Grid-oriented Storage (GOS) was a term used for data storage by a university project during the era when the term grid computing was popular. == Description == GOS was a successor of the term network-attached storage (NAS). GOS systems contained hard disks, often RAIDs (redundant arrays of independent disks), like traditional file servers. GOS was designed to deal with long-distance, cross-domain and single-image file operations, which is typical in Grid environments. GOS behaves like a file server via the file-based GOS-FS protocol to any entity on the grid. Similar to GridFTP, GOS-FS integrates a parallel stream engine and Grid Security Infrastructure (GSI). Conforming to the universal VFS (Virtual Filesystem Switch), GOS-FS can be pervasively used as an underlying platform to best utilize the increased transfer bandwidth and accelerate the NFS/CIFS-based applications. GOS can also run over SCSI, Fibre Channel or iSCSI, which does not affect the acceleration performance, offering both file level protocols and block level protocols for storage area network (SAN) from the same system. In a grid infrastructure, resources may be geographically distant from each other, produced by differing manufacturers, and have differing access control policies. This makes access to grid resources dynamic and conditional upon local constraints. Centralized management techniques for these resources are limited in their scalability both in terms of execution efficiency and fault tolerance. Provision of services across such platforms requires a distributed resource management mechanism and the peer-to-peer clustered GOS appliances allow a single storage image to continue to expand, even if a single GOS appliance reaches its capacity limitations. The cluster shares a common, aggregate presentation of the data stored on all participating GOS appliances. Each GOS appliance manages its own internal storage space. The major benefit of this aggregation is that clustered GOS storage can be accessed by users as a single mount point. GOS products fit the thin-server categorization. Compared with traditional “fat server”-based storage architectures, thin-server GOS appliances deliver numerous advantages, such as the alleviation of potential network/grid bottle-necks, CPU and OS optimized for I/O only, ease of installation, remote management and minimal maintenance, low cost and Plug and Play, etc. Examples of similar innovations include NAS, printers, fax machines, routers and switches. An Apache server has been installed in the GOS operating system, ensuring an HTTPS-based communication between the GOS server and an administrator via a Web browser. Remote management and monitoring makes it easy to set up, manage, and monitor GOS systems. == History == Frank Zhigang Wang and Na Helian proposed a funding proposal to the UK government titled “Grid-Oriented Storage (GOS): Next Generation Data Storage System Architecture for the Grid Computing Era” in 2003. The proposal was approved and granted one million pounds in 2004. The first prototype was constructed in 2005 at Centre for Grid Computing, Cambridge-Cranfield High Performance Computing Facility. The first conference presentation was at IEEE Symposium on Cluster Computing and Grid (CCGrid), 9–12 May 2005, Cardiff, UK. As one of the five best work-in-progress, it was included in the IEEE Distributed Systems Online. In 2006, the GOS architecture and its implementations was published in IEEE Transactions on Computers, titled “Grid-oriented Storage: A Single-Image, Cross-Domain, High-Bandwidth Architecture”. Starting in January 2007, demonstrations were presented at Princeton University, Cambridge University Computer Lab and others. By 2013, the Cranfield Centre still used future tense for the project. Peer-to-peer file sharings use similar techniques.
