Applications

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Applications

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Welford Algorithm

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Immagine
The variance is the expectation of the squared deviation of a random variable form its mean.  Informally, it measures how far a set of (random) numbers are spread out from their average value. The variance is the square of the standard deviation, the second central moment of a distribution and the covariance of the random variable with itself, and it is often represented by Var(X), σ 2 , s 2 . One of the most widely known formula for computing the variance is: where x-bar is the mean of the sample. The definition given above can be converted into an algorithm that computed the variance and the standard deviation in two passes: 1. Compute the mean                            (O(n)) 2. Compute the square differences       (O(n)) Output the variance Even though this algorithm seems working properly, it may become too expensive on some input instances. Just consider a sampling procedu...
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Insight 4 - Kahan and Neumaier Summation

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Immagine
Algorithms for 'online data stream' arithmetic mean Data streams can be denoted as an ordered sequence of points that must be accessed sequentially and can be read only once or a small number of times. Efficient processing over massive data sets has taken an increased importance in the last few decades due to the growth of the data traffic and storage required by even more applications. In particular, monitoring huge and rapidly changing streams of data has emerged as an important data management problem. Some applications are the analysis of network traffic, transaction logs, telephone call records and many more. Since data streams have potentially unbounded size, when the amount of computation memory is limited it may be impossible to produce an exact result. Here the challenge is to produce high quality approximated answers, that is, answers as correct as possible. Let's take the arithmetic mean as an example. Considering a huge data stream and by using the...
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Insight 3 - Bayes' Theorem

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Before giving the Bayes theorem statement is important to define the conditional probability. Once done, the Bayes theorem is easy to derive. Conditional Probability The conditional probability measures the likelihood of an event given that another event has occurred. For instance, assuming that  A  is the event of interest and that  B consists of a different event which has already occurred. The probability of the event  A  to occur, given the occurrence of B , (written as)  P ( A  | B ), can be computed as detailed by the following equality. P(A | B) = P(A ∩ B) / P(B)        P( B ) > 0                               (1) It is defined as the quotient of the probability of the joint events A  and B  and the probability of B . Now consider P(B | A), which is equals to P(B ∩ A) / P(A). Since P(B ∩ A) =  P(A ∩ B), then the fo...
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