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pim.py
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pim.py
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import numpy as np
import tensorflow as tf
# tf.keras.backend.set_floatx('float64')
tf.keras.backend.set_floatx('float32')
from tensorflow.keras import backend as K
from tensorflow.keras.layers import Layer, Input
from tensorflow.keras.models import Model
from tensorflow.keras.initializers import constant, RandomUniform
from tensorflow.keras.callbacks import EarlyStopping
from tensorflow.keras.optimizers import Adam
from scipy.stats import norm, beta
import matplotlib.pyplot as plt
import warnings
### PIM for regression problems (including time series)
class PIM(Layer):
def __init__(self, p, err_type, beta, **kwargs):
super(PIM, self).__init__(**kwargs)
self.p = p
self.err_type = err_type
self.beta = beta
def build(self, input_shape):
# Is the error distribution symmetric or asymmetric?
if self.err_type == 'sym':
self.picp = self.picp_sym
shape = (input_shape[1],)
elif self.err_type == 'asym':
self.picp = self.picp_asym
shape = (input_shape[1], 2)
# Do we add the conformal correction to p?
if self.conformal:
self.p *= (1 + 1/input_shape[0]) # batch_size = m
# Initialize prediction interval normal z-scores
rad0 = norm.ppf(0.5*(1 + self.p)) /np.sqrt(5000)
self.rad = self.add_weight(shape=shape, name = 'rad',
initializer=constant(value=rad0),
dtype = K.floatx(),
trainable=True)
self.built = True
def picp_sym(self, err):
return K.mean(K.sigmoid(self.beta*(self.rad - K.abs(err))), axis=0)
def picp_asym(self, err):
err_l = tf.where(err <= 0, err, self.rad[:,0] * K.ones_like(err))
err_u = tf.where(err > 0, err, self.rad[:,1] * K.ones_like(err))
is_lower = K.sigmoid(self.beta*(self.rad[:,0] - K.abs(err_l)))
is_upper = K.sigmoid(self.beta*(self.rad[:,1] - K.abs(err_u)))
is_lower = tf.where(K.abs(is_lower-0.5)>1e-8, is_lower, K.zeros_like(err))
is_upper = tf.where(K.abs(is_upper-0.5)>1e-8, is_upper, K.zeros_like(err))
picp_l = K.sum(is_lower,0) / (K.cast(tf.math.count_nonzero(is_lower,0), K.floatx())+1e-8)
picp_u = K.sum(is_upper,0) / (K.cast(tf.math.count_nonzero(is_upper,0), K.floatx())+1e-8)
return K.concatenate((picp_l, picp_u))
def call(self, inputs):
return self.picp(inputs)
@property
def loss(self):
''' ypred is the picp computed by PIM '''
return lambda _, picp: K.square(picp - self.p)
@property
def radius(self):
''' Radii of lower and upper ends of PI'''
rad = K.eval(self.rad)
if self.err_type == 'sym':
return np.squeeze(rad), np.squeeze(rad)
else:
return np.squeeze(rad[:,0]), np.squeeze(rad[:,1])
def get_config(self):
config = super(PIM, self).get_config()
config.update({'p': self.p})
class PredictionInterval(Model):
def __init__(self, p, err_type='sym', beta=1e3, **kwargs):
super(PredictionInterval, self).__init__()
self.pim = PIM(p, err_type, beta, **kwargs)
self.early_stop = EarlyStopping(monitor='loss',
mode='min',
patience=5)
def call(self, inputs):
return self.pim(inputs)
### PIM for classification problems
class PIMc(Layer):
def __init__(self, p, num_classes, beta, threshold, **kwargs):
self.p = p
self.beta = beta
