Split (1)

train · 17.8k rows

text stringlengths 12 14.7k
Information gain (decision tree) : In general terms, the expected information gain is the reduction in information entropy Η from a prior state to a state that takes some information as given: I G ( T , a ) = H ( T ) − H ( T \| a ) , -\mathrm , where H ( T \| a ) is the conditional entropy of T given the value of attribute a . This is intuitively plausible when interpreting entropy Η as a measure of uncertainty of a random variable T : by learning (or assuming) a about T , our uncertainty about T is reduced (i.e. I G ( T , a ) is positive), unless of course T is independent of a , in which case H ( T \| a ) = H ( T ) (T\|a)=\mathrm (T) , meaning I G ( T , a ) = 0 .
Information gain (decision tree) : Let T denote a set of training examples, each of the form ( x , y ) = ( x 1 , x 2 , x 3 , . . . , x k , y ) ,y)=(x_,x_,x_,...,x_,y) where x a ∈ v a l s ( a ) \in \mathrm (a) is the value of the a th attribute or feature of example x and y is the corresponding class label. The information gain for an attribute a is defined in terms of Shannon entropy H ( − ) (-) as follows. For a value v taken by attribute a, let S a ( v ) = =\\in T\|x_=v\ be defined as the set of training inputs of T for which attribute a is equal to v. Then the information gain of T for attribute a is the difference between the a priori Shannon entropy H ( T ) (T) of the training set and the conditional entropy H ( T \| a ) . H ( T \| a ) = ∑ v ∈ v a l s ( a ) \| S a ( v ) \| \| T \| ⋅ H ( S a ( v ) ) . (T\|a)=\sum _ (a)\|\cdot \mathrm \left(S_\right). I G ( T , a ) = H ( T ) − H ( T \| a ) (T)-\mathrm (T\|a) The mutual information is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case, the relative entropies subtracted from the total entropy are 0. In particular, the values v ∈ v a l s ( a ) defines a partition of the training set data T into mutually exclusive and all-inclusive subsets, inducing a categorical probability distribution P a ( v ) on the values v ∈ v a l s ( a ) of attribute a. The distribution is given P a ( v ) := \| S a ( v ) \| \| T \| :=\| . In this representation, the information gain of T given a can be defined as the difference between the unconditional Shannon entropy of T and the expected entropy of T conditioned on a, where the expectation value is taken with respect to the induced distribution on the values of a. I G ( T , a ) = H ( T ) − ∑ v ∈ v a l s ( a ) P a ( v ) H ( S a ( v ) ) = H ( T ) − E P a [ H ( S a ( v ) ) ] = H ( T ) − H ( T \| a ) . IG(T,a)&=\mathrm (T)-\sum _ (a)\mathrm \left(S_\right)\\&=\mathrm (T)-\mathbb _ )\right]\\&=\mathrm (T)-\mathrm .\end
Information gain (decision tree) : For a better understanding of information gain, let us break it down. As we know, information gain is the reduction in information entropy, what is entropy? Basically, entropy is the measure of impurity or uncertainty in a group of observations. In engineering applications, information is analogous to signal, and entropy is analogous to noise. It determines how a decision tree chooses to split data. The leftmost figure below is very impure and has high entropy corresponding to higher disorder and lower information value. As we go to the right, the entropy decreases, and the information value increases. Now, it is clear that information gain is the measure of how much information a feature provides about a class. Let's visualize information gain in a decision tree as shown in the right: The node t is the parent node, and the sub-nodes tL and tR are child nodes. In this case, the parent node t has a collection of cancer and non-cancer samples denoted as C and NC respectively. We can use information gain to determine how good the splitting of nodes is in a decision tree. In terms of entropy, information gain is defined as: To understand this idea, let's start by an example in which we create a simple dataset and want to see if gene mutations could be related to patients with cancer. Given four different gene mutations, as well as seven samples, the training set for a decision can be created as follows: In this dataset, a 1 means the sample has the mutation (True), while a 0 means the sample does not (False). A sample with C denotes that it has been confirmed to be cancerous, while NC means it is non-cancerous. Using this data, a decision tree can be created with information gain used to determine the candidate splits for each node. For the next step, the entropy at parent node t of the above simple decision tree is computed as:H(t) = −[pC,t log2(pC,t) + pNC,t log2(pNC,t)] where, probability of selecting a class ‘C’ sample at node t, pC,t = n(t, C) / n(t), probability of selecting a class ‘NC’ sample at node t, pNC,t = n(t, NC) / n(t), n(t), n(t, C), and n(t, NC) are the number of total samples, ‘C’ samples and ‘NC’ samples at node t respectively.Using this with the example training set, the process for finding information gain beginning with H ( t ) for Mutation 1 is as follows: pC, t = 4/7 pNC, t = 3/7 H ( t ) = −(4/7 × log2(4/7) + 3/7 × log2(3/7)) = 0.985 Note: H ( t ) will be the same for all mutations at the root. The relatively high value of entropy H ( t ) = 0.985 =0.985 (1 is the optimal value) suggests that the root node is highly impure and the constituents of the input at the root node would look like the leftmost figure in the above Entropy Diagram. However, such a set of data is good for learning the attributes of the mutations used to split the node. At a certain node, when the homogeneity of the constituents of the input occurs (as shown in the rightmost figure in the above Entropy Diagram), the dataset would no longer be good for learning. Moving on, the entropy at left and right child nodes of the above decision tree is computed using the formulae:H(tL) = −[pC,L log2(pC,L) + pNC,L log2(pNC,L)]H(tR) = −[pC,R log2(pC,R) + pNC,R log2(pNC,R)]where, probability of selecting a class ‘C’ sample at the left child node, pC,L = n(tL, C) / n(tL), probability of selecting a class ‘NC’ sample at the left child node, pNC,L = n(tL, NC) / n(tL), probability of selecting a class ‘C’ sample at the right child node, pC,R = n(tR, C) / n(tR), probability of selecting a class ‘NC’ sample at the right child node, pNC,R = n(tR, NC) / n(tR), n(tL), n(tL, C), and n(tL, NC) are the total number of samples, ‘C’ samples and ‘NC’ samples at the left child node respectively, n(tR), n(tR, C), and n(tR, NC) are the total number of samples, ‘C’ samples and ‘NC’ samples at the right child node respectively. Using these formulae, the H(tL) and H(tR) for Mutation 1 is shown below: H(tL) = −(3/4 × log2(3/4) + 1/4 × log2(1/4)) = 0.811 H(tR) = −(1/3 × log2(1/3) + 2/3 × log2(2/3)) = 0.918 Following this, average entropy of the child nodes due to the split at node t of the above decision tree is computed as:H(s,t) = PLH(tL) + PRH(tR) where, probability of samples at the left child, PL = n(tL) / n(t), probability of samples at the right child, PR = n(tR) / n(t),Finally, H(s,t) along with PL and PR for Mutation 1 is as follows: PL = 4/7 PR = 3/7 H(s, t) = (4/7 × 0.811) + (3/7 × 0.918) = 0.857 Thus, by definition from equation (i):(Information gain) = H(t) - H(s,t) After all the steps, gain(s), where s is a candidate split for the example is: gain(s) = 0.985 – 0.857 = 0.128 Using this same set of formulae with the other three mutations leads to a table of the candidate splits, ranked by their information gain: The mutation that provides the most useful information would be Mutation 3, so that will be used to split the root node of the decision tree. The root can be split and all the samples can be passed though and appended to the child nodes. A tree describing the split is shown on the left. The samples that are on the left node of the tree would be classified as cancerous by the tree, while those on the right would be non-cancerous. This tree is relatively accurate at classifying the samples that were used to build it (which is a case of overfitting), but it would still classify sample C2 incorrectly. To remedy this, the tree can be split again at the child nodes to possibly achieve something even more accurate. To split the right node, information gain must again be calculated for all the possible candidate splits that were not used for previous nodes. So, the only options this time are Mutations 1, 2, and 4. Note: H ( t ) is different this time around since there are only four samples at the right child. PC, t = 1/4 PNC, t = 3/4 H ( t ) = -(1/4 × log2(1/4) + 3/4 × log2(3/4)) = 0.811 From this new H ( t ) , the candidate splits can be calculated using the same formulae as the root node: Thus, the right child will be split with Mutation 4. All the samples that have the mutation will be passed to the left child and the ones that lack it will be passed to the right child. To split the left node, the process would be the same, except there would only be 3 samples to check. Sometimes a node may not need to be split at all if it is a pure set, where all samples at the node are just cancerous or non-cancerous. Splitting the node may lead to the tree being more inaccurate and in this case it will not be split. The tree would now achieve 100% accuracy if the samples that were used to build it are tested. This isn't a good idea, however, since the tree would overfit the data. The best course of action is to try testing the tree on other samples, of which are not part of the original set. Two outside samples are below: By following the tree, NC10 was classified correctly, but C15 was classified as NC. For other samples, this tree would not be 100% accurate anymore. It could be possible to improve this though, with options such as increasing the depth of the tree or increasing the size of the training set.
Information gain (decision tree) : Information gain is the basic criterion to decide whether a feature should be used to split a node or not. The feature with the optimal split i.e., the highest value of information gain at a node of a decision tree is used as the feature for splitting the node. The concept of information gain function falls under the C4.5 algorithm for generating the decision trees and selecting the optimal split for a decision tree node. Some of its advantages include: It can work with both continuous and discrete variables. Due to the factor –[p ∗ log(p)] in the entropy definition as given above, leaf nodes with a small number of instances are assigned less weight and it favors dividing rest of the data into bigger but homogeneous groups. And thus, as we dig deeper into the depth of the tree, the dataset becomes more homogeneous. This approach is usually more stable and chooses the most impactful features on the nodes.
Information gain (decision tree) : Although information gain is usually a good measure for deciding the relevance of an attribute, it is not perfect. A notable problem occurs when information gain is applied to attributes that can take on a large number of distinct values. For example, suppose that one is building a decision tree for some data describing the customers of a business. Information gain is often used to decide which of the attributes are the most relevant, so they can be tested near the root of the tree. One of the input attributes might be the customer's membership number, if they are a member of the business's membership program. This attribute has a high mutual information, because it uniquely identifies each customer, but we do not want to include it in the decision tree. Deciding how to treat a customer based on their membership number is unlikely to generalize to customers we haven't seen before (overfitting). This issue can also occur if the samples that are being tested have multiple attributes with many distinct values. In this case, it can cause the information gain of each of these attributes to be much higher than those without as many distinct values. To counter this problem, Ross Quinlan proposed to instead choose the attribute with highest information gain ratio from among the attributes whose information gain is average or higher. This biases the decision tree against considering attributes with a large number of distinct values, while not giving an unfair advantage to attributes with very low information value, as the information value is higher or equal to the information gain.
