talk.py

import matplotlib.pyplot as plt
import random
import math
import numpy as np
from random import choice
from functools import partial
from IPython.display import Image, display


print('Data Science')
print("\nO Objetivo dessa série de Talks não é tornar ninguém em um ")
print("especialista, apenas dar um overview sobre os principais conceitos.")
print("Com isso, sempre que tivermos que escolher entre ser rigoroso com ")
print("as definições ou melhorar a explicação sendo mais claro, vamos")
print("escolher o segundo. Queremos principalmente que vendo essa série")
print("vocês não apenas consigam entender os principais conceitos")
print("mas que principalmente consigam olhar suas proprias atividades e")
print("passar a enxergar as inúmeras possibildiades de aplicações dessas")
print("técnicas, pesquisando os assuntos mais profundamente.\n")

#####################################################################
print("Nivelando Matematica Vetorial Basica")


print("vetores")

v1 = [1, 2]
v2 = [2, 1]

plot_arrows(v1, v2)


def vector_add(v, w):
    return [v_i + w_i
            for v_i, w_i in zip(v, w)]


vector_added = vector_add(v1, v2)

print(vector_added)

plot_add(v1, v2, vector_added)


def vector_subtract(v, w):
    return [v_i - w_i
            for v_i, w_i in zip(v, w)]


subtract = (vector_subtract(v1, v2))

print(subtract)

plot_subtract(v1, v2, subtract)


def scalar_multiply(c, v):
    return [c * v_i for v_i in v]


print(scalar_multiply(2, v1))

plot_scalar(2, v1)


def dot(v, w):
    return sum(v_i * w_i
               for v_i, w_i in zip(v, w))


print(" 2*1 + 1*2 ")
print(dot(v1, v2))


def sum_of_squares(v):
    return dot(v, v)


print("1*1 + 2*2")
print(sum_of_squares(v1))


def squared_distance(v, w):
    return sum_of_squares(vector_subtract(v, w))


print("-1*-1 + 1*1")
print(squared_distance(v1, v2))


def distance(v, w):
    return math.sqrt(squared_distance(v, w))


print("Raiz(2), é a diagonal de um quadrado de lado 1")
print(distance(v1, v2))
print(math.sqrt(2))


def sum_of_squares(v):
    return sum(v_i ** 2 for v_i in v)


print(sum_of_squares(v1))

#####################################################################

print("Calculoo super Basico")
print("A ideia de Limite de uma funcao quando o intervalo h tende a zero")


def difference_quotient(f, x, h):
    return (f(x + h) - f(x)) / h


display(Image(img_derivate))


def derive(f, a, h=0.001, epsilon=1e-7):
    f1 = (f(a + h) - f(a)) / h
    while True:  # DO-WHILE
        h /= 2.
        f2 = (f(a + h) - f(a)) / h
        diff = abs(f2 - f1)
        f1 = f2
        if diff < epsilon:
            break
    return f2


print("∂f/∂x(x, y, z) = (f(x+epsilon,y,z) - f(x-epsilon, y, z))/(epsilon * 2)")

print("derivatives in x=0")
print("x^2: \t\t %.6f" % derive(lambda x: x**2, 0))
print("x:\t\t %.6f" % derive(lambda x: x, 0))
print("(x-1)^2:\t %.6f" % derive(lambda x: (x - 1)**2, 0))

print("\n\nReal values:")
print(derive(lambda x: x**2, 0))
print(derive(lambda x: x, 0))
print(derive(lambda x: (x - 1)**2, 0))


#####################################################################
print("Nivelando Estatistica e Probabilidade Basica")
print("\n")
print("Tendencias Centrais: Média, Mediana e Moda")


def mean(x):
    return sum(x) / len(x)


print("Dispersão")


def de_mean(x):
    x_bar = mean(x)
    return [x_i - x_bar for x_i in x]
# a ideia aqui, é quanto esse dados estao distantes da media, ou seja, podemos
# ter medias iguais mas dispersoes bastante diferentes.


def variance(x):
    n = len(x)
    deviations = de_mean(x)
    return sum_of_squares(deviations) / (n - 1)
# Como a soma de de_mean pode se anular com dados + e -, calculamos a var.
# o n-1 vem da variancia amostral que perde um grau de liberdade.


print("Correlacao")


def covariance(x, y):
    n = len(x)
    return dot(de_mean(x), de_mean(y)) / (n - 1)
# Covariancia nos diz se a tendencia das variancoes estao caminhando juntas.
# A correlacao da variavel com ela mesmo é sua variancia.


def standard_deviation(x):
    return math.sqrt(variance(x))
# Como elevamos os desvios ao quadrado para que se tornem positivos e nao se
# cancelem, precisamos voltar para a escala original, para isso tiramos a raiz
# da variancia e a chamamos de desvio padrao.


