The crepes.martingales module#

class crepes.martingales.EpsilonBettingFunction(E=[-1, -0.5, 0, 0.5, 1])[source]#

An epsilon betting function generates rewards for a sequence of p-values.

Methods

apply(p_values)

Obtain rewards from p-values.
Parameters:

E (list of floats or integers, default=[-1, -0.5, 0, 0.5, 1]) – epsilons, where each epsilon is in the range (-2, 2)

Examples

An epsilon betting function using default values is obtained by:

bf = EpsilonBettingFunction()

An epsilon betting function using the values -1, 0 and 1 is obtained by:

bf = EpsilonBettingFunction(E = [-1, 0, 1])

Note

A runtime error is reported if not all values in E are within the range (-2, 2).

Methods

apply(p_values)

Obtain rewards from p-values.
apply(p_values)[source]#

Obtain rewards from p-values.

Parameters:

p_values (array-like of shape (n_values,)) – p-values

Returns:

rewards – rewards for each p-value and epsilon, where n_epsilons is the number of values in E (provided during initialization)

Return type:

ndarray of shape (n_values, n_epsilons)

Examples

Assuming that p_values is a vector of p-values, then rewards using the default epsilon values are generated by:

rewards = EpsilonBettingFunction().apply(p_values)

Rewards using the epsilon values [-1, 0 and 1] are obtained by:

rewards = EpsilonBettingFunction(E = [-1, 0, 1]).apply(p_values)
class crepes.martingales.StepBettingFunction(G=10, Gs=None, A=None, s=None)[source]#

A step betting function generates rewards for a sequence of p-values.

Methods

apply(p_values[, s])

Obtain rewards from p-values.
Parameters:

  • G (integer, default=10) – grid size, ignored if Gs is not None

  • Gs (list of floats, default=None) – grid values, set to [1/G, …, (G-1)/G] if Gs is None

  • A (real or array-like of shape (a_values,), default=None) – threshold values, set to Gs if A is None

  • s (integer, default=None) – start index, used for step betting function with drift

Examples

A step betting function using default values is obtained by:

bf = StepBettingFunction()

A step betting function using the threshold values [0.2, 0.4, 0.6] is obtained by:

bf = EpsilonBettingFunction(A = [0.2, 0.4, 0.6])

Methods

apply(p_values[, s])

Obtain rewards from p-values.
apply(p_values, s=None)[source]#

Obtain rewards from p-values.

Parameters:

p_values (array-like of shape (n_values,)) – p-values

Returns:

rewards – rewards for each p-value and epsilon, where grid_size is the number pairs in the grid (specified during initialization)

Return type:

ndarray of shape (n_values, grid_size)

Examples

Assuming that p_values is a vector of p-values, then rewards using the default step betting function are generated by:

rewards = StepBettingFunction().apply(p_values)

where rewards will be an ndarray with the same number of rows as the length of p_values and with 81 columns, which follows from that the default grid size (G) is 10, resulting in a grid that consists of the pairs formed from the grid values [1/G, …, (G-1)/G].

class crepes.martingales.SimpleJumper(J=0.01, bf=<crepes.martingales.EpsilonBettingFunction object>)[source]#

SimpleJumper computes martingale values given a vector of p-values.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
Parameters:

  • J (float, 0 < J < 1) – jumping rate

  • bf (Martingale) – betting function

Examples

SimpleJumper with default settings can be obtained by:

m = SimpleJumper()

We may initialize SimpleJumper with a specific jumping rate and betting function:

m = SimpleJumper(J = 0.1, bf = StepBettingFunction())

Note

The specified betting function must have a method apply that given a vector of p-values (ndarray of shape (n_values,)), returns an ndarray of shape (n_values, n_accounts), where n_accounts >= 1.

Note

SimpleJumper is a generalization of Alg. 8.1: Simple Jumper betting martingale in ALRW, 2ed, p., 232.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
apply(p_values, c=None)[source]#

Generate a vector of martingale values, given a vector of p-values.

Parameters:

  • p_values (array-like of shape (n_values,)) – p-values

  • c (integer, default=None) – threshold value

Returns:

  • martingales (ndarray of shape (n_values,)) – martingale values, returned only if c is None

  • index (integer, 0 <= index <= n_values) – index of the first martingale value that is equal to or greater than c, if any, and index = n_values, otherwise, returned only if c is not None

Examples

Assuming that p_values is a vector of p-values, then martingale values can be generated by:

v = SimpleJumper().apply(p_values)

The following gives the index of the first martingale value that is equal to or exceeds 100:

i =  SimpleJumper().apply(p_values, c=100)

Note

If a threshold (c) is specified, the computation is terminated as soon as a martingale value reaches this, which may save some time compared to computing the full sequence.

class crepes.martingales.SleeperStayer(R=0.001, bf=<crepes.martingales.StepBettingFunction object>)[source]#

SleeperStayer computes martingale values given a vector of p-values.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
Parameters:

  • R (float, 0 < R < 1) – investment rate

  • bf (Martingale) – betting function

Examples

SleeperStayer with default settings can be obtained by:

m = SleeperStayer()

We may initialize SleeperStayer with a specific investment rate and betting function:

m = SleeperStayer(J = 0.1, bf = EpsilonBettingFunction())

Note

The specified betting function must have a method apply that given a vector of p-values (ndarray of shape (n_values,)), returns an ndarray of shape (n_values, n_accounts), where n_accounts >= 1.

