In 1886, a scientist studying human genetics
realized something that must have seemed kind
of terrifying at the time. Oh, definitely. He
was looking at family trees and noticed humanity
basically appeared to be shrinking toward the
average. Right. Like unusually tall parents were
consistently having shorter kids. Exactly. And
unusually short parents were having taller kids.
It was like every extreme was just getting pulled
right back to the middle. And he called this
phenomenon regression toward mediocrity. Which
is quite a phrase. It really is. So welcome to
today's deep dive. We are exploring a really
comprehensive Wikipedia article on a subject
that actually grew out of that exact observation
linear regression. Yeah, and I know that probably
sounds like a dusty high school math topic that
you, you know, tried to forget the second you
graduated. Yeah, totally. But consider this,
it is literally the hidden engine powering our
modern world. It really is. I mean, it's the
invisible architecture behind financial markets,
epidemiological research, and maybe most importantly
right now, it's the absolute bedrock of modern
artificial intelligence. Right, because when
you strip away all the layers of a complex neural
network, you often just find linear regression
quietly doing the heavy lifting in the background.
Exactly. So our mission for you today is to demystify
this statistical powerhouse. The goal here isn't
to force you to memorize formulas or anything
like that. No, definitely not. It's really to
understand how humanity uses math to predict
the future and, you know, make sense of incredibly
chaotic data. So, okay, let's unpack this because
the idea of just drawing a straight line through
a bunch of messy data points, It seems way too
simple to be this important. Well, yeah, that
simplicity is kind of an illusion. The most compelling
part of linear regression isn't actually the
mathematics itself. Oh. Yeah, I mean the math
is just a tool. The truly fascinating part is
the philosophical assumptions we have to make
when we try to draw that straight orderly line
through a deeply messy, unpredictable reality.
We're imposing order on chaos. Exactly. And that
comes with some really profound implications.
So before we can understand how modern AI uses
this tool to impose that order, we kind of have
to look at why it was invented in the first place.
Right. Which actually takes us on a journey from
looking up at the stars all the way to looking
right back at ourselves. Because long before
that geneticist was looking at human heights
in the 1880s, astronomers were dealing with a
massive data problem. Yeah, early scientists
were essentially drowning in observational data
that simply wouldn't line up. Which makes sense.
Right. I mean, if we connect this to the bigger
picture, imagine looking through a 17th century
telescope. Your lenses are imperfect, right?
The atmosphere distorts the light. And human
error creeps into literally every measurement
you make. Exactly. The source actually mentions
Isaac Newton dealing with this. In the year 1700,
Newton scribbled down what we now call the normal
equations to study the equinoxes. Wow. Yeah.
He's basically trying to find the mathematical
equations that would minimize all those observation
errors to find the optimal true path of the celestial
bodies. That makes sense. And then... In the
early 1800s, mathematicians like Legendre and
Gauss formalized this into a method called least
squares to predict planetary movement. Right,
because they were trying to figure out exactly
where a planet would be, even though their telescopes
were giving them slightly conflicting readings
every single night. They were hunting for the
true signal hidden within all that noise. But
as you mentioned at the start, the term regression
didn't actually come from astronomy. No, it came
from Francis Galton in 1886, observing those
human heights. The regression toward mediocrity?
Which, again, sounds so harsh to our modern ears.
It does, but in a statistical sense, it really
just meant returning to the center. Today, we
call it regression toward the mean. Right. Galton
plotted the parents' heights against the children's
heights, drew a line through the scattered dots
to represent that relationship, and the name
regression just permanently stuck to the mathematical
process of fitting that line. To ground this
in how it actually works, the text gives this
really brilliant analogy. Imagine you're throwing
a small ball up into the air. If you track its
height at various fractions of a second, physics
strictly dictates its path. Right. Gravity is
pulling it down. The initial velocity is pushing
it up. Exactly. But when you actually measure
it, Your measurements are going to be slightly
off, you know, your thumb slips on the stopwatch
or you blink. There's always human error. Yeah.
So linear regression is basically a way to estimate
those hidden true forces, like the exact initial
velocity of the ball from the messy, imprecise
data points we actually observe. And what's really
fascinating here is the fundamental shift in
how we view the numbers themselves. How so? Well,
from a pure mathematical perspective, your variables,
like time and height, are just unknown variables.
And the parameters, like the force of gravity,
are fixed constants. Gravity isn't changing.
