import matplotlib
import matplotlib.pyplot as plt
matplotlib.rcParams['figure.figsize'] = (15, 8)
import numpy as np
import pandas as pd
rng = np.random.default_rng(1234)
$$\renewcommand{\P}{\mathbb{P}} \newcommand{\E}{\mathbb{E}} \newcommand{\var}{\text{var}} \newcommand{\sd}{\text{sd}} \newcommand{\cov}{\text{cov}}$$
This is here so we can use \P and \E and \var and \cov and \sd in LaTeX below.
When there's lots of variables¶
Suppose we have a bunch of data, like $$ \begin{bmatrix} X_{11} & X_{12} & \cdots & X_{1k} \\ X_{21} & X_{22} & \cdots & X_{2k}\\ \vdots & \vdots & \ddots & \vdots \\ X_{n1} & \cdots & \cdots &X_{nk} \end{bmatrix} $$ where $$\begin{aligned} X_{i \cdot} &= \text{(one observation)} \\ X_{\cdot j} &= \text{(one variable)} . \end{aligned}$$
How do we "look" at it? First, we look at all the variables separately and in pairs. ... what next?
One way to look at this is that we have $n$ points in $\mathbb{R}^k$.
Then, PCA answers the question: "What are the most important axes of variation (of the data in $\mathbb{R}^k$)?"
Let's start with an example.
Manatee Morphometrics¶
Manatees come in different shapes and sizes. In our morphometric study of manatees, we will take 20 different measurements from each of 250 manatees: head length, tail width, nostril spacing, belly diamter, etcetera. Here's a rough model of the data:
- some manatees are bigger than others; this underlying "size" affects all variables roughly equally
- there are average shape differences between sexes in some variables
So, we'll:
- choose the mean female minus male difference for each variable
- generate the sex and relative size of each manatee
- generate all measurements from independent Normals, adjusted by sex and size
k = 20
n = 250
sexes = np.array(['f', 'm'])[1 * (rng.random(size=n) < 0.5)]
sizes = rng.normal(loc=1.0, scale=0.2, size=n)
mean_measurements = rng.gamma(shape=10, scale=10, size=k)
sex_diffs = np.sort(rng.normal(loc=10, scale=25, size=k)) # female minus male
sd_measurements = 20
X = rng.normal(loc=mean_measurements, scale=sd_measurements, size=(n, k))
X[sexes == 'f', :] += sex_diffs
X *= sizes[:, np.newaxis]
What'd we just do? $$\begin{aligned} \mu_j &= \text{(mean measurement for variable $j$)} \\ d_j &= \text{(mean female minus male difference for variable $j$)} \\ S_i &= \text{("size" of manatee $i$)} \\ &\sim \text{Normal}(\text{mean}=1, \text{sd}=0.1) \\ F_i &= \begin{cases} 1 \qquad &\text{if manatee $i$ is female} \\ 0 \qquad &\text{otherwise} \end{cases} \\ \epsilon_{ij} &\sim \text{Normal}(0, \sigma) \\ X_{ij} &= \text{(measurement $j$ for manatee $i$)} \\ &= S_i \left( \mu_j + d_j F_i + \epsilon_{ij} \right) . \end{aligned}$$
As a result, the two sexes are only slightly offset on any axis (seen in pairwise scatter plots):
# PCA fig 1
colors = np.array(['r', 'b'])[1 * (sexes == 'f')]
fig, axes = plt.subplots(3, 3, figsize=(12,12))
for i in range(3):
for j in range(3):
ax = axes[i][j]
if i == j:
ax.hist(X[:,i])
else:
ax.scatter(X[:,i], X[:,j], c=colors)
First, we do PCA (explanation later):
covmat = np.cov(X.T)
evals, evecs = np.linalg.eig(covmat)
eord = np.argsort(evals)[::-1]
evals = evals[eord]
evecs = evecs[:,eord]
pcs = X.dot(evecs)
Scree plot¶
Larger eigenvalues are more important (in a way we'll explain later). It turns out that the eigenvalue, squared, is proportional to "the percent variation explained" by the corresponding PC. Plotting these is referred as a "scree plot".