self.nc = num_classes if num_classes !=2 else num_classes-1
# Confidence interval for the mean of U(a,b) with b-a = 1/2
self.mpiw_unif = 0.5 * self.p
self.mpiw0 = np.random.uniform(1e-8, self.mpiw_unif,(4,self.nc))
if threshold == 'fix':
self.thresh = 0.5
self.picp = self.picp_fix_thresh
else:
thresholds = np.arange(0.1,0.9,0.1)
self.thresh = K.expand_dims(K.cast(thresholds, K.floatx()),0)
self.picp = self.picp_var_thresh
super(PIMc, self).__init__(**kwargs)
def build(self, input_shape):
self.mpiw = self.add_weight(shape=(4, self.nc), name = 'ci_mpiw',
initializer=constant(value=self.mpiw0),
dtype = K.floatx(),
trainable=True)
self.mpiw_not_found = np.array(4*[True])
self.built = True
def picp_fix_thresh(self, ytrue, ypred):
dist2true = K.abs(ytrue - ypred)
ones, zeros = K.ones_like(ytrue), K.zeros_like(ytrue)
# False or true predictions
ft = tf.where(dist2true < 0.5, ones, zeros)
# Map to indices [0,1,2,3] corresponding to [FN,FP,TN,TP]: bin->dec
S = tf.one_hot(K.cast(2*ft + ytrue, 'int32'), 4, axis=1, dtype=K.floatx())
# S is a ~ one-hot-encoded tensor of shape (batch, 2*num_classes)
# Use it to select the active weights per sample
active_mpiw = K.sum(K.expand_dims(self.mpiw, 0) * S, 1)
# False negative or false positive
dist_ftp = ft * dist2true + (1-ft) * K.abs(0.5 - ypred)
in_or_out = K.sigmoid(self.beta*(active_mpiw - dist_ftp))
# Keep count of in_or_out for each of FN, FP, TN, TP
in_or_out = K.tile(K.expand_dims(in_or_out, 1), (1,4,1)) * S
# Compute probability. If self.mpiw are optimal, we must have p ~ self.p
return K.sum(in_or_out, 0) / (K.sum(S, 0) + K.epsilon())
def picp_var_thresh(self, ytrue, ypred):
ytrue = K.expand_dims(K.squeeze(ytrue,-1), 1)
ypred = K.expand_dims(K.squeeze(ypred,-1), 1)
# False or true predictions
tp = (ypred > self.thresh) & (K.cast(ytrue, 'int32') == 1)
tn = (ypred < self.thresh) & (K.cast(ytrue, 'int32') == 0)
ft = tf.where(tp | tn, K.ones_like(ytrue), K.zeros_like(ytrue))
# Map to indices [0,1,2,3] corresponding to [FN,FP,TN,TP]: bin->dec
S = tf.one_hot(K.cast(2*ft + ytrue, 'int32'), 4, axis=2, dtype=K.floatx())
# S is a ~ one-hot-encoded tensor of shape (batch, 2*num_classes)
# Use it to select the active weights per sample
active_mpiw = K.sum(K.reshape(self.mpiw, (1,1,4)) * S, -1)
# Error distance from preds to targets
dist2targ = ft * K.abs(ytrue - ypred) + (1-ft) * K.abs(self.thresh - ypred)
in_or_out = K.sigmoid(self.beta*(active_mpiw - dist2targ))
# Keep count of in_or_out for each of FN, FP, TN, TP
in_or_out = K.tile(K.expand_dims(in_or_out, -1), (1,1,4)) * S
# Compute probability. If self.mpiw are optimal, we must have p ~ self.p
return K.sum(in_or_out, (0,1)) / (K.sum(S, (0,1)) + K.epsilon())
def call(self, inputs):
# Count ytrues in confidence interval
ytrue, ypred = inputs
return self.picp(ytrue, ypred)
@property
def loss(self):
''' ypred is the picp computed by PIM '''
return lambda ytrue, ypred: K.square(ypred - self.p)
@property
def urates(self):
# piw of rates: FN, FP, TN, TP
mpiw = K.eval(self.mpiw)
mpiw0 = K.eval(self.mpiw0)
eps = K.eval(K.epsilon())
self.mpiw_not_found = np.abs(mpiw - mpiw0) <= eps
if self.mpiw_not_found.any():
msg = f'PIM did not evolve in some cases!'
