torchsurv.loss.weibull#

Functions

`cumulative_hazard`(log_params, time[, all_times])	Cumulative hazard for the Weibull Accelerated Time Failure (AFT) survival model.
`log_hazard`(log_params, time[, all_times])	Log hazard of the Weibull Accelerated Time Failure (AFT) survival model.
`neg_log_likelihood`(log_params, event, time)	Negative of the log likelihood for the Weibull Accelerated Time Failure (AFT) survival model.
`survival_function`(log_params, time[, all_times])	Survival function for the Weibull Accelerated Time Failure (AFT) survival model.

torchsurv.loss.weibull.neg_log_likelihood(log_params: Tensor, event: Tensor, time: Tensor, reduction: str = 'mean', checks: bool = True) → Tensor[source]#

Negative of the log likelihood for the Weibull Accelerated Time Failure (AFT) survival model.

Parameters:

log_params (torch.Tensor, float) – Parameters of the Weibull distribution of shape = (n_samples, 1) or (n_samples, 2). The first column corresponds to the log scale parameter. The second column corresponds to the log shape parameter. If the log shape parameter is missing, it is imputed with 0.
event (torch.Tensor, bool) – Event indicator of length n_samples (= True if event occured).
time (torch.Tensor, float) – Time-to-event or censoring of length n_samples.
reduction (str) – Method to reduce losses. Defaults to “mean”. Must be one of the following: “sum”, “mean”.
checks (bool) – Whether to perform input format checks. Enabling checks can help catch potential issues in the input data. Defaults to True.

Returns:

Negative of the log likelihood.

Return type:

(torch.Tensor, float)

Note

For each subject \(i \in \{1, \cdots, N\}\), denote \(X_i\) as the survival time and \(D_i\) as the censoring time. Survival data consist of the event indicator, \(\delta_i=1(X_i\leq D_i)\) (argument event) and the time-to-event or censoring, \(T_i = \min(\{ X_i,D_i \})\) (argument time).

The log hazard function for the Weibull AFT survival model [Car03] of subject \(i\) at time \(t\) has the form:

\[\log h_i(t) = \log{\rho_i} - \log{\lambda_i} + (\rho_i -1) \left( \log{t} - \log{\lambda_i}\right)\]

where \(\log{\lambda_i}\) is the log scale parameter (first column of argument log_params) and \(\log{\rho_i}\) is the log shape parameter (second column of argument log_params). The cumulative hazard for the Weibull survival model of subject \(i\) at time \(t\) has the form:

\[H_i(t) = \left(\frac{t}{\lambda_i}\right)^{\rho_i}\]

The survival function for the Weibull survival model of subject \(i\) at time \(t\) has the form:

\[S_i(t) = 1 - F(t | \lambda_i, \rho_i)\]

where \(F(t | \lambda, \rho)\) is the cumulative distribution function (CDF) of the Weibull distribution given scale parameter \(\lambda\) and shape parameter \(\rho\).

The log likelihood of the Weibull survival model is

\[ll = \sum_{i: \delta_i = 1} \log h_i(T_i) - \sum_{i = 1}^N H_i(T_i)\]

Examples

>>> _ = torch.manual_seed(42)
>>> n = 4
>>> log_params = torch.randn((n, 2))
>>> event = torch.randint(low=0, high=2, size=(n,), dtype=torch.bool)
>>> time = torch.randint(low=1, high=100, size=(n,))
>>> neg_log_likelihood(log_params, event, time) # Default: mean of log likelihoods across subject
tensor(47.5035)
>>> neg_log_likelihood(log_params, event, time, reduction = 'sum') # Sum of log likelihoods across subject
tensor(190.0141)
>>> neg_log_likelihood(torch.randn((n, 1)), event, time)  # Missing shape: exponential decrease
tensor(66.7203)

References

[Car03]

Kevin J. Carroll. On the use and utility of the weibull model in the analysis of survival data. Controlled Clinical Trials, 24(6):682–701, December 2003.

torchsurv.loss.weibull.survival_function(log_params: Tensor, time: Tensor, all_times: bool = True) → Tensor[source]#

Survival function for the Weibull Accelerated Time Failure (AFT) survival model.

Parameters:

log_params (torch.Tensor, float) – Parameters of the Weibull distribution of shape = (n_samples, 1) or (n_samples, 2). The first column corresponds to the log scale parameter. The second column corresponds to the log shape parameter. If the log shape parameter is missing, it is imputed with 0.
time (torch.Tensor, float) – Time at which to evaluate the survival function. Should be of length n_samples to evaluate the survival function at observed time-to-event or censoring, or of length one to evaluate the survival function at a new time.
all_times (bool) – If True, subject-specific survival function is evaluated at all time (used for evaluation metrics). If False, subject-specific survival function is evaluated at respective time. Defaults is True. Ignored if time is of length one.

Returns:

Subject-specific survival function evaluated at time.

