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What is the Poisson process and how is it used in data science?

Various probability theories enable us to calculate and interpret the distribution of randomly selected variables. We mainly find the use of the Poisson process and distribution when the number of upcoming events is large and their probability of occurring is very low. 

Mathematics and data science share an unbreakable connection and different mathematical functions are being used in the operations of data science. The probability theory is a major part of mathematics and not only does it help us in measuring models but also helps in modeling data. The Poisson process is also a part of a mathematical and probabilistic theory that has many important applications. There are various Poisson process use cases in data science. In this article, we are going to discuss the Poisson process use cases in data science. The major points to be discussed in the article are listed below.

Table of content 

  1. What is the Poisson paradigm 
  2. What is the Poisson distribution 
  3. Implementation of Poisson distribution 
  4. The Poisson process use cases in data science    

Let’s first discuss the paradigm of Poisson.

What is the Poisson paradigm?

Probability theory and statistics are the parts of mathematics and the Poisson paradigm can be considered as the part of probabilistic theory and statistics. Various probability theories enable us to calculate and interpret the distribution of randomly selected variables. We mainly find the use of the Poisson process and distribution when the number of upcoming events is large and their probability of occurring is very low. 

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Mathematically we convert this scenario as the number of events n that has the tendency to go toward infinity and the occurring probability is p that tends to go toward zero. We can also consider this paradigm as the updated version of the binomial paradigm which means the approximation of binomial distribution is Poisson distribution. 

Under this paradigm, it is assumed that the events have independent features or they are dependent in a time interval manner like monthly dependent. The independent word means that any of the events that have occurred will not give any information about any upcoming event; both of them are not correlated. In some cases, we also consider this paradigm as an approximation of a binomial to a Poisson distribution. Let’s go deeper into this concept.


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What is the Poisson process?

The Poisson process can be considered as a counting process where the whole process gives the results as the counting of occurrences of a certain event that has a random structure and that has the probability of happening at a certain rate. We can understand this by taking the example of earthquakes in a certain area where the frequency of earthquakes is 3 per year but the timings of earthquakes are completely random. In such situations, the Poisson process can be a better fit model for us. 

We can apply the Poisson process if the situation is fulfilling the following criteria

  • All the different events in the situation or scenario are independent of each other.
  • The rate of occurrence is constant which means the number of occurring events in a time interval should be a constant number.
  • There should not be the occurrence of two events at the same time.    

One of the major things to notice here is that the events can be compared to the Bernoulli trials which means they are asynchronous or discrete that is events are either success or failure. For the above-given example, the interval we are choosing is 1 year but the sub-interval is the time when the earthquake is activated or deactivated. We can visualize the Poisson process in the following way.

Image source 

In the above image, we can see that the Poisson process is starting at 0 and some increments occur continuously but independently and the rate is λ.

What is a Poisson distribution?

In the above, we have seen that the Poisson process is a model that can be utilized to describe the occurrence of random events, and this model works mainly based on the theory of Poisson distribution. So it becomes a necessity for us to understand the Poisson distribution. Talking in mathematical terms we can consider this distribution as the discrete probability distribution that helps in representing the probability of occurring events in a fixed time distance, area, or volume interval where the rate of occurrence is constant and independent of other events.   

The probability mass function under this distribution can be given as follows:

Whare, 

  • X = random variable have Poisson distribution 
  • k = number of occurrence 
  • e = Euler’s constant      
  • 𝜆 = expected value of random variable or variance of random variable

= E(X) = Var(X)

If the constant value of occurrences of event is not given then  we can adapt the following equation 

Where r is the average rate of occurrence of events. The below image is a representation of the probability mass function of the Poisson distribution.

Image source

In the image, k represents the number of occurrences, and P(x = k) is the probability of k occurrences when the value of 𝜆 is given.  We can utilize Poisson distribution to model the following example events.

  • The number of earthquakes happening in a year in a certain area. 
  • The number of calls coming into a call centre in a certain interval of time.
  • The number of buses coming into the station in a certain interval of time.

Here we can now understand what are the situations that can be modeled using the Poisson process. Let’s see how we can implement the Poisson distribution using the python language.

Implementation of Poisson distribution

For the implementation of Poisson distribution, we can utilize the functions from the scipy library. Let’s see how we can do that 

from scipy.stats import poisson
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
mu = 0.6
mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')
x = np.arange(poisson.ppf(0.01, mu),
              poisson.ppf(0.99, mu))
ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf')
ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5)
rv = poisson(mu)
ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
        label='frozen pmf')
ax.legend(loc='best', frameon=False)
plt.show()

Output:

In the above example, we have generated 100 randomly generated samples, and then using the mu variable we have drawn the Poisson PMF  in the graph.  After implementation let’s see where we can need this theory in the data science journey.

The Poisson process uses cases in data science

There are various use cases of the Poisson process that can be found in the field of data science. Some of them are as follows:

  • We can easily understand that this mathematics part is very important in the data science field b because it is related to probability theory and such mathematics can be found in statistics operations such as parameter estimation, finding confidence intervals, and Bayesian inference.
  • We can also find its usage in the negative binomial regression because the sub-interval of the occurrence of events in Poisson can be compared to the trials of Bernoulli. 
  • It is also being used in various probabilistic models, for example, we can see in linear models we use a Poisson process to model the target variable’s distribution.
  • There are various problems in machine learning dependent on the response variable and mostly the response variable is a counting variable and can be modelled using the Poisson process. 
  • We can use this distribution in time series modeling to model the anomalies of time series.

Here we have seen the use cases of the passion process in real-life problems such as detecting whether,  number of calls on the phone, the number of times when a phone or laptop is being used, etc. 

Final words

In this article, we have understood the Poisson process in which we have discussed the paradigm and distribution of the passion process. Along with this, we have seen how we can implement and use it in different cases.   

References

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Yugesh Verma
Yugesh is a graduate in automobile engineering and worked as a data analyst intern. He completed several Data Science projects. He has a strong interest in Deep Learning and writing blogs on data science and machine learning.

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