Category: Machine learning

Machine Learning Basics: A Comprehensive Introduction for Beginners

Key focus: machine learning, introduction, basics, beginners, algorithms, applications, concepts

Introduction

Machine learning has emerged as a groundbreaking technology that is transforming industries and reshaping our interaction with technology. From personalized recommendations to autonomous vehicles, machine learning algorithms play a pivotal role in these advancements. If you’re new to the field, this comprehensive beginner’s guide will provide you with a solid introduction to machine learning, covering its fundamental concepts, practical applications, and key techniques.

Machine Learning Basics: A Comprehensive Introduction for Beginners

Understanding Machine Learning:

Machine learning is a subset of artificial intelligence (AI) focused on developing algorithms and models that can learn from data and make predictions or decisions without explicit programming. Instead of relying on fixed instructions, these algorithms extract patterns and insights from available data, enabling them to generalize and make accurate predictions on unseen examples.

The core machine learning algorithms are categorized into three types:

  • Supervised learning
  • Unsupervised learning
  • Reinforcement learning

Supervised Learning:

Supervised learning is a crucial branch of machine learning. In this approach, algorithms are trained on labeled datasets, where each example consists of input features and corresponding target labels. By learning from these labeled examples, algorithms can map inputs to correct outputs, identifying underlying patterns. Linear regression, decision trees, and support vector machines are common supervised learning algorithms.

Unsupervised Learning:

Unsupervised learning tackles unlabeled data. Its goal is to discover hidden patterns, structures, or relationships without prior knowledge of labels. Clustering and dimensionality reduction are prominent techniques within unsupervised learning. Clustering algorithms group similar data points, while dimensionality reduction methods aim to reduce feature dimensions while retaining relevant information.

Reinforcement Learning:

Reinforcement learning takes inspiration from how humans learn through trial and error. In this learning paradigm, an agent interacts with an environment and learns to maximize a reward signal by taking appropriate actions. Through repeated interactions, the agent explores the environment, receives feedback, and adjusts its actions to optimize its performance. Reinforcement learning has been successfully applied in areas such as robotics, gaming, and autonomous systems.

Key Steps in Machine Learning:

  1. Data Collection: Machine learning relies on quality data. Gathering relevant and representative data is a crucial initial step. It can come from various sources, including structured databases, APIs, or unstructured text and images.
  2. Data Preprocessing: Raw data often contains noise, missing values, or inconsistencies. Data preprocessing involves cleaning, transforming, and normalizing the data to ensure it is suitable for analysis and model training.
  3. Feature Engineering: Feature engineering involves selecting, extracting, or creating meaningful features from the available data. Good features can significantly impact the performance of a machine learning model.
  4. Model Training: This step involves feeding the prepared data into a machine learning algorithm to create a model. The algorithm learns from the data and adjusts its internal parameters to make accurate predictions or decisions.
  5. Model Evaluation: Evaluating the performance of a trained model is essential to assess its accuracy and generalization capabilities. Various metrics, such as accuracy, precision, recall, and F1 score, are used to measure the model’s performance.
  6. Model Deployment and Monitoring: Once the model is deemed satisfactory, it can be deployed in real-world applications. Continuous monitoring is crucial to ensure the model’s performance remains optimal and to address any issues that may arise.

Business use cases:

Businesses are increasingly leveraging machine learning to gain a competitive edge, improve operational efficiency, and enhance decision-making processes. Here are some common ways in which businesses are using machine learning:

  • Customer Insights and Personalization: Machine learning enables businesses to analyze customer data, such as purchase history, browsing behavior, and demographic information, to gain valuable insights. This information can be used to personalize marketing campaigns, recommend relevant products or services, and improve customer experiences.
  • Fraud Detection and Risk Management: Machine learning algorithms can identify patterns and anomalies in large volumes of transactional data, helping businesses detect fraudulent activities and mitigate risks. These algorithms learn from historical data to spot fraudulent patterns and predict potential risks, enabling proactive measures to safeguard businesses and their customers.
  • Demand Forecasting and Inventory Management: By analyzing historical sales data, market trends, and external factors, machine learning algorithms can predict future demand for products or services. This helps businesses optimize inventory levels, minimize stock-outs, reduce costs, and improve overall supply chain management.
  • Predictive Maintenance: Machine learning models can analyze sensor data from machinery and equipment to detect patterns indicating potential failures or maintenance needs. By identifying issues before they occur, businesses can schedule maintenance proactively, minimize downtime, and optimize equipment performance.
  • Natural Language Processing (NLP) for Customer Support: NLP techniques powered by machine learning are employed in chatbots and virtual assistants to automate customer support processes. These systems can understand and respond to customer queries, provide relevant information, and assist with common issues, improving response times and enhancing customer satisfaction.
  • Sentiment Analysis and Social Media Monitoring: Machine learning algorithms can analyze social media data and other online sources to gauge public sentiment and monitor brand reputation. This information helps businesses understand customer opinions, identify emerging trends, and respond effectively to customer feedback.
  • Supply Chain Optimization: Machine learning algorithms optimize supply chain operations by analyzing data related to logistics, transportation, and inventory management. These models can identify bottlenecks, streamline routes, optimize scheduling, and reduce costs, ultimately improving the overall efficiency of the supply chain.
  • Credit Scoring and Risk Assessment: Financial institutions employ machine learning algorithms to assess creditworthiness, predict default probabilities, and automate the loan approval process. By analyzing a range of variables, such as credit history, income, and demographics, these algorithms provide more accurate risk assessments and streamline lending processes.
  • Image and Speech Recognition: Machine learning models have advanced image and speech recognition capabilities. Businesses can leverage these technologies for various applications, such as facial recognition for security purposes, automatic image tagging, voice-controlled virtual assistants, and automated document analysis.
  • Data Analytics and Business Intelligence: Machine learning algorithms assist in analyzing large volumes of data to extract insights, identify patterns, and make data-driven decisions. By leveraging machine learning techniques, businesses can uncover hidden trends, gain a deeper understanding of their operations, and drive informed strategies.

These are just a few examples of how businesses are utilizing machine learning to improve their operations and decision-making processes. As machine learning continues to evolve, its applications across various industries and business functions are expected to expand, unlocking even greater opportunities for organizations.

Products/Services using machine learning

Machine learning has been integrated into a wide range of products and services across various industries. Here is a list of products that utilize machine learning:

  • Virtual Assistants: Virtual assistants like Amazon Alexa, Google Assistant, and Apple Siri use machine learning to understand and respond to user queries, perform tasks, and provide personalized recommendations.
  • Recommendation Systems: Platforms such as Netflix, Amazon, and Spotify leverage machine learning to analyze user preferences and behavior, providing personalized recommendations for movies, products, and music.
  • Fraud Detection Systems: Financial institutions and online payment processors employ machine learning algorithms to detect and prevent fraudulent activities by analyzing patterns and anomalies in transactions and user behavior.
  • Autonomous Vehicles: Self-driving cars and autonomous vehicles rely on machine learning algorithms to perceive and interpret the environment, make real-time decisions, and navigate safely on the roads.
  • Image and Speech Recognition: Products like Google Photos, Facebook’s automatic photo tagging, and voice assistants utilize machine learning algorithms for image and speech recognition tasks, enabling features such as automatic tagging and voice-controlled interactions.
  • Language Translation: Machine learning plays a significant role in language translation tools like Google Translate and Microsoft Translator, enabling accurate and automated translation between different languages.
  • Social Media News Feed Ranking: Social media platforms like Facebook, Twitter, and Instagram employ machine learning algorithms to rank and personalize users’ news feeds, showing relevant content based on their interests and preferences.
  • Customer Service Chatbots: Many companies use machine learning-powered chatbots to provide automated customer support, answer common queries, and assist with basic tasks without the need for human intervention.
  • Email Filtering: Email service providers such as Gmail utilize machine learning algorithms to automatically filter and categorize incoming emails, separating spam from legitimate messages and prioritizing important emails.
  • Medical Diagnosis Systems: Machine learning is applied in medical diagnosis systems to analyze patient data, medical images, and electronic health records, aiding in accurate disease diagnosis and treatment planning.
  • Smart Home Devices: Smart home devices like smart thermostats, security systems, and voice-controlled assistants incorporate machine learning to learn user preferences, automate tasks, and optimize energy consumption.
  • E-commerce Product Search and Recommendations: E-commerce platforms like Amazon and eBay employ machine learning to enhance product search capabilities, provide personalized recommendations, and optimize product listings.
  • Predictive Maintenance Systems: Industrial equipment and machinery are monitored using machine learning algorithms to predict maintenance needs, detect anomalies, and minimize downtime through proactive maintenance.
  • Financial Trading Systems: Machine learning algorithms are utilized in financial trading systems to analyze market data, identify patterns, and make automated trading decisions.
  • Online Advertising: Platforms such as Google Ads and Facebook Ads leverage machine learning to optimize ad targeting, personalize advertisements, and improve campaign performance.

