value': 200,  # The numerical quantity that was measured
 'unit': 'Some measure',  # A description of the measurement units
 'period': 7,  # The observation period, in days
 'source': 'Some system',  # A description of the data source
}

An observation stream is represented as a chronologically sorted list of observations (oldest first). The particle filter accepts any number of observation streams, which must be provided as a list (i.e., a list of observation lists); see forecast() and run().

Observations can be read from external files:

def obs_from_file(filename, year):
    """
    Read observations from a file with the following format:

    year date       value
    2009 2009-01-02 0.680762021741
    2009 2009-01-03 0.62923826359
    2009 2009-01-04 0.621926239641
    2009 2009-01-05 0.618802422847
    ...
    """
    col_types = [('year', np.int32), ('date', '|O4'), ('value', np.float32)]
    col_convs = {1: lambda s: datetime.datetime.strptime(s, '%Y-%m-%d')}
    df = np.loadtxt(filename, skiprows=1, dtype=col_types,
                    converters=col_convs)
    df = df[df['year'] == year]
    nrows = df.shape[0]
    # Note that counts are assumed to be reported daily (period = 1).
    return [{'date': df['date'][i],
             'value': df['value'][i],
             'unit': 'x',
             'period': 1,
             'source': 'file: {}'.format(filename)}
            for i in range(nrows)]

They can also be generated synthetically:

def generate_obs(x0, alpha, sdev, start, days, seed=42):
    """Generate noisy observations from a known truth."""
    rng = np.random.RandomState(seed)
    time = np.array(range(1, days + 1))
    # Here, x(t) = x(0) * e^{- alpha * t} + error
    xs = x0 * np.exp(- alpha * time)
    if sdev > 0:
        xs += rng.normal(scale=sdev, size=days)
    return [{'date': start + datetime.timedelta(days=i + 1),
             'value': xs[i], 'unit': 'x', 'period': 1,
             'source': 'synthetic'}
            for i in range(days)]

System models¶

The model of the underlying system must inherit from pypfilt.model.Base. Here is a simulation model for the example system:

class Model(pypfilt.model.Base):
    """
    A model of the following system:

        dx/dt = - alpha

    The state vector is ``[x(t), alpha]``.
    """

    @staticmethod
    def state_size():
        """Return the length of a single state vector."""
        return 2

    @staticmethod
    def priors():
        """Return a dictionary of model parameter priors."""
        return {
            'alpha': lambda r, size=None: r.uniform(0.01, 0.10, size=size),
        }

    @staticmethod
    def init(params, vec):
        """Initialise any number of state vectors."""
        rnd = params['resample']['rnd']
        rnd_size = vec[..., 0].shape
        # Assume that x(0) is somewhere between 0.5 and 1.
        vec[..., 0] = rnd.uniform(0.5, 1.0, size=rnd_size)
        # Select alpha according to the prior.
        vec[..., 1] = params['prior']['alpha'](rnd, size=rnd_size)

    @staticmethod
    def update(params, step_date, dt, is_fs, prev, curr):
        """Perform a single time-step for any number of state vectors."""
        rnd = params['resample']['rnd']
        rnd_size = curr[..., 0].shape
        # Calculate the deterministic change for x.
        dx = prev[..., 0] * prev[..., 1] * dt
        # Add stochastic noise to the rate.
        noise = params['sys']['noise_dx']
        noise *= rnd.normal(size=rnd_size) * dt
        dx += noise * np.sqrt(dx / dt)
        # Add stochastic noise to alpha.
        noise_alpha = params['sys']['noise_alpha']
        noise_alpha *= rnd.normal(size=rnd_size) * dt
        # Update the state vectors, ensuring alpha remains strictly positive.
        curr[..., 0] = prev[..., 0] - dx
        curr[..., 1] = np.clip(prev[..., 1] + noise_alpha, 0.01, 0.1)

    @classmethod
    def state_info(cls):
        """Describe each state variable."""
        return [("x", 0)]

    @classmethod
    def param_info(cls):
        """Describe each model parameter."""
        return [("alpha", 1)]

    @classmethod
    def stat_info(cls):
        """Describe each statistic that can be calculated by this model."""
        return []

Observation models¶

The observation model must be stored in params['log_llhd_fn'] and have the following form:

def log_llhd(params, obs_list, curr, prev_dict):
    """
    Calculate the log-likelihood of obtaining specific observations from each
    particle.
    """
    log_llhd = np.zeros(curr.shape[:-1])
    # The expected observation is x(t).
    exp = curr[..., 0]
    # The standard deviation of the observation error.
    sdev = params['obs']['sdev']
    # The likelihood distribution for each particle.
    obs_dist = scipy.stats.norm(loc=exp, scale=sdev)
    for o in obs_list:
        # Calculate the likelihood of this observation for each particle.
        log_llhd += obs_dist.logpdf(o['value'])
    return log_llhd

While the argument prev_dict was not used in this example, it can be used to obtain the state vectors at the beginning of an observation period:

def log_llhd(params, obs_list, curr, prev_dict):
    # Obtain the state vectors for one week prior.
    # This is only valid if an observation has a period of 7.
    one_week_ago = prev_dict[7]
    dx = curr[..., 0] - one_week_ago[..., 0]
    ...

