Financial markets have discontinuities: prices jump up or down in a time so short that it can be considered a real discontinuity if the time measured by our clocks were really continuous, real line continuous.

Those discontinuities create problems for many forms of mathematical modelling, since their modelling are based upon continuous functions. For instance, many price oscillations look like periodic functions, but when a discontinuity is found, any harmonic analysis becomes troublesome.

Actually, a trend can also be troublesome to the fitting of periodic functions to financial data. But in this case, fitting a polynomial of low grade to the data can filter the trend and then a periodic function series can be fitted to the residuals, the difference between the fitted polynomial and the original data.

The purpose of this suite of articles is to present a simple method to eliminate jumps from the observed data. Of course, when reconstructing the fitted data, the discontinuity will be added back.

<h2>
 
 
 Roadmap
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This one, the 1st of 4 small articles will present the general concepts for the solution, the 2nd one will present the implementation of the solution in Python, the 3rd article will add a sinoidal decomposition of the data after the filtering of the solution, and the 4th and last one will use all the elements to attack a real problem in cryptocoins.

<h2>
 
 
 Cluster analysis as a means to group similar data
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Cluster analysis is a means to group data elements that are similar between themselves. In a metric space, similarity means closeness. There are several ways to devise the groups or clusters of data, and one of the simplest is called k-means. In very few words, it creates clusters by assigning a mean average of the coordinates to a point, that's a centroid.

In two dimensions, that's a typical representation of two groups in two dimensions.

<a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F7hwo0a10z279kdqhxm5j.jpg" class="article-body-image-wrapper"><img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F7hwo0a10z279kdqhxm5j.jpg" alt=" " width="640" height="480"></a>

The points are in blue, and the centroids are in red.

<h2>
 
 
 Cluster analysis in the line
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Since the k-means clustering is based upon distance of the points, an interesting effect will happen when the points are in a line, therefore much closer than in the a cloud, like in the previous image.

Consider the following image that shows a time series with a discontinuity.

<a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fkstvt229tjvlhsq3vjwg.jpg" class="article-body-image-wrapper"><img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fkstvt229tjvlhsq3vjwg.jpg" alt=" " width="500" height="900"></a>

When a cluster analysis is applied to it, the centroids of the clusters are shown in red.

<a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Flwo8qcq00qmeq48cxthl.jpg" class="article-body-image-wrapper"><img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Flwo8qcq00qmeq48cxthl.jpg" alt=" " width="500" height="900"></a>

It's easy to see that the two groups are in different levels, as shown in the following image:

<a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fhbged8p8v6je7bglmr9m.jpg" class="article-body-image-wrapper"><img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fhbged8p8v6je7bglmr9m.jpg" alt=" " width="500" height="900"></a>

Then to eliminate the discontinuity, it's enough to lower the Group 1 to the level of the Group 2. That is: to subtract from the points <code>y</code> coordinate the difference between the level of the two groups.

That can be shown in the following image:

<a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fawtwphv9zuxhnjsx4427.jpg" class="article-body-image-wrapper"><img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fawtwphv9zuxhnjsx4427.jpg" alt=" " width="500" height="900"></a>

Now the two groups are indifferenced.

يتناول المقال التحديات التي تطرحها الانقطاعات في الأسواق المالية، حيث يمكن أن تتغير الأسعار بشكل مفاجئ في فترات زمنية قصيرة جدًا. وهذا يشكل مشاكل كبيرة للنمذجة الرياضية، التي تعتمد عادةً على الدوال المستمرة. يبرز الكاتب كيف يمكن أن تعقد هذه الانقطاعات التحليل التوافقي، الذي يُستخدم غالبًا لفهم تذبذبات الأسعار. إن فهم هذه الديناميكيات أمر بالغ الأهمية للمحللين والمتداولين الذين يعتمدون على نماذج دقيقة للتنبؤ بسلوك السوق.

El artículo discute los desafíos que presentan las discontinuidades en los mercados financieros, donde los precios pueden cambiar abruptamente en períodos de tiempo muy cortos. Esto plantea problemas significativos para la modelización matemática, que generalmente se basa en funciones continuas. El autor destaca cómo estas discontinuidades pueden complicar el análisis armónico, que a menudo se utiliza para comprender las oscilaciones de precios. Comprender estas dinámicas es crucial para los analistas y comerciantes que dependen de modelos precisos para predecir el comportamiento del mercado.

L'article aborde les défis posés par les discontinuités sur les marchés financiers, où les prix peuvent changer brusquement en très peu de temps. Cela pose des problèmes importants pour la modélisation mathématique, qui repose généralement sur des fonctions continues. L'auteur souligne comment ces discontinuités peuvent compliquer l'analyse harmonique, souvent utilisée pour comprendre les oscillations des prix. Comprendre ces dynamiques est crucial pour les analystes et les traders qui s'appuient sur des modèles précis pour prédire le comportement du marché.

The article discusses the challenges posed by discontinuities in financial markets, where prices can change abruptly in very short time frames. This poses significant issues for mathematical modeling, which typically relies on continuous functions. The author highlights how these discontinuities can complicate harmonic analysis, which is often used to understand price oscillations. Understanding these dynamics is crucial for analysts and traders who rely on accurate models to predict market behavior.

Leveling with cluster analysis in Python

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