Uniform convergence in probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family uniformly converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. Specifically, the Glivenko-Cantelli theorem and the homonymous classes of functions are fundamentally related to uniform convergence. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class S {\displaystyle S} from one and the same sample. Moreover, it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events S {\displaystyle S} . The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds. == Definitions == For a class of predicates H {\displaystyle H} defined on a set X {\displaystyle X} and a set of samples x = ( x 1 , x 2 , … , x m ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{m})} , where x i ∈ X {\displaystyle x_{i}\in X} , the empirical frequency of h ∈ H {\displaystyle h\in H} on x {\displaystyle x} is Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | . {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|.} The theoretical probability of h ∈ H {\displaystyle h\in H} is defined as Q P ( h ) = P { y ∈ X : h ( y ) = 1 } . {\displaystyle Q_{P}(h)=P\{y\in X:h(y)=1\}.} The Uniform Convergence Theorem states, roughly, that if H {\displaystyle H} is "simple" and we draw samples independently (with replacement) from X {\displaystyle X} according to any distribution P {\displaystyle P} , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability. Here "simple" means that the Vapnik–Chervonenkis dimension of the class H {\displaystyle H} is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole. The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis using the concept of growth function. == Uniform Convergence Theorem == The statement of the Uniform Convergence Theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for ε > 0 {\displaystyle \varepsilon >0} and m {\displaystyle m} a positive integer, we have: P m { | Q P ( h ) − Q x ^ ( h ) | ≥ ε for some h ∈ H } ≤ 4 Π H ( 2 m ) e − ε 2 m / 8 . {\displaystyle P^{m}\{|Q_{P}(h)-{\widehat {Q_{x}}}(h)|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.} In the above, for any x ∈ X m , {\displaystyle x\in X^{m},} Q P ( h ) = P { ( y ∈ X : h ( y ) = 1 } , {\displaystyle Q_{P}(h)=P\{(y\in X:h(y)=1\},} Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|} and | x | = m . {\displaystyle |x|=m.} P m {\displaystyle P^{m}} indicates that the probability is taken over x {\displaystyle x} consisting of m {\displaystyle m} i.i.d. draws from the distribution P . {\displaystyle P.} Finally, the growth function Π H {\displaystyle \Pi _{H}} is defined in the following way, for any { 0 , 1 } {\displaystyle \{0,1\}} -valued functions H {\displaystyle H} over X {\displaystyle X} and for any natural number m {\displaystyle m} : Π H ( m ) = max | { h ∩ D : D ⊆ X , | D | = m , h ∈ H } | . {\displaystyle \Pi _{H}(m)=\max |\{h\cap D:D\subseteq X,|D|=m,h\in H\}|.} From the point of view of Learning Theory one can consider H {\displaystyle H} to be the Concept/Hypothesis class defined over the instance set X {\displaystyle X} . Crucially, the Sauer–Shelah lemma implies that Π H ( m ) ≤ m d {\displaystyle \Pi _{H}(m)\leq m^{d}} , where d {\displaystyle d} is the VC dimension of H {\displaystyle H} . == Proof of the Uniform Convergence Theorem == and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) − Q ^ x ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon } into the problem of analyzing | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} , where r {\displaystyle r} and s {\displaystyle s} are i.i.d samples of size m {\displaystyle m} drawn according to the distribution P {\displaystyle P} . One can view r {\displaystyle r} as the original randomly drawn sample of length m {\displaystyle m} , while s {\displaystyle s} may be thought as the testing sample which is used to estimate Q P ( h ) {\displaystyle Q_{P}(h)} . Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically and independently, so swapping elements between them will not change the probability distribution on r {\displaystyle r} and s {\displaystyle s} . So, we will try to bound the probability of | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} for some h ∈ H {\displaystyle h\in H} by considering the effect of a specific collection of permutations of the joint sample x = r | | s {\displaystyle x=r||s} . Specifically, we consider permutations σ ( x ) {\displaystyle \sigma (x)} which swap x i {\displaystyle x_{i}} and x m + i {\displaystyle x_{m+i}} in some subset of 1 , 2 , . . . , m {\displaystyle {1,2,...,m}} . The symbol r | | s {\displaystyle r||s} means the concatenation of r {\displaystyle r} and s {\displaystyle s} . Reduction to a finite class: We can now restrict the function class H {\displaystyle H} to a fixed joint sample and hence, if H {\displaystyle H} has finite VC Dimension, it reduces to the problem to one involving a finite function class. We present the technical details of the proof. It should be stressed that this proof glosses over details like the measurability of the events V {\displaystyle V} and R {\displaystyle R} ; measurability is granted in the case of H {\displaystyle H} being finite or countable, but this is not normally the case in standard applications of the theorem (e.g. for statistical learning theory or to prove the Glivenko-Cantelli theorem). To get measurability, one needs to use a notion of separability of the underlying space, possibly related to H {\displaystyle H} . === Symmetrization === Lemma: Let V = { x ∈ X m : | Q P ( h ) − Q ^ x ( h ) | ≥ ε for some h ∈ H } {\displaystyle V=\{x\in X^{m}:|Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon {\text{ for some }}h\in H\}} and R = { ( r , s ) ∈ X m × X m : | Q r ^ ( h ) − Q ^ s ( h ) | ≥ ε / 2 for some h ∈ H } . {\displaystyle R=\{(r,s)\in X^{m}\times X^{m}:|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2{\text{ for some }}h\in H\}.