Extremely Randomized (Extra) Survival Trees model
The Extra Survival Trees model is an extension of the Extremely Randomized trees model, introduced by Geurts et al in 2005, that can take into account censoring.
Instance
To create an instance, use pysurvival.models.survival_forest.ExtraSurvivalTreesModel
.
Attributes
max_features
: str or int  The number of features randomly chosen at each split.num_trees
: int  number of trees contained in the foresttimes
: arraylike  representation of the time axis for the modeltime_buckets
: arraylike  representation of the time axis of the model using time bins, which are represented byvariable_importance
: dict  importance of each feature (the higher, the more important the feature is). The importance is the difference between the perturbed and unperturbed error rate for each feature.
Methods
__init__
 Initialize the estimator
ExtraSurvivalTreesModel(num_trees = 10)
Parameters:
num_trees
: int (default=10)  number of trees that will be built in the forest.
fit
 Fit the estimator based on the given parameters
fit(X, T, E, max_features = 'sqrt', max_depth = 5, min_node_size = 10, num_random_splits = 100, num_threads = 1, weights = None, sample_size_pct = 0.63, importance_mode = 'normalized_permutation', seed = None, save_memory=False )
Parameters:

X
: arraylike  input samples; where the rows correspond to an individual sample and the columns represent the features (shape=[n_samples, n_features]). 
T
: arraylike  target values describing the time when the event of interest or censoring occurred. 
E
: arraylike  values that indicate if the event of interest occurred i.e.: E[i]=1 corresponds to an event, and E[i] = 0 means censoring, for all i. 
max_features
: int, float or string (default="sqrt")  number of features to consider when looking for the best split: If int, then consider the given value at each split.
 If float, then
max_features
is a fraction andint(max_features * n_features)
features are considered at each split.  If "sqrt", then
max_features=sqrt(n_features)
 If "log2", then
max_features=log2(n_features)
.  If "all", then
max_features=n_features
.

min_node_size
: int (default=10)  minimum number of samples required to be at a leaf node 
num_random_splits
: int (default=100)  number of random splits to consider for each candidate splitting variable. 
num_threads
: int (default= 1)  number of jobs to run in parallel during training. If 1, then the number of jobs is set to the total number of available cores. 
weights
: arraylike (default = None)  weights for sampling of training observations. Observations with larger weights will be selected with higher probability in the bootstrap. The sum of the weights needs to be 1. 
sample_size_pct
: double (default = 0.63)  percentage of original samples used in each tree building 
importance_mode
: str (default='impurity_corrected')  variable importance mode. Here are the available options:impurity
orimpurity_corrected
: it's the unbiased heterogeneity reduction developed by Sandri & Zuccolotto (2008)permutation
it's unnormalized as recommended by Nicodemus et al.normalized_permutation
it's normalized version of the permutation importance computations by Breiman et al.

seed
: int (default=None)  seed used by the random number generator. If None, the current timestamp converted in UNIX is used. 
save_memory
: bool (default=False)  Use memory saving splitting mode. This will slow down the model training. So, only set toTrue
if you encounter memory problems.
Returns:
self
: object
predict_hazard
 Predicts the hazard function
predict_hazard(x, t = None)
Parameters:

x
: arraylike  input samples; where the rows correspond to an individual sample and the columns represent the features (shape=[n_samples, n_features]). 
t
: double (default=None)  time at which the prediction should be performed. If None, then return the function for all available t.
Returns:
hazard
: numpy.ndarray  arraylike representing the prediction of the hazard function
predict_survival
 Predicts the survival function
predict_survival(x, t = None)
Parameters:

x
: arraylike  input samples; where the rows correspond to an individual sample and the columns represent the features (shape=[n_samples, n_features]). 
t
: double (default=None)  time at which the prediction should be performed. If None, then return the function for all available t.
Returns:
survival
: numpy.ndarray  arraylike representing the prediction of the survival function
predict_risk
 Predicts the risk score
predict_risk(x)
Parameters:
x
: arraylike  input samples; where the rows correspond to an individual sample and the columns represent the features (shape=[n_samples, n_features]).
Returns:
risk_score
: numpy.ndarray  arraylike representing the prediction of the risk score
Example
Let's now take a look at how to use the Extra Survival Trees (XST) model on a simulation dataset generated from a parametric model.
#### 1  Importing packages import numpy as np import pandas as pd from matplotlib import pyplot as plt from sklearn.model_selection import train_test_split from pysurvival.models.simulations import SimulationModel from pysurvival.models.survival_forest import ExtraSurvivalTreesModel from pysurvival.utils.metrics import concordance_index from pysurvival.utils.display import integrated_brier_score %pylab inline #### 2  Generating the dataset from a Exponential parametric model # Initializing the simulation model sim = SimulationModel( survival_distribution = 'exponential', risk_type = 'linear', censored_parameter = 1, alpha = 3) # Generating N random samples N = 1000 dataset = sim.generate_data(num_samples = N, num_features=4) # Showing a few datapoints dataset.head(2)
x_1  x_2  x_3  x_4  time  event 

0.061498  7.065547  0.004457  0.131379  15.412209  0. 
0.079149  6.732271  0.008654  0.090398  0.000700  1. 
PySurvival also displays the Base Survival function of the Simulation model:
from pysurvival.utils.display import display_baseline_simulations display_baseline_simulations(sim, figure_size=(20, 6))
#### 3  Creating the modeling dataset # Defining the features features = sim.features # Building training and testing sets # index_train, index_test = train_test_split( range(N), test_size = 0.2) data_train = dataset.loc[index_train].reset_index( drop = True ) data_test = dataset.loc[index_test].reset_index( drop = True ) # Creating the X, T and E input X_train, X_test = data_train[features], data_test[features] T_train, T_test = data_train['time'].values, data_test['time'].values E_train, E_test = data_train['event'].values, data_test['event'].values #### 4  Creating an instance of the Conditional model and fitting the data. # Building the model xst = ExtraSurvivalTreesModel(num_trees=200) xst.fit(X_train, T_train, E_train, max_features="sqrt", max_depth=5, min_node_size=20, num_random_splits = 1000) #### 5  Cross Validation / Model Performances c_index = concordance_index(xst, X_test, T_test, E_test) #0.81 print('Cindex: {:.2f}'.format(c_index)) ibs = integrated_brier_score(xst, X_test, T_test, E_test, t_max=30, figure_size=(20, 6.5) ) print('IBS: {:.2f}'.format(ibs))
We can see that the cindex is well above 0.5 and that the Prediction error curve is below the 0.25 limit, thus the model yields great performances.
We can show this by randomly selecting datapoints and comparing the actual and predicted survival functions, computed by the simulation model and the XST respectively.
# Initializing the figure fig, ax = plt.subplots(figsize=(8, 4)) # Randomly extracting a datapoint that experienced an event choices = np.argwhere((E_test==1.)&(T_test>=1)).flatten() k = np.random.choice( choices, 1)[0] # Saving the time of event t = T_test[k] # Computing the Survival function for all times t survival = xst.predict_survival(X_test.values[k, :]).flatten() actual = sim.predict_survival(X_test.values[k, :]).flatten() # Displaying the functions plt.plot(xst.times, survival, color = 'blue', label='predicted', lw=4, ls = '.') plt.plot(sim.times, actual, color = 'red', label='actual', lw=2) # Actual time plt.axvline(x=t, color='black', ls ='') ax.annotate('T={:.1f}'.format(t), xy=(t, 0.5), xytext=(t, 0.5), fontsize=12) # Show everything title = "Comparing Survival functions between Actual and Predicted" plt.legend(fontsize=12) plt.title(title, fontsize=15) plt.ylim(0, 1.05) plt.show()