Note

SleeperStayer is a generalization of Alg. 9.4: Sleeper/Stayer in ALRW, 2ed, p., 283.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
apply(p_values, c=None)[source]#

Generate a vector of martingale values, given a vector of p-values.

Parameters:

  • p_values (array-like of shape (n_values,)) – p-values

  • c (integer, default=None) – threshold value

Returns:

  • martingales (ndarray of shape (n_values,)) – martingale values, returned only if c is None

  • index (integer, 0 <= index <= n_values) – index of the first martingale value that is equal to or greater than c, if any, and index = n_values, otherwise, returned only if c is not None

Examples

Assuming that p_values is a vector of p-values, then martingale values can be generated by:

v = SleeperStayer().apply(p_values)

The following gives the index of the first martingale value that is equal to or exceeds 100:

i =  SleeperStayer().apply(p_values, c=100)

Note

If a threshold (c) is specified, the computation is terminated as soon as a martingale value reaches this, which may save some time compared to computing the full sequence.

class crepes.martingales.SleeperDrifter(R=0.1, M=100, bf=<crepes.martingales.StepBettingFunction object>, drift=True)[source]#

SleeperDrifter computes martingale values given a vector of p-values.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
Parameters:

  • R (float, 0 < R < 1) – investment rate

  • M (integer, 0 < M, default=100) – wake-up frequency

  • bf (Martingale) – betting function

Examples

SleeperDrifter with default settings can be obtained by:

m = SleeperDrifter()

We may initialize SleeperDrifter with a specific investment rate, wake-up frequency and betting function:

m = SleeperStayer(J = 0.1, M = 50, bf = EpsilonBettingFunction())

Note

The specified betting function must have a method apply that given a vector of p-values (ndarray of shape (n_values,)), returns an ndarray of shape (n_values, n_accounts), where n_accounts >= 1.

Note

SleeperDrifter is a generalization of Alg. 9.5: Sleeper/Drifter in ALRW, 2ed, p., 284 with a slightly modified initialization, by which rewards greater than 1 may be obtained for the first M p-values.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
apply(p_values, c=None)[source]#

Generate a vector of martingale values, given a vector of p-values.

Parameters:

  • p_values (array-like of shape (n_values,)) – p-values

  • c (integer, default=None) – threshold value

Returns:

  • martingales (ndarray of shape (n_values,)) – martingale values, returned only if c is None

  • index (integer, 0 <= index <= n_values) – index of the first martingale value that is equal to or greater than c, if any, and index = n_values, otherwise, returned only if c is not None

Examples

Assuming that p_values is a vector of p-values, then martingale values can be generated by:

v = SleeperDrifter().apply(p_values)

The following gives the index of the first martingale value that is equal to or exceeds 100:

i =  SleeperDrifter().apply(p_values, c=100)

Note

If a threshold (c) is specified, the computation is terminated as soon as a martingale value reaches this, which may save some time compared to computing the full sequence.

class crepes.martingales.CompositeMartingale(Martingales)[source]#

A composite martingale averages the output of multiple conformal test martingales given at initialization.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
Parameters:

Martingales (list of conformal test martingales) – conformal test martingales, used for averaging

Examples

An composite martingale can be obtained by:

cm = CompositeMartingale([SimpleJumper(), SleeperStayer()])

Note

Each object in the provided list of conformal test martingales is required to have a method apply that given a vector of p-values returns an equally long vector of martingale values.

Methods

apply(p_values[, c])

Generate a vector of martingale values, given a vector of p-values.
apply(p_values, c=None)[source]#

Generate a vector of martingale values, given a vector of p-values.

Parameters:

  • p_values (array-like of shape (n_values,)) – p-values

  • c (integer, default=None) – threshold value

Returns:

  • martingales (ndarray of shape (n_values,)) – martingale values, returned only if c is None

  • index (integer, 0 <= index <= n_values) – index of the first martingale value that is equal to or greater than c, if any, and index = n_values, otherwise, returned only if c is not None

Examples

Assuming that p_values is a vector of p-values, then martingale values can be generated by:

cm = CompositeMartingale([SimpleJumper(), SleeperStayer()])
v = cm.apply(p_values)

The following gives the index of the first martingale value that is equal to or exceeds 100:

i = cm.apply(p_values, c=100)
crepes.martingales.semi_online_p_values(alphas, seed=None)[source]#

Computes semi-online p-values from a vector of non-conformity scores.

Parameters:

  • alphas (array-like of shape (n_values,)) – non-conformity scores

  • seed (int, default=None) – set random seed

Returns:

p_values – p-values

Return type:

array-like of shape (n_values,)

Examples

Assuming that alphas is a vector of non-conformity scores, p-values are obtained by:

from crepes.martingales import semi_online_p_values

p_values = semi_online_p_values(alphas)

The above results in that p_values is assigned a vector of the same length as alphas and where the p-value for each non-conformity score is computed with respect to the preceding non-conformity scores only.

Note

The output p-values are smoothed and generated in the semi-online mode according to the terminology in in ALRW, 2ed, p., 111.