Right. But from a statistical perspective, which
is what regression is, the focus completely flips.
Once we substitute our actual observed data for
the variables, the model becomes a function of
the parameters. I see. The parameters become
the unknowns that we have to estimate. We are
basically using the messy data to work backwards
and guess the hidden rules of the game. We are
essentially reverse engineering reality. Yes,
exactly. Okay, so if we're going to estimate
those hidden forces, we need to peek under the
hood at the anatomy of this line. The core mechanism
doing the heavy lifting here is ordinarily squares,
or OLS, and something called the mean squared
error. Right. The basic formula for this is usually
written as y equals beta zero plus beta one times
x plus epsilon. Okay. Let's apply that formula
directly to Galton's family so you can really
visualize it. Y is the dependent variable. That's
the outcome we want to guess. In this case, the
child's height. Yeah. We are trying to predict
that based on x, our independent variable, which
is the parent's height. And those beta symbols
are the parameters. They represent the intercept
and the slope of our line. Right. They physically
define the angle and position of the straight
line we're drawing across the graph. But the
most crucial part of that entire equation is
actually the very last term, epsilon. Epsilon.
The error term. The noise. Yeah. Epsilon acknowledges
that our line will literally never be perfect.
A child's height isn't purely dictated by their
parent's height. Right. There's a lot of other
stuff going on. Exactly. It captures all the
other hidden factors, childhood diet, illness,
random genetic mutation, that influence Y but
just aren't included in our single X variable.
So the goal of ordinary least squares is to find
the specific beta values, the exact line that
makes those epsilon errors as small as possible
overall. Yep. And it does that by drawing a line
that minimizes the sum of the squared errors.
You take the distance from every single messy
data point to your proposed perfect line, you
square that distance, and you add them all up.
And you want that final number to be as tiny
as possible. Exactly. But wait, why are we squaring
the errors though? I mean, if we just want the
line that is closest to all the dots, why not
just measure the absolute distance from the dot
to the line and minimize that? That is a great
question. Mathematically, squaring the error
does something very specific. It penalizes large
errors exponentially more than small errors.
Exponentially. Yeah. If a data point is one unit
away from your line, the squared error is one.
But if a point is four units away, the squared
error isn't four, it's 16. Wow. So the model
becomes absolutely terrified of being really
far off from any single point. It will actively
twist and warp the angle of the line just to
avoid those massive exponential penalties. But
that seems incredibly vulnerable. Like, what
if you have a hundred normal data points and
one point that is wildly, ridiculously far away?
Like an outlier. Yeah. Like maybe a data entry
typo where someone typed a height of 600 inches
instead of 60. The model will assign so much
importance to that one huge squared error that
it will pull the entire regression line away
from the true data just to accommodate the typo.
You're exactly right. One bad piece of data can
completely derail ordinary least squares. That's
crazy. The source text actually notes that alternative
methods do exist to counter this. There's a technique
called least absolute deviation, which does exactly
what you suggested earlier. It minimizes the
absolute distance without squaring it. Yeah,
it's much more robust to massive outliers, but
historically it was computationally harder and
just less statistically efficient to calculate
by hand, which is why the vulnerable ordinary
least squares became the gold standard. Okay,
but if the standard math is this fragile, if
one bad data point can warp the whole line, How
could statisticians possibly trust it for high
stakes interpretations of reality? Well, they
have to be very careful. This actually introduces
a major theme in our sources. the danger of blind
math. Statistics can easily lie to you if you
just blindly trust the formula without looking
at the real world context. Yes. Take Anscombe's
Quartet, which is a famous statistical trap mentioned
in the text. Anscombe's Quartet is a perfect
example. It consists of four entirely different
data sets. If you graph them, honestly, a child
could tell you they represent four different
phenomena. Right. One is a messy blob of dots.
One is a perfect sweeping curve. One is a tight
straight line with one massive outlier pole.
the math and one is just a vertical stack of
dots with one dot off to the side. So visually
they literally have nothing in common. Nothing.
But if you run a simple linear regression on
all four of those data sets, the math produces
the exact same regression line. Yep. It outputs
the exact same mean. It outputs the same standard
deviation, meaning the math calculates that the
data spreads out from the average by the exact
same amount overall. It's wild. And it gives
the same correlation coefficient. meaning the
formula thinks the variables are moving together
in the exact same way across all four graphs.
