bio_rtd.pdf¶

Wrappers for probability distribution functions.

This module contains subclasses of bio_rtd.core.PDF.

The integral of a probability distribution function (PDF) over time has a value of 1.

Examples

>>> t = _np.linspace(0, 100, 1001)
>>> dt = t[1]
>>> pdf = GaussianFixedDispersion(t, 0.2)
>>> pdf.trim_and_normalize = False
>>> pdf.update_pdf(rt_mean=40)
>>> p = pdf.get_p()
>>> print(round(p.sum() * dt, 8))
1.0
>>> t[p.argmax()]
40.0

GaussianFixedDispersion¶

class bio_rtd.pdf.GaussianFixedDispersion(t, dispersion_index, cutoff=0.0001, pdf_id='GaussianFixedDispersion')[source]¶

Bases: bio_rtd.core.PDF

Gaussian PDF with fixed dispersion.

Parameters

dispersion_index (float) –
Dispersion index.

Dispersion index is defined as sigma * sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation on time scale (not volume).
cutoff –
Cutoff limit for trimming front and end tailing.

Cutoff limit is relative to the peak max value.
pdf_id (str) – Unique identifier. Default = “GaussianFixedDispersion”.

See also

rtd_lib.core.PDF

Examples

>>> t = _np.linspace(0, 100, 1001)
>>> dt = t[1]
>>> pdf = GaussianFixedDispersion(t, dispersion_index=0.2)
>>> pdf.trim_and_normalize = False
>>> pdf.update_pdf(rt_mean=40)
>>> p = pdf.get_p()
>>> print(round(p.sum() * dt, 4))
1.0
>>> t[p.argmax()]
40.0

POSSIBLE_KEY_GROUPS = [['f', 'v_void'], ['rt_mean']]¶

OPTIONAL_KEYS = []¶

dispersion_index¶

Dispersion index.

Dispersion index is defined as sigma * sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation.

cutoff_relative_to_max¶

Cutoff limit for trimming front and end tailing.

Cutoff limit is relative to the peak max value.

assert_and_get_provided_kv_pairs(**kwargs)¶

Parameters: **kwargs – Inputs to calc_pdf(**kwargs) function
Returns: Filtered **kwargs so the keys contain first possible key group in POSSIBLE_KEY_GROUPS and any number of optional keys from OPTIONAL_KEYS.
Return type: dict
Raises: ValueError – If **kwargs do not contain keys from any of the groups in POSSIBLE_KEY_GROUPS.

get_p()¶

Get probability distribution.

Returns

p – Evaluated probability distribution function.

Corresponding time axis starts with 0 and has a fixed step size (_dt).

If trim_and_normalize == 1 then sum(p * _dt) == 1.

Return type

ndarray

property log¶

Reference of the RtdLogger instance.

Setter also plants instance data tree into passed logger.

If logger is requested, but not yet set, then a bio_rtd.logger.DefaultLogger is instantiated.

Return type: RtdLogger

set_logger_from_parent(parent_id, logger)¶

Inherit logger from parent.

Parameters

parent_id (str) – Unique identifier of parent instance.
logger (RtdLogger) – Logger from parent instance.

update_pdf(**kwargs)¶

Re-calculate PDF based on specified parameters.

The calculated probability distribution can be obtained by get_p()

Parameters: **kwargs – Should contain keys from one of the group in POSSIBLE_KEY_GROUPS. It may contain additional keys from OPTIONAL_KEYS.

trim_and_normalize¶

Trim edges of the pdf and normalize it afterwards.

Default = True.

Relative threshold value is specified by cutoff_relative_to_max.

Normalization is performed after the trimming. The area of pd == 1.

GaussianFixedRelativeWidth¶

class bio_rtd.pdf.GaussianFixedRelativeWidth(t, relative_sigma, cutoff=0.0001, pdf_id='GaussianFixedRelativeWidth')[source]¶

Bases: bio_rtd.core.PDF

Gaussian PDF with fixed relative peak width.

Parameters

relative_sigma (float) –
Relative sigma.

Relative sigma is defined as sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation.
cutoff –
Cutoff limit for trimming front and end tailing.

Cutoff limit is relative to the peak max value.
pdf_id (str) – Unique identifier. Default = “GaussianFixedRelativeWidth”.