} Then for m ≥ 2 ε 2 {\displaystyle m\geq {\frac {2}{\varepsilon ^{2}}}} , P m ( V ) ≤ 2 P 2 m ( R ) {\displaystyle P^{m}(V)\leq 2P^{2m}(R)} . Proof: By the triangle inequality, if | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2} then | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} . Therefore, P 2 m ( R ) ≥ P 2 m { ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } = ∫ V P m { s : ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } d P m ( r ) = A {\displaystyle {\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\\[5pt]={}&\int _{V}P^{m}\{s:\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}} since r {\displaystyle r} and s {\displaystyle s} are independent. Now for r ∈ V {\displaystyle r\in V} fix an h ∈ H {\displaystyle h\in H} such that | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } . For this h {\displaystyle h} , we shall
Batch cryptography
Batch cryptography is a field of cryptology focused on the design of cryptographic protocols that perform operations—such as encryption, decryption, key exchange, and authentication—on multiple inputs simultaneously, rather than processing each input individually. Batching cryptographic operations can significantly reduce the marginal cost of handling individual inputs—a principle that was first introduced by Amos Fiat in 1989.
Sex differences in social media use
Men and women use social media in different ways and with different frequencies. In general, several researchers have found that women tend to use social network services (SNSs) more than men and primiarly to socialize. == Differences == === Predilection for usage === Many studies have found that women are more likely to use either specific SNSs such as Facebook or MySpace or SNSs in general. In 2015, 73% of online men and 80% of online women used social networking sites. The gap in gender differences has become less apparent in LinkedIn. In 2015 about 26 percent of online men and 25% of online women used the business-and employee-oriented networking site. Researchers who have examined the gender of users of multiple SNSs have found contradictory results. Hargittai's groundbreaking 2007 study examining race, gender, and other differences between undergraduate college student users of SNSs found that women were not only more likely to have used SNSes than men but that they were also more likely to have used many different services, including Facebook, MySpace, and Friendster; these differences persisted in several models and analyses. Although she only surveyed students at one institution – the University of Illinois at Chicago – Hargittai selected that institution intentionally as "an ideal location for studies of how different kinds of people use online sites and services." In contrast, data collected by the Pew Internet & American Life Project found that men were more likely to have multiple SNS profiles. Although the sample sizes of the two surveys are comparable – 1,650 Internet users in the Pew survey compared with 1,060 in Hargittai's survey – the data from the Pew survey are newer and arguably more representative of the entire adult United States population. Pinterest, Facebook, and Instagram attract more females. Picture sharing sites overall are very popular among women. Pinterest alone attracts three times as many female users than male. However, use of Pinterest by men has increased from 5% in 2012. Facebook attracts about 77% of women online. Instagram is also more likely to attract women. Men are more likely to participate in online forums like Reddit, Digg or Slashdot. One in five men claim to be a part of an online forum. === Uses === In general, women seem to use SNSs more to explicitly foster social connections. A study conducted by Pew research centers found that women were more avid users of social media. In November 2010, the gap between men and women was as high as 15%. Female participants in a multi-stage study conducted in 2007 to discover the motivations of Facebook users scored higher on scales for social connection and posting of photographs. Studies have also been conducted on the differences between females and males with regards to blogging. The Pew Research Center found that younger females are more likely to blog than males their own age, even males that are older than them. Similarly, in a study of blogs maintained in MySpace, women were found to be more likely to not only write blogs but also write about family, romantic relationships, friendships, and health in those blogs. A study of Swedish SNS users found that women were more likely to have expressions of friendship, specifically in the areas of (a) publishing photos of their friends, (b) specifically naming their best friends, and (c) writing poems to and about their friends. Women were also more likely to have expressions related to family relationships and romantic relationships. One of the key findings of this research is that those men who do have expressions of romantic relationships in their profile had expressions just as strong as the women. However, the researcher speculated that this may be in part due to a desire to publicly express heterosexual behaviors and mannerisms instead of merely expressing romantic feelings. A large-scale study of gender differences in MySpace found that both men and women tended to have a majority of female Friends, and both men and women tended to have a majority of female "Top" Friends in the site. A later study found women to author disproportionately many (public) comments in MySpace, but an investigation into the role of emotion in public MySpace comments found that women both give and receive stronger positive emotion. It was hypothesised that women are simply more effective at