def error(xi, yi, h_theta):
'''
Difference between predicted and observed value for a training example
Parameters
xi: x vector (length j+1) for training example i
yi: y observation for training example i
h_theta: vector of parameters (theta0...thetaj)
Returns
error (predicted - observed)
'''
return dot(h_theta, xi) - yi
def error(xi, yi, h_theta):
'''
Difference between predicted and observed value for a training example
Parameters
xi: x vector (length j+1) for training example i
yi: y observation for training example i
h_theta: vector of parameters (theta0...thetaj)
Returns
error (predicted - observed)
'''
return dot(h_theta, xi) - yi
def gradJ(X, y, h_theta):
'''
Gradient of Cost function for batch gradient descent for
Multiple linear regression
Parameters
X: matrix of independent variables (i rows of observations and j cols of variables). x0=1 for all i
y: dependent variable (i rows)
h_theta: coefficients (j cols)
Returns
Gradient of cost function (j cols, one for each h_thetaj)
Will be used to update h_theta i gradient descent
'''
return [dot(errors(X, y, h_theta), xj) / len(y) for xj in T(X)]
def gradJ(X, y, h_theta):
'''
Gradient of Cost function for batch gradient descent for
Multiple linear regression
Parameters
X: matrix of independent variables (i rows of observations and j cols of variables). x0=1 for all i
y: dependent variable (i rows)
h_theta: coefficients (j cols)
Returns
Gradient of cost function (j cols, one for each h_thetaj)
Will be used to update h_theta i gradient descent
'''
return [dot(errors(X, y, h_theta), xj) / len(y) for xj in T(X)]
def grad_logistic(X, y, h_theta):
errors =[logistic(dot(h_theta, xi)) - yi for (xi, yi) in zip(X, y)]
return [dot(errors, xj) for xj in T(X)]
def logistic_log_partial_ij(x_i, y_i, h_theta, j):
"""here i is the index of the data point,
j the index of the derivative"""
return (logistic(dot(x_i, h_theta)) - y_i) * x_i[j]
def logistic_log_likelihood_i(x_i, y_i, h_theta):
return y_i * log(logistic(dot(x_i, h_theta))) + (1 - y_i) * log(1 - logistic(dot(x_i, h_theta)))
def R(h_theta, alpha):
return 0.5 * alpha * dot(h_theta[1:], h_theta[1:])
def predict(X, h_theta):
return [dot(h_theta, xi) for xi in X]
def R(h_theta, alpha): return 0.5 * alpha * dot(h_theta[1:], h_theta[1:])
def predict(X, h_theta):
return [dot(h_theta, xi) for xi in X]
def predict(f, X, h_theta):
return [f(dot(h_theta, xi)) for xi in X]