Inference

Comparison and optimization of model spectra to data.

Anscombe_Poisson_residual(model, data, mask=None)

Return the Anscombe Poisson residuals between model and data.

mask sets the level in model below which the returned residual array is masked. This excludes very small values where the residuals are not normal. 1e-2 seems to be a good default for the NIEHS human data. (model = 1e-2, data = 0, yields a residual of ~1.5.)

Residuals defined in this manner are more normally distributed than the linear residuals when the mean is small. See this reference below for justification: Pierce DA and Schafer DW, "Residuals in generalized linear models" Journal of the American Statistical Association, 81(396)977-986 (1986).

Note that I tried implementing the "adjusted deviance" residuals, but they always looked very biased for the cases where the data was 0.

Source code in dadi/Inference.py

def Anscombe_Poisson_residual(model, data, mask=None):
    """
    Return the Anscombe Poisson residuals between model and data.

    mask sets the level in model below which the returned residual array is
    masked. This excludes very small values where the residuals are not normal.
    1e-2 seems to be a good default for the NIEHS human data. (model = 1e-2,
    data = 0, yields a residual of ~1.5.)

    Residuals defined in this manner are more normally distributed than the
    linear residuals when the mean is small. See this reference below for
    justification: Pierce DA and Schafer DW, "Residuals in generalized linear
    models" Journal of the American Statistical Association, 81(396)977-986
    (1986).

    Note that I tried implementing the "adjusted deviance" residuals, but they
    always looked very biased for the cases where the data was 0.
    """
    if hasattr(data, 'folded') and data.folded and not model.folded:
        model = model.fold()
    if hasattr(data, 'folded_ancestral') and data.folded_ancestral\
       and not model.folded_ancestral:
        model = model.fold_ancestral()
    if hasattr(data, 'folded_major') and data.folded_major\
       and not model.folded_major:
        model = model.fold_major()
    # Because my data have often been projected downward or averaged over many
    # iterations, it appears better to apply the same transformation to the data
    # and the model.
    # For some reason data**(-1./3) results in entries in data that are zero
    # becoming masked. Not just the result, but the data array itself. We use
    # the power call to get around that.
    # This seems to be a common problem, that we want to use numpy.ma functions
    # on masked arrays, because otherwise the mask on the input itself can be
    # changed. Subtle and annoying. If we need to create our own functions, we
    # can use numpy.ma.core._MaskedUnaryOperation.
    datatrans = data**(2./3) - numpy.ma.power(data,-1./3)/9
    modeltrans = model**(2./3) - numpy.ma.power(model,-1./3)/9
    resid = 1.5*(datatrans - modeltrans)/model**(1./6)
    if mask is not None:
        tomask = numpy.logical_and(model <= mask, data <= mask)
        tomask = numpy.logical_or(tomask, data == 0)
        resid = numpy.ma.masked_where(tomask, resid)
    # It makes more sense to me to have a minus sign here... So when the
    # model is high, the residual is positive. This is opposite of the
    # Pierce and Schafner convention.
    return -resid

add_misid_param(func)

Add parameter to model ancestral state misidentification.

The returned function will have an additional element of the params list, which is the proportion of segregating sites whose ancestral state were misidentified.

Source code in dadi/Inference.py

def add_misid_param(func):
    """
    Add parameter to model ancestral state misidentification.

    The returned function will have an additional element of the params
    list, which is the proportion of segregating sites whose ancestral state
    were misidentified.
    """
    warnings.warning("Inference.add_misid_param is deprecated. Please use the updated Numerics.make_anc_state_misd_func instead.\n", FutureWarning)
    def misid_func(params, *args, **kwargs):
        misid = params[-1]
        fs = func(params[:-1], *args, **kwargs)
        return (1-misid)*fs + misid*Numerics.reverse_array(fs)
    misid_func.__name__ = func.__name__
    misid_func.__doc__ = func.__doc__
    return misid_func

linear_Poisson_residual(model, data, mask=None)

Return the Poisson residuals, (model - data)/sqrt(model), of model and data.

mask sets the level in model below which the returned residual array is masked. The default of 0 excludes values where the residuals are not defined.

In the limit that the mean of the Poisson distribution is large, these residuals are normally distributed. (If the mean is small, the Anscombe residuals are better.)

Source code in dadi/Inference.py

def linear_Poisson_residual(model, data, mask=None):
    """
    Return the Poisson residuals, (model - data)/sqrt(model), of model and data.

    mask sets the level in model below which the returned residual array is
    masked. The default of 0 excludes values where the residuals are not 
    defined.

    In the limit that the mean of the Poisson distribution is large, these
    residuals are normally distributed. (If the mean is small, the Anscombe
    residuals are better.)
    """
    if hasattr(data, 'folded') and data.folded and not model.folded:
        model = model.fold()
    if hasattr(data, 'folded_ancestral') and data.folded_ancestral\
       and not model.folded_ancestral:
        model = model.fold_ancestral()
    if hasattr(data, 'folded_major') and data.folded_major\
       and not model.folded_major:
        model = model.fold_major()

    resid = (model - data)/numpy.ma.sqrt(model)
    if mask is not None:
        tomask = numpy.logical_and(model <= mask, data <= mask)
        resid = numpy.ma.masked_where(tomask, resid)
    return resid

ll(model, data)

The log-likelihood of the data given the model sfs.

Evaluate the log-likelihood of the data given the model. This is based on Poisson statistics, where the probability of observing k entries in a cell given that the mean number is given by the model is P(k) = exp(-model) * model**k / k!

Note

If either the model or the data is a masked array, the return ll will ignore any elements that are masked in either the model or the data.

Source code in dadi/Inference.py

def ll(model, data):
    """
    The log-likelihood of the data given the model sfs.

    Evaluate the log-likelihood of the data given the model. This is based on
    Poisson statistics, where the probability of observing k entries in a cell
    given that the mean number is given by the model is 
    P(k) = exp(-model) * model**k / k!

    Note: 
        If either the model or the data is a masked array, the return ll will
          ignore any elements that are masked in *either* the model or the data.
    """
    ll_arr = ll_per_bin(model, data)
    return ll_arr.sum()

ll_multinom(model, data)

Log-likelihood of the data given the model, with optimal rescaling.

Evaluate the log-likelihood of the data given the model. This is based on Poisson statistics, where the probability of observing k entries in a cell given that the mean number is given by the model is P(k) = exp(-model) * model**k / k!

model is optimally scaled to maximize ll before calculation.

