Added doctest, docstring and typehint for sigmoid_function & cost_function (#10828)

* Added doctest for sigmoid_function & cost_function

* Update logistic_regression.py

* Update logistic_regression.py

* Minor formatting changes in doctests

* Apply suggestions from code review

* Made requested changes in logistic_regression.py

* Apply suggestions from code review

---------

Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

machine_learning/logistic_regression.py

@@ -27,7 +27,7 @@ from sklearn import datasets
 # classification problems
 
 
-def sigmoid_function(z):
+def sigmoid_function(z: float | np.ndarray) -> float | np.ndarray:
     """
     Also known as Logistic Function.
 
@@ -42,11 +42,63 @@ def sigmoid_function(z):
 
     @param z:  input to the function
     @returns: returns value in the range 0 to 1
+
+    Examples:
+    >>> sigmoid_function(4)
+    0.9820137900379085
+    >>> sigmoid_function(np.array([-3, 3]))
+    array([0.04742587, 0.95257413])
+    >>> sigmoid_function(np.array([-3, 3, 1]))
+    array([0.04742587, 0.95257413, 0.73105858])
+    >>> sigmoid_function(np.array([-0.01, -2, -1.9]))
+    array([0.49750002, 0.11920292, 0.13010847])
+    >>> sigmoid_function(np.array([-1.3, 5.3, 12]))
+    array([0.21416502, 0.9950332 , 0.99999386])
+    >>> sigmoid_function(np.array([0.01, 0.02, 4.1]))
+    array([0.50249998, 0.50499983, 0.9836975 ])
+    >>> sigmoid_function(np.array([0.8]))
+    array([0.68997448])
     """
     return 1 / (1 + np.exp(-z))
 
 
-def cost_function(h, y):
+def cost_function(h: np.ndarray, y: np.ndarray) -> float:
+    """
+    Cost function quantifies the error between predicted and expected values.
+    The cost function used in Logistic Regression is called Log Loss
+    or Cross Entropy Function.
+
+    J(θ) = (1/m) * Σ [ -y * log(hθ(x)) - (1 - y) * log(1 - hθ(x)) ]
+
+    Where:
+       - J(θ) is the cost that we want to minimize during training
+       - m is the number of training examples
+       - Σ represents the summation over all training examples
+       - y is the actual binary label (0 or 1) for a given example
+       - hθ(x) is the predicted probability that x belongs to the positive class
+
+    @param h: the output of sigmoid function. It is the estimated probability
+    that the input example 'x' belongs to the positive class
+
+    @param y: the actual binary label associated with input example 'x'
+
+    Examples:
+    >>> estimations = sigmoid_function(np.array([0.3, -4.3, 8.1]))
+    >>> cost_function(h=estimations,y=np.array([1, 0, 1]))
+    0.18937868932131605
+    >>> estimations = sigmoid_function(np.array([4, 3, 1]))
+    >>> cost_function(h=estimations,y=np.array([1, 0, 0]))
+    1.459999655669926
+    >>> estimations = sigmoid_function(np.array([4, -3, -1]))
+    >>> cost_function(h=estimations,y=np.array([1,0,0]))
+    0.1266663223365915
+    >>> estimations = sigmoid_function(0)
+    >>> cost_function(h=estimations,y=np.array([1]))
+    0.6931471805599453
+
+    References:
+       - https://en.wikipedia.org/wiki/Logistic_regression
+    """
     return (-y * np.log(h) - (1 - y) * np.log(1 - h)).mean()
 
 
@@ -75,6 +127,10 @@ def logistic_reg(alpha, x, y, max_iterations=70000):
 # In[68]:
 
 if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
     iris = datasets.load_iris()
     x = iris.data[:, :2]
     y = (iris.target != 0) * 1