An in-depth tutorial on principal component analysis (PCA) with mathematics and Python coding examples
This tutorial’s code is available on Github and its full implementation as well on Google Colab.
When implementing machine learning algorithms, the inclusion of more features might lead to worsening performance issues. Increasing the number of features will not always improve classification accuracy, which is also known as the curse of dimensionality. Hence, we apply dimensionality reduction to improve classification accuracy by selecting the optimal set of lower dimensionality features.
Principal component analysis (PCA) is essential for data science, machine learning, data visualization, statistics, and other quantitative fields.
Figure 1: Curse of dimensionality.
There are two techniques to make dimensionality reduction:
It is essential to know about vector, matrix, and transpose matrix, eigenvalues, eigenvectors, and others to understand the concept of dimensionality reduction.
Dimensionality in a dataset becomes a severe impediment to achieve a reasonable efficiency for most algorithms. Increasing the number of features does not always improve accuracy. When data does not have enough features, the model is likely to underfit, and when data has too many features, it is likely to overfit. Hence it is called the curse of dimensionality. The curse of dimensionality is an astonishing paradox for data scientists, based on the exploding amount of n-dimensional spaces — as the number of dimensions, n, increases.
The sparseness of data is the property of being scanty or scattered. It lacks denseness, and its high percentage of the variable’s cells do not contain actual data. Fundamentally full of “empty” or “N/A” values.
Points in an n-dimensional space frequently become sparse as the number of dimensions grows. The distance between points will extend to grow as the number of dimensions increases.
Figure 2: Data sparseness.
There are few implications of the curse of dimensionality:
Dimensionality reduction eliminates some features of the dataset and creates a restricted set of features that contains all of the information needed to predict the target variables more efficiently and accurately.
Reducing the number of features normally also reduces the output variability and complexity of the learning process. The covariance matrix is an important step in the dimensionality reduction process. It is a critical process to check the correlation between different features.
There is a concept of correlation in machine learning that is called multicollinearity. Multicollinearity exists when one or more independent variables highly correlate with each other. Multicollinearity makes variables highly correlated to one another, which makes the variables’ coefficients highly unstable .
The coefficient is a significant part of regression, and if this is unstable, then there will be a poor outcome of the regression result. Multicollinearity is confirmed by using Variance Inflation Factors (VIF). Therefore, if multicollinearity is suspected, it can be checked using the variance inflation factor (VIF).
Figure 3: VIF equation.
Rules from VIF:
Heatmap also plays a crucial role in understanding the correlation between variables.
The type of relationship between any two quantities varies over a period of time.
Correlation varies from **-1 **to +1
To be precise,
Below is the heatmap to show how we will correlate which features are highly dependent on the target feature and consider them.
The covariance matrix is the first step in dimensionality reduction because it gives an idea of the number of features that strongly relate, and it is usually the first step in dimensionality reduction because it gives an idea of the number of strongly related features so that those features can be discarded.
It also gives the detail of all independent features. It provides an idea of the correlation between all the different pairs of features.
Identification of features in Iris dataset that are strongly correlated
Import all the required packages:
import numpy as np import pandas as pd from sklearn import datasets import matplotlib.pyplot as plt
Load Iris dataset:
iris = datasets.load_iris() iris.data
Figure 4: Iris dataset.
List all features:
Figure 5: Features of the Iris dataset.
Create a covariance matrix:
cov_data = np.corrcoef(iris.data.T)cov_data
Figure 6: Covariance matrix of the Iris dataset.
Plot the covariance matrix to identify the correlation between features using a heatmap:
img = plt.matshow(cov_data, cmap=plt.cm.rainbow) plt.colorbar(img, ticks = [-1, 0, 1], fraction=0.045)for x in range(cov_data.shape): for y in range(cov_data.shape): plt.text(x, y, "%0.2f" % cov_data[x,y], size=12, color='black', ha="center", va="center") plt.show()
Figure 7: Heatmap of the correlation matrix.
A correlation from the representation of the heatmap:
Here the correlation matrix and its pictorial representation have given the idea about the potential number of features reduction. Therefore, two features can be kept, and other features can be reduced apart from those two features.
