from datascience import *
import numpy as np
import matplotlib
%matplotlib inline
import matplotlib.pyplot as plots
plots.style.use('fivethirtyeight')

# Predefined functions; they should look familiar to functions you've coded in assignments!
def standard_units(arr):
    return (arr - np.average(arr))/np.std(arr)

def correlation(t, x, y):
    x_standard = standard_units(t.column(x))
    y_standard = standard_units(t.column(y))
    return np.average(x_standard * y_standard)

def slope(t, x, y):
    r = correlation(t, x, y)
    y_sd = np.std(t.column(y))
    x_sd = np.std(t.column(x))
    return r * y_sd / x_sd

def intercept(t, x, y):
    x_mean = np.mean(t.column(x))
    y_mean = np.mean(t.column(y))
    return y_mean - slope(t, x, y)*x_mean

def fitted_values(t, x, y):
    """Return an array of the regression estimates at all the x values"""
    a = slope(t, x, y)
    b = intercept(t, x, y)
    return a*t.column(x) + b

def residuals(t, x, y):
    """Return an array of all the residuals"""
    predictions = fitted_values(t, x, y)
    return t.column(y) - predictions

Regression Model (Demo Slide 10)

# Ignore this code; it produces plots for demonstrating the regression model
def draw_and_compare(true_slope, true_int, sample_size):
    x = np.random.normal(50, 5, sample_size)
    xlims = np.array([np.min(x), np.max(x)])
    errors = np.random.normal(0, 6, sample_size)
    y = (true_slope * x + true_int) + errors
    sample = Table().with_columns('x', x, 'y', y)

    sample.scatter('x', 'y')
    plots.plot(xlims, true_slope*xlims + true_int, lw=2, color='green')
    plots.title('True Line, and Points Created')

    sample.scatter('x', 'y')
    plots.title('What We Get to See')

    sample.scatter('x', 'y', fit_line=True)
    plots.title('Regression Line: Estimate of True Line')

    sample.scatter('x', 'y', fit_line=True)
    plots.plot(xlims, true_slope*xlims + true_int, lw=2, color='green')
    plots.title("Regression Line and True Line")

draw_and_compare(2, -5, 10)

draw_and_compare(2, -5, 10)

draw_and_compare(2, -5, 100)