These are just a few examples of the many products and services that incorporate machine learning to provide enhanced functionalities, intelligent automation, and personalized experiences across various industries.

Implementing Markov Chain in Python

Keywords: Markov Chain, Python, probability, data analysis, data science

Markov Chain

Markov chain is a probabilistic models that describe a sequence of observations whose occurrence are statistically dependent only on the previous ones. This article is about implementing Markov chain in Python

Markov chain is described in one of the earlier posts. For better understanding of the concept, review the post before proceeding further.

We will model a car’s behavior using the same transition matrix and starting probabilities described in the earlier post for modeling the corresponding Markov chain model (refer Figure 1). The matrix defines the probabilities of transitioning between different states, including accelerating, maintaining a constant speed, idling, and braking.

Figure 1: Modeling a car’s behavior using Markov chain model

The starting probabilities indicate that the car starts in the break state with probability 1, which means it is already stopped and not moving.

Python implementation

Here’s the sample code in Python that implements the above model:

import random

# Define a transition matrix for the Markov chain
transition_matrix = {
    'accelerate': {'accelerate': 0.3, 'constant speed': 0.2, 'idling': 0 , 'break': 0.5 },
    'constant speed': {'accelerate': 0.1, 'constant speed': 0.4, 'idling': 0 , 'break': 0.5 },
    'idling': {'accelerate': 0.8, 'constant speed': 0, 'idling': 0.2 , 'break': 0 },
    'break': {'accelerate': 0.4, 'constant speed': 0.05, 'idling': 0.5 , 'break': 0.05 },
}

# Define starting probabilities for each state
starting_probabilities = {'accelerate': 0, 'constant speed': 0, 'idling': 0, 'break': 1}

# Choose the starting state randomly based on the starting probabilities
current_state = random.choices(
    population=list(starting_probabilities.keys()),
    weights=list(starting_probabilities.values())
)[0]

# Generate a sequence of states using the transition matrix
num_iterations = 10
for i in range(num_iterations):
    print(current_state)
    next_state = random.choices(
        population=list(transition_matrix[current_state].keys()),
        weights=list(transition_matrix[current_state].values())
    )[0]
    current_state = next_state

In this example, we use the random.choices() function to choose the starting state randomly based on the starting probabilities. We then generate a sequence of 10 states using the transition matrix, and print out the sequence of states as they are generated. A sample output of the program is given below.

>>> exec(open('markov_chain.py').read()) #Python 3 syntax
break
idling
accelerate
break
accelerate
break
accelerate
constant speed
break
accelerate

The Most Important Topics to Learn in Machine Learning

Keywords: machine learning, topics, probability, statistics, linear algebra, data preprocessing, supervised learning, unsupervised learning, deep learning, reinforcement learning, model evaluation, cross-validation, hyperparameter tuning.

Why the buzz ?

Machine learning has been generating a lot of buzz in recent years due to its ability to automate tasks that were previously thought to be impossible or required human-level intelligence. Here are some reasons why there is so much buzz in machine learning:

  1. Improved Data Processing: Machine learning algorithms can process vast amounts of data quickly and accurately. With the advent of big data, there is now more data available than ever before, and machine learning algorithms can analyze this data to extract meaningful insights.
  2. Automation: Machine learning can automate tasks that were previously done by humans, such as image recognition, natural language processing, and even decision making. This has the potential to increase efficiency and reduce costs in many industries.
  3. Personalization: Machine learning can be used to personalize experiences for users. For example, recommendation systems can use machine learning algorithms to suggest products or services that are relevant to a user’s interests.
  4. Predictive Analytics: Machine learning can be used to make predictions about future events based on historical data. This is particularly useful in industries like finance, healthcare, and marketing.
  5. Advancements in Technology: Advancements in technology have made it easier to collect and store data, which has made it possible to train more complex machine learning models. Additionally, the availability of cloud computing has made it easier for companies to implement machine learning solutions.

Overall, the buzz in machine learning is due to its ability to automate tasks, process vast amounts of data, and make predictions about future events. As machine learning continues to evolve, it has the potential to transform many industries and change the way we live and work.