Note

If either the model or the data is a masked array, the return ll will ignore any elements that are masked in either the model or the data.

Source code in dadi/Inference.py

def ll_multinom(model, data):
    """
    Log-likelihood of the data given the model, with optimal rescaling.

    Evaluate the log-likelihood of the data given the model. This is based on
    Poisson statistics, where the probability of observing k entries in a cell
    given that the mean number is given by the model is 
    P(k) = exp(-model) * model**k / k!

    model is optimally scaled to maximize ll before calculation.

    Note:
        If either the model or the data is a masked array, the return ll will
          ignore any elements that are masked in *either* the model or the data.
    """
    ll_arr = ll_multinom_per_bin(model, data)
    return ll_arr.sum()

ll_multinom_per_bin(model, data)

Mutlinomial log-likelihood of each entry in the data given the model.

Scales the model sfs to have the optimal theta for comparison with the data.

Source code in dadi/Inference.py

def ll_multinom_per_bin(model, data):
    """
    Mutlinomial log-likelihood of each entry in the data given the model.

    Scales the model sfs to have the optimal theta for comparison with the data.
    """
    theta_opt = optimal_sfs_scaling(model, data)
    return ll_per_bin(theta_opt*model, data)

ll_per_bin(model, data, missing_model_cutoff=1e-06)

The Poisson log-likelihood of each entry in the data given the model sfs.

Parameters:

Name	Type	Description	Default
model	Spectrum	Model spectrum.	required
data	Spectrum	Data spectrum.	required
missing_model_cutoff	float	Due to numerical issues, there may be entries in the FS that cannot be stable calculated. If these entries involve a fraction of the data larger than missing_model_cutoff, a warning is printed.	1e-06

Source code in dadi/Inference.py

def ll_per_bin(model, data, missing_model_cutoff=1e-6):
    """
    The Poisson log-likelihood of each entry in the data given the model sfs.

    Args:
        model (Spectrum): Model spectrum.
        data (Spectrum): Data spectrum.
        missing_model_cutoff (float): Due to numerical issues, there may be entries in the
                            FS that cannot be stable calculated. If these entries
                            involve a fraction of the data larger than
                            missing_model_cutoff, a warning is printed.
    """
    if hasattr(data, 'folded') and data.folded and not model.folded:
        model = model.fold()
    if hasattr(data, 'folded_ancestral') and data.folded_ancestral\
       and not model.folded_ancestral:
        model = model.fold_ancestral()
    if hasattr(data, 'folded_major') and data.folded_major\
       and not model.folded_major:
        model = model.fold_major()

    # Using numpy.ma.log here ensures that any negative or nan entries in model
    # yield masked entries in result. We can then check for correctness of
    # calculation by simply comparing masks.
    # Note: Using .data attributes directly saves a little computation time. We
    # use model and data as a whole at least once, to ensure masking is done
    # properly.
    result = -model.data + data.data*model.log() - gammaln(data + 1.)
    if numpy.all(result.mask == data.mask):
        return result

    not_data_mask = logical_not(data.mask)
    data_sum = data.sum()

    missing = logical_and(model < 0, not_data_mask)
    if numpy.any(missing)\
       and data[missing].sum()/data.sum() > missing_model_cutoff:
        logger.warning('Model is < 0 where data is not masked.')
        logger.warning('Number of affected entries is %i. Sum of data in those '
                'entries is %g:' % (missing.sum(), data[missing].sum()))

    # If the data is 0, it's okay for the model to be 0. In that case the ll
    # contribution is 0, which is fine.
    missing = logical_and(model == 0, logical_and(data > 0, not_data_mask))
    if numpy.any(missing)\
       and data[missing].sum()/data_sum > missing_model_cutoff:
        logger.warning('Model is 0 where data is neither masked nor 0.')
        logger.warning('Number of affected entries is %i. Sum of data in those '
                    'entries is %g:' % (missing.sum(), data[missing].sum()))

    missing = numpy.logical_and(model.mask, not_data_mask)
    if numpy.any(missing)\
       and data[missing].sum()/data_sum > missing_model_cutoff:
        print(data[missing].sum(), data_sum)
        logger.warning('Model is masked in some entries where data is not.')
        logger.warning('Number of affected entries is %i. Sum of data in those '
                    'entries is %g:' % (missing.sum(), data[missing].sum()))

    missing = numpy.logical_and(numpy.isnan(model), not_data_mask)
    if numpy.any(missing)\
       and data[missing].sum()/data_sum > missing_model_cutoff:
        logger.warning('Model is nan in some entries where data is not masked.')
        logger.warning('Number of affected entries is %i. Sum of data in those '
                    'entries is %g:' % (missing.sum(), data[missing].sum()))

    return result

minus_ll(model, data)

The negative of the log-likelihood of the data given the model sfs.

Source code in dadi/Inference.py

def minus_ll(model, data):
    """
    The negative of the log-likelihood of the data given the model sfs.
    """
    return -ll(model, data)

minus_ll_multinom(model, data)

The negative of the log-likelihood of the data given the model sfs.

Return a double that is -(log-likelihood)

Source code in dadi/Inference.py

def minus_ll_multinom(model, data):
    """
    The negative of the log-likelihood of the data given the model sfs.

    Return a double that is -(log-likelihood)
    """
    return -ll_multinom(model, data)

optimal_sfs_scaling(model, data)

Optimal multiplicative scaling factor between model and data.

This scaling is based on only those entries that are masked in neither model nor data.

Source code in dadi/Inference.py

def optimal_sfs_scaling(model, data):
    """
    Optimal multiplicative scaling factor between model and data.

    This scaling is based on only those entries that are masked in neither
    model nor data.
    """
    if hasattr(data, 'folded') and data.folded and not model.folded:
        model = model.fold()
    if hasattr(data, 'folded_ancestral') and data.folded_ancestral\
       and not model.folded_ancestral:
        model = model.fold_ancestral()
    if hasattr(data, 'folded_major') and data.folded_major\
       and not model.folded_major:
        model = model.fold_major()

    model, data = Numerics.intersect_masks(model, data)
    return data.sum()/model.sum()

optimally_scaled_sfs(model, data)

Optimially scale model sfs to data sfs.

Returns a new scaled model sfs.

Source code in dadi/Inference.py

def optimally_scaled_sfs(model, data):
    """
    Optimially scale model sfs to data sfs.