using social networking sites because they are better able to harness positive emotion. A study focused on the influence of gender and personality on individuals' use of online social networking websites such as Facebook, reported that men use social networking sites with the intention of forming new relationships, whereas, women use them more for relationship maintenance. In addition to this, women are more likely to use Facebook or MySpace to compare themselves to others and also to search for information. Men, however, are more likely to look at other people's profiles with in the intention to find friends. Women were less successful at actually finding new friends, but more successful at "maintaining existing relationships, making new relationships, using for academic purposes and following specific agenda". Similarly, men also self-reported this motivation "while women reported using them more for relationship maintenance". === Personality === OCEAN personality traits are known to systematically vary between human males and females. In one study, the same women were more extraverted and agreeable, such as less neurotic while on social media than offline. Other studies associated neuroticism with female use of social media. === Privacy === Privacy has been the primary topic of many studies of SNS users, and many of these studies have found differences between male and female SNS users, although some studies have found results contradictory to those found in other studies. Some researchers have found that women are more protective of their personal information and more likely to have private profiles. Other researchers have found that women are less likely to post some types of information. Acquisti and Gross found that women in their sample were less likely to reveal their sexual orientation, personal address, or cell phone number. This is similar to Pew Internet & American Life research of children users of SNSs that found that boys and girls presented different views of privacy and behaviors, with girls being more concerned about and restrictive of information such as city, town, last name, and cell phone number that could be used to locate them. At least one group of researchers has found that women are less likely to share information that "identifies them directly – last name, cell phone number, and address or home phone number," linking that resistance to women's greater concerns about "cyberstalking", "cyberbullying", and security problems. Despite these concerns about privacy, researchers have found that women are more likely to maintain up-to-date photos of themselves. Further, Kolek and Saunders found in their sample of college student Facebook users that women were more likely to not only post a photograph of themselves in their profile but that they were more likely to have a publicly viewable Facebook account (a contradictory finding compared to many other studies), post photos, and post photo albums. Women were more likely to have: (a) a publicly viewable Facebook account, (b) more photo albums, (c) more photos, (d) a photo of themselves as their profile picture, (e) positive references to alcohol, partying, or drugs, and (f) more positive references to or about the institution or institution-related activities. In general, women were more likely to disclose information about themselves in their Facebook profile, with the primary exception of sharing their telephone number. Similarly, female respondents to Strano's study were more likely to keep their profile photo recent and choose a photo that made them appear attractive, happy, and fun-loving. Citing several examples, Strano opined that there may also be a difference in how men and women Facebook users display and interpret profile photos depicting relationships. Privacy has also been a concern for the SnapChat app, which allows you to send messages either text or photo or video which then disappear. One study has shown that security is not a major concern for the majority of users and that most do not use Snapchat to send sensitive content (although up to 25% may do so experimentally). As part of their research almost no statistically significant gender differences were found. === Cyberbullying === Past research carried out to investigate if there are any gender differences in cyber-bullying has found that boys commit more cyber verbal bullying, cyber forgery and more violence based on hidden identity or presenting themselves as other person. === Mansplaining === A 2021 article found that mansplaining could be seen more prominent online rather than offl
Group key
In cryptography, a group key is a cryptographic key that is shared between a group of users. Typically, group keys are distributed by sending them to individual users, either physically, or encrypted individually for each user using either that user's pre-distributed private key. A common use of group keys is to allow a group of users to decrypt a broadcast message that is intended for that entire group of users, and no one else. For example, in the Second World War, group keys (known as "iodoforms", a term invented by a classically educated non-chemist, and nothing to do with the chemical of the same name) were sent to groups of agents by the Special Operations Executive. These group keys allowed all the agents in a particular group to receive a single coded message. In present-day applications, group keys are commonly used in conditional access systems, where the key is the common key used to decrypt the broadcast signal, and the group in question is the group of all paying subscribers. In this case, the group key is typically distributed to the subscribers' receivers using a combination of a physically distributed secure cryptoprocessor in the form of a smartcard and encrypted over-the-air messages.