Information gain (decision tree) : Information gain more broadly Decision tree learning Information content, the starting point of information theory and the basis of Shannon entropy Information gain ratio ID3 algorithm C4.5 algorithm Surprisal analysis
Information gain (decision tree) : Nowozin, Sebastion (2012-06-18). "Improved Information Gain Estimates for Decision Tree Induction". arXiv:1206.4620v1 [cs.LG]. Shouman, Mai (2011). "Using decision tree for diagnosing heart disease patients" (PDF). Proceedings of the Ninth Australasian Data Mining Conference. 121: 23–30. Mitchell, Tom M. (1997). Machine Learning. The Mc-Graw-Hill Companies, Inc. ISBN 978-0070428072.
Information gain ratio : In decision tree learning, information gain ratio is a ratio of information gain to the intrinsic information. It was proposed by Ross Quinlan, to reduce a bias towards multi-valued attributes by taking the number and size of branches into account when choosing an attribute. Information gain is also known as mutual information.
Information gain ratio : Information gain is the reduction in entropy produced from partitioning a set with attributes a and finding the optimal candidate that produces the highest value: IG ( T , a ) = H ( T ) − H ( T \| a ) , (T,a)=\mathrm -\mathrm , where T is a random variable and H ( T \| a ) is the entropy of T given the value of attribute a . The information gain is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case the relative entropies subtracted from the total entropy are 0.
Information gain ratio : The split information value for a test is defined as follows: SplitInformation ( X ) = − ∑ i = 1 n N ( x i ) N ( x ) ∗ log ⁡ 2 N ( x i ) N ( x ) (X)=-\sum _^ (x_) (x)*\log (x_) (x) where X is a discrete random variable with possible values x 1 , x 2 , . . . , x i ,x_,...,x_ and N ( x i ) ) being the number of times that x i occurs divided by the total count of events N ( x ) where x is the set of events. The split information value is a positive number that describes the potential worth of splitting a branch from a node. This in turn is the intrinsic value that the random variable possesses and will be used to remove the bias in the information gain ratio calculation.
Information gain ratio : The information gain ratio is the ratio between the information gain and the split information value: IGR ( T , a ) = IG ( T , a ) / SplitInformation ( T ) (T,a)=(T,a)/(T) IGR ( T , a ) = − ∑ i = 1 n P ( T ) log ⁡ P ( T ) − ( − ∑ i = 1 n P ( T \| a ) log ⁡ P ( T \| a ) ) − ∑ i = 1 n N ( t i ) N ( t ) ∗ log ⁡ 2 N ( t i ) N ( t ) (T,a)=^ (T)\log \mathrm (T)-(-\sum _^ (T\|a)\log \mathrm (T\|a))^ (t_) (t)*\log (t_) (t)
Information gain ratio : Using weather data published by Fordham University, the table was created below: Using the table above, one can find the entropy, information gain, split information, and information gain ratio for each variable (outlook, temperature, humidity, and wind). These calculations are shown in the tables below: Using the above tables, one can deduce that Outlook has the highest information gain ratio. Next, one must find the statistics for the sub-groups of the Outlook variable (sunny, overcast, and rainy), for this example one will only build the sunny branch (as shown in the table below): One can find the following statistics for the other variables (temperature, humidity, and wind) to see which have the greatest effect on the sunny element of the outlook variable: Humidity was found to have the highest information gain ratio. One will repeat the same steps as before and find the statistics for the events of the Humidity variable (high and normal): Since the play values are either all "No" or "Yes", the information gain ratio value will be equal to 1. Also, now that one has reached the end of the variable chain with Wind being the last variable left, they can build an entire root to leaf node branch line of a decision tree. Once finished with reaching this leaf node, one would follow the same procedure for the rest of the elements that have yet to be split in the decision tree. This set of data was relatively small, however, if a larger set was used, the advantages of using the information gain ratio as the splitting factor of a decision tree can be seen more.
Information gain ratio : Information gain ratio biases the decision tree against considering attributes with a large number of distinct values. For example, suppose that we are building a decision tree for some data describing a business's customers. Information gain ratio is used to decide which of the attributes are the most relevant. These will be tested near the root of the tree. One of the input attributes might be the customer's telephone number. This attribute has a high information gain, because it uniquely identifies each customer. Due to its high amount of distinct values, this will not be chosen to be tested near the root.
Information gain ratio : Although information gain ratio solves the key problem of information gain, it creates another problem. If one is considering an amount of attributes that have a high number of distinct values, these will never be above one that has a lower number of distinct values.
Information gain ratio : Information gain's shortcoming is created by not providing a numerical difference between attributes with high distinct values from those that have less. Example: Suppose that we are building a decision tree for some data describing a business's customers. Information gain is often used to decide which of the attributes are the most relevant, so they can be tested near the root of the tree. One of the input attributes might be the customer's credit card number. This attribute has a high information gain, because it uniquely identifies each customer, but we do not want to include it in the decision tree: deciding how to treat a customer based on their credit card number is unlikely to generalize to customers we haven't seen before. Information gain ratio's strength is that it has a bias towards the attributes with the lower number of distinct values. Below is a table describing the differences of information gain and information gain ratio when put in certain scenarios.
Information gain ratio : Information gain in decision trees Entropy (information theory) == References ==
Jackknife variance estimates for random forest : In statistics, jackknife variance estimates for random forest are a way to estimate the variance in random forest models, in order to eliminate the bootstrap effects.
Jackknife variance estimates for random forest : The sampling variance of bagged learners is: V ( x ) = V a r [ θ ^ ∞ ( x ) ] ^(x)] Jackknife estimates can be considered to eliminate the bootstrap effects. The jackknife variance estimator is defined as: V ^ j = n − 1 n ∑ i = 1 n ( θ ^ ( − i ) − θ ¯ ) 2 _=\sum _^(_-)^ In some classification problems, when random forest is used to fit models, jackknife estimated variance is defined as: V ^ j = n − 1 n ∑ i = 1 n ( t ¯ ( − i ) ⋆ ( x ) − t ¯ ⋆ ( x ) ) 2 _=\sum _^(_^(x)-^(x))^ Here, t ⋆ denotes a decision tree after training, t ( − i ) ⋆ ^ denotes the result based on samples without i t h observation.
Jackknife variance estimates for random forest : E-mail spam problem is a common classification problem, in this problem, 57 features are used to classify spam e-mail and non-spam e-mail. Applying IJ-U variance formula to evaluate the accuracy of models with m=15,19 and 57. The results shows in paper( Confidence Intervals for Random Forests: The jackknife and the Infinitesimal Jackknife ) that m = 57 random forest appears to be quite unstable, while predictions made by m=5 random forest appear to be quite stable, this results is corresponding to the evaluation made by error percentage, in which the accuracy of model with m=5 is high and m=57 is low. Here, accuracy is measured by error rate, which is defined as: E r r o r R a t e = 1 N ∑ i = 1 N ∑ j = 1 M y i j , \sum _^\sum _^y_, Here N is also the number of samples, M is the number of classes, y i j is the indicator function which equals 1 when i t h observation is in class j, equals 0 when in other classes. No probability is considered here. There is another method which is similar to error rate to measure accuracy: l o g l o s s = 1 N ∑ i = 1 N ∑ j = 1 M y i j l o g ( p i j ) \sum _^\sum _^y_log(p_) Here N is the number of samples, M is the number of classes, y i j is the indicator function which equals 1 when i t h observation is in class j, equals 0 when in other classes. p i j is the predicted probability of i t h observation in class j .This method is used in Kaggle These two methods are very similar.
Jackknife variance estimates for random forest : When using Monte Carlo MSEs for estimating V I J ∞ ^ and V J ∞ ^ , a problem about the Monte Carlo bias should be considered, especially when n is large, the bias is getting large: E [ V ^ I J B ] − V ^ I J ∞ ≈ n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B _^]-_^\approx ^(t_^-^)^ To eliminate this influence, bias-corrected modifications are suggested: V ^ I J − U B = V ^ I J B − n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B _^=_^-^(t_^-^)^ V ^ J − U B = V ^ J B − ( e − 1 ) n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B _^=_^-(e-1)^(t_^-^)^ == References ==
Large margin nearest neighbor : Large margin nearest neighbor (LMNN) classification is a statistical machine learning algorithm for metric learning. It learns a pseudometric designed for k-nearest neighbor classification. The algorithm is based on semidefinite programming, a sub-class of convex optimization. The goal of supervised learning (more specifically classification) is to learn a decision rule that can categorize data instances into pre-defined classes. The k-nearest neighbor rule assumes a training data set of labeled instances (i.e. the classes are known). It classifies a new data instance with the class obtained from the majority vote of the k closest (labeled) training instances. Closeness is measured with a pre-defined metric. Large margin nearest neighbors is an algorithm that learns this global (pseudo-)metric in a supervised fashion to improve the classification accuracy of the k-nearest neighbor rule.
Large margin nearest neighbor : The main intuition behind LMNN is to learn a pseudometric under which all data instances in the training set are surrounded by at least k instances that share the same class label. If this is achieved, the leave-one-out error (a special case of cross validation) is minimized. Let the training data consist of a data set D = ⊂ R d × C _,y_),\dots ,(_,y_)\\subset R^\times C , where the set of possible class categories is C = . The algorithm learns a pseudometric of the type d ( x → i , x → j ) = ( x → i − x → j ) ⊤ M ( x → i − x → j ) _,_)=(_-_)^\mathbf (_-_) . For d ( ⋅ , ⋅ ) to be well defined, the matrix M needs to be positive semi-definite. The Euclidean metric is a special case, where M is the identity matrix. This generalization is often (falsely) referred to as Mahalanobis metric. Figure 1 illustrates the effect of the metric under varying M . The two circles show the set of points with equal distance to the center x → i _ . In the Euclidean case this set is a circle, whereas under the modified (Mahalanobis) metric it becomes an ellipsoid. The algorithm distinguishes between two types of special data points: target neighbors and impostors.