    Returns a new scaled model sfs.
    """
    return optimal_sfs_scaling(model,data) * model

optimize(p0, data, model_func, pts, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, epsilon=0.001, gtol=1e-05, multinom=True, maxiter=None, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1, output_file=None)

Optimize params to fit model to data using the BFGS method.

This optimization method works well when we start reasonably close to the optimum. It is best at burrowing down a single minimum.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0.	None
verbose	int	If > 0, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
epsilon	float	Step-size to use for finite-difference derivatives.	0.001
gtol	float	Convergence criterion for optimization. For more info, see help(scipy.optimize.fmin_bfgs)	1e-05
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
maxiter	int	Maximum iterations to run for.	None
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_bfgs)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation.	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. (See help(dadi.Inference.optimize_log for examples of func_args and fixed_params usage.)	None
ll_scale	float	The bfgs algorithm may fail if your initial log-likelihood is too large. (This appears to be a flaw in the scipy implementation.) To overcome this, pass ll_scale > 1, which will simply reduce the magnitude of the log-likelihood. Once in a region of reasonable likelihood, you'll probably want to re-optimize with ll_scale=1.	1

Source code in dadi/Inference.py

def optimize(p0, data, model_func, pts, lower_bound=None, upper_bound=None,
             verbose=0, flush_delay=0.5, epsilon=1e-3, 
             gtol=1e-5, multinom=True, maxiter=None, full_output=False,
             func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1,
             output_file=None):
    """
    Optimize params to fit model to data using the BFGS method.

    This optimization method works well when we start reasonably close to the
    optimum. It is best at burrowing down a single minimum.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
                        (params, (n1,n2...), pts)
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
                    length as p0.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
                    length as p0.
        verbose (int): If > 0, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        epsilon (float): Step-size to use for finite-difference derivatives.
        gtol (float): Convergence criterion for optimization. For more info, 
            see help(scipy.optimize.fmin_bfgs)
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        maxiter (int): Maximum iterations to run for.
        full_output (bool): If True, return full outputs as in described in 
                    help(scipy.optimize.fmin_bfgs)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored.
                    (See help(dadi.Inference.optimize_log for examples of func_args and 
                    fixed_params usage.)
        ll_scale (float): The bfgs algorithm may fail if your initial log-likelihood is
                too large. (This appears to be a flaw in the scipy
                implementation.) To overcome this, pass ll_scale > 1, which will
                simply reduce the magnitude of the log-likelihood. Once in a
                region of reasonable likelihood, you'll probably want to
                re-optimize with ll_scale=1.
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, lower_bound, upper_bound, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 
            ll_scale, output_stream)

    p0 = _project_params_down(p0, fixed_params)
    outputs = scipy.optimize.fmin_bfgs(_object_func, p0, 
                                       epsilon=epsilon,
                                       args = args, gtol=gtol, 
                                       full_output=True,
                                       disp=False,
                                       maxiter=maxiter)
    xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflag = outputs
    xopt = _project_params_up(xopt, fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflag

optimize_cons(p0, data, model_func, pts, eq_constraint=None, ieq_constraint=None, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, epsilon=0.0001, gtol=1e-05, multinom=True, maxiter=None, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1, output_file=None)

Optimize params to fit model to data using constrainted optimization.

This method will ensure parameter constraints are satisfied. For example, you might constrain than one parameter is larger than other, or that the sum of several parameters is a particular value.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
eq_constraint	func	Function that returns a 1D array in which each element must equal to 0 in a successful result. For example, to constraint p[1] + p[2] = 1 and p[3] - p[2] = 3, def eq_constraint(p): return [p[1]+p[2]-1, p[3]-p[2]-3]	None
ieq_constraint	func	Function that returns a 1D array in which each element must greater than or equal to 0 in a successful result.	None
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0.	None
verbose	int	If > 0, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
epsilon	int	Step-size to use for finite-difference derivatives.	0.0001
gtol	float	Convergence criterion for optimization. For more info, see help(scipy.optimize.fmin_bfgs)	1e-05
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
maxiter	int	Maximum iterations to run for.	None
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_slsqp)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation. Using func_args. For example, you could define your model function as def func((p1,p2), ns, f1, f2, pts): .... If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you would pass func_args = [0.1,0.2] (and ignore the fixed_params argument).	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. For example, suppose your model function is def func((p1,f1,p2,f2), ns, pts): .... If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you would pass fixed_params = [None,0.1,None,0.2] (and ignore the func_args argument).	None
ll_scale	float	The bfgs algorithm may fail if your initial log-likelihood is too large. (This appears to be a flaw in the scipy implementation.) To overcome this, pass ll_scale > 1, which will simply reduce the magnitude of the log-likelihood. Once in a region of reasonable likelihood, you'll probably want to re-optimize with ll_scale=1.	1

Source code in dadi/Inference.py

def optimize_cons(p0, data, model_func, pts,
                  eq_constraint=None, ieq_constraint=None,
                  lower_bound=None, upper_bound=None, 
                  verbose=0, flush_delay=0.5, epsilon=1e-4,
                  gtol=1e-5, multinom=True, maxiter=None,
                  full_output=False, func_args=[], func_kwargs={},
                  fixed_params=None, ll_scale=1, output_file=None):
    """
    Optimize params to fit model to data using constrainted optimization.

    This method will ensure parameter constraints are satisfied.
    For example, you might constrain than one parameter is larger than other,
     or that the sum of several parameters is a particular value.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
                        (params, (n1,n2...), pts)
        eq_constraint (func): Function that returns a 1D array in which each element must
                    equal to 0 in a successful result.
                    For example, to constraint p[1] + p[2] = 1 and p[3] - p[2] = 3,
                    def eq_constraint(p): return [p[1]+p[2]-1, p[3]-p[2]-3]
        ieq_constraint (func): Function that returns a 1D array in which each element must
                        greater than or equal to 0 in a successful result.
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
                    length as p0.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
                    length as p0.
        verbose (int): If > 0, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        epsilon (int): Step-size to use for finite-difference derivatives.
        gtol (float): Convergence criterion for optimization. For more info, 
            see help(scipy.optimize.fmin_bfgs)
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        maxiter (int): Maximum iterations to run for.
        full_output (bool): If True, return full outputs as in described in 
                    help(scipy.optimize.fmin_slsqp)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
                Using func_args.
                For example, you could define your model function as
                def func((p1,p2), ns, f1, f2, pts):
                    ....
                If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you
                would pass func_args = [0.1,0.2] (and ignore the fixed_params 
                argument).
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored.
                    For example, suppose your model function is 
                    def func((p1,f1,p2,f2), ns, pts):
                        ....
                    If you wanted to fix f1=0.1 and f2=0.2 in the optimization, 
                    you would pass fixed_params = [None,0.1,None,0.2] (and ignore
                    the func_args argument).
        ll_scale (float): The bfgs algorithm may fail if your initial log-likelihood is
                too large. (This appears to be a flaw in the scipy
                implementation.) To overcome this, pass ll_scale > 1, which will
                simply reduce the magnitude of the log-likelihood. Once in a
                region of reasonable likelihood, you'll probably want to
                re-optimize with ll_scale=1.
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, None, None, verbose, 
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 
            ll_scale, output_stream)