EM algorithm and GMM model
In statistics, EM (expectation maximization) algorithm handles latent variables, while GMM is the Gaussian mixture model. == Background == In the picture below, are shown the red blood cell hemoglobin concentration and the red blood cell volume data of two groups of people, the Anemia group and the control group (i.e. the group of people without Anemia). As expected, people with Anemia have lower red blood cell volume and lower red blood cell hemoglobin concentration than those without Anemia. x {\displaystyle x} is a random vector such as x := ( red blood cell volume , red blood cell hemoglobin concentration ) {\displaystyle x:={\big (}{\text{red blood cell volume}},{\text{red blood cell hemoglobin concentration}}{\big )}} , and from medical studies it is known that x {\displaystyle x} are normally distributed in each group, i.e. x ∼ N ( μ , Σ ) {\displaystyle x\sim {\mathcal {N}}(\mu ,\Sigma )} . z {\displaystyle z} is denoted as the group where x {\displaystyle x} belongs, with z i = 0 {\displaystyle z_{i}=0} when x i {\displaystyle x_{i}} belongs to the Anemia group and z i = 1 {\displaystyle z_{i}=1} when x i {\displaystyle x_{i}} belongs to the control group. Also z ∼ Categorical ( k , ϕ ) {\displaystyle z\sim \operatorname {Categorical} (k,\phi )} where k = 2 {\displaystyle k=2} , ϕ j ≥ 0 , {\displaystyle \phi _{j}\geq 0,} and ∑ j = 1 k ϕ j = 1 {\displaystyle \sum _{j=1}^{k}\phi _{j}=1} . See Categorical distribution. The following procedure can be used to estimate ϕ , μ , Σ {\displaystyle \phi ,\mu ,\Sigma } . A maximum likelihood estimation can be applied: ℓ ( ϕ , μ , Σ ) = ∑ i = 1 m log ( p ( x ( i ) ; ϕ , μ , Σ ) ) = ∑ i = 1 m log ∑ z ( i ) = 1 k p ( x ( i ) ∣ z ( i ) ; μ , Σ ) p ( z ( i ) ; ϕ ) {\displaystyle \ell (\phi ,\mu ,\Sigma )=\sum _{i=1}^{m}\log(p(x^{(i)};\phi ,\mu ,\Sigma ))=\sum _{i=1}^{m}\log \sum _{z^{(i)}=1}^{k}p\left(x^{(i)}\mid z^{(i)};\mu ,\Sigma \right)p(z^{(i)};\phi )} As the z i {\displaystyle z_{i}} for each x i {\displaystyle x_{i}} are known, the log likelihood function can be simplified as below: ℓ ( ϕ , μ , Σ ) = ∑ i = 1 m log p ( x ( i ) ∣ z ( i ) ; μ , Σ ) + log p ( z ( i ) ; ϕ ) {\displaystyle \ell (\phi ,\mu ,\Sigma )=\sum _{i=1}^{m}\log p\left(x^{(i)}\mid z^{(i)};\mu ,\Sigma \right)+\log p\left(z^{(i)};\phi \right)} Now the likelihood function can be maximized by making partial derivative over μ , Σ , ϕ {\displaystyle \mu ,\Sigma ,\phi } , obtaining: ϕ j = 1 m ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \phi _{j}={\frac {1}{m}}\sum _{i=1}^{m}1\{z^{(i)}=j\}} μ j = ∑ i = 1 m 1 { z ( i ) = j } x ( i ) ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \mu _{j}={\frac {\sum _{i=1}^{m}1\{z^{(i)}=j\}x^{(i)}}{\sum _{i=1}^{m}1\left\{z^{(i)}=j\right\}}}} Σ j = ∑ i = 1 m 1 { z ( i ) = j } ( x ( i ) − μ j ) ( x ( i ) − μ j ) T ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \Sigma _{j}={\frac {\sum _{i=1}^{m}1\{z^{(i)}=j\}(x^{(i)}-\mu _{j})(x^{(i)}-\mu _{j})^{T}}{\sum _{i=1}^{m}1\{z^{(i)}=j\}}}} If z i {\displaystyle z_{i}} is known, the estimation of the parameters results to be quite simple with maximum likelihood estimation. But if z i {\displaystyle z_{i}} is unknown it is much more complicated. Being z {\displaystyle z} a latent variable (i.e. not observed), with unlabeled scenario, the expectation maximization algorithm is needed to estimate z {\displaystyle z} as well as other parameters. Generally, this problem is set as a GMM since the data in each group is normally distributed. In machine learning, the latent variable z {\displaystyle z} is considered as a latent pattern lying under the data, which the observer is not able to see very directly. x i {\displaystyle x_{i}} is the known data, while ϕ , μ , Σ {\displaystyle \phi ,\mu ,\Sigma } are the parameter of the model. With the EM algorithm, some underlying pattern