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

Let's Understand Ten Machine Learning Algorithms

Ten Machine Learning Algorithms to Learn Machine Learning Practitioners have different personalities. While some of them are “I am an expert in X and X can train on any type of data”, where X = some algorithm, some others are “Right tool for the right job people”. A lot of them also subscribe to “Jack of all trades. Master of one” strategy, where they have one area of deep expertise and know slightly about different fields of Machine Learning. That said, no one can deny the fact that as practicing Data Scientists, we will have to know basics of some common machine learning algorithms, which would help us engage with a new-domain problem we come across. This is a whirlwind tour of common machine learning algorithms and quick resources about them which can help you get started on them. 1. Principal Component Analysis(PCA)/SVD PCA is an unsupervised method to understand global properties of a dataset consisting of vectors. Covariance Matrix of data points is analyzed here to un...
Post a Comment
Read more

gRPC with Java : Build Fast & Scalable Modern API & Microservices using Protocol Buffers

gRPC Java Master Class : Build Fast & Scalable Modern API for your Microservice using gRPC Protocol Buffers gRPC is a revolutionary and modern way to define and write APIs for your microservices. The days of REST, JSON and Swagger are over! Now writing an API is easy, simple, fast and efficient. gRPC is created by Google and Square, is an official CNCF project (like Docker and Kubernetes) and is now used by the biggest tech companies such as Netflix, CoreOS, CockRoachDB, and so on! gRPC is very popular and has over 15,000 stars on GitHub (2 times what Kafka has!). I am convinced that gRPC is the FUTURE for writing API for microservices so I want to give you a chance to learn about it TODAY. Amongst the advantage of gRPC: 1) All your APIs and messages are simply defined using Protocol Buffers 2) All your server and client code for any programming language gets generated automatically for free! Saves you hours of programming 3) Data is compact and serialised 4) API ...
Post a Comment
Read more

What is Big Data ?

What is Big Data ? It is now time to answer an important question – What is Big Data? Big data, as defined by Wikipedia, is this: “Big data is a broad term for  data sets  so large or complex that traditional  data processing  applications are inadequate. Challenges include  analysis , capture,  data curation , search,  sharing ,  storage , transfer ,  visualization ,  querying  and  information privacy . The term often refers simply to the use of  predictive analytics  or certain other advanced methods to extract value from data, and seldom to a particular size of data set.” In simple terms, Big Data is data that has the 3 characteristics that we mentioned in the last section – • It is big – typically in terabytes or even petabytes • It is varied – it could be a traditional database, it could be video data, log data, text data or even voice data • It keeps increasing as new data keeps flowing in This kin...
Post a Comment
Read more