It is the ultimate warning label on the side
of the regression box. Always graph your data
first. Always. Do not just trust the mathematical
summary because the math cannot see the shape
of reality. It only sees the numbers you feed
it. Right. And honestly, the dangers go far beyond
visual illusions. There is a major interpretation
issue with something called the held fixed fallacy.
Oh, yeah. When you move from simple regression
to multiple regression, meaning you have many
x variables predicting one y. The way you interpret
the results gets really tricky. Right. Let's
say you're predicting a house price based on
square footage and the number of bedrooms. The
standard interpretation of the regression coefficient
is, okay, this is the expected change in the
price when the square footage changes by one
unit while all other variables are held fixed.
Exactly. The source uses epidemiology to explain
why this held fixed concept is necessary. Early
studies used observational regression to link
tobacco smoking to mortality. If you just regress
lifespan against smoking, your results will be
skewed by confounding variables. You have to
include other variables like education and income
in the exact same mathematical model. Because
otherwise you're not getting the real picture.
Right. That way you ensure the shorter lifespan
you are seeing is actually isolated to the smoking
and not just reflecting broader socioeconomic
factors. You are trying to find the unique isolated
effect of smoking mathematically assuming education
and income are held fixed in place. But here's
where it gets really interesting. How can you
possibly hold a variable fixed in the real world
if things naturally move together? You often
can't. Like using the house example. You can't
realistically increase the square footage of
a house by 2 ,000 feet while holding the number
of bedrooms fixed at one. Those two things are
deeply correlated in reality. You've hit on one
of the most significant vulnerabilities of multiple
regression. It's a problem known as multicollinearity.
Multicollinearity. Yeah. When predictor variables
are strongly correlated in the real world, the
mathematical assumption that you can increase
one while holding the others perfectly constant
becomes a complete fantasy. The source text emphasizes
that in these highly entangled situations, relying
on the individual isolated effects of a single
variable becomes problematic and often entirely
meaningless. So the math will confidently give
you a number for what happens when you isolate
the variable. The reality it describes just doesn't
actually exist. Precisely. To solve this, statisticians
have to stop looking at isolated variables and
instead look at group effects. Instead of asking
what happens when one variable changes alone
in a vacuum, they have to calculate what happens
when the whole group of correlated variables
changes together. Because, well, that is how
nature actually operates. The real world doesn't
play by isolated rules. And it certainly doesn't
always move in straight lines. Yeah, it doesn't.
Because reality is full of non -linear relationships,
this basic tool actually had to adapt and evolve
to survive, which requires clearing up a massive,
very common misconception about the word linear.
Oh, this is a big one. When we say linear regression,
everyone pictures a perfectly straight line on
a graph. But linear regression doesn't strictly
mean the physical shape on the graph has to be
a straight line. Right. It goes back to our earlier
distinction between variables and parameters.
The model is linear in its unknown parameters,
not necessarily in its variables. OK, unpack
that a bit. So the beta coefficients, the parameters
we are trying to estimate, must be linear. They
are only raised to the power of 1. but our x
variables, the raw data we feed into the machine,
we can transform those however we want before
we run the regression. So we can square the x
variable. Yes. We can take the logarithm of the
x variable. The formula could be y equals beta
0 plus beta 1 times x squared. Exactly. The x
is squared, which physically creates a sweeping
curve on your visual graph, but the beta, the
parameter of the regression is actually estimating
to fit that curve, is still just a simple linear
multiplier. So technically, fitting a curve to
data this way is still linear regression. Yeah.
You are fitting a linear model to nonlinear data
simply by transforming the inputs. That's incredibly
clever. And the evolution didn't stop at curves
either. The framework expanded into what are
called generalized linear models or GLMs. These
were developed for when the real world doesn't
give you simple, continuous, easily measurable
numbers to predict. Right, because not everything
is a fluid measurement like a price or a height.
Sometimes you're counting discrete events. Like
how many people are in a room. Yeah. The source
mentions Poisson regression. Which is a type
of GLM used for modeling positive quantities
that scale wildly, like counting the population
of a city or the number of cars passing through
an intersection. Or consider predicting a binary
choice, like a coin flip. or whether a patient
has a specific disease or not, a straight line
going off into infinity just doesn't make any
sense for a simple yes or no question. Right.