z {\displaystyle z} in the data x i {\displaystyle x_{i}} can be found, along with the estimation of the parameters. The wide application of this circumstance in machine learning is what makes EM algorithm so important. == EM algorithm in GMM == The EM algorithm consists of two steps: the E-step and the M-step. Firstly, the model parameters and the z ( i ) {\displaystyle z^{(i)}} can be randomly initialized. In the E-step, the algorithm tries to guess the value of z ( i ) {\displaystyle z^{(i)}} based on the parameters, while in the M-step, the algorithm updates the value of the model parameters based on the guess of z ( i ) {\displaystyle z^{(i)}} of the E-step. These two steps are repeated until convergence is reached. The algorithm in GMM is: Repeat until convergence: 1. (E-step) For each i , j {\displaystyle i,j} , set w j ( i ) := p ( z ( i ) = j | x ( i ) ; ϕ , μ , Σ ) {\displaystyle w_{j}^{(i)}:=p\left(z^{(i)}=j|x^{(i)};\phi ,\mu ,\Sigma \right)} 2. (M-step) Update the parameters ϕ j := 1 m ∑ i = 1 m w j ( i ) {\displaystyle \phi _{j}:={\frac {1}{m}}\sum _{i=1}^{m}w_{j}^{(i)}} μ j := ∑ i = 1 m w j ( i ) x ( i ) ∑ i = 1 m w j ( i ) {\displaystyle \mu _{j}:={\frac {\sum _{i=1}^{m}w_{j}^{(i)}x^{(i)}}{\sum _{i=1}^{m}w_{j}^{(i)}}}} Σ j := ∑ i = 1 m w j ( i ) ( x ( i ) − μ j ) ( x ( i ) − μ j ) T ∑ i = 1 m w j ( i ) {\displaystyle \Sigma _{j}:={\frac {\sum _{i=1}^{m}w_{j}^{(i)}\left(x^{(i)}-\mu _{j}\right)\left(x^{(i)}-\mu _{j}\right)^{T}}{\sum _{i=1}^{m}w_{j}^{(i)}}}} With Bayes' rule, the following result is obtained by the E-step: p ( z ( i ) = j | x ( i ) ; ϕ , μ , Σ ) = p ( x ( i ) | z ( i ) = j ; μ , Σ ) p ( z ( i ) = j ; ϕ ) ∑ l = 1 k p ( x ( i ) | z ( i ) = l ; μ , Σ ) p ( z ( i ) = l ; ϕ ) {\displaystyle p\left(z^{(i)}=j|x^{(i)};\phi ,\mu ,\Sigma \right)={\frac {p\left(x^{(i)}|z^{(i)}=j;\mu ,\Sigma \right)p\left(z^{(i)}=j;\phi \right)}{\sum _{l=1}^{k}p\left(x^{(i)}|z^{(i)}=l;\mu ,\Sigma \right)p\left(z^{(i)}=l;\phi \right)}}} According to GMM setting, these following formulas are obtained: p ( x ( i ) | z ( i ) = j ; μ , Σ ) = 1 ( 2 π ) n / 2 | Σ j | 1 / 2 exp ( − 1 2 ( x ( i ) − μ j ) T Σ j − 1 ( x ( i ) − μ j ) ) {\displaystyle p\left(x^{(i)}|z^{(i)}=j;\mu ,\Sigma \right)={\frac {1}{(2\pi )^{n/2}\left|\Sigma _{j}\right|^{1/2}}}\exp \left(-{\frac {1}{2}}\left(x^{(i)}-\mu _{j}\right)^{T}\Sigma _{j}^{-1}\left(x^{(i)}-\mu _{j}\right)\right)} p ( z ( i ) = j ; ϕ ) = ϕ j {\displaystyle p\left(z^{(i)}=j;\phi \right)=\phi _{j}} In this way, a switch between the E-step and the M-step is possible, according to the randomly initialized parameters.
Data grid
A data grid is an architecture or set of services that allows users to access, modify and transfer extremely large amounts of geographically distributed data for research purposes. Data grids make this possible through a host of middleware applications and services that pull together data and resources from multiple administrative domains and then present it to users upon request. The data in a data grid can be located at a single site or multiple sites where each site can be its own administrative domain governed by a set of security restrictions as to who may access the data. Likewise, multiple replicas of the data may be distributed throughout the grid outside their original administrative domain and the security restrictions placed on the original data for who may access it must be equally applied to the replicas. Specifically developed data grid middleware is what handles the integration between users and the data they request by controlling access while making it available as efficiently as possible. == Middleware == Middleware provides all the services and applications necessary for efficient management of datasets and files within the data grid while providing users quick access to the datasets and files. There is a number of concepts