    if lower_bound is None:
        lower_bound = [None] * len(p0)
    if upper_bound is None:
        upper_bound = [None] * len(p0)
    lower_bound = _project_params_down(lower_bound, fixed_params)
    upper_bound = _project_params_down(upper_bound, fixed_params)
    if (lower_bound is not None) and (upper_bound is not None):
        bnds = tuple(zip(lower_bound,upper_bound))

    p0 = _project_params_down(p0, fixed_params)

    outputs = scipy.optimize.fmin_slsqp(_object_func, 
        p0, bounds = bnds, args = args,
        f_eqcons = eq_constraint, f_ieqcons = ieq_constraint,
        epsilon = epsilon,
        iter = maxiter, full_output = True,
        disp = False)
    xopt, fopt, func_calls, grad_calls, warnflag = outputs

    xopt = _project_params_up(xopt, fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, func_calls, grad_calls, warnflag

optimize_grid(data, model_func, pts, grid, verbose=0, flush_delay=0.5, multinom=True, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, output_file=None)

Optimize params to fit model to data using brute force search over a grid.

Search grids are specified using a dadi.Inference.index_exp object (which is an alias for numpy.index_exp). The grid is specified by passing a range of values for each parameter. For example, index_exp[0:1.1:0.3, 0.7:0.9:11j] will search over parameter 1 with values 0,0.3,0.6,0.9 and over parameter 2 with 11 points between 0.7 and 0.9 (inclusive). (Notice the 11j in the second parameter range specification.) Note that the grid list should include only parameters that are optimized over, not fixed parameter values.

Parameters:

Name	Type	Description	Default
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
pts	list[int]	Grid points list for evaluating likelihoods	required
grid	list[int]	Grid of parameter values over which to evaluate likelihood. See below for specification instructions.	required
verbose	float	If > 0, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
full_output	bool	If True, return popt, llopt, grid, llout, thetas. Here popt is the best parameter set found and llopt is the corresponding (composite) log-likelihood. grid is the array of parameter values tried, llout is the corresponding log-likelihoods, and thetas is the corresponding thetas. Note that the grid includes only the parameters optimized over, and that the order of indices is such that grid[:,0,2] would be a set of parameters if two parameters were optimized over. (Note the : in the first index.)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation.	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. (See help(dadi.Inference.optimize_log for examples of func_args and fixed_params usage.)	None

Source code in dadi/Inference.py

def optimize_grid(data, model_func, pts, grid,
                  verbose=0, flush_delay=0.5,
                  multinom=True, full_output=False,
                  func_args=[], func_kwargs={}, fixed_params=None,
                  output_file=None):
    """
    Optimize params to fit model to data using brute force search over a grid.

    Search grids are specified using a dadi.Inference.index_exp object (which
    is an alias for numpy.index_exp). The grid is specified by passing a range
    of values for each parameter. For example, index_exp[0:1.1:0.3,
    0.7:0.9:11j] will search over parameter 1 with values 0,0.3,0.6,0.9 and
    over parameter 2 with 11 points between 0.7 and 0.9 (inclusive). (Notice
    the 11j in the second parameter range specification.) Note that the grid
    list should include only parameters that are optimized over, not fixed
    parameter values.

    Args:
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
                    (params, (n1,n2...), pts)
        pts (list[int]): Grid points list for evaluating likelihoods
        grid (list[int]): Grid of parameter values over which to evaluate likelihood. See
            below for specification instructions.
        verbose (float): If > 0, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        full_output (bool): If True, return popt, llopt, grid, llout, thetas. Here popt is
                    the best parameter set found and llopt is the corresponding
                    (composite) log-likelihood. grid is the array of parameter
                    values tried, llout is the corresponding log-likelihoods, and
                    thetas is the corresponding thetas. Note that the grid includes
                    only the parameters optimized over, and that the order of
                    indices is such that grid[:,0,2] would be a set of parameters
                    if two parameters were optimized over. (Note the : in the
                    first index.)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored.
                    (See help(dadi.Inference.optimize_log for examples of func_args and 
                    fixed_params usage.)
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, None, None, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 1.0,
            output_stream, full_output)

    if full_output:
        global _theta_store
        _theta_store = {}

    outputs = scipy.optimize.brute(_object_func, ranges=grid,
                                   args=args, full_output=full_output,
                                   finish=False)
    if full_output:
        xopt, fopt, grid, fout = outputs
        # Thetas are stored as a dictionary, because we can't guarantee
        # iteration order in brute(). So we have to iterate back over them
        # to produce the proper order to return.
        thetas = numpy.zeros(fout.shape)
        for indices, temp in numpy.ndenumerate(fout):
            # This is awkward, because we need to access grid[:,indices]
            grid_indices = tuple([slice(None,None,None)] + list(indices))
            thetas[indices] = _theta_store[tuple(grid[grid_indices])]
    else:
        xopt = outputs
    xopt = _project_params_up(xopt, fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, grid, fout, thetas

optimize_lbfgsb(p0, data, model_func, pts, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, epsilon=0.001, pgtol=1e-05, multinom=True, maxiter=100000.0, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1, output_file=None)

Optimize log(params) to fit model to data using the L-BFGS-B method.

