Skip to main content

Apache Spark RDD - Sampling With Replacement and Sampling Without Replacement

Sampling is a popular Spark RDD operation. Sampling With Replacement and sampling without replacement are different ways of doing sampling. This article explains the difference between them.

Sampling with replacement:
Consider a population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two with replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I replace it. Then I pick another. Every one of them still has 1/7 probability of being chosen. And there are exactly 49 different possibilities here (assuming we distinguish between the first and second.) They are: (12,12), (12,13), (12, 14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,13), (13,14), etc.
Sampling without replacement:
Consider the same population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two without replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I pick another. At this point, there are only six possibilities: 12, 13, 15, 16, 17, and 18. So there are only 42 different possibilities here (again assuming that we distinguish between the first and the second.) They are: (12,13), (12,14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,14), (13,15), etc.
What’s the Difference?
When we sample with replacement, the two sample values are independent. Practically, this means that what we get on the first one doesn’t affect what we get on the second. Mathematically, this means that the covariance between the two is zero.
In sampling without replacement, the two sample values aren’t independent. Practically, this means that what we got on the for the first one affects what we can get for the second one. Mathematically, this means that the covariance between the two isn’t zero. That complicates the computations. In particular, if we have a SRS (simple random sample) without replacement, from a population with variance , then the covariance of two of the different sample values is , where N is the population size. (A brief summary of some formulas is provided here. For a discussion of this in a textbook for a course at the level of M378K, see the chapter on Survey Sampling in Mathematical Statistics and Data Analysis by John A. Rice, published by Wadsworth & Brooks/Cole Publishers. There is an outline of an slick, simple, interesting, but indirect, proof in the problems at the end of the chapter.)
Population size — Leading to a discussion of “infinite” populations.
When we sample without replacement, and get a non-zero covariance, the covariance depends on the population size. If the population is very large, this covariance is very close to zero. In that case, sampling with replacement isn’t much different from sampling without replacement. In some discussions, people describe this difference as sampling from an infinite population (sampling with replacement) versus sampling from a finite population (without replacement).

Comments

Post a Comment

Popular posts from this blog

EVENT DRIVEN MICROSERVICES

EVENT BASED MICROSERVICES - Event Sourcing In a Microservice Architecture, especially with Database per Microservice, the Microservices need to exchange data. For resilient, highly scalable, and fault-tolerant systems, they should communicate asynchronously by exchanging Events. In such a case, you may want to have Atomic operations, e.g., update the Database and send the message. If you have SQL databases and want to have distributed transactions for a high volume of data, you cannot use the two-phase locking (2PL) as it does not scale. If you use NoSQL Databases and want to have a distributed transaction, you cannot use 2PL as many NoSQL databases do not support two-phase locking. In such scenarios, use Event based Architecture with Event Sourcing. In traditional databases, the Business Entity with the current “state” is directly stored. In Event Sourcing, any state-changing event or other significant events are stored instead of the entities. It means the modifications of a Busines...
Post a Comment
Read more

KAFKA - Architecture

Kafka - Architecture What is Kafka? Kafka is an event-streaming platform that is designed to process high volumes of data in real-time. Developed by LinkedIn in 2011, it has quickly become the infrastructural backbone of companies like Netflix, Twitter, and Spotify. Why do we need Kafka? In today’s data-driven world, tracking information like user clicks, recommendations, and shopping carts can be invaluable for a company’s growth. With these analytics, companies can make the product improvements needed to boost user engagement and conversion rates. However, on sites with millions of daily users, collecting and analyzing this data is nontrivial. Kafka was des i gned to streamline this operation, acting as a robust tool that maintains efficient, real-time processing capabilities with incredible quantities of data. For instance, as of late 2019, LinkedIn’s Kafka deployments were managing more than 7 trillion messages per day. How does Kafka work? Kafka provides a structured architecture ...
Post a Comment
Read more

Tapping Into the “Long Tail” of Big Data

Variety, not volume or velocity, drives big-data investments !!! Gartner defines big data as the three Vs: high-volume, high-velocity, high-variety information assets. While all three Vs are growing, variety is becoming the single biggest driver of big-data investments, as seen in the results of a recent survey by New Vantage Partners. This trend will continue to grow as firms seek to integrate more sources and focus on the “long tail” of big data. From schema-free JSON to nested types in other databases (relational and NoSQL), to non-flat data (Avro, Parquet, XML), data formats are multiplying and connectors are becoming crucial. In 2017, analytics platforms will be evaluated based on their ability to provide live direct connectivity to these disparate sources. Tapping Into the “Long Tail” of Big Data When asked about drivers of Big Data success, 69% of corporate executives named greater data variety as the most important factor, followed by volume (25%), with ...
Post a Comment
Read more