Large margin nearest neighbor : Large margin nearest neighbors optimizes the matrix M with the help of semidefinite programming. The objective is twofold: For every data point x → i _ , the target neighbors should be close and the impostors should be far away. Figure 1 shows the effect of such an optimization on an illustrative example. The learned metric causes the input vector x → i _ to be surrounded by training instances of the same class. If it was a test point, it would be classified correctly under the k = 3 nearest neighbor rule. The first optimization goal is achieved by minimizing the average distance between instances and their target neighbors ∑ i , j ∈ N i d ( x → i , x → j ) d(_,_) . The second goal is achieved by penalizing distances to impostors x → l _ that are less than one unit further away than target neighbors x → j _ (and therefore pushing them out of the local neighborhood of x → i _ ). The resulting value to be minimized can be stated as: ∑ i , j ∈ N i , l , y l ≠ y i [ d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ] + ,l,y_\neq y_[d(_,_)+1-d(_,_)]_ With a hinge loss function [ ⋅ ] + = max ( ⋅ , 0 ) =\max(\cdot ,0) , which ensures that impostor proximity is not penalized when outside the margin. The margin of exactly one unit fixes the scale of the matrix M . Any alternative choice c > 0 would result in a rescaling of M by a factor of 1 / c . The final optimization problem becomes: min M ∑ i , j ∈ N i d ( x → i , x → j ) + λ ∑ i , j , l ξ i j l \sum _d(_,_)+\lambda \sum _\xi _ ∀ i , j ∈ N i , l , y l ≠ y i ,l,y_\neq y_ d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ≤ ξ i j l _,_)+1-d(_,_)\leq \xi _ ξ i j l ≥ 0 \geq 0 M ⪰ 0 \succeq 0 The hyperparameter λ > 0 is some positive constant (typically set through cross-validation). Here the variables ξ i j l (together with two types of constraints) replace the term in the cost function. They play a role similar to slack variables to absorb the extent of violations of the impostor constraints. The last constraint ensures that M is positive semi-definite. The optimization problem is an instance of semidefinite programming (SDP). Although SDPs tend to suffer from high computational complexity, this particular SDP instance can be solved very efficiently due to the underlying geometric properties of the problem. In particular, most impostor constraints are naturally satisfied and do not need to be enforced during runtime (i.e. the set of variables ξ i j l is sparse). A particularly well suited solver technique is the working set method, which keeps a small set of constraints that are actively enforced and monitors the remaining (likely satisfied) constraints only occasionally to ensure correctness.
Large margin nearest neighbor : LMNN was extended to multiple local metrics in the 2008 paper. This extension significantly improves the classification error, but involves a more expensive optimization problem. In their 2009 publication in the Journal of Machine Learning Research, Weinberger and Saul derive an efficient solver for the semi-definite program. It can learn a metric for the MNIST handwritten digit data set in several hours, involving billions of pairwise constraints. An open source Matlab implementation is freely available at the authors web page. Kumal et al. extended the algorithm to incorporate local invariances to multivariate polynomial transformations and improved regularization.
Large margin nearest neighbor : Matlab Implementation ICML 2010 Tutorial on Metric Learning
Latent class model : In statistics, a latent class model (LCM) is a model for clustering multivariate discrete data. It assumes that the data arise from a mixture of discrete distributions, within each of which the variables are independent. It is called a latent class model because the class to which each data point belongs is unobserved, or latent. Latent class analysis (LCA) is a subset of structural equation modeling, used to find groups or subtypes of cases in multivariate categorical data. These subtypes are called "latent classes". Confronted with a situation as follows, a researcher might choose to use LCA to understand the data: Imagine that symptoms a-d have been measured in a range of patients with diseases X, Y, and Z, and that disease X is associated with the presence of symptoms a, b, and c, disease Y with symptoms b, c, d, and disease Z with symptoms a, c and d. The LCA will attempt to detect the presence of latent classes (the disease entities), creating patterns of association in the symptoms. As in factor analysis, the LCA can also be used to classify case according to their maximum likelihood class membership. Because the criterion for solving the LCA is to achieve latent classes within which there is no longer any association of one symptom with another (because the class is the disease which causes their association), and the set of diseases a patient has (or class a case is a member of) causes the symptom association, the symptoms will be "conditionally independent", i.e., conditional on class membership, they are no longer related.
Latent class model : Within each latent class, the observed variables are statistically independent. This is an important aspect. Usually the observed variables are statistically dependent. By introducing the latent variable, independence is restored in the sense that within classes variables are independent (local independence). We then say that the association between the observed variables is explained by the classes of the latent variable (McCutcheon, 1987). In one form, the latent class model is written as p i 1 , i 2 , … , i N ≈ ∑ t T p t ∏ n N p i n , t n , ,i_,\ldots ,i_\approx \sum _^p_\,\prod _^p_,t^, where T is the number of latent classes and p t are the so-called recruitment or unconditional probabilities that should sum to one. p i n , t n ,t^ are the marginal or conditional probabilities. For a two-way latent class model, the form is p i j ≈ ∑ t T p t p i t p j t . \approx \sum _^p_\,p_\,p_. This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization. The probability model used in LCA is closely related to the Naive Bayes classifier. The main difference is that in LCA, the class membership of an individual is a latent variable, whereas in Naive Bayes classifiers the class membership is an observed label.
Latent class model : There are a number of methods with distinct names and uses that share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data, and assumes that such data arise from a mixture of distributions: imagine a set of heights arising from a mixture of men and women. If a multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution, it is termed latent profile analysis. Modified to handle discrete data, this constrained analysis is known as LCA. Discrete latent trait models further constrain the classes to form from segments of a single dimension: essentially allocating members to classes on that dimension: an example would be assigning cases to social classes on a dimension of ability or merit. As a practical instance, the variables could be multiple choice items of a political questionnaire. The data in this case consists of a N-way contingency table with answers to the items for a number of respondents. In this example, the latent variable refers to political opinion and the latent classes to political groups. Given group membership, the conditional probabilities specify the chance certain answers are chosen.
Latent class model : LCA may be used in many fields, such as: collaborative filtering, Behavior Genetics and Evaluation of diagnostic tests.
Latent class model : Linda M. Collins; Stephanie T. Lanza (2010). Latent class and latent transition analysis for the social, behavioral, and health sciences. New York: Wiley. ISBN 978-0-470-22839-5. Allan L. McCutcheon (1987). Latent class analysis. Quantitative Applications in the Social Sciences Series No. 64. Thousand Oaks, California: SAGE Publications. ISBN 978-0-521-59451-6. Leo A. Goodman (1974). "Exploratory latent structure analysis using both identifiable and unidentifiable models". Biometrika. 61 (2): 215–231. doi:10.1093/biomet/61.2.215. Paul F. Lazarsfeld, Neil W. Henry (1968). Latent Structure Analysis.
Latent class model : Statistical Innovations, Home Page, 2016. Website with latent class software (Latent GOLD 5.1), free demonstrations, tutorials, user guides, and publications for download. Also included: online courses, FAQs, and other related software. The Methodology Center, Latent Class Analysis, a research center at Penn State, free software, FAQ John Uebersax, Latent Class Analysis, 2006. A web-site with bibliography, software, links and FAQ for latent class analysis
Linear classifier : In machine learning, a linear classifier makes a classification decision for each object based on a linear combination of its features. Such classifiers work well for practical problems such as document classification, and more generally for problems with many variables (features), reaching accuracy levels comparable to non-linear classifiers while taking less time to train and use.
Linear classifier : If the input feature vector to the classifier is a real vector x → , then the output score is y = f ( w → ⋅ x → ) = f ( ∑ j w j x j ) , \cdot )=f\left(\sum _w_x_\right), where w → is a real vector of weights and f is a function that converts the dot product of the two vectors into the desired output. (In other words, w → is a one-form or linear functional mapping x → onto R.) The weight vector w → is learned from a set of labeled training samples. Often f is a threshold function, which maps all values of w → ⋅ x → \cdot above a certain threshold to the first class and all other values to the second class; e.g., f ( x ) = )=1&\ \mathbf ^\cdot \mathbf >\theta ,\\0&\end The superscript T indicates the transpose and θ is a scalar threshold. A more complex f might give the probability that an item belongs to a certain class. For a two-class classification problem, one can visualize the operation of a linear classifier as splitting a high-dimensional input space with a hyperplane: all points on one side of the hyperplane are classified as "yes", while the others are classified as "no". A linear classifier is often used in situations where the speed of classification is an issue, since it is often the fastest classifier, especially when x → is sparse. Also, linear classifiers often work very well when the number of dimensions in x → is large, as in document classification, where each element in x → is typically the number of occurrences of a word in a document (see document-term matrix). In such cases, the classifier should be well-regularized.
Linear classifier : There are two broad classes of methods for determining the parameters of a linear classifier w → . They can be generative and discriminative models. Methods of the former model joint probability distribution, whereas methods of the latter model conditional density functions P ( c l a s s \| x → ) \|) . Examples of such algorithms include: Linear Discriminant Analysis (LDA)—assumes Gaussian conditional density models Naive Bayes classifier with multinomial or multivariate Bernoulli event models. The second set of methods includes discriminative models, which attempt to maximize the quality of the output on a training set. Additional terms in the training cost function can easily perform regularization of the final model. Examples of discriminative training of linear classifiers include: Logistic regression—maximum likelihood estimation of w → assuming that the observed training set was generated by a binomial model that depends on the output of the classifier. Perceptron—an algorithm that attempts to fix all errors encountered in the training set Fisher's Linear Discriminant Analysis—an algorithm (different than "LDA") that maximizes the ratio of between-class scatter to within-class scatter, without any other assumptions. It is in essence a method of dimensionality reduction for binary classification. Support vector machine—an algorithm that maximizes the margin between the decision hyperplane and the examples in the training set. Note: Despite its name, LDA does not belong to the class of discriminative models in this taxonomy. However, its name makes sense when we compare LDA to the other main linear dimensionality reduction algorithm: principal components analysis (PCA). LDA is a supervised learning algorithm that utilizes the labels of the data, while PCA is an unsupervised learning algorithm that ignores the labels. To summarize, the name is a historical artifact. Discriminative training often yields higher accuracy than modeling the conditional density functions. However, handling missing data is often easier with conditional density models. All of the linear classifier algorithms listed above can be converted into non-linear algorithms operating on a different input space φ ( x → ) ) , using the kernel trick.