Return type:

(torch.Tensor, float)

Examples

>>> _ = torch.manual_seed(42)
>>> time = torch.randint(low=1, high=100, size=(4,))
>>> log_params = torch.randn((4, 2))
>>> survival_function(log_params, time, all_times = False)  # Survival at respective time
tensor([0.0002, 0.0000, 0.0299, 0.0000])
>>> survival_function(log_params, time, all_times = True)  # Default. Survival at all observed time
tensor([[1.7941e-04, 0.0000e+00, 0.0000e+00, 0.0000e+00],
        [2.8610e-06, 0.0000e+00, 0.0000e+00, 0.0000e+00],
        [4.1870e-01, 3.1040e-02, 2.9881e-02, 6.8224e-02],
        [9.5576e-04, 0.0000e+00, 0.0000e+00, 0.0000e+00]])
>>> survival_function(log_params, time=torch.tensor(10.0))  # Survival at one new time (e.g., 10 years)
tensor([1.3709e-06, 5.9605e-08, 3.4954e-01, 1.5438e-05])
>>> for t in torch.tensor([100.0, 150.0]): survival_function(log_params, time=t)  # Subject-specific survival at multiple new times
tensor([0.0000, 0.0000, 0.0288, 0.0000])
tensor([0.0000, 0.0000, 0.0123, 0.0000])

torchsurv.loss.weibull.log_hazard(log_params: Tensor, time: Tensor, all_times: bool = True) → Tensor[source]#

Log hazard of the Weibull Accelerated Time Failure (AFT) survival model.

Parameters:

log_params (torch.Tensor, float) – Parameters of the Weibull distribution of shape = (n_samples, 1) or (n_samples, 2). The first column corresponds to the log scale parameter. The second column corresponds to the log shape parameter. If the log shape parameter is missing, it is imputed with 0.
time (torch.Tensor, float) – Time at which to evaluate the log hazard. Should be of length n_samples to evaluate the log hazard at observed time-to-event or censoring, or of length one to evaluate the log hazard at a new time.
all_times (bool) – If True, subject-specific log hazard is evaluated at all time (used for evaluation metrics). If False, subject-specific log hazard is evaluated at respective time. Defaults is True. Ignored if time is of length one.

Returns:

Subject-specific log hazard evaluated at time.

Return type:

(torch.Tensor, float)

Examples

>>> _ = torch.manual_seed(42)
>>> time = torch.randint(low=1, high=100, size=(4,))
>>> log_params = torch.randn((4, 2))
>>> log_hazard(log_params, time, all_times = False)  # Log hazard at respective time
tensor([ 0.4392, -0.0303, -3.9672,  0.9140])
>>> log_hazard(log_params, time, all_times = True)  # Default. Log hazard at all time
tensor([[ 0.4392,  1.1174,  1.1227,  0.9913],
        [ 0.4148, -0.0303, -0.0338,  0.0525],
        [-2.7225, -3.9575, -3.9672, -3.7279],
        [ 0.2606,  1.0632,  1.0695,  0.9140]])
>>> log_hazard(log_params, time=torch.tensor(10.0))  # Log hazard at one new time (e.g., 10 years)
tensor([ 0.5316,  0.3542, -2.8907,  0.3699])
>>> for t in torch.tensor([100.0, 150.0]): log_hazard(log_params, time=t)  # Subject-specific log hazard at multiple new times
tensor([ 1.1280, -0.0372, -3.9767,  1.0757])
tensor([ 1.2330, -0.1062, -4.1680,  1.1999])

torchsurv.loss.weibull.cumulative_hazard(log_params: Tensor, time: Tensor, all_times: bool = True) → Tensor[source]#

Cumulative hazard for the Weibull Accelerated Time Failure (AFT) survival model.

Parameters:

log_params (torch.Tensor, float) – Parameters of the Weibull distribution of shape = (n_samples, 1) or (n_samples, 2). The first column corresponds to the log scale parameter. The second column corresponds to the log shape parameter. If the log shape parameter is missing, it is imputed with 0.
time (torch.Tensor, float) – Time-to-event or censoring of length n_samples.
all_times (bool) – If True, subject-specific cumulative hazard is evaluated at all time (used for evaluation metrics). If False, subject-specific cumulative hazard is evaluated at respective time. Defaults is True.

Returns:

Subject-specific cumulative hazard evaluated at time.

Return type:

(torch.Tensor, float)

Examples

>>> _ = torch.manual_seed(42)
>>> time = torch.randint(low=1, high=100, size=(4,))
>>> log_params = torch.randn((4, 2))
>>> cumulative_hazard(log_params, time, all_times=False) # Cumulative hazard at respective time
tensor([  8.6257, 112.2115,   3.5105, 112.6339])
>>> cumulative_hazard(log_params, time, all_times=True) # Default. Cumulative hazard at all time
tensor([[  8.6257, 233.0865, 239.2167, 126.2805],
        [ 12.7698, 112.2115, 114.1484,  74.9134],
        [  0.8706,   3.4725,   3.5105,   2.6850],
        [  6.9530, 212.7592, 218.5687, 112.6339]])

torchsurv.loss.weibull

Contents

torchsurv.loss.weibull#