Lab 04 — Data as Points

1. The Big Idea

A dataset is more than a table. After we choose features, each row becomes a point in a feature space. Many rows become a data cloud.

Main message: Data becomes geometry. Linear algebra gives us tools for studying the shape of that geometry.

One student, house, song, image, document, or customer.

A numerical description of one object.

Many vectors collected together.

The geometric picture made by all points.

In many data science settings, rows are objects and columns are features.

Student	Hours	Sleep	Score
A	2	7	70
B	5	6	85
C	1	8	65
D	7	5	92

X = [[2, 7, 70], [5, 6, 85], [1, 8, 65], [7, 5, 92]]

Choose a row

Choose a column

Selected row vector

Selected feature vector

Use the sliders to change the slope, noise, and number of points. Watch the shape of the cloud change.

Slope:

Noise:

Points:

The same houses can look geometrically different when price is measured in dollars, thousands of dollars, or standardized units.

Price scale

Warning: A distance computation is only as meaningful as the feature representation behind it.

Clusters are groups of nearby points. Change the separation and spread to see when clusters become clear or confusing.

Separation:

Spread:

An outlier is a point far from the main cloud. It may be an error, a special case, or an important discovery.

Move the new point. It is classified by whichever class center is closer.

New x:

New y:

Draw a small 8 by 8 image. The image is also a vector in R^64.

Each square is one coordinate. A full-size image can have thousands or millions of coordinates.

Random points tend to get farther apart as dimension increases. This is one reason high-dimensional intuition is different.

Dimension:

Write a short response after using the page.

feature spacedata cloudscalingclustersoutliershigh dimension