Linear classifier : Backpropagation Linear regression Perceptron Quadratic classifier Support vector machines Winnow (algorithm)
Linear classifier : Y. Yang, X. Liu, "A re-examination of text categorization", Proc. ACM SIGIR Conference, pp. 42–49, (1999). paper @ citeseer R. Herbrich, "Learning Kernel Classifiers: Theory and Algorithms," MIT Press, (2001). ISBN 0-262-08306-X
Linear discriminant analysis : Linear discriminant analysis (LDA), normal discriminant analysis (NDA), canonical variates analysis (CVA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification. LDA is closely related to analysis of variance (ANOVA) and regression analysis, which also attempt to express one dependent variable as a linear combination of other features or measurements. However, ANOVA uses categorical independent variables and a continuous dependent variable, whereas discriminant analysis has continuous independent variables and a categorical dependent variable (i.e. the class label). Logistic regression and probit regression are more similar to LDA than ANOVA is, as they also explain a categorical variable by the values of continuous independent variables. These other methods are preferable in applications where it is not reasonable to assume that the independent variables are normally distributed, which is a fundamental assumption of the LDA method. LDA is also closely related to principal component analysis (PCA) and factor analysis in that they both look for linear combinations of variables which best explain the data. LDA explicitly attempts to model the difference between the classes of data. PCA, in contrast, does not take into account any difference in class, and factor analysis builds the feature combinations based on differences rather than similarities. Discriminant analysis is also different from factor analysis in that it is not an interdependence technique: a distinction between independent variables and dependent variables (also called criterion variables) must be made. LDA works when the measurements made on independent variables for each observation are continuous quantities. When dealing with categorical independent variables, the equivalent technique is discriminant correspondence analysis. Discriminant analysis is used when groups are known a priori (unlike in cluster analysis). Each case must have a score on one or more quantitative predictor measures, and a score on a group measure. In simple terms, discriminant function analysis is classification - the act of distributing things into groups, classes or categories of the same type.
Linear discriminant analysis : The original dichotomous discriminant analysis was developed by Sir Ronald Fisher in 1936. It is different from an ANOVA or MANOVA, which is used to predict one (ANOVA) or multiple (MANOVA) continuous dependent variables by one or more independent categorical variables. Discriminant function analysis is useful in determining whether a set of variables is effective in predicting category membership.
Linear discriminant analysis : Consider a set of observations x → (also called features, attributes, variables or measurements) for each sample of an object or event with known class y . This set of samples is called the training set in a supervised learning context. The classification problem is then to find a good predictor for the class y of any sample of the same distribution (not necessarily from the training set) given only an observation x → .: 338 LDA approaches the problem by assuming that the conditional probability density functions p ( x → \| y = 0 ) \|y=0) and p ( x → \| y = 1 ) \|y=1) are both the normal distribution with mean and covariance parameters ( μ → 0 , Σ 0 ) _,\Sigma _\right) and ( μ → 1 , Σ 1 ) _,\Sigma _\right) , respectively. Under this assumption, the Bayes-optimal solution is to predict points as being from the second class if the log of the likelihood ratios is bigger than some threshold T, so that: 1 2 ( x → − μ → 0 ) T Σ 0 − 1 ( x → − μ → 0 ) + 1 2 ln ⁡ \| Σ 0 \| − 1 2 ( x → − μ → 1 ) T Σ 1 − 1 ( x → − μ → 1 ) − 1 2 ln ⁡ \| Σ 1 \| > T (-_)^ \Sigma _^(-_)+\ln \|\Sigma _\|-(-_)^ \Sigma _^(-_)-\ln \|\Sigma _\|\ >\ T Without any further assumptions, the resulting classifier is referred to as quadratic discriminant analysis (QDA). LDA instead makes the additional simplifying homoscedasticity assumption (i.e. that the class covariances are identical, so Σ 0 = Σ 1 = Σ =\Sigma _=\Sigma ) and that the covariances have full rank. In this case, several terms cancel: x → T Σ 0 − 1 x → = x → T Σ 1 − 1 x → ^ \Sigma _^=^ \Sigma _^ x → T Σ i − 1 μ → i = μ → i T Σ i − 1 x → ^ ^_=_^ ^ because Σ i is Hermitian and the above decision criterion becomes a threshold on the dot product w → T x → > c ^ >c for some threshold constant c, where w → = Σ − 1 ( μ → 1 − μ → 0 ) =\Sigma ^(_-_) c = 1 2 w → T ( μ → 1 + μ → 0 ) \,^ (_+_) This means that the criterion of an input x → being in a class y is purely a function of this linear combination of the known observations. It is often useful to see this conclusion in geometrical terms: the criterion of an input x → being in a class y is purely a function of projection of multidimensional-space point x → onto vector w → (thus, we only consider its direction). In other words, the observation belongs to y if corresponding x → is located on a certain side of a hyperplane perpendicular to w → . The location of the plane is defined by the threshold c .
Linear discriminant analysis : The assumptions of discriminant analysis are the same as those for MANOVA. The analysis is quite sensitive to outliers and the size of the smallest group must be larger than the number of predictor variables. Multivariate normality: Independent variables are normal for each level of the grouping variable. Homogeneity of variance/covariance (homoscedasticity): Variances among group variables are the same across levels of predictors. Can be tested with Box's M statistic. It has been suggested, however, that linear discriminant analysis be used when covariances are equal, and that quadratic discriminant analysis may be used when covariances are not equal. Independence: Participants are assumed to be randomly sampled, and a participant's score on one variable is assumed to be independent of scores on that variable for all other participants. It has been suggested that discriminant analysis is relatively robust to slight violations of these assumptions, and it has also been shown that discriminant analysis may still be reliable when using dichotomous variables (where multivariate normality is often violated).
Linear discriminant analysis : Discriminant analysis works by creating one or more linear combinations of predictors, creating a new latent variable for each function. These functions are called discriminant functions. The number of functions possible is either N g − 1 -1 where N g = number of groups, or p (the number of predictors), whichever is smaller. The first function created maximizes the differences between groups on that function. The second function maximizes differences on that function, but also must not be correlated with the previous function. This continues with subsequent functions with the requirement that the new function not be correlated with any of the previous functions. Given group j , with R j _ sets of sample space, there is a discriminant rule such that if x ∈ R j _ , then x ∈ j . Discriminant analysis then, finds “good” regions of R j _ to minimize classification error, therefore leading to a high percent correct classified in the classification table. Each function is given a discriminant score to determine how well it predicts group placement. Structure Correlation Coefficients: The correlation between each predictor and the discriminant score of each function. This is a zero-order correlation (i.e., not corrected for the other predictors). Standardized Coefficients: Each predictor's weight in the linear combination that is the discriminant function. Like in a regression equation, these coefficients are partial (i.e., corrected for the other predictors). Indicates the unique contribution of each predictor in predicting group assignment. Functions at Group Centroids: Mean discriminant scores for each grouping variable are given for each function. The farther apart the means are, the less error there will be in classification.
Linear discriminant analysis : Maximum likelihood: Assigns x to the group that maximizes population (group) density. Bayes Discriminant Rule: Assigns x to the group that maximizes π i f i ( x ) f_(x) , where πi represents the prior probability of that classification, and f i ( x ) (x) represents the population density. Fisher's linear discriminant rule: Maximizes the ratio between SSbetween and SSwithin, and finds a linear combination of the predictors to predict group.
Linear discriminant analysis : An eigenvalue in discriminant analysis is the characteristic root of each function. It is an indication of how well that function differentiates the groups, where the larger the eigenvalue, the better the function differentiates. This however, should be interpreted with caution, as eigenvalues have no upper limit. The eigenvalue can be viewed as a ratio of SSbetween and SSwithin as in ANOVA when the dependent variable is the discriminant function, and the groups are the levels of the IV. This means that the largest eigenvalue is associated with the first function, the second largest with the second, etc..
Linear discriminant analysis : Some suggest the use of eigenvalues as effect size measures, however, this is generally not supported. Instead, the canonical correlation is the preferred measure of effect size. It is similar to the eigenvalue, but is the square root of the ratio of SSbetween and SStotal. It is the correlation between groups and the function. Another popular measure of effect size is the percent of variance for each function. This is calculated by: (λx/Σλi) X 100 where λx is the eigenvalue for the function and Σλi is the sum of all eigenvalues. This tells us how strong the prediction is for that particular function compared to the others. Percent correctly classified can also be analyzed as an effect size. The kappa value can describe this while correcting for chance agreement.Kappa normalizes across all categorizes rather than biased by a significantly good or poorly performing classes.
Linear discriminant analysis : Canonical discriminant analysis (CDA) finds axes (k − 1 canonical coordinates, k being the number of classes) that best separate the categories. These linear functions are uncorrelated and define, in effect, an optimal k − 1 space through the n-dimensional cloud of data that best separates (the projections in that space of) the k groups. See “Multiclass LDA” for details below. Because LDA uses canonical variates, it was initially often referred as the "method of canonical variates" or canonical variates analysis (CVA).
Linear discriminant analysis : The terms Fisher's linear discriminant and LDA are often used interchangeably, although Fisher's original article actually describes a slightly different discriminant, which does not make some of the assumptions of LDA such as normally distributed classes or equal class covariances. Suppose two classes of observations have means μ → 0 , μ → 1 _,_ and covariances Σ 0 , Σ 1 ,\Sigma _ . Then the linear combination of features w → T x → ^ will have means w → T μ → i ^ _ and variances w → T Σ i w → ^ \Sigma _ for i = 0 , 1 . Fisher defined the separation between these two distributions to be the ratio of the variance between the classes to the variance within the classes: S = σ between 2 σ within 2 = ( w → ⋅ μ → 1 − w → ⋅ μ → 0 ) 2 w → T Σ 1 w → + w → T Σ 0 w → = ( w → ⋅ ( μ → 1 − μ → 0 ) ) 2 w → T ( Σ 0 + Σ 1 ) w → ^^=\cdot _-\cdot _)^^ \Sigma _+^ \Sigma _=\cdot (_-_))^^ (\Sigma _+\Sigma _) This measure is, in some sense, a measure of the signal-to-noise ratio for the class labelling. It can be shown that the maximum separation occurs when w → ∝ ( Σ 0 + Σ 1 ) − 1 ( μ → 1 − μ → 0 ) \propto (\Sigma _+\Sigma _)^(_-_) When the assumptions of LDA are satisfied, the above equation is equivalent to LDA. Be sure to note that the vector w → is the normal to the discriminant hyperplane. As an example, in a two dimensional problem, the line that best divides the two groups is perpendicular to w → . Generally, the data points to be discriminated are projected onto w → ; then the threshold that best separates the data is chosen from analysis of the one-dimensional distribution. There is no general rule for the threshold. However, if projections of points from both classes exhibit approximately the same distributions, a good choice would be the hyperplane between projections of the two means, w → ⋅ μ → 0 \cdot _ and w → ⋅ μ → 1 \cdot _ . In this case the parameter c in threshold condition w → ⋅ x → > c \cdot >c can be found explicitly: c = w → ⋅ 1 2 ( μ → 0 + μ → 1 ) = 1 2 μ → 1 T Σ 1 − 1 μ → 1 − 1 2 μ → 0 T Σ 0 − 1 μ → 0 \cdot (_+_)=_^ \Sigma _^_-_^ \Sigma _^_ . Otsu's method is related to Fisher's linear discriminant, and was created to binarize the histogram of pixels in a grayscale image by optimally picking the black/white threshold that minimizes intra-class variance and maximizes inter-class variance within/between grayscales assigned to black and white pixel classes.