See also

rtd_lib.core.PDF

Examples

>>> t = _np.linspace(0, 100, 1001)
>>> dt = t[1]
>>> pdf = GaussianFixedRelativeWidth(t, relative_sigma=0.15)
>>> pdf.trim_and_normalize = False
>>> pdf.update_pdf(rt_mean=40)
>>> p = pdf.get_p()
>>> print(round(p.sum() * dt, 4))
1.0
>>> t[p.argmax()]
40.0

POSSIBLE_KEY_GROUPS = [['f', 'v_void'], ['rt_mean']]¶

OPTIONAL_KEYS = []¶

relative_sigma¶

Relative sigma.

Relative sigma is defined as sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation.

cutoff_relative_to_max¶

Cutoff limit for trimming front and end tailing.

Cutoff limit is relative to the peak max value.

assert_and_get_provided_kv_pairs(**kwargs)¶

Parameters: **kwargs – Inputs to calc_pdf(**kwargs) function
Returns: Filtered **kwargs so the keys contain first possible key group in POSSIBLE_KEY_GROUPS and any number of optional keys from OPTIONAL_KEYS.
Return type: dict
Raises: ValueError – If **kwargs do not contain keys from any of the groups in POSSIBLE_KEY_GROUPS.

get_p()¶

Get probability distribution.

Returns

p – Evaluated probability distribution function.

Corresponding time axis starts with 0 and has a fixed step size (_dt).

If trim_and_normalize == 1 then sum(p * _dt) == 1.

Return type

ndarray

property log¶

Reference of the RtdLogger instance.

Setter also plants instance data tree into passed logger.

If logger is requested, but not yet set, then a bio_rtd.logger.DefaultLogger is instantiated.

Return type: RtdLogger

set_logger_from_parent(parent_id, logger)¶

Inherit logger from parent.

Parameters

parent_id (str) – Unique identifier of parent instance.
logger (RtdLogger) – Logger from parent instance.

update_pdf(**kwargs)¶

Re-calculate PDF based on specified parameters.

The calculated probability distribution can be obtained by get_p()

Parameters: **kwargs – Should contain keys from one of the group in POSSIBLE_KEY_GROUPS. It may contain additional keys from OPTIONAL_KEYS.

trim_and_normalize¶

Trim edges of the pdf and normalize it afterwards.

Default = True.

Relative threshold value is specified by cutoff_relative_to_max.

Normalization is performed after the trimming. The area of pd == 1.

ExpModGaussianFixedDispersion¶

class bio_rtd.pdf.ExpModGaussianFixedDispersion(t, dispersion_index, skew, pdf_id='ExpModGaussianFixedDispersion')[source]¶

Bases: bio_rtd.core.PDF

Exponentially Modified Gaussian PDF with fixed dispersion.

Parameters

dispersion_index (float) –
Dispersion index of Gaussian part.

Dispersion index is defined as sigma * sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation on time scale (not volume).
skew (float) – Rate of exponential part.
pdf_id (str) – Unique identifier. Default = “ExpModGaussianFixedDispersion”.

See also

rtd_lib.core.PDF

Examples

>>> t = _np.linspace(0, 100, 1001)
>>> dt = t[1]
>>> dispersion_index = 0.2
>>> skew = 0.5
>>> pdf = ExpModGaussianFixedDispersion(t, dispersion_index, skew)
>>> pdf.trim_and_normalize = False
>>> pdf.update_pdf(rt_mean=40)
>>> p = pdf.get_p()
>>> print(round(p.sum() * dt, 8))
1.0
>>> t[p.argmax()]  # position of peak max
39.6
>>> print(round((p * t[:p.size]).sum() * dt, 3))  # 1st momentum
40.0

POSSIBLE_KEY_GROUPS = [['f', 'v_void'], ['rt_mean']]¶

OPTIONAL_KEYS = ['skew']¶

dispersion_index¶

Dispersion index for Gaussian part.

Dispersion index is defined as sigma * sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation.

skew¶: Rate of exponential part.

assert_and_get_provided_kv_pairs(**kwargs)¶

Parameters: **kwargs – Inputs to calc_pdf(**kwargs) function
Returns: Filtered **kwargs so the keys contain first possible key group in POSSIBLE_KEY_GROUPS and any number of optional keys from OPTIONAL_KEYS.
Return type: dict
Raises: ValueError – If **kwargs do not contain keys from any of the groups in POSSIBLE_KEY_GROUPS.

get_p()¶

Get probability distribution.

