Introduction to Density Estimation

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來自 University of Illinois at Urbana-Champaign 的課程

Data Analytics Foundations for Accountancy I

3 個評分

University of Illinois at Urbana-Champaign

Data Analytics Foundations for Accountancy I

3 個評分

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Module 8: Introduction to Density Estimation

Often, as part of exploratory data analysis, a histogram is used to understand how data are distributed, and in fact this technique can be used to compute a probability mass function (or PMF) from a data set as was shown in an earlier module. However, the binning approach has issues, including a dependance on the number and width of the bins used to compute the histogram. One approach to overcome these issues is to fit a function to the binned data, which is known as parametric estimation. Alternatively, we can construct an approximation to the data by employing a non-parametric density estimation. The most commonly used non-parametric technique is kernel density estimation (or KDE). In this module, you will learn about density estimation and specifically how to employ KDE. One often overlooked aspect of density estimation is the model representation that is generated for the data, which can be used to emulate new data. This concept is demonstrated by applying density estimation to images of handwritten digits, and sampling from the resulting model.

Introduction to Density Estimation6:49

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Robert Brunner
Professor
Accountancy

[MUSIC]

Hello.

This lesson is going to introduce the basic concepts behind density estimation,

in particular, kernel density estimation.

You'll see why is density estimation important.

Why does it add value to the types of visualization that we've been doing,

and then you'll learn how we can employ density estimation on

a one-dimensional visualization to transform a box pllot into a violin plot,

where we see more than just the quantiles, or the quartiles of the distribution,

and outliers, but we actually get a feeling for

how the data is distributed throughout the entire one dimensional data set.

This notebook will be used for this particular lesson,

is introduction to density estimation.

First we setup our notebook to make sure that all of our plots, visualizations,

are shown inline.

We then do our standard imports.

And, finally, we're going to load the Iris Data set and

use that to demonstrate concepts.

Now we've seen this before,

but I wanted to quickly mention the Iris Data set has four dimensions.

And there are 150 roads.

So there's four features, 150 different instances, and

those 150 are broken into three different types of flowers.

Now the first thing you do is we look Look at Rug Plot to see how this data points

are distributed.

And as you can see when we just make a standard Rug Plot, you just see this.

Now, I just told you there were 150 instances, that's not 150 lines.

And the reason is that there are defined values in this data set.

In other words, there's a limited precision with which the measurements

were made.

So in order to see all of the data, we have to generate the points, which is

shifting them slightly in the x direction, so that we can see the full breadth.

So here you can see there's clumps of points around 6 centimeters.

And around 5.5, and around 5.

We didn't see that in the unjittered, so that's an important demonstration of,

sometimes you want to make sure your points get a little jitters,

that you don't have this over lap.

Now we gotta round that by also using histograms,

which basically counts the number of values, or instances, in each bend.

So we can do that with the histogram, and we can see here's our data bend up.

So there's our supple length, and you can see there's a peak right around 6,

around 5.5 and around 5, just like we saw once we jittered the points.

But histograms have another problem and

that is, what is the bin size we should use?

Here is one bin size and here is the same data but with a different bin size.

And you notice how spiky it's become.

What does that mean?

Is that physically important, or

is it just simply an artifact of the measurement process?

And that's a challenge with using histograms.

We can compare histograms by using the same number, and just connect the points,

and say is there some kind of generative model at work here.

But as you can see this is just kind of up and down, up and down,

that's really hard to see what is, fundamentally, going on.

In other words, is there some sort of physical model for

the appearance of an iris flower?

If so we really want to try to understand that, to pull that out of the data.

You'll be doing this in your career, as you try to analyze data and

understand what is the driving mechanism here?

Can I model it?

Because once you can model something, you have a much better

ability to understand what's going on, and how to control it, and how to use it.

So that's where Kernal Density Estimation comes into play.

Rather than just histograming the data and building some connection between the bins,

we can actually try to approximate the underlying function, and

we do that with density estimation.

I won't, in this video, go through all of the the counts, but the basic idea is we

put little functions over each data point, and then we sum those functions up.

This next figure here should show that better, where we have our rug plot.

So here's all our little data points, and then over each one we put a little

Gaussian, which is a normal curve, and we sum all of these curves up.

And when we do that you can see there is not a lot out here,

because there's only one or two out at the edges, but

there's a lot in the center here, and there's a lot here.

These correspond to those spikes in the histogram.

So that when we add these curves together, we're going to nice smooth feature,

to go through and show the functional form at the end.

And that's what we see here, it rises up and there's a little bit of wiggle here,

and then it comes down.

And, overall, we could then start to think about interpreting this kernel density

estimation plot in terms of a particular theoretical distribution function.

So you could start with a Gaussian, or a normal curve, and you might switch over to

a Log-normal, or a Chi-Square, based on some sort of physical intuition.

That was a pretty interesting plot, but

let's look at another way of using density estimation.

And in this case, it automatically comes from C-Born.

Here is a box plot that shows us the distribution of points for Sepal Length.

You notice that it's broken into core tiles, here's the 25th, here's the 50,

here's the 75.

The whiskers show the extent of the data, the notch shows the median, that's nice.

But we have no idea how data are distributed inside this box, or

outside it.

By simply changing the box plot to a violin plot, we get this,

were now we see our full range of our data.

And this is effectively a KDE, a kernel density estimation of the data, so

rather than a square box we get this more curvature added to our data.

Inside we actually have the data shown, along with the median.

So this is a nice way of looking at the data, you can actually compare multiple

data sets in the same way as we did with box plots.

You can break out the data set and split it up by some other variable.

So, for instance, you could do this with the tips data set and

compare the total bill for different genders, or for

different days of the week, with a violin plot.

Hopefully you've gotten a feel for what density estimation is,

after going through this notebook, and how to apply it in different ways,

as well as the ability to build a functional representation of your data,

that you can then think about interpreting, in terms of a model, or

in terms of some ability to generate new data.

We'll see more of this in the advanced density estimation.

If you have any questions, let us know in the course forums, and good luck.

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