The most important topics to learn in machine learning

There are several important topics to learn in machine learning that are crucial for building effective machine learning models. Here are some of the most important topics to learn:

  1. Probability and Statistics: Probability and statistics are the foundation of machine learning. It is important to have a solid understanding of concepts like probability distributions, statistical inference, hypothesis testing, and Bayesian methods.
  2. Linear Algebra: Linear algebra is used extensively in machine learning algorithms, especially in deep learning. Topics like matrices, vectors, eigenvectors, and eigenvalues are important to understand.
  3. Data Preprocessing: Data preprocessing is the process of cleaning and transforming raw data into a format that can be used by machine learning algorithms. It includes tasks like feature scaling, feature selection, data normalization, and data augmentation.
  4. Supervised Learning: Supervised learning is a type of machine learning where the model learns from labeled data to make predictions or classifications on new, unseen data. This includes topics like regression, classification, decision trees, and support vector machines.
  5. Unsupervised Learning: Unsupervised learning is a type of machine learning where the model learns from unlabeled data to discover patterns and relationships in the data. This includes topics like clustering, dimensionality reduction, and anomaly detection.
  6. Deep Learning: Deep learning is a subset of machine learning that involves training artificial neural networks with multiple layers. It is used for tasks like image recognition, natural language processing, and speech recognition.
  7. Reinforcement Learning: Reinforcement learning is a type of machine learning where an agent learns to take actions in an environment to maximize a reward signal. It is used for tasks like game playing, robotics, and autonomous driving.
  8. Model Evaluation and Selection: Model evaluation and selection is the process of selecting the best machine learning model for a given task. It includes topics like cross-validation, bias-variance tradeoff, and hyperparameter tuning.

Overall, these are some of the most important topics to learn in machine learning. However, it is important to note that the field of machine learning is constantly evolving, and there may be new topics and techniques to learn in the future.

Hidden Markov Models (HMM) – Simplified !!!

Markov chains are useful in computing the probability of events that are observable. However, in many real world applications, the events that we are interested in are usually hidden, that is we don’t observe them directly. These hidden events need to be inferred. For example, given a sentence in a natural language we only observe the words and characters directly. The parts-of-speech from the sentence are hidden, they have to be inferred. This bring us to the following topic– the hidden Markov models.

Hidden Markov models enables us to visualize both observations and the associated hidden events. Let’s consider an example for understanding the concept.

The cheating casino and the gullible gambler

Consider a dishonest casino that deceives it player by using two types of die : a fair die (F) and a loaded die (L). For a fair die, each of the faces has the same probability of landing facing up. For the loaded die, the probabilities of the faces are skewed as given next

When the gambler throws the die, numbers land facing up. These are our observations at a given time t (denoted as Ot = {1,2,3,4,5,6}). At any given time t, whether these number are rolled from a fair die (state St = F) or a loaded die (St = L), is unknown to an observer and therefore they are the hidden events.

Emission probabilities

The probabilities associated with the observations are the observation likelihoods, also called emission probabilities (B).

Initial probabilities

The initial probability of starting (at time = 0) with any of fair die or loaded die (hidden event) is 50%.

Transition probabilities

A gullible gambler switches from the fair die to loaded die with 10% probability. He switches back from loaded die to fair die with 5% probability.

The probabilities of transitioning from one hidden event to another is described by the transition probability matrix (A). The elements of the probability transition matrix, are the transition probabilities (pij) of moving from one hidden state i to another hidden state j.

The transition probabilities from time t-1 to t, for the hidden events are

Therefore, the transition probability matrix is

Based on the given information so far, a probability model is constructed. This is the Hidden Markov Model (HMM) for the given problem.

Figure 1: Hidden Markov Model for the cheating Casino problem

Assumptions

We saw, in previous article, that the Markov models come with assumptions. Similarly, HMMs models also have such assumptions.

1. Assumption on probability of hidden states

In the model given here, the probability of a given hidden state depends only on the previous hidden state. This is a typical first order Markov chain assumption.

2. Assumption on Output

The probability of any observation (output) depends on the hidden state that produce it and not on any other hidden state or output observations.

Problems and Algorithms

Let’s briefly discuss the different problems and the related algorithms for HMMs. The algorithms will be explained in detail in the future articles.

In the dishonest casino, the gambler rolls the following numbers:

Figure 2: Sample Observations

1. Evaluation

Given the model of the dishonest casino, what is the probability of obtaining the above sequence ? This is a typical evaluation problem in HMMs. Forward algorithm is applied for such evaluation problems.

2. Decoding

What is the most likely sequence of die (hidden states) given the above sequence ? Such problems are addressed by Viterbi decoding.

What is the probability of fourth die being loaded, given the above sequence ? Forward-backward algorithm to our rescue.

