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)
Related
Can someone give me an understandable explanation of the parameter Alpha in SKlearn's Ridge Regression? How does it influence the function etc.?
Examples would be helpful :)
Ridge regression minimizes the objective function:
||y - Xw||^2_2 + alpha * ||w||^2_2
This model solves a regression model where the loss function is the linear least squares function and regularization is given by the l2-norm. In simple words, alpha is a parameter of how much should ridge regression tries to prevent overfitting!
Let say you have three parameter W = [w1, w2, w3]. In overfitting situation, the loss function can fit a model with W=[0.95, 0.001, 0.0004] which means it is highly biased to the first parameter. However, alpha * ||w||^2_2 increases the loss function in those cases and tries to keep all parameters in some sort of boundaries to prevent overfitting. For instance, with a regularizer, the W could be W=[0.5, 0.2, 0.33]. When you increase alpha you are pushing the Ridge regression to be more robust against overfitting, but might be getting larger training error.
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'm trying to write my own logistic regressor (using batch/mini-batch gradient descent) for practice purposes.
I generated a random dataset (see below) with normally distributed inputs, and the output is binary (0,1). I manually used coefficients for the input and was hoping to be able to reproduce them (see below for the code snippet). However, to my surprise, neither my own code, nor sklearn LogisticRegression were able to reproduce the actual numbers (although the sign and order of magnitude are in line). Moreso, the coefficients my algorithm produced are different than the one produced by sklearn.
Am I misinterpreting what the coefficients for a logistic regression are?
I will appreciate any insight into this discrepancy.
Thank you!
edit: I tried using statsmodels Logit and got yet a third set of slightly different values for the coefficients
Some more info that might be relevant:
I wrote a linear regressor using an almost identical code and it worked perfectly, so I am fairly confident this is not a problem in the code. Also my regressor actually outperformed the sklearn one on the training set, and they have the exact same accuracy on the test set, so I have no reason to believe the regressors are wrong.
Code snippets for the generation of the dataset:
o1 = 2
o2 = -3
x[:,1]=np.random.rand(size)*2
x[:,2]=np.random.rand(size)*3
y = np.vectorize(sigmoid)(x[:,1]*o1+x[:,2]*o2 + np.random.normal(size=size))
so as can be seen, input coefficients are +2 and -3 (intercept 0);
sklearn coefficients were ~2.8 and ~-4.8;
my coefficients were ~1.7 and ~-2.6
and of the regressor (the most relevant parts of it):
for j in range(bin_size):
xs = x[i]
y_real = y[i]
z = np.dot(self.coeff,xs)
h = sigmoid(z)
dc+= (h-y_real)*xs
self.coeff-= dc * (learning_rate/n)
What was the intercept learned? It really should not be a surprise, as your y is polynomial of 3rd degree, while your model has only two coefficients, while 3 + y-intercept would be needed to model the response variable from predictors.
Furthermore, values may be different due to SGD for example.
Not really sure, but the coefficients could be different and return correct y for finite set of points. What are the metrics on each model? Do those differ?
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.
I think I understand the paper of Auto-Encoding Variational Bayes. And I am reading some tensorflow codes implementing this paper. But I don't understand their loss function in those codes. Since lots of codes are written in same way, probably I am wrong.
The problem is like this. The following equation is from AEVB paper.
The loss function is like this equation. This equation can be divided into two: Regularization term and Reconstruction term. Therefore, it becomes
Loss_function = Regularization_term + Reconstruction_term
However, lots of codes implement this Regularization term in a negative sign, like
Loss_function = -Regularization_term + Reconstruction_term
For example, in this code, 79th line shows Regularization term as
KLD = -.5 * tf.reduce_sum(1. + enc_logsd - tf.pow(enc_mu, 2) - tf.exp(enc_logsd), reduction_indices=1)
And then, it just adds to Reconstruction term.
loss = tf.reduce_mean(KLD + BCE)
I don't understand. The sign of KLD is opposite from equation from the paper. There are lots of codes like this. I think I am wrong but I don't know what is wrong. Can you explain why it should be like this?
Reference codes: code1, code2, code3
Equation (10) is the log-likelihood loss we want to maximize. It is equivalent to minimizing the negative log-likelihood (NLL). This is what optimization functions do in practice. Note that the Reconstruction_term is already negated in tf.nn.sigmoid_cross_entropy_with_logits (see https://github.com/tegg89/VAE-Tensorflow/blob/master/model.py#L96). We need to negate the Regularization_term as well.
So the code implements Loss_function = -Regularization_term + -Reconstruction_term.