Michio JP

Michio JP


How to code Gaussian Mixture Models from scratch in Python

They are parametric generative models that attempt to learn the true data distribution. Hence, once we learn the Gaussian parameters, we can generate data from the same distribution as the source.

We can think of GMMs as the soft generalization of the K-Means clustering algorithm. Like K-means, GMMs also demand the number of clusters K as an input to the learning algorithm. However, there is a key difference between the two. K-means can only learn clusters with a circular form. GMMs, on the other hand, can learn clusters with any elliptical shape.

Also, K-means only allows for an observation to belong to one, and only one cluster. Differently, GMMs give probabilities that relate each example with a given cluster. In other words, it allows for an observation to belong to more than one cluster — with a level of uncertainty. For each observation, GMMs learn the probabilities of that example to belong to each cluster k.

In general, GMMs try to learn each cluster as a different Gaussian distribution. It assumes the data is generated from a limited mixture of Gaussians.

Assuming one-dimensional data and the number of clusters K equals 3, GMMs attempt to learn 9 parameters.

  • 3 parameters for the means
  • 3 parameters for the variances
  • 3 scaling parameters

Here, each cluster is represented by an individual Gaussian distribution (for this example, 3 in total). For each Gaussian, it learns one mean and one variance parameters from data. The 3 scaling parameters, 1 for each Gaussian, are only used for density estimation.

To learn such parameters, GMMs use the expectation-maximization (EM) algorithm to optimize the maximum likelihood. In the process, GMM uses Bayes Theorem to calculate the probability of a given observation xᵢ to belong to each clusters k, for k = 1,2,…, K.

Let’s dive into an example. For the sake of simplicity, let’s consider a synthesized 1-dimensional data. But, as we are going to see later, the algorithm is easily expanded to high dimensional data with D > 1. You can follow along using this jupyter notebook.

To build a toy dataset, we start by sampling points from K different Gaussian distributions. Each one (with its own mean and variance) represents a different cluster in our synthesized data. To make things clearer, let’s use K equals 2.

Below, you can see the resulting synthesized data. We are going to use it as training data to learn these clusters (from data) using GMMs. Note that some of the values do overlap at some point.

We can think of GMMs as a weighted sum of Gaussian distributions. The number of clusters K defines the number of Gaussians we want to fit.

As we said, the number of clusters needs to be defined beforehand. For simplicity, let’s assume we know the number of clusters and define K as 2. In this situation, GMMs will try to learn 2 Gaussian distributions. For 1-dim data, we need to learn a mean and a variance parameter for each Gaussian.

Before we start running EM, we need to give initial values for the learnable parameters. We can guess the values for the means and variances, and initialize the weight parameters as 1/k.

Then, we can start maximum likelihood optimization using the EM algorithm. EM can be simplified in 2 phases: The E (expectation) and M (maximization) steps.

In the E step, we calculate the likelihood of each observation xᵢ using the estimated parameters.

1-d gaussian distribution equation

For each cluster k = 1,2,3,…,K, we calculate the probability density (pdf) of our data using the estimated values for the mean and variance. At this point, these values are mere random guesses.

Then, we can calculate the likelihood of a given example xᵢ to belong to the kᵗʰ cluster.

Using Bayes Theorem, we get the posterior probability of the kth Gaussian to explain the data. That is the likelihood that the observation xᵢ was generated by kᵗʰ Gaussian. Note that the parameters Φ act as our prior beliefs that an example was drawn from one of the Gaussians we are modeling. Since we do not have any additional information to favor a Gaussian over the other, we start by guessing an equal probability that an example would come from each Gaussian. However, at each iteration, we refine our priors until convergence.

Then, in the maximization, or M step, we re-estimate our learning parameters as follows.

Here, for each cluster, we update the mean (μₖ), variance (σ₂²), and the scaling parameters Φₖ. To update the mean, note that we weight each observation using the conditional probabilities bₖ.

We may repeat these steps until converge. That could be up to a point where parameters’ updates are smaller than a given tolerance threshold. At each iteration, we update our parameters so that it resembles the true data distribution.

Gaussian Mixture Models for 1D data using K equals 2

For high-dimensional data (D>1), only a few things change. Instead of estimating the mean and variance for each Gaussian, now we estimate the mean and the covariance. The covariance is a squared matrix of shape (D, D) — where D represents the data dimensionality. Below, I show a different example where a 2-D dataset is used to fit a different number of mixture of Gaussians.

Check the jupyter notebook for 2-D data here.

Gaussian Mixture Models for 2D data using K equals 2

Gaussian Mixture Models for 2D data using K equals 3

Gaussian Mixture Models for 2D data using K equals 4

Note that the synthesized dataset above was drawn from 4 different gaussian distributions. Nevertheless, GMMs make a good case for twothree, and four different clusters.

That is it for Gaussian Mixture Models. These are some key points to take from this piece.

  • GMMs are a family of generative parametric unsupervised models that attempt to cluster data using Gaussian distributions.
  • Like K-Mean, you still need to define the number of clusters K you want to learn.
  • Different from K-Means, GMMs represent clusters as probability distributions. This allows for one data points to belong to more than one cluster with a level of uncertainty.

Thanks for reading.

Originally published by Thalles Silva at towardsdatascience.com


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How to code Gaussian Mixture Models from scratch in Python
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Lambda, Map, Filter functions in python

Welcome to my Blog, In this article, we will learn python lambda function, Map function, and filter function.

Lambda function in python: Lambda is a one line anonymous function and lambda takes any number of arguments but can only have one expression and python lambda syntax is

Syntax: x = lambda arguments : expression

Now i will show you some python lambda function examples:

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Shardul Bhatt

Shardul Bhatt


Why use Python for Software Development

No programming language is pretty much as diverse as Python. It enables building cutting edge applications effortlessly. Developers are as yet investigating the full capability of end-to-end Python development services in various areas. 