3. Learning

Learning problems involve parametrization of the model. In learning problems, we attempt to find the various parameters (transition probabilities, emission probabilities) of the HMM, given the observation. Baum-Welch algorithm helps us to find the unknown parameters of a HMM.

Some real-life examples

Here are some real-life examples of HMM applications:

  1. Speech recognition: HMMs are widely used in speech recognition systems to model the variability of speech sounds. In this application, the observable events are the acoustic features of the speech signal, while the hidden states represent the phonemes or words that generate the speech signal.
  2. Handwriting recognition: HMMs can be used to recognize handwritten characters by modeling the temporal variability of the pen strokes. In this application, the observable events are the coordinates of the pen on the writing surface, while the hidden states represent the letters or symbols that generate the handwriting.
  3. Stock price prediction: HMMs can be used to model the behavior of stock prices and predict future price movements. In this application, the observable events are the daily price movements, while the hidden states represent the different market conditions that generate the price movements.
  4. Gene prediction: HMMs can be used to identify genes in DNA sequences. In this application, the observable events are the nucleotides in the DNA sequence, while the hidden states represent the different regions of the genome that generate the sequence.
  5. Natural language processing: HMMs are used in many natural language processing tasks, such as part-of-speech tagging and named entity recognition. In these applications, the observable events are the words in the text, while the hidden states represent the grammatical structures or semantic categories that generate the text.
  6. Image and video analysis: HMMs can be used to analyze images and videos, such as for object recognition and tracking. In this application, the observable events are the pixel values in the image or video, while the hidden states represent the object or motion that generates the pixel values.
  7. Bio-signal analysis: HMMs can be used to analyze physiological signals, such as electroencephalograms (EEGs) and electrocardiograms (ECGs). In this application, the observable events are the signal measurements, while the hidden states represent the physiological states that generate the signal.
  8. Radar signal processing: HMMs can be used to process radar signals and detect targets in noisy environments. In this application, the observable events are the radar measurements, while the hidden states represent the presence or absence of targets.

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 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
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 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
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 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
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 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
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● Choosing a filter : FIR or IIR : understanding the design perspective
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Markov Chains – Simplified !!

Key focus: Markov chains are a probabilistic models that describe a sequence of observations whose occurrence are statistically dependent only on the previous ones.

● Time-series data like speech, stock price movements.
● Words in a sentence.
● Base pairs on the rung of a DNA ladder.

States and transitions

Assume that we want to model the behavior of a driver behind the wheel. The possible behaviors are

● accelerate (state 1)
● constant speed (state 2)
● idling (engine running slowly but the vehicle is not moving – (state 3))
● brake (state 4)

Let’s refer each of these behaviors as a state. In the given example, there are N=4 states, refer them as Q = {q1,q2,q3,q4}.

We observe the following pattern in the driver’s behavior (Figure 1). That is, the driver operates the vehicle through a certain sequence of states. In the graph shown in Figure 1, the states are represented as nodes and the transitions as edges.

Figure 1: Driver’s behavior – operating the vehicle through a sequence of states

We see that, sometimes, the driver changes the state of the vehicle from one state to another and sometimes stays in the same state (as indicated by the arrows).

We also note that either the vehicle stays in the same state or changes to the next state. Therefore, from this model, if we want to predict the future state, all that matters is the current state of the vehicle. The past states has no bearing on the future state except through the current state. Take note of this important assumption for now.

Probabilistic model

We know that we cannot be certain about the driver’s behavior at any given point in time. Therefore, to model this uncertainty, the model is turned into a probabilistic model. A probabilistic model allows us to account for the likelihood of the behaviors or change of states.

An example for a probabilistic model for the given problem is given in Figure 2.

Figure 2: Driver’s behavior – a probabilistic model (transition matrix shown)

In this probabilistic model, we have assigned probability values to the transitions.These probabilities are collectively called transition probabilities. For example, considering the state named “idling”, the probability of the car to transition from this state to the next state (accelerate) is 0.8. In probability mathematics this is expressed as a conditional probability conditioned on the previous state.

p(state = “accelerate” | previous state = “idling” ) = 0.8

Usually, the transition probabilities are formulated in the form of matrix called transition probability matrix. The transition probability matrix is shown in Figure 2. In a transition matrix, denoted as A, each element aij represent the probability of transitioning from state i to state j. The elements of the transition matrix satisfy the following property.

That is, the sum of transition probabilities leaving any given state is 1.