Scree plot:
# PCA fig 2
perc_var = 100 * evals**2 / np.sum(evals**2)
plt.plot(perc_var)
plt.xlabel("principal component")
plt.ylabel("percent variance explained")
plt.xticks(np.arange(0, 20, 2));
Manatees in PC space¶
Here is a plot of the (projections of each manatee on to the) first two principal components, colored by sex of the manatee. The two sexes form nearly separated clouds, both spread out along PC1 but with females below males on PC2.
# PCA fig 3
plt.scatter(pcs[:,0], pcs[:,1], c=np.array(['red', 'blue'])[1*(sexes == 'f')])
plt.scatter(pcs[0,0], pcs[0,1], marker="*", s=1000) # manatee number 0
plt.xlabel(f"PC 1 ({perc_var[0]:.0f}%)"); plt.ylabel(f"PC 2 ({perc_var[1]:.0f}%)");
"PC1 is size"¶
What is PC1 telling us? PC1 value and the avarage measurement (as a proxy for "size") are highly (negatively) correlated:
# PCA fig 4
plt.scatter(pcs[:,0], np.mean(X, axis=1))
plt.xlabel("PC1"); plt.ylabel("average measurement");
How variables contribute to each PC¶
Each PC is a linear combination of the 20 measurements; these coefficients are called the "loadings". Here these are for the first two PCs: PC1 is nearly constant (and negative), while PC2 starts positive and decreases to be negative (which makes sense because we sorted the sex effects of each variable).
# PCA fig 5
for k in np.arange(2):
plt.plot(evecs[:, k], label=f"PC{k+1}")
plt.xlabel("variable number")
plt.ylabel("coefficient in the PC")
plt.legend();
What's happening: the math¶
The Principal Components¶
Suppose that $X$ is $k$-dimensional with $\E[X] = 0$ and $$ \cov[X] = C .$$
Say we want to combine a bunch of these variables into a new, composite variable: take coefficients $a = (a_1, \ldots, a_{k})$ and define $Y = a \cdot X = \sum_{i=1}^k a_i X_i$.
What is $\var[Y]$?
Since $\E[X] = \E[Y] = 0$, $$\begin{aligned} \var[Y] &= \E[Y^2] \\ &= \E\left[ \left(\sum_{i=1}^k a_i X_i \right)^2 \right] \\ &= \E\left[ \sum_{i=1}^k \sum_{j=1}^k a_i a_j X_i X_j \right] \\ &= \sum_{i=1}^k \sum_{j=1}^k a_i a_j \E\left[ X_i X_j \right] \\ &= \sum_{i=1}^k \sum_{j=1}^k a_i a_j C_{ij} \\ &= a^T C a . \end{aligned}$$
Which choice of coefficients, $a$, maximizes the variance of $Y = a \cdot X$?
The top eigenvector of the covariance matrix, $C$.
Which choice of coefficients $b$ maximizes the variance of $Z = b \cdot X$, for $b$ such that $\cov[Y, Z] = 0$?
The second eigenvector of the covariance matrix.
Linear algebra fact: If $\lambda_i$ and $v_i$ are the $i^\text{th}$ eigenvalue and eigenvector of $C$, with $\|v_i\| = 1$, then $$ C = \sum_{i=1}^k \lambda_i v_i v_i^T . $$ Furthermore, since $C$ is symmetric, $v_i \cdot v_j = \delta_{ij}$, and $$ \sum_{ij} C_{ij}^2 = \sum_i \lambda_i^2 . $$
Said another way: If we change into the basis given by the eigenvectors, $C$ is diagonal, and this change of basis is orthogonal (i.e., it preserves distances).
Terminology:
- $(v_i \cdot X)$: the $i^\text{th}$ principal component, PC$i$
- $v_i$: the loadings of PC$i$ on each variable
- $\lambda_i^2$: the amount of variance explained by PC$i$
So, what'd we do?