Linear discriminant analysis : In the case where there are more than two classes, the analysis used in the derivation of the Fisher discriminant can be extended to find a subspace which appears to contain all of the class variability. This generalization is due to C. R. Rao. Suppose that each of C classes has a mean μ i and the same covariance Σ . Then the scatter between class variability may be defined by the sample covariance of the class means Σ b = 1 C ∑ i = 1 C ( μ i − μ ) ( μ i − μ ) T =\sum _^(\mu _-\mu )(\mu _-\mu )^ where μ is the mean of the class means. The class separation in a direction w → in this case will be given by S = w → T Σ b w → w → T Σ w → ^ \Sigma _^ \Sigma This means that when w → is an eigenvector of Σ − 1 Σ b \Sigma _ the separation will be equal to the corresponding eigenvalue. If Σ − 1 Σ b \Sigma _ is diagonalizable, the variability between features will be contained in the subspace spanned by the eigenvectors corresponding to the C − 1 largest eigenvalues (since Σ b is of rank C − 1 at most). These eigenvectors are primarily used in feature reduction, as in PCA. The eigenvectors corresponding to the smaller eigenvalues will tend to be very sensitive to the exact choice of training data, and it is often necessary to use regularisation as described in the next section. If classification is required, instead of dimension reduction, there are a number of alternative techniques available. For instance, the classes may be partitioned, and a standard Fisher discriminant or LDA used to classify each partition. A common example of this is "one against the rest" where the points from one class are put in one group, and everything else in the other, and then LDA applied. This will result in C classifiers, whose results are combined. Another common method is pairwise classification, where a new classifier is created for each pair of classes (giving C(C − 1)/2 classifiers in total), with the individual classifiers combined to produce a final classification.
Linear discriminant analysis : The typical implementation of the LDA technique requires that all the samples are available in advance. However, there are situations where the entire data set is not available and the input data are observed as a stream. In this case, it is desirable for the LDA feature extraction to have the ability to update the computed LDA features by observing the new samples without running the algorithm on the whole data set. For example, in many real-time applications such as mobile robotics or on-line face recognition, it is important to update the extracted LDA features as soon as new observations are available. An LDA feature extraction technique that can update the LDA features by simply observing new samples is an incremental LDA algorithm, and this idea has been extensively studied over the last two decades. Chatterjee and Roychowdhury proposed an incremental self-organized LDA algorithm for updating the LDA features. In other work, Demir and Ozmehmet proposed online local learning algorithms for updating LDA features incrementally using error-correcting and the Hebbian learning rules. Later, Aliyari et al. derived fast incremental algorithms to update the LDA features by observing the new samples.
Linear discriminant analysis : In practice, the class means and covariances are not known. They can, however, be estimated from the training set. Either the maximum likelihood estimate or the maximum a posteriori estimate may be used in place of the exact value in the above equations. Although the estimates of the covariance may be considered optimal in some sense, this does not mean that the resulting discriminant obtained by substituting these values is optimal in any sense, even if the assumption of normally distributed classes is correct. Another complication in applying LDA and Fisher's discriminant to real data occurs when the number of measurements of each sample (i.e., the dimensionality of each data vector) exceeds the number of samples in each class. In this case, the covariance estimates do not have full rank, and so cannot be inverted. There are a number of ways to deal with this. One is to use a pseudo inverse instead of the usual matrix inverse in the above formulae. However, better numeric stability may be achieved by first projecting the problem onto the subspace spanned by Σ b . Another strategy to deal with small sample size is to use a shrinkage estimator of the covariance matrix, which can be expressed mathematically as Σ = ( 1 − λ ) Σ + λ I where I is the identity matrix, and λ is the shrinkage intensity or regularisation parameter. This leads to the framework of regularized discriminant analysis or shrinkage discriminant analysis. Also, in many practical cases linear discriminants are not suitable. LDA and Fisher's discriminant can be extended for use in non-linear classification via the kernel trick. Here, the original observations are effectively mapped into a higher dimensional non-linear space. Linear classification in this non-linear space is then equivalent to non-linear classification in the original space. The most commonly used example of this is the kernel Fisher discriminant. LDA can be generalized to multiple discriminant analysis, where c becomes a categorical variable with N possible states, instead of only two. Analogously, if the class-conditional densities p ( x → ∣ c = i ) \mid c=i) are normal with shared covariances, the sufficient statistic for P ( c ∣ x → ) ) are the values of N projections, which are the subspace spanned by the N means, affine projected by the inverse covariance matrix. These projections can be found by solving a generalized eigenvalue problem, where the numerator is the covariance matrix formed by treating the means as the samples, and the denominator is the shared covariance matrix. See “Multiclass LDA” above for details.
Linear discriminant analysis : In addition to the examples given below, LDA is applied in positioning and product management.
Linear discriminant analysis : Discriminant function analysis is very similar to logistic regression, and both can be used to answer the same research questions. Logistic regression does not have as many assumptions and restrictions as discriminant analysis. However, when discriminant analysis’ assumptions are met, it is more powerful than logistic regression. Unlike logistic regression, discriminant analysis can be used with small sample sizes. It has been shown that when sample sizes are equal, and homogeneity of variance/covariance holds, discriminant analysis is more accurate. Despite all these advantages, logistic regression has none-the-less become the common choice, since the assumptions of discriminant analysis are rarely met.
Linear discriminant analysis : Geometric anomalies in higher dimensions lead to the well-known curse of dimensionality. Nevertheless, proper utilization of concentration of measure phenomena can make computation easier. An important case of these blessing of dimensionality phenomena was highlighted by Donoho and Tanner: if a sample is essentially high-dimensional then each point can be separated from the rest of the sample by linear inequality, with high probability, even for exponentially large samples. These linear inequalities can be selected in the standard (Fisher's) form of the linear discriminant for a rich family of probability distribution. In particular, such theorems are proven for log-concave distributions including multidimensional normal distribution (the proof is based on the concentration inequalities for log-concave measures) and for product measures on a multidimensional cube (this is proven using Talagrand's concentration inequality for product probability spaces). Data separability by classical linear discriminants simplifies the problem of error correction for artificial intelligence systems in high dimension.
Linear discriminant analysis : Data mining Decision tree learning Factor analysis Kernel Fisher discriminant analysis Logit (for logistic regression) Linear regression Multiple discriminant analysis Multidimensional scaling Pattern recognition Preference regression Quadratic classifier Statistical classification
Linear discriminant analysis : Duda, R. O.; Hart, P. E.; Stork, D. H. (2000). Pattern Classification (2nd ed.). Wiley Interscience. ISBN 978-0-471-05669-0. MR 1802993. Hilbe, J. M. (2009). Logistic Regression Models. Chapman & Hall/CRC Press. ISBN 978-1-4200-7575-5. Mika, S.; et al. (1999). "Fisher discriminant analysis with kernels". Neural Networks for Signal Processing IX: Proceedings of the 1999 IEEE Signal Processing Society Workshop (Cat. No.98TH8468). pp. 41–48. CiteSeerX 10.1.1.35.9904. doi:10.1109/NNSP.1999.788121. ISBN 978-0-7803-5673-3. S2CID 8473401. McFarland, H. Richard; Donald, St. P. Richards (2001). "Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. I. The Equal-Means Case". Journal of Multivariate Analysis. 77 (1): 21–53. doi:10.1006/jmva.2000.1924. McFarland, H. Richard; Donald, St. P. Richards (2002). "Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. II. The Heterogeneous Case". Journal of Multivariate Analysis. 82 (2): 299–330. doi:10.1006/jmva.2001.2034. Haghighat, M.; Abdel-Mottaleb, M.; Alhalabi, W. (2016). "Discriminant Correlation Analysis: Real-Time Feature Level Fusion for Multimodal Biometric Recognition". IEEE Transactions on Information Forensics and Security. 11 (9): 1984–1996. doi:10.1109/TIFS.2016.2569061. S2CID 15624506.
Linear discriminant analysis : Discriminant Correlation Analysis (DCA) of the Haghighat article (see above) ALGLIB contains open-source LDA implementation in C# / C++ / Pascal / VBA. LDA in Python- LDA implementation in Python LDA tutorial using MS Excel Biomedical statistics. Discriminant analysis StatQuest: Linear Discriminant Analysis (LDA) clearly explained on YouTube Course notes, Discriminant function analysis by G. David Garson, NC State University Discriminant analysis tutorial in Microsoft Excel by Kardi Teknomo Course notes, Discriminant function analysis by David W. Stockburger, Missouri State University Archived 2016-03-03 at the Wayback Machine Discriminant function analysis (DA) by John Poulsen and Aaron French, San Francisco State University
Margin classifier : In machine learning (ML), a margin classifier is a type of classification model which is able to give an associated distance from the decision boundary for each data sample. For instance, if a linear classifier is used, the distance (typically Euclidean, though others may be used) of a sample from the separating hyperplane is the margin of that sample. The notion of margins is important in several ML classification algorithms, as it can be used to bound the generalization error of these classifiers. These bounds are frequently shown using the VC dimension. The generalization error bound in boosting algorithms and support vector machines is particularly prominent.