As we know, in this example, the driver cannot start car in any state (example, it is impossible to start the car in “constant speed” state). He can only start the car from at rest (i.e, brake state). To model this uncertainty, we introduce πi – the probability that the Markov chain starts in a given state i. The set of starting probabilities for all the N states are called initial probability distribution (π = π1, π2, …, πN). In Figure 3, the starting probabilities are denoted by green arrows.

Figure 3: Markov Chain model for driver’s behavior

Markov Assumption

As noted in the definition, the Markov chain in this example, assumes that the occurrence of each event/observation is statistically dependent only on the previous one. This is a first order Markov chain (or termed as bigram language model in natural language processing application). For the states Q = {q1, …, qn}, predicting the probability of a future state depends only on the current observation, all other previous observations do not matter. In probabilistic terms, this first order Markov chain assumption is denoted as

Extending the assumption for mth order Markov chain, predicting the probability of a future observation depends only on the previous m observations. This is an m-gram model.

Given a set of n arbitrary random variables/observations Q = {q1, …, qn}, their joint probability distribution is usually computed by applying the following chain rule.

However, if the random observations in Q are of sequential in nature and follows the generic mth order Markov chain model, then the computation of joint probability gets simplified.

The Markov assumptions for first and second order of Markov models are summarized in Figure 4.Figure 4: Assumptions for 1st order and 2nd order Markov chains

Hidden Markov Model (HMM)

Markov chains are useful in computing the probability of events that are observable. However, in many real world applications, the events that we are interested in are usually hidden, that is we don’t observe them directly. These hidden events need to be inferred. For example, given a sentence in a natural language we only observe the words and characters directly. The parts-of-speech from the sentence are hidden, they have to be inferred. This brings us to the next topic of discussion – the hidden Markov models.

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Second Edition(PDF)
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(PDF ebook)
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(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
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Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Similar topics

Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

Linear regression using python – demystified

Key focus: Let’s demonstrate basics of univariate linear regression using Python SciPy functions. Train the model and use it for predictions.

Linear regression model

Regression is a framework for fitting models to data. At a fundamental level, a linear regression model assumes linear relationship between input variables () and the output variable (). The input variables are often referred as independent variables, features or predictors. The output is often referred as dependent variable, target, observed variable or response variable.

If there are only one input variable and one output variable in the given dataset, this is the simplest configuration for coming up with a regression model and the regression is termed as univariate regression. Multivariate regression extends the concept to include more than one independent variables and/or dependent variables.

Univariate regression example

Let us start by considering the following example of a fictitious dataset. To begin we construct the fictitious dataset by our selves and use it to understand the problem of linear regression which is a supervised machine learning technique. Let’s consider linear looking randomly generated data samples.

import numpy as np
import matplotlib.pyplot as plt #for plotting

np.random.seed(0) #to generate predictable random numbers

m = 100 #number of samples
x = np.random.rand(m,1) #uniformly distributed random numbers
theta_0 = 50 #intercept
theta_1 = 35 #coefficient
noise_sigma = 3

noise = noise_sigma*np.random.randn(m,1) #gaussian random noise

y = theta_0 + theta_1*x + noise #noise added target
 
plt.ion() #interactive plot on
fig,ax = plt.subplots(nrows=1,ncols=1)
plt.plot(x,y,'.',label='training data')
plt.xlabel(r'Feature $x_1$');plt.ylabel(r'Target $y$')
plt.title('Feature vs. Target')
Figure 1: Simulated data for linear regression problem

In this example, the data samples represent the feature and the corresponding targets . Given this dataset, how can we predict target as a function of ? This is a typical regression problem.

Linear regression

Let be the pair that forms one training example (one point on the plot above). Assuming there are such sample points as training examples, then the set contains all the pairs .

In the univariate linear regression problem, we seek to approximate the target as a linear function of the input , which implies the equation of a straight line (example in Figure 2) as given by

where, is the intercept, is the slope of the straight line that is sought and is always . The approximated target serves as a guideline for prediction. The approximated target is denoted by

Using all the samples from the training set , we wish to find the parameters that well approximates the relationship between the given target samples and the straight line function .