Returns

p – Evaluated probability distribution function.

Corresponding time axis starts with 0 and has a fixed step size (_dt).

If trim_and_normalize == 1 then sum(p * _dt) == 1.

Return type

ndarray

property log¶

Reference of the RtdLogger instance.

Setter also plants instance data tree into passed logger.

If logger is requested, but not yet set, then a bio_rtd.logger.DefaultLogger is instantiated.

Return type: RtdLogger

set_logger_from_parent(parent_id, logger)¶

Inherit logger from parent.

Parameters

parent_id (str) – Unique identifier of parent instance.
logger (RtdLogger) – Logger from parent instance.

update_pdf(**kwargs)¶

Re-calculate PDF based on specified parameters.

The calculated probability distribution can be obtained by get_p()

Parameters: **kwargs – Should contain keys from one of the group in POSSIBLE_KEY_GROUPS. It may contain additional keys from OPTIONAL_KEYS.

trim_and_normalize¶

Trim edges of the pdf and normalize it afterwards.

Default = True.

Relative threshold value is specified by cutoff_relative_to_max.

Normalization is performed after the trimming. The area of pd == 1.

cutoff_relative_to_max¶

Cutoff as a share of max value of the pdf (default 0.0001).

It is defined to avoid very long tails of the distribution.

Cutoff is enabled if trim_and_normalize == True.

ExpModGaussianFixedRelativeWidth¶

class bio_rtd.pdf.ExpModGaussianFixedRelativeWidth(t, sigma_relative, tau_relative, pdf_id='ExpModGaussianFixedRelativeWidth')[source]¶

Bases: bio_rtd.core.PDF

Exponentially Modified Gaussian PDF with fixed relative sigma.

Parameters

sigma_relative (float) –
Relative sigma for Gaussian part.

Relative sigma is defined as sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation.
tau_relative (float) –
Relative characteristic time of exponential part.

It is defined as 1 / (skew * rt_mean).
pdf_id (str) – Unique identifier. Default = “ExpModGaussianFixedRelativeWidth”.

See also

rtd_lib.core.PDF

Examples

>>> t = _np.linspace(0, 100, 1001)
>>> dt = t[1]
>>> sigma_relative = 0.15
>>> skew = 0.5
>>> pdf = ExpModGaussianFixedDispersion(t, sigma_relative, skew)
>>> pdf.trim_and_normalize = False
>>> pdf.update_pdf(rt_mean=40)
>>> p = pdf.get_p()
>>> print(round(p.sum() * dt, 8))
1.0
>>> t[p.argmax()]  # position of peak max
39.5
>>> print(round((p * t[:p.size]).sum() * dt, 2))  # 1st momentum
40.0

POSSIBLE_KEY_GROUPS = [['rt_mean'], ['f', 'v_void']]¶

OPTIONAL_KEYS = ['skew']¶

sigma_relative¶

Relative sigma for Gaussian part.

Relative sigma is defined as sigma / rt_mean. Where rt_mean is a mean residence time and sigma is a standard deviation.

tau_relative¶

Relative characteristic time of exponential part.

Relative characteristic time is defined as 1 / (skew * rt_mean). Where rt_mean is a mean residence time and skew is the rate of the exponential part.

assert_and_get_provided_kv_pairs(**kwargs)¶

Parameters: **kwargs – Inputs to calc_pdf(**kwargs) function
Returns: Filtered **kwargs so the keys contain first possible key group in POSSIBLE_KEY_GROUPS and any number of optional keys from OPTIONAL_KEYS.
Return type: dict
Raises: ValueError – If **kwargs do not contain keys from any of the groups in POSSIBLE_KEY_GROUPS.

get_p()¶

Get probability distribution.

Returns

p – Evaluated probability distribution function.

Corresponding time axis starts with 0 and has a fixed step size (_dt).

If trim_and_normalize == 1 then sum(p * _dt) == 1.

Return type

ndarray

property log¶

Reference of the RtdLogger instance.

Setter also plants instance data tree into passed logger.

If logger is requested, but not yet set, then a bio_rtd.logger.DefaultLogger is instantiated.

Return type: RtdLogger

set_logger_from_parent(parent_id, logger)¶

Inherit logger from parent.

Parameters

parent_id (str) – Unique identifier of parent instance.
logger (RtdLogger) – Logger from parent instance.

update_pdf(**kwargs)¶

Re-calculate PDF based on specified parameters.