Margin classifier : The margin for an iterative boosting algorithm given a dataset with two classes can be defined as follows: the classifier is given a sample pair ( x , y ) , where x ∈ X is a domain space and y ∈ Y = is the sample's label. The algorithm then selects a classifier h j ∈ C \in C at each iteration j where C is a space of possible classifiers that predict real values. This hypothesis is then weighted by α j ∈ R \in R as selected by the boosting algorithm. At iteration t , the margin of a sample x can thus be defined as y ∑ j t α j h j ( x ) ∑ \| α j \| . ^\alpha _h_(x)\|. By this definition, the margin is positive if the sample is labeled correctly, or negative if the sample is labeled incorrectly. This definition may be modified and is not the only way to define the margin for boosting algorithms. However, there are reasons why this definition may be appealing.
Margin classifier : Many classifiers can give an associated margin for each sample. However, only some classifiers utilize information of the margin while learning from a dataset. Many boosting algorithms rely on the notion of a margin to assign weight to samples. If a convex loss is utilized (as in AdaBoost or LogitBoost, for instance) then a sample with a higher margin will receive less (or equal) weight than a sample with a lower margin. This leads the boosting algorithm to focus weight on low-margin samples. In non-convex algorithms (e.g., BrownBoost), the margin still dictates the weighting of a sample, though the weighting is non-monotone with respect to the margin.
Margin classifier : One theoretical motivation behind margin classifiers is that their generalization error may be bound by the algorithm parameters and a margin term. An example of such a bound is for the AdaBoost algorithm. Let S be a set of m data points, sampled independently at random from a distribution D . Assume the VC-dimension of the underlying base classifier is d and m ≥ d ≥ 1 . Then, with probability 1 − δ , we have the bound: P D ( y ∑ j t α j h j ( x ) ∑ \| α j \| ≤ 0 ) ≤ P S ( y ∑ j t α j h j ( x ) ∑ \| α j \| ≤ θ ) + O ( 1 m d log 2 ⁡ ( m / d ) / θ 2 + log ⁡ ( 1 / δ ) ) \left(^\alpha _h_(x)\|\leq 0\right)\leq P_\left(^\alpha _h_(x)\|\leq \theta \right)+O\left((m/d)/\theta ^+\log(1/\delta )\right) for all θ > 0 . == References ==
Margin-infused relaxed algorithm : Margin-infused relaxed algorithm (MIRA) is a machine learning algorithm, an online algorithm for multiclass classification problems. It is designed to learn a set of parameters (vector or matrix) by processing all the given training examples one-by-one and updating the parameters according to each training example, so that the current training example is classified correctly with a margin against incorrect classifications at least as large as their loss. The change of the parameters is kept as small as possible. A two-class version called binary MIRA simplifies the algorithm by not requiring the solution of a quadratic programming problem (see below). When used in a one-vs-all configuration, binary MIRA can be extended to a multiclass learner that approximates full MIRA, but may be faster to train. The flow of the algorithm looks as follows: The update step is then formalized as a quadratic programming problem: Find m i n ‖ w ( i + 1 ) − w ( i ) ‖ -w^\\| , so that s c o r e ( x t , y t ) − s c o r e ( x t , y ′ ) ≥ L ( y t , y ′ ) ∀ y ′ ,y_)-score(x_,y')\geq L(y_,y')\ \forall y' , i.e. the score of the current correct training y must be greater than the score of any other possible y ′ by at least the loss (number of errors) of that y ′ in comparison to y .
Margin-infused relaxed algorithm : adMIRAble – MIRA implementation in C++ Miralium – MIRA implementation in Java MIRA implementation for Mahout in Hadoop
Multi-label classification : In machine learning, multi-label classification or multi-output classification is a variant of the classification problem where multiple nonexclusive labels may be assigned to each instance. Multi-label classification is a generalization of multiclass classification, which is the single-label problem of categorizing instances into precisely one of several (greater than or equal to two) classes. In the multi-label problem the labels are nonexclusive and there is no constraint on how many of the classes the instance can be assigned to. The formulation of multi-label learning was first introduced by Shen et al. in the context of Semantic Scene Classification, and later gained popularity across various areas of machine learning. Formally, multi-label classification is the problem of finding a model that maps inputs x to binary vectors y; that is, it assigns a value of 0 or 1 for each element (label) in y.
Multi-label classification : Several problem transformation methods exist for multi-label classification, and can be roughly broken down into:
Multi-label classification : Some classification algorithms/models have been adapted to the multi-label task, without requiring problem transformations. Examples of these including for multi-label data are k-nearest neighbors: the ML-kNN algorithm extends the k-NN classifier to multi-label data. decision trees: "Clare" is an adapted C4.5 algorithm for multi-label classification; the modification involves the entropy calculations. MMC, MMDT, and SSC refined MMDT, can classify multi-labeled data based on multi-valued attributes without transforming the attributes into single-values. They are also named multi-valued and multi-labeled decision tree classification methods. kernel methods for vector output neural networks: BP-MLL is an adaptation of the popular back-propagation algorithm for multi-label learning.
Multi-label classification : Based on learning paradigms, the existing multi-label classification techniques can be classified into batch learning and online machine learning. Batch learning algorithms require all the data samples to be available beforehand. It trains the model using the entire training data and then predicts the test sample using the found relationship. The online learning algorithms, on the other hand, incrementally build their models in sequential iterations. In iteration t, an online algorithm receives a sample, xt and predicts its label(s) ŷt using the current model; the algorithm then receives yt, the true label(s) of xt and updates its model based on the sample-label pair: (xt, yt).
Multi-label classification : Data streams are possibly infinite sequences of data that continuously and rapidly grow over time. Multi-label stream classification (MLSC) is the version of multi-label classification task that takes place in data streams. It is sometimes also called online multi-label classification. The difficulties of multi-label classification (exponential number of possible label sets, capturing dependencies between labels) are combined with difficulties of data streams (time and memory constraints, addressing infinite stream with finite means, concept drifts). Many MLSC methods resort to ensemble methods in order to increase their predictive performance and deal with concept drifts. Below are the most widely used ensemble methods in the literature: Online Bagging (OzaBagging)-based methods: Observing the probability of having K many of a certain data point in a bootstrap sample is approximately Poisson(1) for big datasets, each incoming data instance in a data stream can be weighted proportional to Poisson(1) distribution to mimic bootstrapping in an online setting. This is called Online Bagging (OzaBagging). Many multi-label methods that use Online Bagging are proposed in the literature, each of which utilizes different problem transformation methods. EBR, ECC, EPS, EBRT, EBMT, ML-Random Rules are examples of such methods. ADWIN Bagging-based methods: Online Bagging methods for MLSC are sometimes combined with explicit concept drift detection mechanisms such as ADWIN (Adaptive Window). ADWIN keeps a variable-sized window to detect changes in the distribution of the data, and improves the ensemble by resetting the components that perform poorly when there is a drift in the incoming data. Generally, the letter 'a' is used as a subscript in the name of such ensembles to indicate the usage of ADWIN change detector. EaBR, EaCC, EaHTPS are examples of such multi-label ensembles. GOOWE-ML-based methods: Interpreting the relevance scores of each component of the ensemble as vectors in the label space and solving a least squares problem at the end of each batch, Geometrically-Optimum Online-Weighted Ensemble for Multi-label Classification (GOOWE-ML) is proposed. The ensemble tries to minimize the distance between the weighted prediction of its components and the ground truth vector for each instance over a batch. Unlike Online Bagging and ADWIN Bagging, GOOWE-ML utilizes a weighted voting scheme where better performing components of the ensemble are given more weight. The GOOWE-ML ensemble grows over time, and the lowest weight component is replaced by a new component when it is full at the end of a batch. GOBR, GOCC, GOPS, GORT are the proposed GOOWE-ML-based multi-label ensembles. Multiple Windows : Here, BR models that use a sliding window are replaced with two windows for each label, one for relevant and one for non-relevant examples. Instances are oversampled or undersampled according to a load factor that is kept between these two windows. This allows concept drifts that are independent for each label to be detected, and class-imbalance (skewness in the relevant and non-relevant examples) to be handled.
Multi-label classification : Considering Y i to be a set of labels for i t h data sample (do not confuse it with a one-hot vector; it is simply a collection of all of the labels that belong to this sample), the extent to which a dataset is multi-label can be captured in two statistics: Label cardinality is the average number of labels per example in the set: 1 N ∑ i = 1 N \| Y i \| \sum _^\|Y_\| where N is the total number of data samples; Label density is the number of labels per sample divided by the total number of labels, averaged over the samples: 1 N ∑ i = 1 N \| Y i \| \| L \| \sum _^\| where L = ⋃ i = 1 N Y i ^Y_ , the total number of available classes (which is the maximum number of elements that can make up Y i ). Evaluation metrics for multi-label classification performance are inherently different from those used in multi-class (or binary) classification, due to the inherent differences of the classification problem. If T denotes the true set of labels for a given sample, and P the predicted set of labels, then the following metrics can be defined on that sample: Hamming loss: the fraction of the wrong labels to the total number of labels, i.e. 1 \| N \| ⋅ \| L \| ∑ i = 1 \| N \| ∑ j = 1 \| L \| xor ⁡ ( y i , j , z i , j ) \sum _^\sum _^\operatorname (y_,z_) , where y i , j is the target, z i , j is the prediction, and xor ⁡ ( ⋅ ) (\cdot ) is the "Exclusive, or" operator that returns zero when the target and prediction are identical and one otherwise. This is a loss function, so the optimal value is zero and its upper bound is one. The closely related Jaccard index, also called Intersection over Union in the multi-label setting, is defined as the number of correctly predicted labels divided by the union of predicted and true labels, \| T ∩ P \| \| T ∪ P \| , where P and T are sets of predicted labels and true labels respectively. Precision, recall and F 1 score: precision is \| T ∩ P \| \| P \| , recall is \| T ∩ P \| \| T \| , and F 1 is their harmonic mean. Exact match (also called Subset accuracy): is the most strict metric, indicating the percentage of samples that have all their labels classified correctly. Cross-validation in multi-label settings is complicated by the fact that the ordinary (binary/multiclass) way of stratified sampling will not work; alternative ways of approximate stratified sampling have been suggested.
