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**What is this?**

*This jupyter notebook is part of a collection of notebooks on various topics discussed during the Time Domain Astrophysics course delivered by Stefano Covino at the [Università dell'Insubria](https://www.uninsubria.eu/) in Como (Italy). Please direct questions and suggestions to [stefano.covino@inaf.it](mailto:stefano.covino@inaf.it).*

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**This is a `julia` notebook**

%% Cell type:code id:41d298ca-3a5c-457d-ab57-308a7de2d3c0 tags:

``` julia

import Pkg; Pkg.activate(".")

```

%% Output

      Activating project at `/mnt/chromeos/GoogleDrive/MyDrive/Teaching/Insubria/Docs_2025_26/Lectures/Lecture - Singular Spectrum Analysis`

%% Cell type:code id:6fff2b26-f553-42dd-a7fd-292dbcdbfbed tags:

``` julia

Pkg.instantiate()

```

%% Output

       Installed FFMPEG_jll ─ v8.0.0+0

       Installed Unitful ──── v1.26.0

    Precompiling packages...

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      43 dependencies successfully precompiled in 252 seconds. 252 already precompiled.

%% Cell type:code id:8fa2633c-c139-4196-8a1e-3273bfe990a4 tags:

``` julia

using CairoMakie

```

%% Cell type:markdown id:53194e25 tags:

![Time Domain Astrophysics](Pics/TimeDomainBanner.jpg)

%% Cell type:markdown id:a7b36f9a tags:

# Principal Component Analysis (PCA)

***

- Principal component analysis (PCA) is a dimensionality reduction technique that transforms a data set into a set of orthogonal components, called *principal components*, which capture the maximum variance in the data.

- PCA simplifies complex data sets while preserving their most important structures.

%% Cell type:markdown id:8781bea3 tags:

## What Are Principal Components?

***

- Principal components are new variables that are constructed as linear combinations or mixtures of the initial variables.

- These combinations are done in such a way that the new variables (i.e., principal components) are uncorrelated and most of the information within the initial variables is squeezed or compressed into the first components.

- So, the idea is 10-dimensional data gives you 10 principal components, but PCA tries to put maximum possible information in the first component, then maximum remaining information in the second and so on, until having something like shown below:

![PCAComps](Pics/PCAComponents.jpg)

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- Organizing information in principal components will allow one to reduce dimensionality without losing much information, and this by discarding the components with low information and considering the remaining components as your new variables.

- Principal components are often less interpretable and do not (necessarily) have any real physical meaning since they are constructed as linear combinations of the initial variables.

- Geometrically speaking, principal components represent the directions of the data that explain a maximal amount of variance.

> To put all this simply, just think of principal components as new axes that provide the best angle to see and evaluate the data, so that the differences between the observations are better visible.

%% Cell type:markdown id:15b3a2a7-1cbc-4615-8c20-c8c35e5c92b7 tags:

- For example, let us assume that the scatter plot of our data set is as shown below, the first principal component is approximately the line that matches the purple marks because it goes through the origin and it’s the line in which the projection of the points (red dots) is the most spread out.

- Or mathematically speaking, it’s the line that maximizes the variance (the average of the squared distances from the projected points (red dots) to the origin).

![PCA1_2](Pics/PCA1_2.gif)

- The second principal component is calculated in the same way, with the condition that it is uncorrelated with (i.e., perpendicular to) the first principal component and that it accounts for the next highest variance.

- This continues until a total of $p$ principal components have been calculated, equal to the original number of variables.

%% Cell type:markdown id:6311910b-04ac-4c56-bdc5-b77191949ebb tags:

## How Principal Component Analysis Works: 5 Steps

***

- Principal component analysis can be broken down into five steps. We’ll go through each step, providing logical explanations of what PCA is doing.

### Step 1: Standardization and Centering Data

***

- The aim of this step is to standardize the range of the continuous initial variables so that each one of them contributes equally to the analysis.

- More specifically, the reason why it is critical to perform standardization prior to PCA, is that the latter is quite sensitive regarding the variances of the initial variables. That is, if there are large differences between the ranges of initial variables, those variables with larger ranges will dominate over those with small ranges (for example, a variable that ranges between 0 and 100 will dominate over a variable that ranges between 0 and 1), which will lead to biased results. So, transforming the data to comparable scales can prevent this problem.

- Mathematically, this can be done by subtracting the mean and dividing by the standard deviation for each value of each variable. E.g., if $x$ is the considered variable, the standardized variable, $z$, turns out to be:

$$ z = \frac{x - <x>}{\sigma_x} $$

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### Credits

***

This notebook contains material obtained by https://towardsdatascience.com/a-proof-of-the-central-limit-theorem-8be40324da83.

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## Course Flow

***

<table>

  <tr>

    <td>Previous lecture</td>

    <td>Next lecture</td>

  </tr>

  <tr>

      <td><a href="Lecture-StatisticsReminder.ipynb">Reminder of frequentist statistics</a></td>

    <td><a href="Lecture-StatisticsReminder.ipynb">Reminder of frequentist statistics</a></td>

  </tr>

 </table>

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**Copyright**

This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the examples, unless obtained from other properly quoted sources, under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Stefano Covino, Time Domain Astrophysics - Lecture notes featuring computational examples, 2026*.