This is useful for situations where the observation depends on the change in the state vector over the observation period.

Summary objects¶

Simulations typically comprise a large number of both particles and time steps, and so it is generally preferable to record statistics that summarise the particles than to store the entire state history of each simulation.

This functionality is provided by pypfilt.summary.HDF5, which allows any number of summary tables to be recorded. Once all of the estimation and forecasting simulations have been performed, save_forecasts() will save the results to disk.

Here is an example of how to record fixed-probability central credible intervals for the state variable \(x\) and model parameter \(\alpha\) of the example system, using the ModelCIs summary table:

def main(args=None):
    """Generate forecasts against noisy synthetic data."""
    params = get_params()
    # Define the simulation period.
    year = 2009
    days = 42
    start = datetime.datetime(year, 1, 1)
    until = start + datetime.timedelta(days=days)
    # Generate noisy (synthetic) observations.
    true_x0 = 0.7
    true_alpha = 0.05
    sdev = 0.03
    obs_list = generate_obs(true_x0, true_alpha, sdev, start, days)
    streams = [obs_list]
    # Define the summary tables to be saved to disk.
    summary = pypfilt.summary.HDF5(params, obs_list)
    summary.add_tables(pypfilt.summary.ModelCIs(probs=[0, 50, 95]))
    # Define the forecasting dates.
    fs = [datetime.datetime(year, 1, 2), datetime.datetime(year, 1, 3),
          datetime.datetime(year, 1, 5), datetime.datetime(year, 1, 9),
          datetime.datetime(year, 1, 16), datetime.datetime(year, 1, 23)]
    # Run the model estimation and forecasting simulations.
    data_file = os.path.join(os.path.dirname(__file__), 'example.hdf5')
    pypfilt.forecast(params, start, until, streams, fs, summary, data_file)
    return 0

Forecasting¶

Model estimations and subsequent forecasts are generated by pypfilt.forecast(), as illustrated in the example above.

This function takes the following arguments:

• A parameter dictionary;
• The start and end of the simulation period (datetime.date instances);
• Any number of observation streams;
• The dates at which forecasts should be generated (a datetime.date list);
• A summary object to calculate relevant statistics; and
• The output file, if desired, otherwise set to None.

Forecast plots¶

The code presented here is available in the doc/example directory. To generate and plot forecasts for this system, run the following commands from the root directory of the pypfilt repository:

./doc/example/run.py
./doc/example/plot.R

This will generate forecasts (stored in example.hdf5) and plot the credible intervals for \(x(t)\) and \(\alpha\). Important: the plotting script requires a working version of R and the following packages: ggplot2, rhdf5, and scales.

Credible intervals for \(\alpha\) (top row) and \(x\) (bottom row, plotted against the observations), shown for each forecast (identified by date at the top of the plot). Note that the right-most plots (“2009–2–12”) show the credible intervals obtained by using all of the observations.

Generating a series of forecasts¶

Model estimation and forecasting is provided as a single function:

pypfilt.forecast(params, start, end, streams, dates, summary, filename)¶

Generate forecasts from various dates during a simulation.

Parameters:	params (dict) – The simulation parameters. start (datetime.date) – The start of the simulation period. end (datetime.date) – The (exclusive) end of the simulation period. streams – A list of observation streams. dates – The dates at which forecasts should be generated. summary – An object that generates summaries of each simulation. filename – The output file to generate (can be None).
Returns:	The simulation state for each forecast date.

This function returns a dictionary that contains the following keys:

• 'obs': a (flattened) list of every observation;
• 'complete': the simulation state obtained by assimilating every observation; and
• datetime.datetime instances: the simulation state obtained for each forecast, identified by the forecasting date.

The simulation states are generated by pypfilt.run() and contain the following keys:

• 'params': the simulation parameters;
• 'summary': the dictionary of summary statistics; and
• 'hist': the matrix of particle state vectors, including individual particle weights (hist[..., -2]) and the index of each particle at the previous time-step (hist[..., -1]), since these can change due to resampling.

The matrix has dimensions \(N_{Steps} \times N_{Particles} \times (N_{SV} + 2)\) for state vectors of size \(N_{SV}\).