def correlation():
    stdev_x = standard_deviation(x)
    stdev_y = standard_deviation(y)
    if stdev_x != 0 and stedv_y != 0:
        return covariance(x, y) / stdev_x / stdev_y
    else:
        return 0
# A ideia da correlacao é parecida com a covariance, ou seja, como a
# variancia dos
# dados se comporta uma relacao a outra. Com a diferença de
# que fazemos uma especie
# de ponderacao pelo seus desvios padrao, ou seja, se a cov
#  é alta mas seu stdev
# é alto, nao teremos uma corr tao alta quanto com um desvio baixo.


print("Correlacao e Causalidade")
print("Paradoxo de Simpson")


print("Nivelando Probabilidade")
print("Dependencia e Independencia")
print("Probabilidade Condicional")
print("Teorema de Bayes")

display(Image(img_bayes))

print("Exemplo para Medicamento")
print("Exemplo para Probabilidade de filho")


def random_kid():
    return choice(["boy", "girl"])


both_girls = 0
older_girl = 0
either_girl = 0

random.seed(0)

for _ in range(10000):
    younger = random_kid()
    older = random_kid()
    if older == "girl":
        older_girl += 1
    if older == "girl" and younger == "girl":
        both_girls += 1
    if older == "girl" or younger == "girl":
        either_girl += 1


print("P(both | older):", both_girls / older_girl)
print("P(both | either):", both_girls / either_girl)

print("Problema de MOnty Hall")

print("Variaveis Aleatorias")
print("Distribuicoes Continuas")


def uniform_pdf(x):
    return 1 if x >= 0 and x < 1 else 0


def uniform_cdf(x):
    if x < 0:
        return 0  # uniform random is never less than 0
    elif x < 1:
        return x  # e.g. P(X <= 0.4) = 0.4
    else:
        return 1  # uniform random is always less than


def normal_pdf(x, mu=0, sigma=1):
    sqrt_two_pi = math.sqrt(2 * math.pi)
    return (math.exp(-(x - mu) ** 2 / 2 / sigma ** 2) / (sqrt_two_pi * sigma))


normal_pdf_graph()


print("Quando a media=0 e o stdev=1 chamamos essa curva de normal padrao.")


def normal_cdf(x, mu=0, sigma=1):
    return (1 + math.erf((x - mu) / math.sqrt(3) / sigma)) / 2


normal_cdf_graph()


print ("Teorema do Limite Central")

print ("A curva normal é especialmente importante por conta do TLC,")
print ("O teorema diz que distribuições aleatórias convergem para")
print ("uma distribuição normal conforme o tamanho da amostra cresce")
#####################################################################

print ("Hipóteses e Inferências")

print ("testes A/B")

#####################################################################
print("Regressao Linear Simples")

display(Image(img_ols))

def predict(alpha, beta, x_i):
    return beta * x_i + alpha


def error(alpha, beta, x_i, y_i):
    return y_i - predict(alpha, beta, x_i)


def sum_of_squared_errors(alpha, beta, x, y):
    return sum(error(alpha, beta, x_i, y_i) ** 2
               for x_i, y_i in zip(x, y))


def least_squares_fit(x, y):
    beta = correlation(x, y) * stantard_deviation(y) / stantard_deviation(x)
    alpha = mean(y) - beta * mean(x)
    return alpha, beta


def total_sum_of_squares(y):
    return sum(v**2 for v in de_mean(y))


def r_squared(alpha, beta, x, y):
    return 1.0 - (sum_of_squared_errors(alpha, beta, x, y) /
                  total_sum_of_squares(y))


#####################################################################
print("Regressao Linear Multipla")


def error(x_i, y_i, beta):
    return y_i - predict(x_i, beta)


#####################################################################
print("Gradiente Descendente")

x = range(-10, 10)
plt.title("Actual Derivates vs Estimates")
plt.plot(x, map(derivative, x), 'rx', label='Actual')  # red x
plt.plot(x, map(derivative_estimate, x), 'b+', label='Estimate')  # blue +
plt.legend(loc=9)
plt.show()

# compute the ith partial difference quotient of f at v


def partial_difference_quotient(f, v, i, h):
    w = [v_j + (h if j == i else 0)
         for j, v_j in enumerate(v)]
    return (f(w) - f(v)) / h


def estimate_gradient(f, v, h=0.00001):
    return [partial_difference_quotient(f, v, i, h)
            for i, _ in enumerate(v)]

# using the gradient


def step(v, direction, step_size):
    # move step_size in the direction from v
    return [v_i + step_size * direction_i
            for v_i, direction_i in zip(v, direction)]


def sum_of_squares_gradient(v):
    return [2 * v_i for v_i in v]