Find the covariance matrix:
covmat = np.cov(X.T)
Get the eigenvalues and eigenvectors:
evals, evecs = np.linalg.eig(covmat)
Sort them by size of eigenvalue:
eord = np.argsort(evals)[::-1]
evals = evals[eord]
evecs = evecs[:,eord]
Change the data matrix into the basis given by the eigenvectors:
pcs = X.dot(evecs)
Find the percent variance explained by each PC:
perc_var = 100 * evals**2 / np.sum(evals**2)
Example: text analysis¶
In data/passages.txt
we have a number of short passages from a few different books.
Can we identify the authors of each passage?
The true sources of the passages are in data/passage_sources.tsv.
Turn the data into a matrix¶
from collections import defaultdict
pfile = open("data/passages.txt", "r")
passages = pfile.read().split("\n")[:-1]
sources = pd.read_table("data/passage_sources.tsv")
words = np.unique(" ".join(passages).split(" "))[1:]
def tabwords(x, words):
d = defaultdict(int)
for w in x.split(" "):
d[w] += 1
out = np.array([d[w] for w in words])
return out
wordmat = np.array([tabwords(x, words) for x in passages])
passages[1]
'elinor encouraged her as much as possible to talk of what she felt and before breakfast was ready they had gone through the subject again and again and with the same steady conviction and affectionate counsel on elinors side the same impetuous feelings and varying opinions on mariannes as before sometimes she could believe willoughby to be as unfortunate and as innocent as herself and at others lost every consolation in the impossibility of acquitting him at one moment she was absolutely indifferent to the observation of all the world at another she would seclude herself from it for ever and at a third could resist it with energy in one thing however she was uniform when it came to the point in avoiding where it was possible the presence of mrs jennings and in a determined silence when obliged to endure it jenningss entering into her sorrows with any compassion her kindness is not sympathy her goodnature is not tenderness all that she wants is gossip and she only likes me now because i supply it elinor had not needed this to be assured of the injustice to which her sister was often led in her opinion of others by the irritable refinement of her own mind and the too great importance placed by her on the delicacies of a strong sensibility and the graces of a polished manner like half the rest of the world if more than half there be that are clever and good marianne with excellent abilities and an excellent disposition was neither reasonable nor candid she expected from other people the same opinions and feelings as her own and she judged of their motives by the immediate effect of their actions on herself thus a circumstance occurred while the sisters were together in their own room after breakfast which sunk the heart of mrs jennings still lower in her estimation because through her own weakness it chanced to prove a source of fresh pain to herself though mrs jennings was governed in it by an impulse of the utmost goodwill with a letter in her outstretched hand and countenance gaily smiling from the persuasion of bringing comfort she entered their room saying now my dear i bring you something that i am sure will do you good in one moment her imagination placed before her a letter from willoughby full of tenderness and contrition explanatory of all that had passed satisfactory convincing and instantly followed by willoughby himself rushing eagerly into the room to inforce at her feet by the eloquence of his eyes the assurances of his letter the hand writing of her mother never till then unwelcome was before her and in the acuteness of the disappointment which followed such an ecstasy of more than hope she felt as if till that instant she had never suffered jennings no language within her reach in her moments of happiest eloquence could have expressed and now she could reproach her only by the tears which streamed from her eyes with passionate violencea reproach however so entirely lost on its object that after many expressions of pity she withdrew still referring her to the letter of comfort but the letter when she was calm enough to read it brought little comfort her mother still confident of their engagement and relying as warmly as ever on his constancy had only been roused by elinors application to intreat from marianne greater openness towards them both and this with such tenderness towards her such affection for willoughby and such a conviction of their future happiness in each other that she wept with agony through the whole of it'
for w, x in zip(words[:20], wordmat[:20,:].T):
print(f"{w}: {x}")
a: [15 9 12 8 7 7 5 12 8 3 18 12 13 11 2 2 12 14 11 14] aback: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abaft: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abandon: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abandoned: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abandonment: [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abased: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abasement: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abashed: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abate: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abated: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abatement: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abating: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abbeyland: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abbreviate: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abbreviation: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abeam: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abednego: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abhor: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] abhorred: [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
wordmat.shape
(1000, 19224)
This time we "do PCA" by finding the "singular value decomposition"
(SVD) of the data matrix,
because scipy.sparse.linalg.svds lets us only
find the PCs we're interested in:
finding all would take waaaay too long.