This optimization method works well when we start reasonably close to the optimum. It is best at burrowing down a single minimum. This method is better than optimize_log if the optimum lies at one or more of the parameter bounds. However, if your optimum is not on the bounds, this method may be much slower.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	float	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
verbose	int	If > 0, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
epsilon	float	Step-size to use for finite-difference derivatives.	0.001
pgtol	float	Convergence criterion for optimization. For more info, see help(scipy.optimize.fmin_l_bfgs_b)	1e-05
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
maxiter	int	Maximum algorithm iterations evaluations to run.	100000.0
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_bfgs)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation.	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. (See help(dadi.Inference.optimize_log for examples of func_args and fixed_params usage.)	None
ll_scale	float	The bfgs algorithm may fail if your initial log-likelihood is too large. (This appears to be a flaw in the scipy implementation.) To overcome this, pass ll_scale > 1, which will simply reduce the magnitude of the log-likelihood. Once in a region of reasonable likelihood, you'll probably want to re-optimize with ll_scale=1.	1

Citation

The L-BFGS-B method was developed by Ciyou Zhu, Richard Byrd, and Jorge Nocedal. The algorithm is described in:

• R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound Constrained Optimization, (1995), SIAM Journal on Scientific and Statistical Computing , 16, 5, pp. 1190-1208.

• C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization (1997), ACM Transactions on Mathematical Software, Vol 23, Num. 4, pp. 550-560.

Source code in dadi/Inference.py

def optimize_lbfgsb(p0, data, model_func, pts, 
                    lower_bound=None, upper_bound=None,
                    verbose=0, flush_delay=0.5, epsilon=1e-3, 
                    pgtol=1e-5, multinom=True, maxiter=1e5, full_output=False,
                    func_args=[], func_kwargs={}, fixed_params=None, 
                    ll_scale=1, output_file=None):
    """
    Optimize log(params) to fit model to data using the L-BFGS-B method.

    This optimization method works well when we start reasonably close to the
    optimum. It is best at burrowing down a single minimum. This method is
    better than optimize_log if the optimum lies at one or more of the
    parameter bounds. However, if your optimum is not on the bounds, this
    method may be much slower.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (float): Function to evaluate model spectrum. Should take arguments
                        (params, (n1,n2...), pts)
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
                    length as p0. A parameter can be declared unbound by assigning
                    a bound of None.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
                    length as p0. A parameter can be declared unbound by assigning
                    a bound of None.
        verbose (int): If > 0, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        epsilon (float): Step-size to use for finite-difference derivatives.
        pgtol (float): Convergence criterion for optimization. For more info, 
            see help(scipy.optimize.fmin_l_bfgs_b)
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        maxiter (int): Maximum algorithm iterations evaluations to run.
        full_output (bool): If True, return full outputs as in described in 
                    help(scipy.optimize.fmin_bfgs)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored.
                    (See help(dadi.Inference.optimize_log for examples of func_args and 
                    fixed_params usage.)
        ll_scale (float): The bfgs algorithm may fail if your initial log-likelihood is
                too large. (This appears to be a flaw in the scipy
                implementation.) To overcome this, pass ll_scale > 1, which will
                simply reduce the magnitude of the log-likelihood. Once in a
                region of reasonable likelihood, you'll probably want to
                re-optimize with ll_scale=1.

    Citation:
        The L-BFGS-B method was developed by Ciyou Zhu, Richard Byrd, and Jorge
        Nocedal. The algorithm is described in:

        * R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
            Constrained Optimization, (1995), SIAM Journal on Scientific and
            Statistical Computing , 16, 5, pp. 1190-1208.

        * C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
            FORTRAN routines for large scale bound constrained optimization (1997),
            ACM Transactions on Mathematical Software, Vol 23, Num. 4, pp. 550-560.
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, None, None, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 
            ll_scale, output_stream)

    # Make bounds list. For this method it needs to be in terms of log params.
    if lower_bound is None:
        lower_bound = [None] * len(p0)
    lower_bound = _project_params_down(lower_bound, fixed_params)
    if upper_bound is None:
        upper_bound = [None] * len(p0)
    upper_bound = _project_params_down(upper_bound, fixed_params)
    bounds = list(zip(lower_bound,upper_bound))

    p0 = _project_params_down(p0, fixed_params)

    outputs = scipy.optimize.fmin_l_bfgs_b(_object_func, 
                                           numpy.log(p0), bounds=bounds,
                                           epsilon=epsilon, args=args,
                                           iprint=-1, pgtol=pgtol,
                                           maxfun=maxiter, approx_grad=True)
    xopt, fopt, info_dict = outputs

    xopt = _project_params_up(xopt, fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, info_dict

optimize_log(p0, data, model_func, pts, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, epsilon=0.001, gtol=1e-05, multinom=True, maxiter=None, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1, output_file=None)

Optimize log(params) to fit model to data using the BFGS method.

This optimization method works well when we start reasonably close to the optimum. It is best at burrowing down a single minimum.

Because this works in log(params), it cannot explore values of params < 0. It should also perform better when parameters range over scales.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0.	None
verbose	int	If > 0, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
epsilon	float	Step-size to use for finite-difference derivatives.	0.001
gtol	float	Convergence criterion for optimization. For more info, see help(scipy.optimize.fmin_bfgs)	1e-05
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
maxiter	inf	Maximum iterations to run for.	None
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_bfgs)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation. Using func_args. For example, you could define your model function as def func((p1,p2), ns, f1, f2, pts): .... If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you would pass func_args = [0.1,0.2] (and ignore the fixed_params argument).	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. For example, suppose your model function is def func((p1,f1,p2,f2), ns, pts): .... If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you would pass fixed_params = [None,0.1,None,0.2] (and ignore the func_args argument).	None
ll_scale	float	The bfgs algorithm may fail if your initial log-likelihood is too large. (This appears to be a flaw in the scipy implementation.) To overcome this, pass ll_scale > 1, which will simply reduce the magnitude of the log-likelihood. Once in a region of reasonable likelihood, you'll probably want to re-optimize with ll_scale=1.	1

Source code in dadi/Inference.py

def optimize_log(p0, data, model_func, pts, lower_bound=None, upper_bound=None,
                 verbose=0, flush_delay=0.5, epsilon=1e-3, 
                 gtol=1e-5, multinom=True, maxiter=None, full_output=False,
                 func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1,
                 output_file=None):
    """
    Optimize log(params) to fit model to data using the BFGS method.

    This optimization method works well when we start reasonably close to the
    optimum. It is best at burrowing down a single minimum.

