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Reading list

Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

Estimators – An Introduction to Beginners in Data Science

N
Naman Last Updated : 01 Jul, 2021
6 min read

This article was published as a part of the Data Science Blogathon.

Introduction

Not having much information about the distribution of a random variable can become a major problem for data scientists and statisticians. Consider, a researcher trying to understand the distribution of Choco-chips in a cookie (a very popular example of Poisson distribution). The researcher is well aware that the distribution of Choco-chips follows a Poisson distribution, but does not know how to estimate the parameter λ of the distribution.

A parameter is essentially a numerical characteristic of a distribution (or any statistical model in general). Normal distributions have µ & σ as parameters, uniform distributions have a & b as parameters, and binomial distributions have n & p as parameters. These numerical characteristics are vital for understanding the size, shape, spread, and other properties of a distribution. In the absence of the true value of the parameter, it seems that the researcher may not be able to continue her investigation. But that’s when estimators step in.

Estimators are functions of random variables that can help us find approximate values for these parameters. Think of these estimators like any other function, that takes an input, processes it, and renders an output. So, the process of estimation goes as follows:

1) From the distribution, we take a series of random samples.

2) We input these random samples into the estimator function.

3) The estimator function processes it and gives a set of outputs.

4) The expected value of that set is the approximate value of the parameter.

Example

Let’s take an example. Consider a random variable X showing a uniform distribution. The distribution of X can be represented as U[0, θ]. This has been plotted below:

estimators 1

(Figure A)

We have the random variable X and its distribution. But we don’t know how to determine the value of θ. Let’s use estimators. There are many ways to approach this problem. I’ll discuss two of them:

 

1) Using Sample Mean

We know that for a U[a, b] distribution, the mean µ is given by the following equation:

e(x)

For U[0, θ] distribution, a = 0 & b = θ, we get:

estimators 2

Thus, if we estimate µ, we can estimate θ. To estimate µ, we use a very popular estimator called the sample mean estimator. The sample mean is the sum of the random sample value drawn divided by the size of the sample. For instance, if we have a random sample S = {4, 7, 3, 2}, then the sample mean is (4+7+3+2)/4 = 4 (the average value). In general, the sample mean is defined using the following notation:

estimators 3

Here, µ-hat is the sample mean estimator & n is the size of the random sample that we take from the distribution. A variable with a hat on top of it is the general notation for an estimator. Since our unknown parameter θ is twice of µ, we arrive at the following estimator for θ:

estimators 4

We take a random sample, plug it into the above estimator, and get a number. We repeat this process and get a set of numbers. The following figure illustrates the process:

figure b

(Figure B)

The lines on the x-axes correspond to the values present in the sample taken from the distribution. The red lines in the middle indicate the average value of the sample, and the red lines at the end are twice that average value i.e., the expected value of θ for one sample. Many such samples are taken, and the estimated value of θ for each sample is noted. The expected value/mean of that set of numbers gives the final estimate for θ. It can be mathematically proved (using properties of expectation):

estimators 5

It is seen that the expectation of the estimator is equal to the true value of the parameter. This amazing property that certain estimators have is called unbiasedness, which is a very useful criterion for assessing estimators.

 

2) Maximum Value Method

This time, instead of using mean, we’ll use order statistics, particularly the nth order statistic. The nth order statistic is defined as the nth smallest value of a random sample of size n. In other words, it’s the maximum value of a random sample. For instance, if we have a random sample S = {4, 7, 3, 2}, then the nth order statistic is 7 (the largest value). The estimator is now defined as follows:

MVM

We follow the same procedure- take random samples, input them, collect the output and find the expectation. The following figure illustrates the process:

c fig

(Figure C)

As noted previously, the lines on the x-axes are the values present in one sample. The red lines at the end are the maximum value for that sample i.e., the nth order statistic. Two random samples are shown for reference. However, we need to take much larger samples. Why? To prove it, we’ll use the general expression for the PDF (Probability Distribution Function) of nth order statistics for U[a, b] distribution:

estimators pdf

For U[0, θ] distribution, a = 0 & b = θ, we get:

U[0, θ] distribution, a = 0 & b = θ,

Using the integral form of expectation of a continuous variable,

expectation of a continuous variable

Unlike before, no exact equality has been ascertained between the expectation of θ-hat & θ. This is because the nth order statistic estimator is biased. We can observe this bias on comparing figures B & C. In figure B, both positive (θ-hat > θ) and negative (θ-hat < θ) deviations are possible with equal probability (as shown). These deviations can cancel each other out, making the sample mean estimator unbiased. However, in figure C, only negative deviations are possible. Why? Because the maximum value of a random sample will always be lesser than the range [0, θ] of the distribution. Consequently, negative bias steps in.

Does that mean that we cannot use this estimator? Certainly not. As discussed earlier, the estimator bias can be significantly lowered by taking large n. For large values of n, n = n+1 (approximately). Thus, we get:

appx

The Bottom Line

Hence, we have successfully solved our problem through estimators. We also learned a very important property of estimators- unbiasedness. While this may have been an extensive read, it’s imperative to acknowledge that the study of estimators is not restricted to just the above-explained concepts. Various other properties of estimators such as their efficiency, robustness, mean squared error, and consistency are also vital to deepen our understanding of them.

The media shown in this article are not owned by Analytics Vidhya and is used at the Author’s discretion. 

 

N
Naman

I am currently a first-year undergraduate student at the National University of Singapore (NUS), who is deeply interested in Statistics, Data Science, Economics and Machine Learning. I love working on different Data Science projects to enhance my analytical and inferential skills. I find it important to share my learning with other members of the community by simplifying data science is such a way that young minds can understand and local leaders can implement.

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