from scipy.sparse.linalg import svds
# center and scale the data
x = wordmat - np.mean(wordmat, axis=1)[:,np.newaxis]
x /= np.std(x, axis=1)[:, np.newaxis]
pcs, evals, evecs = svds(x, k=3)
eord = np.argsort(evals)[::-1]
evals = evals[eord]
evecs = evecs[eord,:]
pcs = pcs[:,eord]
The loadings¶
loadings = pd.DataFrame(evecs.T, columns=[f"PC{k}" for k in range(1,4)], index=words)
loadings
| PC1 | PC2 | PC3 | |
|---|---|---|---|
| a | -0.200405 | 0.037541 | -0.055637 |
| aback | 0.000536 | 0.000239 | 0.000244 |
| abaft | 0.000554 | 0.000167 | 0.000109 |
| abandon | 0.000514 | 0.000292 | -0.000012 |
| abandoned | 0.000394 | 0.000976 | 0.000132 |
| ... | ... | ... | ... |
| zephyr | 0.000553 | 0.000176 | 0.000067 |
| zodiac | 0.000460 | 0.000522 | -0.000393 |
| zone | 0.000406 | 0.000765 | -0.000035 |
| zones | 0.000505 | 0.000423 | -0.000012 |
| zoroaster | 0.000543 | 0.000196 | 0.000136 |
19224 rows × 3 columns
The PCs¶
# PCA fig 6 & 7
fig, (ax0, ax1) = plt.subplots(1, 2)
for s in np.unique(sources['source']):
ax0.scatter(pcs[sources['source'] == s,0], pcs[sources['source'] == s,1])
ax1.scatter(pcs[sources['source'] == s,1], pcs[sources['source'] == s,2], label=s)
ax0.set_xlabel("PC 1"); ax0.set_ylabel("PC 2")
ax1.set_xlabel("PC 2"); ax1.set_ylabel("PC 3")
ax1.legend();
PC 1 is length¶
# PCA fig 8
lengths = np.sum(wordmat, axis=1)
for s in np.unique(sources['source']):
plt.scatter(
pcs[sources['source'] == s, 0],
lengths[sources['source'] == s]
)
plt.legend();
plt.xlabel("PC1"); plt.ylabel("passage length");
/tmp/ipykernel_123181/1265099608.py:8: UserWarning: No artists with labels found to put in legend. Note that artists whose label start with an underscore are ignored when legend() is called with no argument. plt.legend();
PC2 is book¶
loadings.sort_values("PC2").head(20)
| PC1 | PC2 | PC3 | |
|---|---|---|---|
| her | -0.160492 | -0.355002 | 0.462193 |
| to | -0.335033 | -0.311159 | 0.019167 |
| i | -0.132836 | -0.256586 | -0.532524 |
| she | -0.104468 | -0.248379 | 0.239013 |
| you | -0.076411 | -0.150797 | -0.331023 |
| not | -0.098052 | -0.131523 | -0.062419 |
| be | -0.093912 | -0.131226 | -0.075934 |
| was | -0.141180 | -0.130879 | 0.204603 |
| had | -0.082233 | -0.094667 | 0.110598 |
| my | -0.047833 | -0.089361 | -0.194434 |
| have | -0.060876 | -0.078142 | -0.118018 |
| it | -0.130834 | -0.071298 | -0.134508 |
| could | -0.038412 | -0.070246 | 0.058351 |
| for | -0.100339 | -0.057560 | -0.043023 |
| your | -0.025251 | -0.054792 | -0.116548 |
| very | -0.033219 | -0.052090 | -0.018889 |
| elinor | -0.019032 | -0.050489 | 0.023545 |
| he | -0.101428 | -0.047711 | -0.022485 |
| would | -0.038035 | -0.047110 | -0.013667 |
| me | -0.030992 | -0.046881 | -0.137308 |
loadings.sort_values("PC2").tail(20)
| PC1 | PC2 | PC3 | |
|---|---|---|---|