So statisticians use another GLM called logistic
regression. OK, but how does that actually map
a straight line to a yes or no question? Well,
instead of the line plotting a raw value from
zero to infinity, the math basically bends the
output into an S -curve. clamping it so the result
is always a percentage between 0 and 1. Oh, we'll
see. It takes the linear math and translates
it into a probability. So a 0 .8 output means
there is an 80 % chance the answer is yes. Wow.
This tool just morphs to fit whatever domain
it enters. It really does. The real -world applications
mentioned in the text are staggering. Like, take
building science. There's an ongoing debate over
thermal comfort scales for designing HVAC systems.
Oh yeah, the temperature debate. Yeah. You ask
people in an office to rate how they feel on
a scale from negative three, which is cold, to
positive three, which is hot. The debate is entirely
about the direction of the regression. Do you
regress the comfort votes against the room temperature,
treating the temperature as the independent variable?
Or do you do the opposite, regressing the temperature
against the comfort votes to find the ideal neutral
setting? And deciding which variable is X and
which is Y completely changes the mathematical
outcome of the model. Wait, really? Just flipping
them? Yes. If you flip the assumption of what
is causing what, you calculate a completely different
optimal temperature, which physically changes
how the architecture of the building's climate
control is calibrated. That is wild. And then
you have finance. The capital asset pricing model,
the CAPM, uses the beta coefficient of a linear
regression to quantify investment risk. Right.
It measures how much a specific stock jumps around
compared to the entire stock market. If a trader
sees a stock with a beta of 1, it means the math
shows the stock moves perfectly with the market.
Right, it tracks it exactly. But if it has a
beta of 2, it's twice as volatile. For every
1 % the market drops, that stock drops 2%. That
single regression coefficient quantifies the
exact premium a trader needs to demand for taking
on that extra risk. Wall Street basically runs
on this simple mathematical relationship. And
of course, we cannot ignore machine learning.
No, absolutely not. Linear regression is one
of the fundamental supervised machine learning
algorithms. So what does this all mean for you
listening? It means that before artificial intelligence
could generate photorealistic art or write code
or drive cars, it first had to master the basic
act of drawing the best possible line through
the noise. It is the absolute foundation of learning
from data. It's the foundational act of taking
known inputs and mapped outputs and finding the
mathematical bridge between them. We have covered
incredible ground today. We started with 18th
century astronomers scribbling equations to track
planets and we followed that exact same mathematical
thread. We saw Galton mapping human heights,
we navigated the visual traps of Anscombe's quartet,
we challenged the illusion of holding variables
fixed in a messy world, all the way to the invisible
architecture that predicts stock risks and trains
modern artificial intelligence. It's really a
testament to the enduring human need to find
the signal in the noise. Definitely. But I actually
want to leave you with a final thought, drawn
from an advanced technique mentioned in the text
called regularized regression. These are methods
like ridge and lasso regression. We talked earlier
about how ordinary least squares tries to find
the absolutely perfect fit for the data it is
given, minimizing that error to zero if possible.
Yeah, that terrifying penalty for outliers. Exactly.
But sometimes a perfect fit on your past data
makes you terrible at predicting the future.
all the random noise and outliers instead of
learning the actual underlying trend. Precisely.
It's a mathematical failure called overfitting.
Your model becomes perfectly adapted to the past
and utterly useless for the future. So regularized
regression does something highly counterintuitive.
It deliberately introduces mathematical bias
into the estimation. Wait, it intentionally makes
its own calculations a little bit wrong on the
training data? Yes. It deliberately accepts a
mathematical bias, a slight deviation from the
pure data in order to drastically reduce its
variance when it faces brand new unseen data.
Wow. It essentially calculates that trying to
be absolutely perfect makes the system too fragile.
By accepting a little bias, the algorithm becomes
vastly more robust and useful when it hits the
messy reality of the real world. The algorithm
is literally programmed to abandon perfection
to survive. Exactly. And this raises an important
question I want you to mull over. OK. If the
most advanced predictive algorithms the world,
the ones managing our finances and driving our
cars, are literally programmed to accept a little
bit of bias just to function better and survive
in reality. What does that say about our own
human pursuit of pure, absolute, unbiased truth?
That is a phenomenal question to end on. Because
whether you're a 1700s astronomer peering through
a foggy telescope, or a modern AI crunching millions
of data points, you're still just trying to draw
a line through a very chaotic reality. Thank
you for joining us on this deep dive. We will
see you next time.