Difference quotients from scratch with Python

Difference quotients from scratch with Python

From the Data Science from Scratch book.

Libraries and helper functions

import pandas as pd
import altair as alt

from typing import List
Vector = List[float]
Vector

typing.List[float]

def dot(vector1: Vector, vector2: Vector) -> float:
    assert len(vector1) == len(vector2)

    return sum(v1 * v2 for v1, v2 in zip(vector1, vector2))


assert dot([1, 2, 3], [4, 5, 6]) == 32

def sum_of_squares(v: Vector) -> Vector:
    return dot(v, v)

assert sum_of_squares([1, 2, 3]) == 14

Difference quotient

from typing import Callable

def difference_quotient(
    f: Callable[[float], float],
    x: float,
    h: float
) -> float :
    return (f(x + h) - f(x)) / h

A simple estimate

def square(x: float) -> float:
    return x * x

def derivative_x2(x: float) -> float:
    return 2 * x

xs = range(-10, 11)
actuals = [derivative_x2(x) for x in xs]
actuals

[-20,
 -18,
 -16,
 -14,
 -12,
 -10,
 -8,
 -6,
 -4,
 -2,
 0,
 2,
 4,
 6,
 8,
 10,
 12,
 14,
 16,
 18,
 20]

estimates = [difference_quotient(square, x, h=0.001) for x in xs]
estimates

[-19.998999999984335,
 -17.998999999988996,
 -15.999000000007868,
 -13.999000000005424,
 -11.99900000000298,
 -9.999000000004088,
 -7.998999999999867,
 -5.998999999999199,
 -3.9989999999994197,
 -1.998999999999973,
 0.001,
 2.0009999999996975,
 4.000999999999699,
 6.000999999999479,
 8.0010000000037,
 10.001000000002591,
 12.001000000005035,
 14.00100000000748,
 16.000999999988608,
 18.000999999983947,
 20.000999999993496]

df = pd.DataFrame({'actuals': actuals, 'estimates': estimates}).reset_index()
df = df.melt(id_vars='index')
df.sample(10)

	index	variable	value
5	5	actuals	-10.000
13	13	actuals	6.000
34	13	estimates	6.001
2	2	actuals	-16.000
19	19	actuals	18.000
0	0	actuals	-20.000
7	7	actuals	-6.000
3	3	actuals	-14.000
24	3	estimates	-13.999
11	11	actuals	2.000

alt.Chart(df).mark_circle(opacity=0.75).encode(
    alt.X('index:Q'),
    alt.Y('value:Q'),
    alt.Size('variable:N'),
    alt.Color('variable:N')
).properties(title='Actual derivatives and estimated quotients')

Calculating an i-th difference quotient

def partial_difference_quotient(
    f: Callable[[Vector], float],
    v: Vector,
    i: int,
    h: float
) -> float:
    """Return i-th parital difference quotient of `f` at a`v`"""
    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: Callable[[Vector], float],
    v: Vector,
    h: float = 0.0001
):
    return [
        partial_difference_quotient(f, v, i, h)
        for i in range(len(v))
    ]