Note: if max_days > 0 was passed to pypfilt.default_params(), only a fraction of the entire simulation period will be available.

Particle filter parameters¶

Default values for the particle filter parameters are provided:

pypfilt.default_params(model, max_days=0, px_count=0)¶

The default particle filter parameters.

Memory usage can reach extreme levels with a large number of particles, and so it may be necessary to keep only a sliding window of the entire particle history matrix in memory.

Parameters:	model – The system model. max_days – The number of contiguous days that must be kept in memory (e.g., the largest observation period). px_count – The number of particles.

The bootstrap particle filter¶

The bootstrap particle filter is exposed as a single-step function, which will update particle weights and perform resampling as necessary:

pypfilt.step(params, hist, hist_ix, step_num, when, step_obs, max_back, is_fs)¶

Perform a single time-step for every particle.

Parameters:

params – The simulation parameters. hist – The particle history matrix. hist_ix – The index of the current time-step in the history matrix. step_num – The time-step number. when – The current simulation time. step_obs – The list of observations for this time-step. max_back – The number of time-steps into the past when the most recent resampling occurred; must be either a positive integer or None (no limit). is_fs – Indicate whether this is a forecasting simulation (i.e., no observations). For deterministic models it is useful to add some random noise when estimating, to allow identical particles to differ in their behaviour, but this is not desirable when forecasting.

Returns:

True if resampling was performed, otherwise False.

Running a single simulation¶

pypfilt.run(params, start, end, streams, summary, state=None, save_when=None, save_to=None)¶

Run the particle filter against any number of data streams.

Parameters:

params (dict) – The simulation parameters. start (datetime.date) – The start of the simulation period. end (datetime.date) – The (exclusive) end of the simulation period. streams – A list of observation streams (see with_observations()). summary – An object that generates summaries of each simulation. state – A previous simulation state as returned by, e.g., this function. save_when – Dates at which to save the particle history matrix. save_to – The filename for saving the particle history matrix.

Returns:

The resulting simulation state: a dictionary that contains the simulation parameters ('params'), the particle history matrix ('hist'), and the summary statistics ('summary').

Simulation models¶

All simulation models should derive the following base class.

class pypfilt.model.Base¶

The base class for simulation models, which defines the minimal set of methods that are required.

static init(params, vec)¶

Initialise a state vector.

Parameters:	params – Simulation parameters. vec – An uninitialised state vector of correct dimensions (see state_size()).
Raises:	NotImplementedError – Derived classes must implement this method.

static state_size()¶

Return the size of the state vector.

Raises:	NotImplementedError – Derived classes must implement this method.

static priors(params)¶

Return a dictionary of model parameter priors. Each key must identify a parameter by name. Each value must be a function that returns samples from the associated prior distribution, and should have the following form:

lambda r, size=None: r.uniform(1.0, 2.0, size=size)

Here, the argument r is a PRNG instance and size specifies the output shape (by default, a single value).

Parameters:	params – Simulation parameters.
Raises:	NotImplementedError – Derived classes must implement this method.

classmethod update(params, step_date, dt, is_fs, prev, curr)¶

Perform a single time-step.

Parameters:	params – Simulation parameters. step_date – The date and time of the current time-step. dt – The time-step size (days). is_fs – Indicates whether this is a forecasting simulation. prev – The state before the time-step. curr – The state after the time-step (destructively updated).
Raises:	NotImplementedError – Derived classes must implement this method.

classmethod state_info()¶

Describe each state variable as a (name, index) tuple, where name is a descriptive name for the variable and index is the index of that variable in the state vector.

Raises:	NotImplementedError – Derived classes must implement this method.

classmethod param_info()¶

Describe each model parameter as a (name, index) tuple, where name is a descriptive name for the parameter and index is the index of that parameter in the state vector.

Raises:	NotImplementedError – Derived classes must implement this method.

classmethod stat_info()¶

Describe each statistic that can be calculated by this model as a (name, stat_fn) tuple, where name is a string that identifies the statistic and stat_fn is a function that calculates the value of the statistic.

Raises:	NotImplementedError – Derived classes must implement this method.

static is_valid(hist)¶

Identify particles whose state and parameters can be inspected. By default, this function returns True for all particles. Override this function to ensure that inchoate particles are correctly ignored.

Weighted statistics¶

The pypfilt.stats module provides functions for calculating weighted statistics across particle populations.

pypfilt.stats.cov_wt(x, wt, cor=False)¶

Estimate the weighted covariance matrix, based on a NumPy pull request.

Equivalent to cov.wt(x, wt, cor, center=TRUE, method="unbiased") as provided by the stats package for R.