    Because this works in log(params), it cannot explore values of params < 0.
    It should also perform better when parameters range over scales.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
                        (params, (n1,n2...), pts)
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
                    length as p0.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
                    length as p0.
        verbose (int): If > 0, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        epsilon (float): Step-size to use for finite-difference derivatives.
        gtol (float): Convergence criterion for optimization. For more info, 
            see help(scipy.optimize.fmin_bfgs)
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        maxiter (inf): Maximum iterations to run for.
        full_output (bool): If True, return full outputs as in described in 
                    help(scipy.optimize.fmin_bfgs)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
                Using func_args.
                For example, you could define your model function as
                def func((p1,p2), ns, f1, f2, pts):
                    ....
                If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you
                would pass func_args = [0.1,0.2] (and ignore the fixed_params 
                argument).
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored.
                    For example, suppose your model function is 
                    def func((p1,f1,p2,f2), ns, pts):
                        ....
                    If you wanted to fix f1=0.1 and f2=0.2 in the optimization, 
                    you would pass fixed_params = [None,0.1,None,0.2] (and ignore
                    the func_args argument).
        ll_scale (float): The bfgs algorithm may fail if your initial log-likelihood is
                too large. (This appears to be a flaw in the scipy
                implementation.) To overcome this, pass ll_scale > 1, which will
                simply reduce the magnitude of the log-likelihood. Once in a
                region of reasonable likelihood, you'll probably want to
                re-optimize with ll_scale=1.
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, lower_bound, upper_bound, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 
            ll_scale, output_stream)

    p0 = _project_params_down(p0, fixed_params)
    outputs = scipy.optimize.fmin_bfgs(_object_func_log, 
                                       numpy.log(p0), epsilon=epsilon,
                                       args = args, gtol=gtol, 
                                       full_output=True,
                                       disp=False,
                                       maxiter=maxiter)
    xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflag = outputs
    xopt = _project_params_up(numpy.exp(xopt), fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflag

optimize_log_fmin(p0, data, model_func, pts, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, multinom=True, maxiter=None, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, output_file=None)

Optimize log(params) to fit model to data using Nelder-Mead.

This optimization method may work better than BFGS when far from a minimum. It is much slower, but more robust, because it doesn't use gradient information.

Because this works in log(params), it cannot explore values of params < 0. It should also perform better when parameters range over large scales.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
verbose	int	If True, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
maxiter	int	Maximum iterations to run for.	None
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_bfgs)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation.	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. (See help(dadi.Inference.optimize_log for examples of func_args and fixed_params usage.)	None

Source code in dadi/Inference.py

def optimize_log_fmin(p0, data, model_func, pts, 
                      lower_bound=None, upper_bound=None,
                      verbose=0, flush_delay=0.5, 
                      multinom=True, maxiter=None, 
                      full_output=False, func_args=[], 
                      func_kwargs={},
                      fixed_params=None, output_file=None):
    """
    Optimize log(params) to fit model to data using Nelder-Mead. 

    This optimization method may work better than BFGS when far from a
    minimum. It is much slower, but more robust, because it doesn't use
    gradient information.

    Because this works in log(params), it cannot explore values of params < 0.
    It should also perform better when parameters range over large scales.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
                        (params, (n1,n2...), pts)
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
                    length as p0. A parameter can be declared unbound by assigning
                    a bound of None.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
                    length as p0. A parameter can be declared unbound by assigning
                    a bound of None.
        verbose (int): If True, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        maxiter (int): Maximum iterations to run for.
        full_output (bool): If True, return full outputs as in described in 
                    help(scipy.optimize.fmin_bfgs)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored.
                    (See help(dadi.Inference.optimize_log for examples of func_args and
                    fixed_params usage.)
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, lower_bound, upper_bound, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 1.0,
            output_stream)

    p0 = _project_params_down(p0, fixed_params)
    outputs = scipy.optimize.fmin(_object_func_log, numpy.log(p0), args = args,
                                  disp=False, maxiter=maxiter, full_output=True)
    xopt, fopt, iter, funcalls, warnflag = outputs
    xopt = _project_params_up(numpy.exp(xopt), fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, iter, funcalls, warnflag

optimize_log_lbfgsb(p0, data, model_func, pts, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, epsilon=0.001, pgtol=1e-05, multinom=True, maxiter=100000.0, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1, output_file=None)

Optimize log(params) to fit model to data using the L-BFGS-B method.

This optimization method works well when we start reasonably close to the optimum. It is best at burrowing down a single minimum. This method is better than optimize_log if the optimum lies at one or more of the parameter bounds. However, if your optimum is not on the bounds, this method may be much slower.

Because this works in log(params), it cannot explore values of params < 0. It should also perform better when parameters range over scales.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
verbose	int	If > 0, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
epsilon	float	Step-size to use for finite-difference derivatives.	0.001
pgtol	float	Convergence criterion for optimization. For more info, see help(scipy.optimize.fmin_l_bfgs_b)	1e-05
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
maxiter	int	Maximum algorithm iterations to run.	100000.0
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_bfgs)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation.	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. (See help(dadi.Inference.optimize_log for examples of func_args and fixed_params usage.)	None
ll_scale	float	The bfgs algorithm may fail if your initial log-likelihood is too large. (This appears to be a flaw in the scipy implementation.) To overcome this, pass ll_scale > 1, which will simply reduce the magnitude of the log-likelihood. Once in a region of reasonable likelihood, you'll probably want to re-optimize with ll_scale=1.	1

Citation

The L-BFGS-B method was developed by Ciyou Zhu, Richard Byrd, and Jorge Nocedal. The algorithm is described in:

• R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound Constrained Optimization, (1995), SIAM Journal on Scientific and Statistical Computing , 16, 5, pp. 1190-1208.

• C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization (1997), ACM Transactions on Mathematical Software, Vol 23, Num. 4, pp. 550-560.

Source code in dadi/Inference.py

def optimize_log_lbfgsb(p0, data, model_func, pts, 
                        lower_bound=None, upper_bound=None,
                        verbose=0, flush_delay=0.5, epsilon=1e-3, 
                        pgtol=1e-5, multinom=True, maxiter=1e5, 
                        full_output=False,
                        func_args=[], func_kwargs={}, fixed_params=None, 
                        ll_scale=1, output_file=None):
    """
    Optimize log(params) to fit model to data using the L-BFGS-B method.

    This optimization method works well when we start reasonably close to the
    optimum. It is best at burrowing down a single minimum. This method is
    better than optimize_log if the optimum lies at one or more of the
    parameter bounds. However, if your optimum is not on the bounds, this
    method may be much slower.