Multi-label classification : Java implementations of multi-label algorithms are available in the Mulan and Meka software packages, both based on Weka. The scikit-learn Python package implements some multi-labels algorithms and metrics. The scikit-multilearn Python package specifically caters to the multi-label classification. It provides multi-label implementation of several well-known techniques including SVM, kNN and many more. The package is built on top of scikit-learn ecosystem. The binary relevance method, classifier chains and other multilabel algorithms with a lot of different base learners are implemented in the R-package mlr A list of commonly used multi-label data-sets is available at the Mulan website.
Multi-label classification : Multiclass classification Multiple-instance learning Structured prediction Life-time of correlation
Multi-label classification : Madjarov, Gjorgji; Kocev, Dragi; Gjorgjevikj, Dejan; Džeroski, Sašo (2012). "An extensive experimental comparison of methods for multi-label learning". Pattern Recognition. 45 (9): 3084–3104. Bibcode:2012PatRe..45.3084M. doi:10.1016/j.patcog.2012.03.004. S2CID 14064264.
Multiclass classification : In machine learning and statistical classification, multiclass classification or multinomial classification is the problem of classifying instances into one of three or more classes (classifying instances into one of two classes is called binary classification). For example, deciding on whether an image is showing a banana, peach, orange, or an apple is a multiclass classification problem, with four possible classes (banana, peach, orange, apple), while deciding on whether an image contains an apple or not is a binary classification problem (with the two possible classes being: apple, no apple). While many classification algorithms (notably multinomial logistic regression) naturally permit the use of more than two classes, some are by nature binary algorithms; these can, however, be turned into multinomial classifiers by a variety of strategies. Multiclass classification should not be confused with multi-label classification, where multiple labels are to be predicted for each instance (e.g., predicting that an image contains both an apple and an orange, in the previous example).
Multiclass classification : The existing multi-class classification techniques can be categorised into transformation to binary extension from binary hierarchical classification.
Multiclass classification : Based on learning paradigms, the existing multi-class classification techniques can be classified into batch learning and online learning. Batch learning algorithms require all the data samples to be available beforehand. It trains the model using the entire training data and then predicts the test sample using the found relationship. The online learning algorithms, on the other hand, incrementally build their models in sequential iterations. In iteration t, an online algorithm receives a sample, xt and predicts its label ŷt using the current model; the algorithm then receives yt, the true label of xt and updates its model based on the sample-label pair: (xt, yt). Recently, a new learning paradigm called progressive learning technique has been developed. The progressive learning technique is capable of not only learning from new samples but also capable of learning new classes of data and yet retain the knowledge learnt thus far.
Multiclass classification : The performance of a multi-class classification system is often assessed by comparing the predictions of the system against reference labels with an evaluation metric. Common evaluation metrics are Accuracy or macro F1.
Multiclass classification : Binary classification One-class classification Multi-label classification Multiclass perceptron Multi-task learning
Multinomial logistic regression : In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.). Multinomial logistic regression is known by a variety of other names, including polytomous LR, multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model.
Multinomial logistic regression : Multinomial logistic regression is used when the dependent variable in question is nominal (equivalently categorical, meaning that it falls into any one of a set of categories that cannot be ordered in any meaningful way) and for which there are more than two categories. Some examples would be: Which major will a college student choose, given their grades, stated likes and dislikes, etc.? Which blood type does a person have, given the results of various diagnostic tests? In a hands-free mobile phone dialing application, which person's name was spoken, given various properties of the speech signal? Which candidate will a person vote for, given particular demographic characteristics? Which country will a firm locate an office in, given the characteristics of the firm and of the various candidate countries? These are all statistical classification problems. They all have in common a dependent variable to be predicted that comes from one of a limited set of items that cannot be meaningfully ordered, as well as a set of independent variables (also known as features, explanators, etc.), which are used to predict the dependent variable. Multinomial logistic regression is a particular solution to classification problems that use a linear combination of the observed features and some problem-specific parameters to estimate the probability of each particular value of the dependent variable. The best values of the parameters for a given problem are usually determined from some training data (e.g. some people for whom both the diagnostic test results and blood types are known, or some examples of known words being spoken).
Multinomial logistic regression : The multinomial logistic model assumes that data are case-specific; that is, each independent variable has a single value for each case. As with other types of regression, there is no need for the independent variables to be statistically independent from each other (unlike, for example, in a naive Bayes classifier); however, collinearity is assumed to be relatively low, as it becomes difficult to differentiate between the impact of several variables if this is not the case. If the multinomial logit is used to model choices, it relies on the assumption of independence of irrelevant alternatives (IIA), which is not always desirable. This assumption states that the odds of preferring one class over another do not depend on the presence or absence of other "irrelevant" alternatives. For example, the relative probabilities of taking a car or bus to work do not change if a bicycle is added as an additional possibility. This allows the choice of K alternatives to be modeled as a set of K − 1 independent binary choices, in which one alternative is chosen as a "pivot" and the other K − 1 compared against it, one at a time. The IIA hypothesis is a core hypothesis in rational choice theory; however numerous studies in psychology show that individuals often violate this assumption when making choices. An example of a problem case arises if choices include a car and a blue bus. Suppose the odds ratio between the two is 1 : 1. Now if the option of a red bus is introduced, a person may be indifferent between a red and a blue bus, and hence may exhibit a car : blue bus : red bus odds ratio of 1 : 0.5 : 0.5, thus maintaining a 1 : 1 ratio of car : any bus while adopting a changed car : blue bus ratio of 1 : 0.5. Here the red bus option was not in fact irrelevant, because a red bus was a perfect substitute for a blue bus. If the multinomial logit is used to model choices, it may in some situations impose too much constraint on the relative preferences between the different alternatives. It is especially important to take into account if the analysis aims to predict how choices would change if one alternative were to disappear (for instance if one political candidate withdraws from a three candidate race). Other models like the nested logit or the multinomial probit may be used in such cases as they allow for violation of the IIA.
Multinomial logistic regression : When using multinomial logistic regression, one category of the dependent variable is chosen as the reference category. Separate odds ratios are determined for all independent variables for each category of the dependent variable with the exception of the reference category, which is omitted from the analysis. The exponential beta coefficient represents the change in the odds of the dependent variable being in a particular category vis-a-vis the reference category, associated with a one unit change of the corresponding independent variable.
Multinomial logistic regression : The observed values y i ∈ \in \ for i = 1 , … , n of the explained variables are considered as realizations of stochastically independent, categorically distributed random variables Y 1 , … , Y n ,\dots ,Y_ . The likelihood function for this model is defined by L = ∏ i = 1 n P ( Y i = y i ) = ∏ i = 1 n ∏ j = 1 K P ( Y i = j ) δ j , y i , ^P(Y_=y_)=\prod _^\prod _^P(Y_=j)^, where the index i denotes the observations 1 to n and the index j denotes the classes 1 to K. δ j , y i = =1,j=y_\\0,\end is the Kronecker delta. The negative log-likelihood function is therefore the well-known cross-entropy: − log ⁡ L = − ∑ i = 1 n ∑ j = 1 K δ j , y i log ⁡ ( P ( Y i = j ) ) = − ∑ j = 1 K ∑ y i = j log ⁡ ( P ( Y i = j ) ) . ^\sum _^\delta _\log(P(Y_=j))=-\sum _^\sum _=j\log(P(Y_=j)).
Multinomial logistic regression : In natural language processing, multinomial LR classifiers are commonly used as an alternative to naive Bayes classifiers because they do not assume statistical independence of the random variables (commonly known as features) that serve as predictors. However, learning in such a model is slower than for a naive Bayes classifier, and thus may not be appropriate given a very large number of classes to learn. In particular, learning in a naive Bayes classifier is a simple matter of counting up the number of co-occurrences of features and classes, while in a maximum entropy classifier the weights, which are typically maximized using maximum a posteriori (MAP) estimation, must be learned using an iterative procedure; see #Estimating the coefficients.
Multinomial logistic regression : Logistic regression Multinomial probit == References ==
Multispectral pattern recognition : Multispectral remote sensing is the collection and analysis of reflected, emitted, or back-scattered energy from an object or an area of interest in multiple bands of regions of the electromagnetic spectrum (Jensen, 2005). Subcategories of multispectral remote sensing include hyperspectral, in which hundreds of bands are collected and analyzed, and ultraspectral remote sensing where many hundreds of bands are used (Logicon, 1997). The main purpose of multispectral imaging is the potential to classify the image using multispectral classification. This is a much faster method of image analysis than is possible by human interpretation.
Multispectral pattern recognition : Remote sensing systems gather data via instruments typically carried on satellites in orbit around the Earth. The remote sensing scanner detects the energy that radiates from the object or area of interest. This energy is recorded as an analog electrical signal and converted into a digital value though an A-to-D conversion. There are several multispectral remote sensing systems that can be categorized in the following way:
Multispectral pattern recognition : A variety of methods can be used for the multispectral classification of images: Algorithms based on parametric and nonparametric statistics that use ratio-and interval-scaled data and nonmetric methods that can also incorporate nominal scale data (Duda et al., 2001), Supervised or unsupervised classification logic, Hard or soft (fuzzy) set classification logic to create hard or fuzzy thematic output products, Per-pixel or object-oriented classification logic, and Hybrid approaches
Multispectral pattern recognition : In this classification method, the identity and location of some of the land-cover types are obtained beforehand from a combination of fieldwork, interpretation of aerial photography, map analysis, and personal experience. The analyst would locate sites that have similar characteristics to the known land-cover types. These areas are known as training sites because the known characteristics of these sites are used to train the classification algorithm for eventual land-cover mapping of the remainder of the image. Multivariate statistical parameters (means, standard deviations, covariance matrices, correlation matrices, etc.) are calculated for each training site. All pixels inside and outside of the training sites are evaluated and allocated to the class with the more similar characteristics.
Multispectral pattern recognition : Unsupervised classification (also known as clustering) is a method of partitioning remote sensor image data in multispectral feature space and extracting land-cover information. Unsupervised classification require less input information from the analyst compared to supervised classification because clustering does not require training data. This process consists in a series of numerical operations to search for the spectral properties of pixels. From this process, a map with m spectral classes is obtained. Using the map, the analyst tries to assign or transform the spectral classes into thematic information of interest (i.e. forest, agriculture, urban). This process may not be easy because some spectral clusters represent mixed classes of surface materials and may not be useful. The analyst has to understand the spectral characteristics of the terrain to be able to label clusters as a specific information class. There are hundreds of clustering algorithms. Two of the most conceptually simple algorithms are the chain method and the ISODATA method.