The calculated probability distribution can be obtained by get_p()

Parameters: **kwargs – Should contain keys from one of the group in POSSIBLE_KEY_GROUPS. It may contain additional keys from OPTIONAL_KEYS.

trim_and_normalize¶

Trim edges of the pdf and normalize it afterwards.

Default = True.

Relative threshold value is specified by cutoff_relative_to_max.

Normalization is performed after the trimming. The area of pd == 1.

cutoff_relative_to_max¶

Cutoff as a share of max value of the pdf (default 0.0001).

It is defined to avoid very long tails of the distribution.

Cutoff is enabled if trim_and_normalize == True.

TanksInSeries¶

class bio_rtd.pdf.TanksInSeries(t, n_tanks, pdf_id='TanksInSeries')[source]¶

Bases: bio_rtd.core.PDF

Tanks in series PDF.

rt_mean means flow-through time through entire unit operation (all tanks).

For n_tanks == 1, the distribution becomes exponential drop.

Parameters

n_tanks (float) – Number of tanks.
pdf_id (str) – Unique identifier. Default = “TanksInSeries”.

See also

rtd_lib.core.PDF

Examples

>>> t = _np.linspace(0, 100, 1001)
>>> dt = t[1]
>>> sigma_relative = 0.15
>>> skew = 0.5
>>> pdf = TanksInSeries(t, n_tanks=5)
>>> pdf.update_pdf(rt_mean=10)
>>> pdf.trim_and_normalize = False
>>> p = pdf.get_p()
>>> print(round(p.sum() * dt, 8))
1.0
>>> t[p.argmax()]  # position of peak max
8.0
>>> print(round((p * t[:p.size]).sum() * dt, 2))  # 1st momentum
10.0
>>> pdf = TanksInSeries(t, n_tanks=1)
>>> pdf.trim_and_normalize = False
>>> pdf.update_pdf(rt_mean=10)
>>> p = pdf.get_p()
>>> print(round(p.sum() * dt, 2))
1.0
>>> t[p.argmax()]  # position of peak max
0.0
>>> print(round((p * t[:p.size]).sum() * dt, 1))  # 1st momentum
10.0

assert_and_get_provided_kv_pairs(**kwargs)¶

Parameters: **kwargs – Inputs to calc_pdf(**kwargs) function
Returns: Filtered **kwargs so the keys contain first possible key group in POSSIBLE_KEY_GROUPS and any number of optional keys from OPTIONAL_KEYS.
Return type: dict
Raises: ValueError – If **kwargs do not contain keys from any of the groups in POSSIBLE_KEY_GROUPS.

get_p()¶

Get probability distribution.

Returns

p – Evaluated probability distribution function.

Corresponding time axis starts with 0 and has a fixed step size (_dt).

If trim_and_normalize == 1 then sum(p * _dt) == 1.

Return type

ndarray

property log¶

Reference of the RtdLogger instance.

Setter also plants instance data tree into passed logger.

If logger is requested, but not yet set, then a bio_rtd.logger.DefaultLogger is instantiated.

Return type: RtdLogger

set_logger_from_parent(parent_id, logger)¶

Inherit logger from parent.

Parameters

parent_id (str) – Unique identifier of parent instance.
logger (RtdLogger) – Logger from parent instance.

update_pdf(**kwargs)¶

Re-calculate PDF based on specified parameters.

The calculated probability distribution can be obtained by get_p()

Parameters: **kwargs – Should contain keys from one of the group in POSSIBLE_KEY_GROUPS. It may contain additional keys from OPTIONAL_KEYS.

trim_and_normalize¶

Trim edges of the pdf and normalize it afterwards.

Default = True.

Relative threshold value is specified by cutoff_relative_to_max.

Normalization is performed after the trimming. The area of pd == 1.

cutoff_relative_to_max¶

Cutoff as a share of max value of the pdf (default 0.0001).

It is defined to avoid very long tails of the distribution.

Cutoff is enabled if trim_and_normalize == True.

POSSIBLE_KEY_GROUPS = [['rt_mean'], ['f', 'v_void']]¶

OPTIONAL_KEYS = ['n_tanks']¶

n_tanks¶: Number of tanks.

allow_open_end: bool¶

Prevent warnings and errors if the pdf does not fit on t.

Default: False

If True, no warnings or errors are reported in case the distribution does not fit on provided time vector.