and tools that must be available to make a data grid operationally viable. However, at the same time not all data grids require the same capabilities and services because of differences in access requirements, security and location of resources in comparison to users. In any case, most data grids will have similar middleware services that provide for a universal name space, data transport service, data access service, data replication and resource management service. When taken together, they are key to the data grids functional capabilities. === Universal namespace === Since sources of data within the data grid will consist of data from multiple separate systems and networks using different file naming conventions, it would be difficult for a user to locate data within the data grid and know they retrieved what they needed based solely on existing physical file names (PFNs). A universal or unified name space makes it possible to create logical file names (LFNs) that can be referenced within the data grid that map to PFNs. When an LFN is requested or queried, all matching PFNs are returned to include possible replicas of the requested data. The end user can then choose from the returned results the most appropriate replica to use. This service is usually provided as part of a management system known as a Storage Resource Broker (SRB). Information about the locations of files and mappings between the LFNs and PFNs may be stored in a metadata or replica catalogue. The replica catalogue would contain information about LFNs that map to multiple replica PFNs. === Data transport service === Another middleware service is that of providing for data transport or data transfer. Data transport will encompass multiple functions that are not just limited to the transfer of bits, to include such items as fault tolerance and data access. Fault tolerance can be achieved in a data grid by providing mechanisms that ensures data transfer will resume after each interruption until all requested data is received. There are multiple possible methods that might be used to include starting the entire transmission over from the beginning of the data to resuming from where the transfer was interrupted. As an example, GridFTP provides for fault tolerance by sending data from the last acknowledged byte without starting the entire transfer from the beginning. The data transport service also provides for the low-level access and connections between hosts for file transfer. The data transport service may use any number of modes to implement the transfer to include parallel data transfer where two or more data streams are used over the same channel or striped data transfer where two or more steams access different blocks of the file for simultaneous transfer to also using the underlying built-in capabilities of the network hardware or specifically developed protocols to support faster transfer speeds. The data transport service might optionally include a network overlay function to facilitate the routing and transfer of data as well as file I/O functions that allow users to see remote files as if they were local to their system. The data transport service hides the complexity of access and transfer between the different systems to the user so it appears as one unified data source. === Data access service === Data access services work hand in hand with the data transfer service to provide security, access controls and management of any data transfers within the data grid. Security services provide mechanisms for authentication of users to ensure they are properly identified. Common forms of security for authentication can include the use of passwords or Kerberos (protocol). Authorization services are the mechanisms that control what the user is able to access after being identified through authentication. Common forms of authorization