Parameters:	x – A 2-D array; columns represent variables and rows represent observations. wt – A 1-D array of observation weights. cor – Whether to return a correlation matrix instead of a covariance matrix.
Returns:	The covariance matrix (or correlation matrix, if cor=True).

pypfilt.stats.avg_var_wt(x, weights, biased=True)¶

Return the weighted average and variance (based on a Stack Overflow answer).

Parameters:	x – The data points. weights – The normalised weights. biased – Use a biased variance estimator.
Returns:	A tuple that contains the weighted average and weighted variance.

pypfilt.stats.qtl_wt(x, weights, probs)¶

Equivalent to wtd.quantile(x, weights, probs, normwt=TRUE) as provided by the Hmisc package for R.

Parameters:	x – The numerical data. weights – The weight of each data point. probs – The quantile(s) to compute.
Returns:	The array of weighted quantiles.

pypfilt.stats.cred_wt(x, weights, creds)¶

Calculate weighted credible intervals.

Parameters:	x – The numerical data. weights – The weight of each data point. creds (List(int)) – The credible interval(s) to compute (0..100, where 0 represents the median and 100 the entire range).
Returns:	A dictionary that maps credible intervals to the lower and upper interval bounds.

Simulation metadata¶

Every simulation data file should include metadata that documents the simulation parameters and working environment. The pypfilt.summary provides a function to automatically generate such metadata:

pypfilt.summary.metadata(params, pkgs=None)¶

Construct a metadata dictionary that documents the simulation parameters and system environment. Note that this should be generated at the start of the simulation, and that the git metadata will only be valid if the working directory is located within a git repository.

Parameters:	params – The simulation parameters. pkgs – A dictionary that maps package names to modules that define appropriate __version__ attributes, used to record the versions of additional relevant packages; see below for an example:

By default, the versions of pypfilt, h5py, numpy and scipy are recorded. The following example demonstrates how to also record the installed version of the epifx package:

import epifx
import pypfilt.summary
params = ...
metadata = pypfilt.summary.metadata(params, {'epifx': epifx})

If the above function isn’t sufficiently flexible, several other utility functions are provided to assist with generating metadata:

pypfilt.summary.metadata_priors(params)¶

Return a dictionary that describes the model parameter priors.

Each key identifies a parameter (by name); the corresponding value is a string representation of the prior distribution, which is typically a numpy.random.RandomState method call.

For example:

{'alpha': "random.uniform(0.1, 1.0)"}

pypfilt.summary.encode_value(value)¶

Encode values in a form suitable for serialisation in HDF5 files.

• Integer values are converted to numpy.int32 values.
• Floating-point values and arrays retain their data type.
• All other (i.e., non-numerical) values are converted to UTF-8 strings.

pypfilt.summary.filter_dict(values, ignore, encode_fn)¶

Recursively filter items from a dictionary, used to remove parameters from the metadata dictionary that, e.g., have no meaningful representation.

Parameters:	values – The original dictionary. ignore – A dictionary that specifies which values to ignore. encode_fn – A function that encodes the remaining values (see encode_value()).

For example, to ignore ['px_range'], ['resample']['rnd'], and 'expect_fn' and 'log_llhd_fn' for every observation system:

ignore = {
    'px_range': None,
    'resample': {'rnd': None},
    # Note the use of ``None`` to match any key under 'obs'.
    'obs': {None: {'expect_fn': None, 'log_llhd_fn': None}}
}
filter_dict(params, ignore, encode_value)

Summary statistics¶

Summary statistics are stored in tables, each of which comprises a set of named columns and a specific number of rows.

Summary data files¶

class pypfilt.summary.HDF5(params, all_obs, meta=None, first_day=False)¶

Save tables of summary statistics to an HDF5 file.

Parameters:	params – The simulation parameters. obs_list – A list of all observations. meta – The simulation metadata; by default the output of metadata() is used. first_day – If False (the default) statistics are calculated from the date of the first observation. If True, statistics are calculated from the very beginning of the simulation period.

add_tables(*tables)¶

Add summary statistic tables that will be included in the output file.

save_forecasts(fs, filename)¶

Save forecast summaries to disk in the HDF5 binary data format.

This function creates the following datasets that summarise the estimation and forecasting outputs:

• 'data/TABLE' for each table.

The provided metadata will be recorded under 'meta/'.

If dataset creation timestamps are enabled, two simulations that produce identical outputs will not result in identical files. Timestamps will be disabled where possible (requires h5py >= 2.2):

• 'hdf5_track_times': Presence of creation timestamps.

Parameters:	fs – Simulation outputs, as returned by pypfilt.forecast(). filename – The filename to which the data will be written.