By areas, we mean FinTech, HealthTech, InsureTech, Cybersecurity, and that's just the beginning. These are New Economy areas, and Python has the ability to serve every one of them. The vast majority of them require massive computational abilities. Python's code is dynamic and powerful - equipped for taking care of the heavy traffic and substantial algorithmic capacities. 

Programming advancement is multidimensional today. Endeavor programming requires an intelligent application with AI and ML capacities. Shopper based applications require information examination to convey a superior client experience. Netflix, Trello, and Amazon are genuine instances of such applications. Python assists with building them effortlessly. 

5 Reasons to Utilize Python for Programming Web Apps 

Python can do such numerous things that developers can't discover enough reasons to admire it. Python application development isn't restricted to web and enterprise applications. It is exceptionally adaptable and superb for a wide range of uses.

Robust frameworks 

Python is known for its tools and frameworks. There's a structure for everything. Django is helpful for building web applications, venture applications, logical applications, and mathematical processing. Flask is another web improvement framework with no conditions. 

Web2Py, CherryPy, and Falcon offer incredible capabilities to customize Python development services. A large portion of them are open-source frameworks that allow quick turn of events. 

Simple to read and compose 

Python has an improved sentence structure - one that is like the English language. New engineers for Python can undoubtedly understand where they stand in the development process. The simplicity of composing allows quick application building. 

The motivation behind building Python, as said by its maker Guido Van Rossum, was to empower even beginner engineers to comprehend the programming language. The simple coding likewise permits developers to roll out speedy improvements without getting confused by pointless subtleties. 

Utilized by the best 

Alright - Python isn't simply one more programming language. It should have something, which is the reason the business giants use it. Furthermore, that too for different purposes. Developers at Google use Python to assemble framework organization systems, parallel information pusher, code audit, testing and QA, and substantially more. Netflix utilizes Python web development services for its recommendation algorithm and media player. 

Massive community support 

Python has a steadily developing community that offers enormous help. From amateurs to specialists, there's everybody. There are a lot of instructional exercises, documentation, and guides accessible for Python web development solutions. 

Today, numerous universities start with Python, adding to the quantity of individuals in the community. Frequently, Python designers team up on various tasks and help each other with algorithmic, utilitarian, and application critical thinking. 

Progressive applications 

Python is the greatest supporter of data science, Machine Learning, and Artificial Intelligence at any enterprise software development company. Its utilization cases in cutting edge applications are the most compelling motivation for its prosperity. Python is the second most well known tool after R for data analytics.

The simplicity of getting sorted out, overseeing, and visualizing information through unique libraries makes it ideal for data based applications. TensorFlow for neural networks and OpenCV for computer vision are two of Python's most well known use cases for Machine learning applications.


Thinking about the advances in programming and innovation, Python is a YES for an assorted scope of utilizations. Game development, web application development services, GUI advancement, ML and AI improvement, Enterprise and customer applications - every one of them uses Python to its full potential. 

The disadvantages of Python web improvement arrangements are regularly disregarded by developers and organizations because of the advantages it gives. They focus on quality over speed and performance over blunders. That is the reason it's a good idea to utilize Python for building the applications of the future.

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50+ Basic Python Code Examples

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1. How to print “Hello World” on Python?

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3. How to add 2 numbers entered on Python?

4. How to find the Average of 2 Entered Numbers on Python?

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6. How to find the Average of 3 Written Grades entered on Python?

7. How to show the Class Pass Status (PASSED — FAILED) of the Student whose Written Average Has Been Entered on Python?

8. How to find out if the entered number is odd or even on Python?

9. How to find out if the entered number is Positive, Negative, or 0 on Python?

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Ray  Patel

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Common Anti-Patterns in Python

Improve and streamline your code by learning about these common anti-patterns that will save you time and effort. Examples of good and bad practices included.

1. Not Using with to Open Files

When you open a file without the with statement, you need to remember closing the file via calling close() explicitly when finished with processing it. Even while explicitly closing the resource, there are chances of exceptions before the resource is actually released. This can cause inconsistencies, or lead the file to be corrupted. Opening a file via with implements the context manager protocol that releases the resource when execution is outside of the with block.

2. Using list/dict/set Comprehension Unnecessarily

3. Unnecessary Use of Generators

4. Returning More Than One Object Type in a Function Call

5. Not Using get() to Return Default Values From a Dictionary

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August  Larson

August Larson


Live Coding of Python in the Eclipse IDE

This live coding extension makes coders/programmers life easier

Let’s get started…

In this article, I’ll be talking about how to use Live Coding features of Python in Eclipse.

Every time the programmer has to spend a lot of time debugging their code. And still, they failed to debug. This extension will help the coders or programmers to reduce their debugging time. This extension is downloadable in Eclipse IDE.

If you are unaware of how to install the extensions or plugins in eclipse.

Don’t worry at all, I’ll help you out.

Follow these simple steps:-

  • Download Eclipse IDE.
  • After downloading Eclipse, install it on your machine.
  • After installing Eclipse, download Python in Eclipse.
  • Open the Eclipse IDE and set up your workspace.
  • Once done with the above steps, navigate to the **Help **menu tab.
  • In the Help menu tab, click on the “Eclipse Marketplace” option.
  • Search for **“Live Coding in Python” (**or use this link)andclick on the **Install **button.
  • After click on the Install button, accept all the** terms and conditions.**
  • The download and installation process will start.
  • Now, Restart the Eclipse IDE.
  • And start using the Live coding feature.

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