If we represent the variables s, the input samples for and the target samples as matrices, then, equation (1) can be expressed as a dot product between the two sequences

It may seem that the solution for finding is straight forward

However, matrix inversion is not defined for matrices that are not square. Moore-Penrose pseudo inverse generalizes the concept of matrix inversion to a matrix. Denoting the Moore-Penrose pseudo inverse for as , the solution for finding is

For coding in Python, we utilize the scipy.linalg.pinv function to compute Moore-Penrose pseudo inverse and estimate .

xMat = np.c_[ np.ones([len(x),1]), x ] #form x matrix
from scipy.linalg import pinv
theta_estimate = pinv(xMat).dot(y)
print(f'theta_0 estimate: {theta_estimate[0]}')
print(f'theta_1 estimate: {theta_estimate[1]}')

The code results in the following estimates for , which are very close to the values used to generate the random data points for this problem.

>> theta_0 estimate: [50.66645323]
>> theta_1 estimate: [34.81080506]

Now, we know the parameters of our example system, the target predictions for new values of feature can be done as follows

x_new = np.array([[-0.2],[0.5],[1.2] ]) #new unseen inputs
x_newmat = np.c_[ np.ones([len(x_new),1]), x_new ] #form xNew matrix
y_predict  = np.dot(x_newmat,theta_estimate)
>>> y_predict #predicted y values for new inputs for x_1
array([[43.70429222],
       [68.07185576],
       [92.43941931]])

The approximated target as a linear function of feature, is plotted as a straight line.

plt.plot(x_new,y_predict,'-',label='prediction')
plt.text(0.7, 55, r'Intercept $\theta_0$ = %0.2f'%theta_estimate[0])
plt.text(0.7, 50, r'Coefficient $\theta_1$ = %0.2f'%theta_estimate[1])
plt.text(0.5, 45, r'y= $\theta_0+ \theta_1 x_1$ = %0.2f + %0.2f $x_1$'%(theta_estimate[0],theta_estimate[1]))
plt.legend() #plot legend
Figure 2: Linear Regression – training samples and prediction

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References

[1] Boyd and Vandenberghe , “Convex Optimization”, ISBN: 978-0521833783, Cambridge University Press, 1 edition, March 2004.↗

Related topics

[1] Introduction to Signal Processing for Machine Learning
[2] Generating simulated dataset for regression problems - sklearn make_regression
[3] Hands-on: Basics of linear regression

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Generating simulated dataset for regression problems

Key focus: Generating simulated dataset for regression problems using sklearn make_regression function (Python 3) is discussed in this article.

Problem statement

Suppose, a survey is conducted among the employees of a company. In that survey, the salary and the years of experience of the employees are collected. The aim of this data collection is to build a regression model that could predict the salary from the given experience (especially for the values not seen by the model).

If you are developer, you often have no access to survey data. In this scenario, you wish you could simulate the data for building a regression model.

Generating the dataset

To construct a simulated dataset for this scenario, the sklearn.dataset.make_regression↗ function available in the scikit-learn library can be used. The function generates the samples for a random regression problem.

The make_regression↗ function generates samples for inputs (features) and output (target) by applying random linear regression model. The values for generated samples have to be scaled to appropriate range for the given problem.

import numpy as np
from sklearn import datasets
import matplotlib.pyplot as plt #for plotting

x, y, coef = datasets.make_regression(n_samples=100,#number of samples
                                      n_features=1,#number of features
                                      n_informative=1,#number of useful features 
                                      noise=10,#bias and standard deviation of the guassian noise
                                      coef=True,#true coefficient used to generated the data
                                      random_state=0) #set for same data points for each run

# Scale feature x (years of experience) to range 0..20
x = np.interp(x, (x.min(), x.max()), (0, 20))

# Scale target y (salary) to range 20000..150000 
y = np.interp(y, (y.min(), y.max()), (20000, 150000))

plt.ion() #interactive plot on
plt.plot(x,y,'.',label='training data')
plt.xlabel('Years of experience');plt.ylabel('Salary $')
plt.title('Experience Vs. Salary')
Figure 1: Simulated dataset for linear regression problem

If you want the data to be presented in pandas dataframe format:

import pandas as pd
df = pd.DataFrame(data={'experience':x.flatten(),'salary':y})
df.head(10)
Figure 2: Generated dataset presented as pandas dataframe

We have successfully completed generating simulated dataset for regression problems in Python3. Let’s move on to build and train a linear regression model using the generated dataset and use it for predictions.

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Related topics

[1] Introduction to Signal Processing for Machine Learning
[2] Generating simulated dataset for regression problems - sklearn make_regression
[3] Hands-on: Basics of linear regression

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Introduction to Signal Processing for Machine Learning

Key focus: Fundamentals of signal processing for machine learning. Speaker identification is taken as an example for introducing supervised learning concepts.