Multispectral pattern recognition : Ball, Geoffrey H., Hall, David J. (1965) Isodata: a method of data analysis and pattern classification, Stanford Research Institute, Menlo Park, United States. Office of Naval Research. Information Sciences Branch Duda, R. O., Hart, P. E., & Stork, D. G. (2001). Pattern Classification. New York: John Wiley & Sons. Jensen, J. R. (2005). Introductory Digital Image Processing: A Remote Sensing Perspective. Upper Saddle River : Pearson Prentice Hall. Belokon, W. F. et al. (1997). Multispectral Imagery Reference Guide. Fairfax: Logicon Geodynamics, Inc.
Naive Bayes classifier : In statistics, naive (sometimes simple or idiot's) Bayes classifiers are a family of "probabilistic classifiers" which assumes that the features are conditionally independent, given the target class. In other words, a naive Bayes model assumes the information about the class provided by each variable is unrelated to the information from the others, with no information shared between the predictors. The highly unrealistic nature of this assumption, called the naive independence assumption, is what gives the classifier its name. These classifiers are some of the simplest Bayesian network models. Naive Bayes classifiers generally perform worse than more advanced models like logistic regressions, especially at quantifying uncertainty (with naive Bayes models often producing wildly overconfident probabilities). However, they are highly scalable, requiring only one parameter for each feature or predictor in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression (simply by counting observations in each group),: 718 rather than the expensive iterative approximation algorithms required by most other models. Despite the use of Bayes' theorem in the classifier's decision rule, naive Bayes is not (necessarily) a Bayesian method, and naive Bayes models can be fit to data using either Bayesian or frequentist methods.
Naive Bayes classifier : Naive Bayes is a simple technique for constructing classifiers: models that assign class labels to problem instances, represented as vectors of feature values, where the class labels are drawn from some finite set. There is not a single algorithm for training such classifiers, but a family of algorithms based on a common principle: all naive Bayes classifiers assume that the value of a particular feature is independent of the value of any other feature, given the class variable. For example, a fruit may be considered to be an apple if it is red, round, and about 10 cm in diameter. A naive Bayes classifier considers each of these features to contribute independently to the probability that this fruit is an apple, regardless of any possible correlations between the color, roundness, and diameter features. In many practical applications, parameter estimation for naive Bayes models uses the method of maximum likelihood; in other words, one can work with the naive Bayes model without accepting Bayesian probability or using any Bayesian methods. Despite their naive design and apparently oversimplified assumptions, naive Bayes classifiers have worked quite well in many complex real-world situations. In 2004, an analysis of the Bayesian classification problem showed that there are sound theoretical reasons for the apparently implausible efficacy of naive Bayes classifiers. Still, a comprehensive comparison with other classification algorithms in 2006 showed that Bayes classification is outperformed by other approaches, such as boosted trees or random forests. An advantage of naive Bayes is that it only requires a small amount of training data to estimate the parameters necessary for classification.
Naive Bayes classifier : Abstractly, naive Bayes is a conditional probability model: it assigns probabilities p ( C k ∣ x 1 , … , x n ) \mid x_,\ldots ,x_) for each of the K possible outcomes or classes C k given a problem instance to be classified, represented by a vector x = ( x 1 , … , x n ) =(x_,\ldots ,x_) encoding some n features (independent variables). The problem with the above formulation is that if the number of features n is large or if a feature can take on a large number of values, then basing such a model on probability tables is infeasible. The model must therefore be reformulated to make it more tractable. Using Bayes' theorem, the conditional probability can be decomposed as: p ( C k ∣ x ) = p ( C k ) p ( x ∣ C k ) p ( x ) \mid \mathbf )=)\ p(\mathbf \mid C_) )\, In plain English, using Bayesian probability terminology, the above equation can be written as posterior = prior × likelihood evidence =\times \, In practice, there is interest only in the numerator of that fraction, because the denominator does not depend on C and the values of the features x i are given, so that the denominator is effectively constant. The numerator is equivalent to the joint probability model p ( C k , x 1 , … , x n ) ,x_,\ldots ,x_)\, which can be rewritten as follows, using the chain rule for repeated applications of the definition of conditional probability: p ( C k , x 1 , … , x n ) = p ( x 1 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) p ( x 3 , … , x n , C k ) = ⋯ = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) ⋯ p ( x n − 1 ∣ x n , C k ) p ( x n ∣ C k ) p ( C k ) p(C_,x_,\ldots ,x_)&=p(x_,\ldots ,x_,C_)\\&=p(x_\mid x_,\ldots ,x_,C_)\ p(x_,\ldots ,x_,C_)\\&=p(x_\mid x_,\ldots ,x_,C_)\ p(x_\mid x_,\ldots ,x_,C_)\ p(x_,\ldots ,x_,C_)\\&=\cdots \\&=p(x_\mid x_,\ldots ,x_,C_)\ p(x_\mid x_,\ldots ,x_,C_)\cdots p(x_\mid x_,C_)\ p(x_\mid C_)\ p(C_)\\\end Now the "naive" conditional independence assumptions come into play: assume that all features in x are mutually independent, conditional on the category C k . Under this assumption, p ( x i ∣ x i + 1 , … , x n , C k ) = p ( x i ∣ C k ) . \mid x_,\ldots ,x_,C_)=p(x_\mid C_)\,. Thus, the joint model can be expressed as p ( C k ∣ x 1 , … , x n ) ∝ p ( C k , x 1 , … , x n ) = p ( C k ) p ( x 1 ∣ C k ) p ( x 2 ∣ C k ) p ( x 3 ∣ C k ) ⋯ = p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) , p(C_\mid x_,\ldots ,x_)\varpropto \ &p(C_,x_,\ldots ,x_)\\&=p(C_)\ p(x_\mid C_)\ p(x_\mid C_)\ p(x_\mid C_)\ \cdots \\&=p(C_)\prod _^p(x_\mid C_)\,,\end where ∝ denotes proportionality since the denominator p ( x ) ) is omitted. This means that under the above independence assumptions, the conditional distribution over the class variable C is: p ( C k ∣ x 1 , … , x n ) = 1 Z p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) \mid x_,\ldots ,x_)=\ p(C_)\prod _^p(x_\mid C_) where the evidence Z = p ( x ) = ∑ k p ( C k ) p ( x ∣ C k ) )=\sum _p(C_)\ p(\mathbf \mid C_) is a scaling factor dependent only on x 1 , … , x n ,\ldots ,x_ , that is, a constant if the values of the feature variables are known.
Naive Bayes classifier : A class's prior may be calculated by assuming equiprobable classes, i.e., p ( C k ) = 1 K )= , or by calculating an estimate for the class probability from the training set: prior for a given class = no. of samples in that class total no. of samples =\, To estimate the parameters for a feature's distribution, one must assume a distribution or generate nonparametric models for the features from the training set. The assumptions on distributions of features are called the "event model" of the naive Bayes classifier. For discrete features like the ones encountered in document classification (include spam filtering), multinomial and Bernoulli distributions are popular. These assumptions lead to two distinct models, which are often confused.
Naive Bayes classifier : Despite the fact that the far-reaching independence assumptions are often inaccurate, the naive Bayes classifier has several properties that make it surprisingly useful in practice. In particular, the decoupling of the class conditional feature distributions means that each distribution can be independently estimated as a one-dimensional distribution. This helps alleviate problems stemming from the curse of dimensionality, such as the need for data sets that scale exponentially with the number of features. While naive Bayes often fails to produce a good estimate for the correct class probabilities, this may not be a requirement for many applications. For example, the naive Bayes classifier will make the correct MAP decision rule classification so long as the correct class is predicted as more probable than any other class. This is true regardless of whether the probability estimate is slightly, or even grossly inaccurate. In this manner, the overall classifier can be robust enough to ignore serious deficiencies in its underlying naive probability model. Other reasons for the observed success of the naive Bayes classifier are discussed in the literature cited below.
Naive Bayes classifier : AODE Anti-spam techniques Bayes classifier Bayesian network Bayesian poisoning Email filtering Linear classifier Logistic regression Markovian discrimination Mozilla Thunderbird mail client with native implementation of Bayes filters Perceptron Random naive Bayes Take-the-best heuristic
Naive Bayes classifier : Domingos, Pedro; Pazzani, Michael (1997). "On the optimality of the simple Bayesian classifier under zero-one loss". Machine Learning. 29 (2/3): 103–137. doi:10.1023/A:1007413511361. Webb, G. I.; Boughton, J.; Wang, Z. (2005). "Not So Naive Bayes: Aggregating One-Dependence Estimators". Machine Learning. 58 (1): 5–24. doi:10.1007/s10994-005-4258-6. Mozina, M.; Demsar, J.; Kattan, M.; Zupan, B. (2004). Nomograms for Visualization of Naive Bayesian Classifier (PDF). Proc. PKDD-2004. pp. 337–348. Maron, M. E. (1961). "Automatic Indexing: An Experimental Inquiry". Journal of the ACM. 8 (3): 404–417. doi:10.1145/321075.321084. hdl:2027/uva.x030748531. S2CID 6692916. Minsky, M. (1961). Steps toward Artificial Intelligence. Proc. IRE. Vol. 49. pp. 8–30.
Naive Bayes classifier : Book Chapter: Naive Bayes text classification, Introduction to Information Retrieval Naive Bayes for Text Classification with Unbalanced Classes
Nearest centroid classifier : In machine learning, a nearest centroid classifier or nearest prototype classifier is a classification model that assigns to observations the label of the class of training samples whose mean (centroid) is closest to the observation. When applied to text classification using word vectors containing tf*idf weights to represent documents, the nearest centroid classifier is known as the Rocchio classifier because of its similarity to the Rocchio algorithm for relevance feedback. An extended version of the nearest centroid classifier has found applications in the medical domain, specifically classification of tumors.
Nearest centroid classifier : Cluster hypothesis k-means clustering k-nearest neighbor algorithm Linear discriminant analysis == References ==
Nearest neighbor search : Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest-neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M, find the closest point in S to q. Donald Knuth in vol. 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a k-NN search, where we need to find the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where dissimilarity is measured using the Euclidean distance, Manhattan distance or other distance metric. However, the dissimilarity function can be arbitrary. One example is asymmetric Bregman divergence, for which the triangle inequality does not hold.
Nearest neighbor search : The nearest neighbor search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic Search Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.