There are two ways of dimensionality reduction:
Dimensionality Reduction can ignore the components of lesser significance.
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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.
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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.
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.
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.
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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Python is awesome, it’s one of the easiest languages with simple and intuitive syntax but wait, have you ever thought that there might ways to write your python code simpler?
In this tutorial, you’re going to learn a variety of Python tricks that you can use to write your Python code in a more readable and efficient way like a pro.
Swapping value in Python
Instead of creating a temporary variable to hold the value of the one while swapping, you can do this instead
>>> FirstName = "kalebu" >>> LastName = "Jordan" >>> FirstName, LastName = LastName, FirstName >>> print(FirstName, LastName) ('Jordan', 'kalebu')
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Today you’re going to learn how to use Python programming in a way that can ultimately save a lot of space on your drive by removing all the duplicates.
In many situations you may find yourself having duplicates files on your disk and but when it comes to tracking and checking them manually it can tedious.
Heres a solution
Instead of tracking throughout your disk to see if there is a duplicate, you can automate the process using coding, by writing a program to recursively track through the disk and remove all the found duplicates and that’s what this article is about.
But How do we do it?
If we were to read the whole file and then compare it to the rest of the files recursively through the given directory it will take a very long time, then how do we do it?
The answer is hashing, with hashing can generate a given string of letters and numbers which act as the identity of a given file and if we find any other file with the same identity we gonna delete it.
There’s a variety of hashing algorithms out there such as
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Magic Methods are the special methods which gives us the ability to access built in syntactical features such as ‘<’, ‘>’, ‘==’, ‘+’ etc…
You must have worked with such methods without knowing them to be as magic methods. Magic methods can be identified with their names which start with __ and ends with __ like init, call, str etc. These methods are also called Dunder Methods, because of their name starting and ending with Double Underscore (Dunder).
Now there are a number of such special methods, which you might have come across too, in Python. We will just be taking an example of a few of them to understand how they work and how we can use them.
class AnyClass: def __init__(): print("Init called on its own") obj = AnyClass()
The first example is _init, _and as the name suggests, it is used for initializing objects. Init method is called on its own, ie. whenever an object is created for the class, the init method is called on its own.
The output of the above code will be given below. Note how we did not call the init method and it got invoked as we created an object for class AnyClass.
Init called on its own
Let’s move to some other example, add gives us the ability to access the built in syntax feature of the character +. Let’s see how,
class AnyClass: def __init__(self, var): self.some_var = var def __add__(self, other_obj): print("Calling the add method") return self.some_var + other_obj.some_var obj1 = AnyClass(5) obj2 = AnyClass(6) obj1 + obj2
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At the end of 2019, Python is one of the fastest-growing programming languages. More than 10% of developers have opted for Python development.
In the programming world, Data types play an important role. Each Variable is stored in different data types and responsible for various functions. Python had two different objects, and They are mutable and immutable objects.
Table of Contents hide
III Built-in data types in Python
The Size and declared value and its sequence of the object can able to be modified called mutable objects.
Mutable Data Types are list, dict, set, byte array
The Size and declared value and its sequence of the object can able to be modified.
Immutable data types are int, float, complex, String, tuples, bytes, and frozen sets.
id() and type() is used to know the Identity and data type of the object
a**=str(“Hello python world”)****#str**
Numbers are stored in numeric Types. when a number is assigned to a variable, Python creates Number objects.
Python supports 3 types of numeric data.
int (signed integers like 20, 2, 225, etc.)
float (float is used to store floating-point numbers like 9.8, 3.1444, 89.52, etc.)
complex (complex numbers like 8.94j, 4.0 + 7.3j, etc.)
A complex number contains an ordered pair, i.e., a + ib where a and b denote the real and imaginary parts respectively).
The string can be represented as the sequence of characters in the quotation marks. In python, to define strings we can use single, double, or triple quotes.
# String Handling
#single (') Quoted String
# Double (") Quoted String
# triple (‘’') (“”") Quoted String
In python, string handling is a straightforward task, and python provides various built-in functions and operators for representing strings.
The operator “+” is used to concatenate strings and “*” is used to repeat the string.
'Output : Python python ’
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