Signal Processing

A signal, mathematically a function, is a mechanism for conveying information. Audio, image, electrocardiograph (ECG) signal, radar signals, stock price movements, electrical current/voltages etc.., are some of the examples.

Signal processing is an engineering discipline that focuses on synthesizing, analyzing and modifying such signals. Some of the applications of signal processing are

● Converting one signal to another – filtering, decomposition, denoising
● Information extraction and interpretation – computer vision, speech recognition, Iris recognition, finger print recognition
● Error control and source coding – low density parity codes (LDPC), turbo coding, linear prediction coding, JPG, PNG
● Detection – SONAR, RADAR

Machine Learning (ML)

Machine learning is a science that deals with the development of algorithms that learn from data. According to Arthur Samuel (1959)[1] machine learning is a “Field of study that gives computers the ability to learn without being explicitly programmed”. Kevin Murphy, in his seminal book [2], defines machine learning as a collection of algorithms that automatically detect patterns in data that use the uncovered patterns to predict future data or other outcomes of interest.

Essentially, a machine learning algorithm may learn from data to
● learn from data to recognize patterns – example: recognizing text patterns in a set of spam emails
● classify data into different categories – example: classifying the emails into spam or non-spam emails.
● predict a future outcome – example: predicting whether the incoming email is spam or not

Machine learning algorithms are divided into three main types
● Supervised learning – a predictive learning approach where the goal is to learn from a labeled set of input-output pairs. The labeled set provides the training examples for further classification or prediction. In machine learning jargon, inputs are called ‘features’ and outputs are called ‘response variables’.
● Unsupervised learning – A kind of less well defined knowledge discovery process, the goal is to learn structured patterns in the data by separating them from pure unstructured noise
● Reinforced learning – is learning by interacting with an environment in order to make decision making tasks

Based on the discussion so far, we can start to recognize how the synergy between the fields of signal processing and machine learning can provide a new perspective to approach many problems.

Speaker identification – an application of ML algorithms in signal processing

Speaker identification (Figure 1) is the identification of a person from the analysis of voice characteristics. In this supervised classification application, a labeled training set of voice samples (from a set of speakers) are used in the learning process.

Figure 1: Speaker recognition using machine learning and signal processing

Voice samples/recordings cannot be used as such in the learning process. For further processing, it may require sampling, cleaning (removal of noise or invalid samples etc..,) or re-formatting the samples to suitable format. This step is called ‘data pre-processing‘.

Also, we may have to transform the data specific to the ML algorithm and the knowledge of the problem. To train the ML model recognize the patterns in the voice samples, feature extraction on voice samples is performed using signal processing. In this case, the features that are used to train the ML model are pitch and Mel-Frequency Cepstrum Coefficients (MFCC) [3] extracted from the voice samples.

Generally, the available dataset (set of input voice samples) is split into two sets: one set for training the model and the other set for testing needs (typically in 75%-25% ratio). The training set is used to train the ML model and the test set is used to evaluate the effectiveness and performance of the ML algorithm.

The training process should attempt to generalize the underlying relationship between the feature vectors (input to the supervised learning algorithm) and the class labels (supervised learner’s output). Cross-validation is one of the verification technique for evaluating the generalization ability of the ML model.

The training process should also avoid overfitting, which may cause poor generalization and erroneous classification in the execution phase. If the performance of the algorithm needs improvement, we need to go back and make changes to the previous steps. Metrics such as accuracy, recall, confusion matrix are typically used to evaluate the effectiveness and performance of the ML algorithm.

After the ML model is adequately trained to provide satisfying performance, we move on to the execution phase. In the execution phase, when an unlabeled instance of an voice sample is presented to the trained classifier, it identifies the person to which it belongs to.

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References

[1] Samuel, Arthur L. “Some Studies in Machine Learning Using the Game of Checkers,” IBM Journal of Research and Development 44:1.2 (1959): 210–229.↗
[2] Kevin P. Murphy, “Machine Learning – A Probabilistic Perspective”, ISBN 978-0262018029, The MIT Press, Cambridge, UK.↗
[3] P. M. Chauhan and N. P. Desai, “Mel Frequency Cepstral Coefficients (MFCC) based speaker identification in noisy environment using wiener filter,” 2014 International Conference on Green Computing Communication and Electrical Engineering (ICGCCEE), Coimbatore, 2014, pp. 1-5.↗

Articles in this series

[1] Introduction to Signal Processing for Machine Learning
[2] Generating simulated dataset for regression problems - sklearn make_regression
[3] Hands-on: Basics of linear regression

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