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2020-10-24-pca-preparations.md

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keywords title nb_path layout
fastai
Principal Component Analysis from scatch - preparations
_notebooks/2020-10-24-pca-preparations.ipynb
notebook
 
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{% endraw %}

Libraries and helper functions

{% raw %} 
import math as m
import random
import pandas as pd
import numpy as np

import altair as alt


</div>
{% endraw %} 
{% raw %}

from typing import List
Vector = List[float]

</div>
{% endraw %} 
{% raw %}

def add(vector1: Vector, vector2: Vector) -> Vector:
    assert len(vector1) == len(vector2)
    return [v1 + v2 for v1, v2 in zip(vector1, vector2)]

</div>
{% endraw %} 
{% raw %}

def subtract(vector1: Vector, vector2:Vector) -> Vector:
    assert len(vector1) == len(vector2)
    return [v1 - v2 for v1, v2 in zip(vector1, vector2)]

</div>
{% endraw %} 
{% raw %}

def vector_sum(vectors: List[Vector]) -> Vector:
    assert vectors

<span class="n">vector_length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">vectors</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">assert</span> <span class="nb">all</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">==</span> <span class="n">vector_length</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">vectors</span><span class="p">)</span>

<span class="n">sums</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">vector_length</span>
<span class="k">for</span> <span class="n">vector</span> <span class="ow">in</span> <span class="n">vectors</span><span class="p">:</span>
    <span class="n">sums</span> <span class="o">=</span> <span class="n">add</span><span class="p">(</span><span class="n">sums</span><span class="p">,</span> <span class="n">vector</span><span class="p">)</span>

<span class="k">return</span> <span class="n">sums</span>




</div>
{% endraw %} 
{% raw %}

def scalar_multiply(c: float, vector: Vector) -> Vector:
    return [c * v for v in vector]

</div>
{% endraw %} 
{% raw %}

def vector_mean(vector: Vector) -> float:
    n = len(vector)
    return scalar_multiply(1/n, vector)

</div>
{% endraw %} 
{% raw %}

def dot(vector1: Vector, vector2: Vector) -> float:
    assert len(vector1) == len(vector2)
    return sum(v1 * v2 for v1, v2 in zip(vector1, vector2))

</div>
{% endraw %} 
{% raw %}

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

</div>
{% endraw %} 
{% raw %}

def magnitude(v: Vector) -> Vector:
    return m.sqrt(sum_of_squares(v))

</div>
{% endraw %} 
{% raw %}

def gradient_step(v: Vector, gradient: Vector, step_size: float) -> Vector:
    """Return vector adjusted with step. Step is gradient times step size.
    """
    step = scalar_multiply(step_size, gradient)
    return add(v, step)

</div>
{% endraw %}

Steps

{% raw %} 
intercept = random.randint(-30, 30)
coefficient = random.uniform(-1, 1)
n = 30

xs = np.random.randint(-50, 10 + 1, 30)
ys = np.random.randint(-20, 50 + 1, 30)
df = pd.DataFrame({'x': xs, 'y': ys})


print(intercept, coefficient)


alt.Chart(df).mark_point().encode(
alt.X('x:Q'), alt.Y('y:Q'), alt.Tooltip(['x', 'y'])
)


</div>

-10 0.9679420748641416
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{% endraw %}

De-meaning

{% raw %} 
def de_mean(data: List[Vector]) -> List[Vector]:
    # mean = vector_mean(data)
    return [vector - np.mean(vector) for vector in data]

</div>
{% endraw %} 
{% raw %}

xs_demean, ys_demean = de_mean([xs, ys])

df = pd.DataFrame({'x': xs_demean, 'y': ys_demean})
alt.Chart(df).mark_point().encode(
alt.X('x:Q'), alt.Y('y:Q'), alt.Tooltip(['x', 'y'])
)


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{% endraw %}

Direction

{% raw %} 
def direction(w: Vector) -> Vector:
    mag = magnitude(w)
    return [w_i / mag for w_i in w]

direction(xs)


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{% endraw %} 
{% raw %}

xs_dir = direction(xs_demean)
ys_dir = direction(ys_demean)

df = pd.DataFrame({'x': xs_dir, 'y': ys_dir})
alt.Chart(df).mark_point().encode(
alt.X('x:Q'), alt.Y('y:Q'), alt.Tooltip(['x', 'y'])
)


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{% endraw %}