    Because this works in log(params), it cannot explore values of params < 0.
    It should also perform better when parameters range over scales.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
                        (params, (n1,n2...), pts)
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
                    length as p0. A parameter can be declared unbound by assigning
                    a bound of None.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
                    length as p0. A parameter can be declared unbound by assigning
                    a bound of None.
        verbose (int): If > 0, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        epsilon (float): Step-size to use for finite-difference derivatives.
        pgtol (float): Convergence criterion for optimization. For more info, 
            see help(scipy.optimize.fmin_l_bfgs_b)
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        maxiter (int): Maximum algorithm iterations to run.
        full_output (bool): If True, return full outputs as in described in 
                    help(scipy.optimize.fmin_bfgs)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored. 
                    (See help(dadi.Inference.optimize_log for examples of func_args and 
                    fixed_params usage.)
        ll_scale (float): The bfgs algorithm may fail if your initial log-likelihood is
                too large. (This appears to be a flaw in the scipy
                implementation.) To overcome this, pass ll_scale > 1, which will
                simply reduce the magnitude of the log-likelihood. Once in a
                region of reasonable likelihood, you'll probably want to
                re-optimize with ll_scale=1.

    Citation:
        The L-BFGS-B method was developed by Ciyou Zhu, Richard Byrd, and Jorge
        Nocedal. The algorithm is described in:

        * R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
            Constrained Optimization, (1995), SIAM Journal on Scientific and
            Statistical Computing , 16, 5, pp. 1190-1208.

        * C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
            FORTRAN routines for large scale bound constrained optimization (1997),
            ACM Transactions on Mathematical Software, Vol 23, Num. 4, pp. 550-560.

    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, None, None, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 
            ll_scale, output_stream)

    # Make bounds list. For this method it needs to be in terms of log params.
    if lower_bound is None:
        lower_bound = [None] * len(p0)
    else:
        lower_bound = numpy.log(lower_bound)
        lower_bound[numpy.isnan(lower_bound)] = None
    lower_bound = _project_params_down(lower_bound, fixed_params)
    if upper_bound is None:
        upper_bound = [None] * len(p0)
    else:
        upper_bound = numpy.log(upper_bound)
        upper_bound[numpy.isnan(upper_bound)] = None
    upper_bound = _project_params_down(upper_bound, fixed_params)
    bounds = list(zip(lower_bound,upper_bound))

    p0 = _project_params_down(p0, fixed_params)

    outputs = scipy.optimize.fmin_l_bfgs_b(_object_func_log, 
                                           numpy.log(p0), bounds = bounds,
                                           epsilon=epsilon, args = args,
                                           iprint = -1, pgtol=pgtol,
                                           maxfun=maxiter, approx_grad=True)
    xopt, fopt, info_dict = outputs

    xopt = _project_params_up(numpy.exp(xopt), fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, info_dict

optimize_log_powell(p0, data, model_func, pts, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, multinom=True, maxiter=None, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, output_file=None)

Optimize log(params) to fit model to data using Powell's method.

Because this works in log(params), it cannot explore values of params < 0.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0. A parameter can be declared unbound by assigning a bound of None.	None
verbose	int	If True, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters. multinom: If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	0.5
maxiter	int	Maximum iterations to run for.	None
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_bfgs)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation.	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. (See help(dadi.Inference.optimize_log for examples of func_args and fixed_params usage.)	None

Source code in dadi/Inference.py

def optimize_log_powell(p0, data, model_func, pts,
                      lower_bound=None, upper_bound=None,
                      verbose=0, flush_delay=0.5,
                      multinom=True, maxiter=None,
                      full_output=False, func_args=[],
                      func_kwargs={},
                      fixed_params=None, output_file=None):
    """
    Optimize log(params) to fit model to data using Powell's method.

    Because this works in log(params), it cannot explore values of params < 0.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
            (params, (n1,n2...), pts)
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
            length as p0. A parameter can be declared unbound by assigning
            a bound of None.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
            length as p0. A parameter can be declared unbound by assigning
            a bound of None.
        verbose (int): If True, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
            standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
            minutes. This is useful to avoid overloading I/O on clusters.
            multinom: If True, do a multinomial fit where model is optimially scaled to
            data at each step. If False, assume theta is a parameter and do
            no scaling.
        maxiter (int): Maximum iterations to run for.
        full_output (bool): If True, return full outputs as in described in
            help(scipy.optimize.fmin_bfgs)
        func_args (list): Additional arguments to model_func. It is assumed that
            model_func's first argument is an array of parameters to
            optimize, that its second argument is an array of sample sizes
            for the sfs, and that its last argument is the list of grid
            points to use in evaluation.
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
            particular values. For example, if the model parameters
            are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
            will hold nu1=0.5 and m=2. The optimizer will only change
            T and m. Note that the bounds lists must include all
            parameters. Optimization will fail if the fixed values
            lie outside their bounds. A full-length p0 should be passed
            in; values corresponding to fixed parameters are ignored.
            (See help(dadi.Inference.optimize_log for examples of func_args and
            fixed_params usage.)
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, pts, lower_bound, upper_bound, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 1.0,
            output_stream)

    p0 = _project_params_down(p0, fixed_params)
    outputs = scipy.optimize.fmin_powell(_object_func_log, numpy.log(p0), args = args,
                                  disp=False, maxiter=maxiter, full_output=True)
    xopt, fopt, direc, iter, funcalls, warnflag = outputs
    xopt = _project_params_up(numpy.exp(xopt), fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, direc, iter, funcalls, warnflag

optimize_log_resid(p0, data, model_func, target_resid, pts, lower_bound=None, upper_bound=None, verbose=0, flush_delay=0.5, epsilon=0.001, gtol=1e-05, multinom=True, maxiter=None, full_output=False, func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1, output_file=None)

Optimize log(params) to fit model to data using the BFGS method.

This optimization method works well when we start reasonably close to the optimum. It is best at burrowing down a single minimum.

Because this works in log(params), it cannot explore values of params < 0. It should also perform better when parameters range over scales.