mechanisms can be as simple as file permissions. However, need for more stringent controlled access to data is done using Access Control Lists (ACLs), Role-Based Access Control (RBAC) and Tasked-Based Authorization Controls (TBAC). These types of controls can be used to provide granular access to files to include limits on access times, duration of access to granular controls that determine which files can be read or written to. The final data access service that might be present to protect the confidentiality of the data transport is encryption. The most common form of encryption for this task has been the use of SSL while in transport. While all of these access services operate within the data grid, access services within the various administrative domains that host the datasets will still stay in place to enforce access rules. The data grid access services must be in step with the administrative domains access services for this to work. === Data replication service === To meet the needs for scalability, fast access and user collaboration, most data grids support replication of datasets to points within the distributed storage architecture. The use of replicas allows multiple users faster access to datasets and the preservation of bandwidth since replicas can often be placed strategically close to or within sites where users need them. However, replication of datasets and creation of replicas is bound by the availability of storage within sites and bandwidth between sites. The replication and creation of replica datasets is controlled by a replica management system. The replica management system determines user needs for replicas based on input requests and creates them based on availability of storage and bandwidth. All replicas are then cataloged or added to a directory based on the data grid as to their location for query by users. In order to perform the tasks undertaken by the replica management system, it needs to be able to manage the underlying storage infrastructure. The data management system will also ensure the timely updates of changes to replicas are propagated to all nodes. ==== Replication update strategy ==== There are a number of ways the replication management system can handle the updates of replicas. The updates may be designed around a centralized model where a single master replica updates all others, or a decentralized model, where all peers update each other. The topology of node placement may also influence the updates of replicas. If a hierarchy topology is used then updates would flow in a tree like structure through specific paths. In a flat topology it is entirely a matter of the peer relationships between nodes as to how updates take place. In a hybrid topology consisting of both flat and hierarchy topologies updates may take place through specific paths and between peers. ==== Replication placement strategy ==== There are a number of ways the replication management system can handle the creation and placement of replicas to best serve the user community. If the storage architecture supports replica placement with sufficient site storage, then it becomes a matter of the needs of the users who access the datasets and a strategy for placement of replicas. There have been numerous strategies proposed and tested on how to best manage replica placement of datasets within the data grid to meet user requirements. There is not one universal strategy that fits every requirement the best. It is a matter of the type of data grid and user community requirements for access that will determine the best strategy to use. Replicas can even be created where the files are encrypted for confidentiality that would be useful in a research project dealing with medical files. The following section contains several strategies for replica placement. ===== Dynamic replication ===== Dynam