| white | -0.003848 | 0.019542 | -0.005490 |
| boat | -0.003445 | 0.020078 | -0.001117 |
| of | -0.347599 | 0.020880 | 0.089325 |
| these | -0.010246 | 0.021112 | -0.008600 |
| now | -0.024838 | 0.022074 | -0.015997 |
| into | -0.018128 | 0.025168 | -0.007119 |
| one | -0.033698 | 0.027119 | -0.037192 |
| whales | -0.005390 | 0.027329 | -0.010196 |
| old | -0.008500 | 0.027787 | -0.017462 |
| his | -0.120435 | 0.028370 | 0.009677 |
| ship | -0.005879 | 0.028600 | -0.006917 |
| sea | -0.006046 | 0.031019 | -0.006752 |
| ahab | -0.006510 | 0.032928 | -0.007920 |
| upon | -0.016007 | 0.035191 | -0.028701 |
| like | -0.014690 | 0.035441 | -0.017307 |
| a | -0.200405 | 0.037541 | -0.055637 |
| this | -0.047997 | 0.042451 | -0.052062 |
| in | -0.196969 | 0.053266 | -0.005499 |
| whale | -0.014152 | 0.069795 | -0.026534 |
| the | -0.523547 | 0.652591 | 0.022868 |
PC3?¶
loadings.sort_values("PC3").head(20)
| PC1 | PC2 | PC3 | |
|---|---|---|---|
| i | -0.132836 | -0.256586 | -0.532524 |
| you | -0.076411 | -0.150797 | -0.331023 |
| my | -0.047833 | -0.089361 | -0.194434 |
| is | -0.065981 | 0.015494 | -0.170012 |
| me | -0.030992 | -0.046881 | -0.137308 |
| it | -0.130834 | -0.071298 | -0.134508 |
| have | -0.060876 | -0.078142 | -0.118018 |
| your | -0.025251 | -0.054792 | -0.116548 |
| that | -0.142322 | -0.020713 | -0.103028 |
| be | -0.093912 | -0.131226 | -0.075934 |
| but | -0.081493 | -0.014145 | -0.075810 |
| will | -0.025469 | -0.030832 | -0.075157 |
| are | -0.023219 | 0.004333 | -0.066721 |
| if | -0.028036 | -0.019286 | -0.064901 |
| not | -0.098052 | -0.131523 | -0.062419 |
| am | -0.015088 | -0.041852 | -0.060722 |
| do | -0.020977 | -0.035387 | -0.059673 |
| can | -0.014108 | -0.016091 | -0.056266 |
| a | -0.200405 | 0.037541 | -0.055637 |
| this | -0.047997 | 0.042451 | -0.052062 |
loadings.sort_values("PC3").tail(20)
| PC1 | PC2 | PC3 | |
|---|---|---|---|
| mother | -0.010312 | -0.025910 | 0.021928 |
| them | -0.037627 | -0.026889 | 0.022182 |
| the | -0.523547 | 0.652591 | 0.022868 |
| elinor | -0.019032 | -0.050489 | 0.023545 |
| they | -0.045059 | -0.018149 | 0.024460 |
| every | -0.022388 | -0.019543 | 0.028844 |
| mrs | -0.012199 | -0.028403 | 0.031719 |
| marianne | -0.013762 | -0.034782 | 0.033626 |
| which | -0.049184 | -0.029560 | 0.034382 |
| by | -0.064756 | -0.005060 | 0.034632 |
| were | -0.042672 | -0.005147 | 0.043909 |
| herself | -0.016034 | -0.036806 | 0.055770 |
| could | -0.038412 | -0.070246 | 0.058351 |
| their | -0.041101 | -0.003621 | 0.065769 |
| of | -0.347599 | 0.020880 | 0.089325 |
| and | -0.331271 | -0.025959 | 0.091769 |
| had | -0.082233 | -0.094667 | 0.110598 |
| was | -0.141180 | -0.130879 | 0.204603 |
| she | -0.104468 | -0.248379 | 0.239013 |
| her | -0.160492 | -0.355002 | 0.462193 |