PyMC3 has excellent functionality for dealing with Bayesian regressions, so I've been trying to leverage that to run a Bayesian Gamma Regression using PyMC3 where the likelihood would be Gamma.
From what I understand, running any sort of Bayesian Regression in PyMC3 requires the pymc3.glm.GLM() function, which takes in a model formula in Patsy form (e.g. y ~ x_1 + x_2 + ... + x_m), the dataframe, and a distribution.
However, the issue is that the pymc3.glm.GLM() function requires a pymc3..families object (https://github.com/pymc-devs/pymc3/blob/master/pymc3/glm/families.py) for the distribution. But the Gamma distribution doesn't show up as one of the families built into the package so I'm stuck. Or is the Gamma function family hidden somewhere? Would appreciate any help in this matter!
For context:
I have a dataframe of features [x_1, x_2, ..., x_m] (call it X) and a target variable (call it y). This is the code I have prepared so far, but just need to figure out how to get the Gamma distribution in as my likelihood.
import pymc3 as pm
# Combine X and y into a single dataframe
patsy_DF = X
patsy_DF['y'] = y
# Get Patsy Formula
all_columns = "+".join(X.columns)
patsy_formula = "y~" + all_columns
# Instantiate model
model = pm.Model()
# Construct Model
with model:
# Fit Bayesian Gamma Regression
pm.glm.GLM(patsy_formula, df_dummied, family=pm.families.Gamma())
# !!! ... but pm.families.Gamma() doesn't exist ... !!!
# Get MAP Estimate and Trace
map_estimate = pm.find_MAP(model=model)
trace = pm.sample(draws=2000, chains=3, start = map_estimate)
# Get regression results summary (coefficient estimates,
pm.summary(trace).round(3)
Related
I'm trying to do a Bayesian regression in Python. I have included multiple priors for the same outcome (y) variable but the posterior mean doesn't seem to change. Could someone comment on my implementation?
import pymc3 as pm
formula = schooling_y_r5 ~ xvariable
with pm.Model() as normal_model:
my_priors= {
'Intercept': pm.Normal.dist(mu=0., sigma=100.),
'schooling_y_r5': pm.Normal.dist(mu=14, sigma=3.8) ,
'schooling_y_r5': pm.Normal.dist(mu=17, sigma=3.8) ,
'schooling_y_r5': pm.Normal.dist(mu=7.8, sigma=3.8) ,
'schooling_y_r5': pm.Normal.dist(mu=7.6, sigma=3.8)
}
# Creating the model requires a formula and data (and optionally a family)
pm.GLM.from_formula(formula, data = X_train, priors=my_priors)
# Perform Markov Chain Monte Carlo sampling letting PyMC3 choose the algorithm
normal_trace = pm.sample(draws=3000, chains = 2, tune = 4000)
From many documents, I have learned the recipe of Ridge regression that is:
loss_Ridge = loss_function + lambda x L2 norm of slope
and the recipe of Lasso regression that is:
loss_Lasso = loss_function + lambda x L1 norm of slope
When I have read topic "Implementing Lasso and Ridge Regression" in "TensorFlow Machine Learning Cookbook", its author explained that:
"...we will use a continuous approximation to a step function, called
the continuous heavy step function..."
and its author also provided lines of code here.
I don't understand about which is called 'the continuous heavy step function' in this context. Please help me.
From the link that you provided,
if regression_type == 'LASSO':
# Declare Lasso loss function
# Lasso Loss = L2_Loss + heavyside_step,
# Where heavyside_step ~ 0 if A < constant, otherwise ~ 99
lasso_param = tf.constant(0.9)
heavyside_step = tf.truediv(1., tf.add(1., tf.exp(tf.multiply(-50., tf.subtract(A, lasso_param)))))
regularization_param = tf.multiply(heavyside_step, 99.)
loss = tf.add(tf.reduce_mean(tf.square(y_target - model_output)), regularization_param)
This heavyside_step function is very close to a logistic function which in turn can be a continuous approximation for a step function.
You use continuous approximation because the loss function needs to be differentiable with respect to the parameters of your model.
To get an intuition about read the constrained formulation section 1.6 in https://www.cs.ubc.ca/~schmidtm/Documents/2005_Notes_Lasso.pdf
You can see that in your code if A < 0.9 then regularization_param vanishes, so optimization will constrain A in that range.
If you want to normalize features using Lasso Regression here you have one example:
from sklearn.feature_selection import SelectFromModel
from sklearn.linear_model import Lasso
estimator = Lasso()
featureSelection = SelectFromModel(estimator)
featureSelection.fit(features_vector, target)
selectedFeatures = featureSelection.transform(features_vector)
print(selectedFeatures)
I was looking at the robust linear regression in statsmodels and I couldn't find a way to specify the "weights" of this regression. For example in least square regression assigning weights to each observation. Similar to what WLS does in statsmodels.
Or is there a way to get around it?
http://www.statsmodels.org/dev/rlm.html
RLM currently does not allow user specified weights. Weights are internally used to implement the reweighted least squares fitting method.
If the weights have the interpretation of variance weights to account for different variances across observations, then rescaling the data, both endog y and exog x, in analogy to WLS will produce the weighted parameter estimates.
WLS used this in the whiten method to rescale y and x
X = np.asarray(X)
if X.ndim == 1:
return X * np.sqrt(self.weights)
elif X.ndim == 2:
return np.sqrt(self.weights)[:, None]*X
I'm not sure whether all extra results that are available will be appropriate for the rescaled model.