warnings.warn(msg)
mpiw[self.mpiw_not_found] = None
# PIM learns mpiw, uncertainty is half of that
return 0.5 * mpiw
def binary_umetrics(self, ypred, ytrue):
# After the model is fit
# Uncertainty to priors
ytrue, ypred = np.squeeze(ytrue), np.squeeze(np.round(ypred))
inds_pos, inds_neg = (ytrue == 1), (ytrue == 0)
sum_pos, sum_neg = np.sum(inds_pos), np.sum(inds_neg)
fnr = np.sum(ypred[inds_pos] == 0) / sum_pos
fpr = np.sum(ypred[inds_neg] == 1) / sum_neg
tnr = 1-fpr
tpr = 1-fnr
ufnr, ufpr, utnr, utpr = np.squeeze(self.urates)
# Uncertainty to posteriors
inds_ppos, inds_pneg = (ypred == 1), (ypred == 0)
npv = np.sum(ytrue[inds_pneg] == 0) / np.sum(inds_pneg)
ppv = np.sum(ytrue[inds_ppos] == 1) / np.sum(inds_ppos)
p_1 = sum_pos / ytrue.shape[0]
p_0 = 1 - p_1
imb = p_0 / p_1
iimb = 1/imb
uppv = imb * (ppv**2) * ((fpr/tpr)*(utpr/tpr) + ufpr/tpr + imb*ppv*((ufpr/tpr)**2))
unpv = iimb * (npv**2) * ((fnr/tnr)*(utnr/tnr) + ufnr/tnr + iimb*npv*((ufnr/tnr)**2))
# Accuracy and uncertainty
acc = tnr * p_0 + tpr * p_1
uacc = utnr * p_0 + utpr * p_1
# F1 score and uncertainty
f1 = 2*(ppv * tpr)/(ppv + tpr)
uf1_dtpr = (f1*(1+imb*(fpr/tpr)*ppv)+0.5*(f1**2/ppv)*(1+imb*fpr*(ppv/tpr)**2))*(utpr/tpr)
uf1_dfpr = imb*f1*ppv*(0.5*f1/tpr-1)*(ufpr/tpr)
uf1_d2fpr = 0.5*(imb**2)*f1*(ppv**2)*((0.5*f1-tpr)**2 +\
0.5*((f1/tpr)**2*(0.5-tpr/f1)-f1/tpr)+1)*(ufpr/tpr)**2
uf1 = abs(uf1_dtpr) + abs(uf1_dfpr) + abs(uf1_d2fpr)
res = {'TPR': [tpr, utpr],
'FPR': [fpr, ufpr],
'PPV': [ppv, uppv],
'NPV': [npv, unpv],
'F1': [f1, uf1],
'ACC': [acc, uacc],
}
return res
def categorical_acc(self, ypred, ytrue):
ypred = np.round(ypred)
def tpr_per_class(k):
inds_pos = (ytrue[:,k] == 1)
tpr_k = np.sum(ypred[inds_pos,k] == 1,0) / np.sum(inds_pos,0)
return tpr_k
tpr = np.array([tpr_per_class(k) for k in range(ytrue.shape[1])])
# Accuracy and uncertainty
p_1 = np.mean(ytrue == 1,0)
acc = np.sum(tpr * p_1)
_, _, _, utpr = self.urates
uacc = np.sum(utpr * p_1)
return acc, uacc
class ConfidenceInterval(Model):
def __init__(self, p, num_classes, beta=1e3, threshold='fix', **kwargs):
super(ConfidenceInterval, self).__init__()
self.pim = PIMc(p, num_classes, beta, threshold, **kwargs)
self.early_stop = EarlyStopping(monitor='loss',
mode='min',
patience=5)
def call(self, inputs):
return self.pim(inputs)
if __name__ == '__main__':
### Regression
# y_true = np.random.normal(loc=0., scale=1.0, size=(1000,1))
# y_pred = np.zeros_like(y_true)
# mu = 0
a, b, mu = 1, 0.5, 3./4
y_true = np.random.beta(a, b, (5000,1))
y_pred = mu * np.ones_like(y_true)
err = y_true - y_pred
umodel = PredictionInterval(0.95, err_type='asym')
umodel.compile(loss=umodel.pim.loss, optimizer=Adam(lr=0.005))
hist = umodel.fit(err, err,
epochs=4000,
batch_size=100,
callbacks=[umodel.early_stop],
verbose=0)
plt.plot(hist.history['loss'])
plt.show()
rad_l, rad_u = umodel.pim.radius
print(f'PI estimated:{(mu-rad_l, mu+rad_u)}')
# print(f'PI theoretical: {norm.interval(0.95)}')
print(f'PI theoretical: {beta.interval(0.95, a, b)}')
print(f'PICP = {umodel.pim.picp(err.astype(np.float32))}')