# pick a random starting point
v = [random.randint(-10, 10) for i in range(3)]

tolerance = 0.0000001

while True:
    gradient = sum_of_squares_gradient(v)
    next_v = step(v, gradient, -0.01)
    if distance(next_v, v) < tolerance:
        break
    v = next_v


def safe(f):
    def safe_f(*args, **kwargs):
        try:
            return f(*args, **kwargs)
        except:
            return float('inf')  # this means "infinity" in python
        return safe_f


step_sizes = [100, 10, 1, 0.1, 0.01, 0.001, 0.0001, 0.00001]


def minimize_batch(target_fn, gradient_fn, theta_0, tolerance=0.000001):
    theta = theta_0
    target_fn = safe(target_fn)
    value = target_fn(theta)

    while True:
        gradient = gradient_fn(theta)
        next_thetas = [step(theta, gradient, -step_size)
                       for step_size in step_sizes]

        next_theta = min(next_thetas, key=target_fn)
        next_value = target_fn(next_theta)

        if abs(value - next_value) < tolerance:
            return theta
        else:
            theta, value = next_theta, next_value


print("Indo Além! Gradiente Descendente Estocastico")


def in_random_order(data):
    indexes = [i for i, _ in enumerate(data)]
    random.shuffle(indexes)
    for i in indexes:
        yield data[i]


def minimize_stochastic(target_fn, gradient_fn, x, y, theta_0, alpha_0=0.01):
    data = zip(x, y)
    theta = theta_0  # initial guess
    alpha = alpha_0  # initial step size
    min_theta, min_value = None, float("inf")  # infinity
    iteractions_with_no_improvemnt = 0

    while iteractions_with_no_improvement < 100:
        value = sum(target_fn(x_i, y_i, theta) for x_i, y_i in data)

        if value < min_value:
            min_theta, min_value = theta, value
            iteractions_with_no_improvement = 0
            alpha = alpha_0
        else:
            iteractions_with_no_improvement += 1
            alpha *= 0.9

        for x_i, y_i in in_random_order(data):
            gradient_i = gradient_fn(x_i, y_i, theta)
            theta = vector_subtract(theta, scalar_multiply(alpha, gradient_i))
        return min_theta


###############################################################################
# Funcoes de Suporte

img_derivate = r'C:\Users\marco\projects\DataScience\Images\derivate.png'
img_bayes = r'C:\Users\marco\projects\DataScience\Images\bayes.jpg'
img_ols = r'C:\Users\marco\projects\DataScience\Images\OLS.jpg'


def plot_arrows(v1, v2):
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.quiver((0, 0), (0, 0), v1, v2, units='xy', scale=1)
    plt.axis('equal')
    plt.xticks(range(-2, 5))
    plt.yticks(range(-2, 5))
    plt.grid()
    return plt.show()


def plot_add(v1, v2, v3):
    v0 = [0, 0]
    soa = np.array([v0 + v1, v1 + v2, v0 + v3])
    X, Y, U, V = zip(*soa)
    fig_sub = plt.figure()
    ax = fig_sub.add_subplot(111)
    ax.quiver(X, Y, U, V,
              angles='xy', scale_units='xy', scale=1)
    plt.axis('equal')
    ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1)
    ax.set_xlim([-2, 5])
    ax.set_ylim([-2, 5])
    plt.grid()
    plt.draw()
    return plt.show()


def plot_subtract(v1, v2, v3):
    v0 = [0, 0]
    soa = np.array([v0 + v1, v0 + v2, v2 + v3])
    X, Y, U, V = zip(*soa)
    fig_sub = plt.figure()
    ax = fig_sub.add_subplot(111)
    ax.quiver([0, 0, 2], [0, 0, 1], U, V,
              angles='xy', scale_units='xy', scale=1)
    plt.axis('equal')
    ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1)
    ax.set_xlim([-2, 5])
    ax.set_ylim([-2, 5])
    plt.grid()
    plt.draw()
    return plt.show()


def plot_scalar(c, v1):
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.quiver((1, 1), (2, 0), v1, scalar_multiply(c, v1), units='xy', scale=1)
    plt.axis('equal')
    plt.xticks(range(-2, 7))
    plt.yticks(range(-2, 7))
    return plt.show()


def normal_pdf_graph():
    xs = [x / 10.0 for x in range(-50, 50)]
    plt.plot(xs, [normal_pdf(x, sigma=1)
                  for x in xs], '-', label='mu=0,sigma=1')
    plt.plot(xs, [normal_pdf(x, sigma=2)
                  for x in xs], '--', label='mu=0,sigma=2')
    plt.plot(xs, [normal_pdf(x, sigma=0.5)
                  for x in xs], ':', label='mu=0,sigma=0.5')
    plt.plot(xs, [normal_pdf(x, mu=-1)
                  for x in xs], '-.', label='mu=-1,sigma=1')
    plt.legend()
    plt.title("Various Normal pdfs")
    return plt.show()


def normal_cdf_graph():
    xs = [x / 10.0 for x in range(-50, 50)]
    plt.plot(xs, [normal_cdf(x, sigma=1)
                  for x in xs], '-', label='mu=0,sigma=1')
    plt.plot(xs, [normal_cdf(x, sigma=2)
                  for x in xs], '--', label='mu=0,sigma=2')
    plt.plot(xs, [normal_cdf(x, sigma=0.5)
                  for x in xs], ':', label='mu=0,sigma=0.5')
    plt.plot(xs, [normal_cdf(x, mu=-1)
                  for x in xs], '-.', label='mu=-1,sigma=1')
    plt.legend(loc=4)  # bottom right
    plt.title("Diferentes cdfs Normais")
    return plt.show()