Edit Followup based on comments
In WLS the equivalence W*( Y_est - Y )^2 = (sqrt(W)*Y_est - sqrt(W)*Y)^2 means that the parameter estimates are the same independent of the interpretation of weights.
In RLM we have a nonlinear objective function g((y - y_est) / sigma) for which this equivalence does not hold in general
fw * g((y - y_est) / sigma) != g((y - y_est) * sw / sigma )
where fw are frequency weights and sw are scale or variance weights and sigma is the estimated scale or standard deviation of the residual. (In general, we cannot find sw that would correspond to the fw.)
That means that in RLM we cannot use rescaling of the data to account for frequency weights.
Aside: The current development in statsmodels is to add different weight categories to GLM to develop the pattern that can be added to other models. The target is to get similar to Stata at least freq_weights, var_weights and prob_weights as options into the models.
There are standard ways of predicting proportions such as logistic regression (without thresholding) and beta regression. There have already been discussions about this:
http://scikit-learn-general.narkive.com/4dSCktaM/using-logistic-regression-on-a-continuous-target-variable
http://scikit-learn-general.narkive.com/lLVQGzyl/beta-regression
I cannot tell if there exists a work-around within the sklearn framework.
There exists a workaround, but it is not intrinsically within the sklearn framework.
If you have a proportional target variable (value range 0-1) you run into two basic difficulties with scikit-learn:
Classifiers (such as logistic regression) deal with class labels as target variables only. As a workaround you could simply threshold your probabilities to 0/1 and interpret them as class labels, but you would lose a lot of information.
Regression models (such as linear regression) do not restrict the target variable. You can train them on proportional data, but there is no guarantee that the output on unseen data will be restricted to the 0/1 range. However, in this situation, there is a powerful work-around (below).
There are different ways to mathematically formulate logistic regression. One of them is the generalized linear model, which basically defines the logistic regression as a normal linear regression on logit-transformed probabilities. Normally, this approach requires sophisticated mathematical optimization because the probabilities are unknown and need to be estimated along with the regression coefficients.
In your case, however, the probabilities are known. This means you can simply transform them with y = log(p / (1 - p)). Now they cover the full range from -oo to oo and can serve as the target variable for a LinearRegression model [*]. Of course, the model output then needs to be transformed again to result in probabilities p = 1 / (exp(-y) + 1).
import numpy as np
from sklearn.linear_model import LinearRegression
class LogitRegression(LinearRegression):
def fit(self, x, p):
p = np.asarray(p)
y = np.log(p / (1 - p))
return super().fit(x, y)
def predict(self, x):
y = super().predict(x)
return 1 / (np.exp(-y) + 1)
if __name__ == '__main__':
# generate example data
np.random.seed(42)
n = 100
x = np.random.randn(n).reshape(-1, 1)
noise = 0.1 * np.random.randn(n).reshape(-1, 1)
p = np.tanh(x + noise) / 2 + 0.5
model = LogitRegression()
model.fit(x, p)
print(model.predict([[-10], [0.0], [1]]))
# [[ 2.06115362e-09]
# [ 5.00000000e-01]
# [ 8.80797078e-01]]
There are also numerous other alternatives. Some non-linear regression models can work naturally in the 0-1 range. For example Random Forest Regressors will never exceed the target variables' range they were trained with. Simply put probabilities in and you will get probabilities out. Neural networks with appropriate output activation functions (tanh, I guess) will also work well with probabilities, but if you want to use those there are more specialized libraries than sklearn.
[*] You could in fact plug in any linear regression model which can make the method more powerful, but then it no longer is exactly equivalent to logistic regression.
I want to implement Logisitic regression from scratch in python. Following are the functions in it:
sigmoid
cost
fminunc
Evaluating Logistic regression
I would like to know, what would be a great start to this to start from scratch in python. Any guidance on how and what would be good. I know the theory of those functions but looking for a better pythonic answer.
I used octave and I got it all right but dont know how to start in python as OCtave already has those packages setup to do the work.
You may want to try to translate your octave code to python and see what's going on. You can also use the python package to do this for you. Check out scikit-learn on logistic regression. There is also an easy example in this blog.
In order to implement Logistic Regression, You may consider the following 2 approaches:
Consider How Linear Regression Works. Apply Sigmoid Function to the Hypothesis of Linear Regression and run gradient Descent until convergence. OR Apply the Exponential based Softmax function to rule out lower possibility of occurrence.
def logistic_regression(x, y,alpha=0.05,lamda=0):
'''
Logistic regression for datasets
'''
m,n=np.shape(x)
theta=np.ones(n)
xTrans = x.transpose()
oldcost=0.0
value=True
while(value):
hypothesis = np.dot(x, theta)
logistic=hypothesis/(np.exp(-hypothesis)+1)
reg = (lamda/2*m)*np.sum(np.power(theta,2))
loss = logistic - y
cost = np.sum(loss ** 2)
#print(cost)
# avg cost per example (the 2 in 2*m doesn't really matter here.
# But to be consistent with the gradient, I include it)
# avg gradient per example
gradient = np.dot(xTrans, loss)/m
# update
if(reg):
cost=cost+reg
theta = (theta - (alpha) * (gradient+reg))
else:
theta=theta -(alpha/m) * gradient
if(oldcost==cost):
value=False
else:
oldcost=cost
print(accuracy(theta,m,y,x))
return theta,accuracy(theta,m,y,x)