Gaussian Distribution

In this video, I'd like to talk about the Gaussian distribution which is also

called the normal distribution.

In case you're already intimately familiar with the Gaussian distribution,

it's probably okay to skip this video, but if you're not sure or

if it has been a while since you've worked with the Gaussian distribution or

normal distribution then please do watch this video all the way to the end.

And in the video after this we'll start applying the Gaussian distribution

to developing an anomaly detection algorithm.

Let's say x is a row value's random variable, so x is a row number.

If the probability distribution of x is Gaussian with mean mu and

variance sigma squared.

Then, we'll write this as x, the random variable.

Tilde, this little tilde, this is distributed as.

And then to denote a Gaussian distribution,

sometimes I'm going to write script N parentheses mu comma sigma script.

So this script N stands for normal since Gaussian and

normal they mean the thing are synonyms.

And the Gaussian distribution is parametarized by two parameters,

by a mean parameter which we denote mu and

a variance parameter which we denote via sigma squared.

If we plot the Gaussian distribution or Gaussian probability density.

It'll look like the bell shaped curve which you may have seen before.

And so

this bell shaped curve is paramafied by those two parameters, mu and sequel.

And the location of the center of this bell shaped curve is the mean mu.

And the width of this bell shaped curve, roughly that,

is this parameter, sigma, is also called one standard deviation,

and so this specifies the probability of x taking on different values.

So, x taking on values here in the middle here it's pretty high,

since the Gaussian density here is pretty high, whereas x taking on values further,

and further away will be diminishing in probability.

Finally just for completeness let me write out the formula for

the Gaussian distribution.

So the probability of x, and I'll sometimes write this as the p (x)

when we write this as P ( x ; mu, sigma squared), and so this denotes that

the probability of X is parameterized by the two parameters mu and sigma squared.

And the formula for the Gaussian density is this 1/ root 2 pi,

sigma e (-(x-mu/g) squared/2 sigma squared.

So there's no need to memorize this formula.

This is just the formula for the bell-shaped curve over here on the left.

There's no need to memorize it,

and if you ever need to use this, you can always look this up.

And so that figure on the left, that is what you get if you take a fixed value of

mu and take a fixed value of sigma, and you plot P(x) so this curve here.

This is really p(x) plotted as a function of X for

a fixed value of Mu and of sigma squared.

And by the way sometimes it's easier to think in terms of sigma squared that's

called the variance.

And sometimes is easier to think in terms of sigma.

So sigma is called the standard deviation, and

so it specifies the width of this Gaussian probability density,

where as the square sigma, or sigma squared, is called the variance.

Let's look at some examples of what the Gaussian distribution looks like.

If mu equals zero, sigma equals one.

Then we have a Gaussian distribution that's centered around zero,

because that's mu and

the width of this Gaussian, so that's one standard deviation is sigma over there.

Let's look at some examples of Gaussians.

If mu is equal to zero and sigma equals one,

then that corresponds to a Gaussian distribution that is centered at zero,

since mu is zero, and the width of this Gaussian

is is controlled by sigma by that variance parameter sigma.

Here's another example.

That same mu is equal to 0 and sigma is equal to .5 so

the standard deviation is .5 and the variance sigma squared would therefore be

the square of 0.5 would be 0.25 and in that case the Gaussian distribution,

the Gaussian probability density goes like this.

Is also sent as zero.

But now the width of this is much smaller because the smaller the area is,

the width of this Gaussian density is roughly half as wide.

But because this is a probability distribution, the area under the curve,

that's the shaded area there.

That area must integrate to one this is a property of probability distributing.

So this is a much taller Gaussian density because this half is Y but

half the standard deviation but it twice as tall.

Another example is sigma is equal to 2 then you get a much fatter a much

wider Gaussian density and so

here the sigma parameter controls that Gaussian distribution has a wider width.

And once again, the area under the curve, that is the shaded area, will always

integrate to one, that's the property of probability distributions and because it's

wider it's also half as tall in order to still integrate to the same thing.

And finally one last example would be if we now change the mu parameters as well.

Then instead of being centered at 0 we now have a Gaussian distribution that's

centered at 3 because this shifts over the entire Gaussian distribution.

Next, let's talk about the Parameter estimation problem.

So what's the parameter estimation problem?

Let's say we have a dataset of m examples so exponents x m and

lets say each of this example is a row number.

Here in the figure I've plotted an example of the dataset so

the horizontal axis is the x axis and either will have a range of examples of x,

and I've just plotted them on this figure here.

And the parameter estimation problem is,

let's say I suspect that these examples came from a Gaussian distribution.

So let's say I suspect that each of my examples, x i, was distributed.

That's what this tilde thing means.

Let's not suspect that each of these examples were distributed according to

a normal distribution, or Gaussian distribution, with some parameter mu and

some parameter sigma square.

But I don't know what the values of these parameters are.

The problem of parameter estimation is, given my data set, I want to try to

figure out, well I want to estimate what are the values of mu and sigma squared.

So if you're given a data set like this,

it looks like maybe if I estimate what Gaussian distribution the data came from,

maybe that might be roughly the Gaussian distribution it came from.

With mu being the center of the distribution, sigma standing for

the deviation controlling the width of this Gaussian distribution.

Seems like a reasonable fit to the data.

Because, you know, looks like the data has a very high probability of

being in the central region, and a low probability of being further out,

even though probability of being further out, and so on.

So maybe this is a reasonable estimate of mu and sigma squared.

That is, if it corresponds to a Gaussian distribution function

that looks like this.

So what I'm going to do is just write out the formula the standard formulas for

estimating the parameters Mu and sigma squared.

Our estimate or the way we're going to estimate mu

is going to be just the average of my example.

So mu is the mean parameter.

Just take my training set, take my m examples and average them.

And that just means the center of this distribution.

How about sigma squared?

Well, the variance, I'll just write out the standard formula again,

I'm going to estimate as sum over one through m of x i minus mu squared.

And so this mu here is actually the mu that I compute over here using

this formula.

And what the variance is, or

one interpretation of the variance is that if you look at this term,

that's the square difference between the value I got in my example minus the mean.

Minus the center, minus the mean of the distribution.

And so in the variance I'm gonna estimate as

just the average of the square differences between my examples, minus the mean.

And as a side comment, only for those of you that are experts in statistics.

If you're an expert in statistics, and if you've heard of

maximum likelihood estimation, then these parameters, these estimates,

are actually the maximum likelihood estimates of the primes of mu and

sigma squared but if you haven't heard of that before don't worry about it,

all you need to know is that these are the two standard formulas for

how to figure out what are mu and Sigma squared given the data set.

Finally one last side comment again only for those of you that have maybe taken

the statistics class before but if you've taken statistics This class before.

Some of you may have seen the formula here where this is M-1 instead of M so

this first term becomes 1/M-1 instead of 1/M.

In machine learning people tend to learn 1/M formula but in practice whether it is

1/M or 1/M-1 it makes essentially no difference assuming M is reasonably large.

a reasonably large training set size.

So just in case you've seen this other version before.

In either version it works just about equally well but in machine learning

most people tend to use 1/M in this formula.And the two versions have

slightly different theoretical properties like these are different math properties.

Bit of practice it really makes makes very little difference, if any.

So, hopefully you now have a good sense of what the Gaussian distribution looks like,

as well as how to estimate the parameters mu and

sigma squared of Gaussian distribution if you're given a training set,

that is if you're given a set of data that you suspect comes from a Gaussian

distribution with unknown parameters, mu and sigma squared.

In the next video, we'll start to take this and

apply it to develop an anomaly detection algorithm.

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