Parameters:

Name	Type	Description	Default
p0	list[float]	Initial parameters.	required
data	Spectrum	Spectrum with data.	required
model_func	func	Function to evaluate model spectrum. Should take arguments (params, (n1,n2...), pts)	required
target_resid	Spectrum	The residual sfs that we want to match, obtained from the synonymous fits.	required
lower_bound	list[float]	Lower bound on parameter values. If not None, must be of same length as p0.	None
upper_bound	list[float]	Upper bound on parameter values. If not None, must be of same length as p0.	None
verbose	int	If > 0, print optimization status every steps.	0
output_file	str	Stream verbose output into this filename. If None, stream to standard out.	None
flush_delay	float	Standard output will be flushed once every minutes. This is useful to avoid overloading I/O on clusters.	0.5
epsilon	float	Step-size to use for finite-difference derivatives.	0.001
gtol	float	Convergence criterion for optimization. For more info, see help(scipy.optimize.fmin_bfgs)	1e-05
multinom	bool	If True, do a multinomial fit where model is optimially scaled to data at each step. If False, assume theta is a parameter and do no scaling.	True
maxiter	int	Maximum iterations to run for.	None
full_output	bool	If True, return full outputs as in described in help(scipy.optimize.fmin_bfgs)	False
func_args	list	Additional arguments to model_func. It is assumed that model_func's first argument is an array of parameters to optimize, that its second argument is an array of sample sizes for the sfs, and that its last argument is the list of grid points to use in evaluation. Using func_args. For example, you could define your model function as def func((p1,p2), ns, f1, f2, pts): .... If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you would pass func_args = [0.1,0.2] (and ignore the fixed_params argument).	[]
func_kwargs	list	Additional keyword arguments to model_func.	{}
fixed_params	list[float]	If not None, should be a list used to fix model parameters at particular values. For example, if the model parameters are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2] will hold nu1=0.5 and m=2. The optimizer will only change T and m. Note that the bounds lists must include all parameters. Optimization will fail if the fixed values lie outside their bounds. A full-length p0 should be passed in; values corresponding to fixed parameters are ignored. For example, suppose your model function is def func((p1,f1,p2,f2), ns, pts): .... If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you would pass fixed_params = [None,0.1,None,0.2] (and ignore the func_args argument).	None
ll_scale	float	The bfgs algorithm may fail if your initial log-likelihood is too large. (This appears to be a flaw in the scipy implementation.) To overcome this, pass ll_scale > 1, which will simply reduce the magnitude of the log-likelihood. Once in a region of reasonable likelihood, you'll probably want to re-optimize with ll_scale=1.	1

Source code in dadi/Inference.py

def optimize_log_resid(p0, data, model_func, target_resid, pts, lower_bound=None, upper_bound=None,
                 verbose=0, flush_delay=0.5, epsilon=1e-3, 
                 gtol=1e-5, multinom=True, maxiter=None, full_output=False,
                 func_args=[], func_kwargs={}, fixed_params=None, ll_scale=1,
                 output_file=None):
    """
    Optimize log(params) to fit model to data using the BFGS method.

    This optimization method works well when we start reasonably close to the
    optimum. It is best at burrowing down a single minimum.

    Because this works in log(params), it cannot explore values of params < 0.
    It should also perform better when parameters range over scales.

    Args:
        p0 (list[float]): Initial parameters.
        data (Spectrum): Spectrum with data.
        model_func (func): Function to evaluate model spectrum. Should take arguments
                        (params, (n1,n2...), pts)
        target_resid (Spectrum): The residual sfs that we want to match, obtained from the 
                    synonymous fits.
        lower_bound (list[float]): Lower bound on parameter values. If not None, must be of same
                    length as p0.
        upper_bound (list[float]): Upper bound on parameter values. If not None, must be of same
                    length as p0.
        verbose (int): If > 0, print optimization status every <verbose> steps.
        output_file (str): Stream verbose output into this filename. If None, stream to
                    standard out.
        flush_delay (float): Standard output will be flushed once every <flush_delay>
                    minutes. This is useful to avoid overloading I/O on clusters.
        epsilon (float): Step-size to use for finite-difference derivatives.
        gtol (float): Convergence criterion for optimization. For more info, 
            see help(scipy.optimize.fmin_bfgs)
        multinom (bool): If True, do a multinomial fit where model is optimially scaled to
                data at each step. If False, assume theta is a parameter and do
                no scaling.
        maxiter (int): Maximum iterations to run for.
        full_output (bool): If True, return full outputs as in described in 
                    help(scipy.optimize.fmin_bfgs)
        func_args (list): Additional arguments to model_func. It is assumed that 
                model_func's first argument is an array of parameters to
                optimize, that its second argument is an array of sample sizes
                for the sfs, and that its last argument is the list of grid
                points to use in evaluation.
                Using func_args.
                For example, you could define your model function as
                def func((p1,p2), ns, f1, f2, pts):
                    ....
                If you wanted to fix f1=0.1 and f2=0.2 in the optimization, you
                would pass func_args = [0.1,0.2] (and ignore the fixed_params 
                argument).
        func_kwargs (list): Additional keyword arguments to model_func.
        fixed_params (list[float]): If not None, should be a list used to fix model parameters at
                    particular values. For example, if the model parameters
                    are (nu1,nu2,T,m), then fixed_params = [0.5,None,None,2]
                    will hold nu1=0.5 and m=2. The optimizer will only change 
                    T and m. Note that the bounds lists must include all
                    parameters. Optimization will fail if the fixed values
                    lie outside their bounds. A full-length p0 should be passed
                    in; values corresponding to fixed parameters are ignored.
                    For example, suppose your model function is 
                    def func((p1,f1,p2,f2), ns, pts):
                        ....
                    If you wanted to fix f1=0.1 and f2=0.2 in the optimization, 
                    you would pass fixed_params = [None,0.1,None,0.2] (and ignore
                    the func_args argument).
        ll_scale (float): The bfgs algorithm may fail if your initial log-likelihood is
                too large. (This appears to be a flaw in the scipy
                implementation.) To overcome this, pass ll_scale > 1, which will
                simply reduce the magnitude of the log-likelihood. Once in a
                region of reasonable likelihood, you'll probably want to
                re-optimize with ll_scale=1.
    """
    if output_file:
        output_stream = open(output_file, 'w')
    else:
        output_stream = sys.stdout

    args = (data, model_func, target_resid, pts, lower_bound, upper_bound, verbose,
            multinom, flush_delay, func_args, func_kwargs, fixed_params, 
            ll_scale, output_stream)

    p0 = _project_params_down(p0, fixed_params)
    outputs = scipy.optimize.fmin_bfgs(_object_func_log_resid, 
                                       numpy.log(p0), epsilon=epsilon,
                                       args = args, gtol=gtol, 
                                       full_output=True,
                                       disp=False,
                                       maxiter=maxiter)
    xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflag = outputs
    xopt = _project_params_up(numpy.exp(xopt), fixed_params)

    if output_file:
        output_stream.close()

    if not full_output:
        return xopt
    else:
        return xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflag