Variance Explained by Principal Components

Once you have fitted PCA and projected your data onto the new axes, a scatter plot of the first two principal components can look clean and informative. The trouble is that the axes themselves rarely come with a label you can read off directly. Each axis points along a direction in the original data space, and that direction is not one of the measurements you originally collected. It is a mixture of them.

This article closes that gap. You will learn where the information describing each axis is stored, how to lay it out so the numbers can be read, and why the mixing of features is the source of most of the difficulty. The goal is not simply to print the numbers. It is to know what they say about the data and to recognize when a reading of them is wrong.

After a PCA object has been fitted, the directions it found are kept in an attribute called components_. You do not need to compute them again. You read them from the fitted object. Each direction is one principal component, and the components are ordered by importance: the first captures the largest share of the variance in the data, the second captures the next largest, and so on.

The layout of the array is the first thing to check. Its shape tells you how it is organized before you look at a single value. In the example the source works through, the first number is the count of components, which is 2, and the second number is the count of original features, which is 30. The array therefore has one row per component and one column per feature. Each row is a direction, and each column is a feature that the direction draws on.

Two questions about the layout come up constantly: how many components there are, and which features a given component covers. The answers come straight from the shape of the array and from the row and column that you are reading.

A principal component is never built from a single measurement. Its direction is spread across all of the original features at once, so every feature has some weight in it. This is why the axes are hard to name. An axis called after one measurement would be easy to explain. An axis built from thirty interacting measurements is not, because its meaning is a pattern across all of them.

In this notation, $x_1$ through $x_{30}$ are the original features, and each weight $w$ is one entry of components_. The first row of the array holds the 30 weights for the first component. Each weight says how strongly that feature pulls the component's direction. A large weight in absolute value means the feature matters more to that direction, and a weight near zero means the feature barely contributes to it.

The weights can also disagree with one another in sign. Some features may push the component in one direction while others push it the other way. A mixed-sign pattern tells you the component is contrasting groups of features rather than measuring one overall quantity. A component whose weights all share one sign is closer to a general measure that rises or falls across the features together.

A table of 60 numbers is hard to scan. A heat map turns each coefficient into a coloured cell, so you can see the sign pattern and the relative size of the weights at a glance. The source demonstrates this with the first two components of a breast cancer dataset, where the columns are the 30 features and the two rows are the components.

In that heat map, the first component has a single sign across its row, so its coloured cells are uniform in tone. That uniformity marks a general correlation among the features. The second component's row is split between two tones, which reflects its mixed signs. Both rows touch every column, which is the visual version of the point that every feature participates.

The heat map is a guide, not the final answer. A colour can suggest a sign and a rough size, but it does not give you the exact weight. The source is explicit that interpretations should always be checked against the printed numbers.

Two features that carry the same sign in a component tend to move together in the data. If one feature is high, the others sharing its sign are likely to be high too. Features with opposite signs trade off against each other along that axis. The worked example below applies this reading to a small set of weights.

When an interpretation disagrees with what you expected, the useful question is where in the chain the mismatch enters. The chain runs from the fitted object, to the components_ attribute, to the printed numbers, to the written description. Most errors happen at one of two links: the orientation of the array, or the text you write about the numbers.

Reading PCA output is a sequence of checks. Confirm the orientation of components_, read across the row for a component, group the features by sign, and then verify the description against the printed numbers. Each step keeps the interpretation tied to the data rather than to the picture.