What is the difference between Big Data and Data Mining?
As Wikpedia states
The overall goal of the data mining process is to extract information from a data set and transform it into an understandable structure for further use
How is this related with Big Data? Is it correct if I say that Hadoop is doing data mining in a parallel manner?
Answer
Big data is everything
Big data is a marketing term, not a technical term. Everything is big data these days. My USB stick is a "personal cloud" now, and my harddrive is big data. Seriously. This is a totally unspecific term that is largely defined by what the marketing departments of various very optimistic companies can sell - and the C*Os of major companies buy, in order to make magic happen. Update: and by now, the same applies to data science. It's just marketing.
Data mining is the old big data
Actually, data mining was just as overused... it could mean anything such as
- collecting data (think NSA)
- storing data
- machine learning / AI (which predates the term data mining)
- non-ML data mining (as in "knowledge discovery", where the term data mining was actually coined; but where the focus is on new knowledge, not on learning of existing knowledge)
- business rules and analytics
- visualization
- anything involving data you want to sell for truckloads of money
It's just that marketing needed a new term. "Business intelligence", "business analytics", ... they still keep on selling the same stuff, it's just rebranded as "big data" now.
Most "big" data mining isn't big
Since most methods - at least those that give interesting results - just don't scale, most data "mined" isn't actually big. It's clearly much bigger than 10 years ago, but not big as in Exabytes. A survey by KDnuggets had something like 1-10 GB being the average "largest data set analyzed". That is not big data by any data management means; it's only large by what can be analyzed using complex methods. (I'm not talking about trivial algorithms such a k-means).
Most "big data" isn't data mining
Now "Big data" is real. Google has Big data, and CERN also has big data. Most others probably don't. Data starts being big, when you need 1000 computers just to store it.
Big data technologies such as Hadoop are also real. They aren't always used sensibly (don't bother to run hadoop clusters less than 100 nodes - as this point you probably can get much better performance from well-chosen non-clustered machines), but of course people write such software.
But most of what is being done isn't data mining. It's Extract, Transform, Load (ETL), so it is replacing data warehousing. Instead of using a database with structure, indexes and accelerated queries, the data is just dumped into hadoop, and when you have figured out what to do, you re-read all your data and extract the information you really need, tranform it, and load it into your excel spreadsheet. Because after selection, extraction and transformation, usually it's not "big" anymore.
Data quality suffers with size
Many of the marketing promises of big data will not hold. Twitter produces much less insights for most companies than advertised (unless you are a teenie rockstar, that is); and the Twitter user base is heavily biased. Correcting for such a bias is hard, and needs highly experienced statisticians.
Bias from data is one problem - if you just collect some random data from the internet or an appliction, it will usually be not representative; in particular not of potential users. Instead, you will be overfittig to the existing heavy-users if you don't manage to cancel out these effects.
The other big problem is just noise. You have spam bots, but also other tools (think Twitter "trending topics" that cause reinforcement of "trends") that make the data much noiser than other sources. Cleaning this data is hard, and not a matter of technology but of statistical domain expertise. For example Google Flu Trends was repeatedly found to be rather inaccurate. It worked in some of the earlier years (maybe because of overfitting?) but is not anymore of good quality.
Unfortunately, a lot of big data users pay too little attention to this; which is probably one of the many reasons why most big data projects seem to fail (the others being incompetent management, inflated and unrealistic expectations, and lack of company culture and skilled people).
Hadoop != data mining
Now for the second part of your question. Hadoop doesn't do data mining. Hadoop manages data storage (via HDFS, a very primitive kind of distributed database) and it schedules computation tasks, allowing you to run the computation on the same machines that store the data. It does not do any complex analysis.
There are some tools that try to bring data mining to Hadoop. In particular, Apache Mahout can be called the official Apache attempt to do data mining on Hadoop. Except that it is mostly a machine learning tool (machine learning != data mining; data mining sometimes uses methods from machine learning). Some parts of Mahout (such as clustering) are far from advanced. The problem is that Hadoop is good for linear problems, but most data mining isn't linear. And non-linear algorithms don't just scale up to large data; you need to carefully develop linear-time approximations and live with losses in accuracy - losses that must be smaller than what you would lose by simply working on smaller data.
A good example of this trade-off problem is k-means. K-means actually is a (mostly) linear problem; so it can be somewhat run on Hadoop. A single iteration is linear, and if you had a good implementation, it would scale well to big data. However, the number of iterations until convergence also grows with data set size, and thus it isn't really linear. However, as this is a statistical method to find "means", the results actually do not improve much with data set size. So while you can run k-means on big data, it does not make a whole lot of sense - you could just take a sample of your data, run a highly-efficient single-node version of k-means, and the results will be just as good. Because the extra data just gives you some extra digits of precision of a value that you do not need to be that precise.
Since this applies to quite a lot of problems, actual data mining on Hadoop doesn't seem to kick off. Everybody tries to do it, and a lot of companies sell this stuff. But it doesn't really work much better than the non-big version. But as long as customers want to buy this, companies will sell this functionality. And as long as it gets you a grant, researchers will write papers on this. Whether it works or not. That's life.
There are a few cases where these things work. Google search is an example, and Cern. But also image recognition (but not using Hadoop, clusters of GPUs seem to be the way to go there) has recently benefited from an increase in data size. But in any of these cases, you have rather clean data. Google indexes everything; Cern discards any non-interesting data, and only analyzes interesting measurements - there are no spammers feeding their spam into Cern... and in image analysis, you train on preselected relevant images, not on say webcams or random images from the internet (and if so, you treat them as random images, not as representative data).