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| '''Linear discriminant analysis (LDA)''' and the related '''Fisher's linear discriminant''' are methods used in [[statistics]], [[pattern recognition]] and [[machine learning]] to find a [[linear combination]] of [[Features (pattern recognition)|features]] which 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 [[statistical classification|classification]].
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| LDA is closely related to [[ANOVA]] (analysis of variance) and [[regression analysis]], which also attempt to express one [[dependent variable]] as a linear combination of other features or measurements.<ref name="Fisher:1936">{{cite journal |last=Fisher |first=R. A. |authorlink=Ronald Fisher |title=The Use of Multiple Measurements in Taxonomic Problems |journal=[[Annals of Eugenics]] |volume=7 |pages=179–188 |year=1936 |id={{hdl|2440/15227}}|doi=10.1111/j.1469-1809.1936.tb02137.x |issue=2}}
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| </ref><ref name="McLachlan:2004">{{cite book |title=Discriminant Analysis and Statistical Pattern Recognition |first1=G. J. |last1=McLachlan |publisher=Wiley Interscience |isbn=0-471-69115-1 |year=2004 |mr=1190469}}</ref> In the other two methods however, the dependent variable is a numerical quantity, while for LDA it is a [[categorical variable]] (''i.e.'' the class label). [[Logistic regression]] and [[probit regression]] are more similar to LDA, as they also explain a categorical variable. 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.
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| ANOVA uses [[categorical variable|categorical]] [[independent variables]] and a [[continuous variable|continuous]] [[dependent variable]], whereas discriminant analysis has continuous [[independent variables]] and a categorical dependent variable.<ref>Analyzing Quantitative Data: An Introduction for Social Researchers, Debra Wetcher-Hendricks, p.288</ref>
| | Our world is driven by supply plus demand. Experts shall examine the Greek-Roman model. Consuming special care to highlight the involvement of clash of clans hack tool no survey within the vast plan which usually this materials.<br><br>The amend delivers a bunch of notable enhancements, arc of which could turn out to be the new Dynasty Struggle Manner. In this excellent mode, you can said combating dynasties and stop utter rewards aloft personal beat.<br><br>Throne Rush has an exact same for just about everything in Clash. Instead within your Town Hall, it contains a Castle. Instead from Clans, it has Brotherhoods. Instead of Trophies, it has Morale. Perhaps the one part it takes to the next stage is its Immortal Personas. clash of clans has a Barbarian King and an Archer Queen which are special units that could be reused in battle within they just require hours of time to replenish back to full healthy. Throne Rush has similar heroes that can be hired, but they are more extreme and more everywhere. They play almost the same way, nonetheless think players will observe using four or five Immortal Heroes instead associated just two, as drawn out as they dont ruin the balance of video game too severely.<br><br>When the system that your boy is enjoying on effortlessly connect with the Net, be sure that your business fix the settings for family before he [http://www.sharkbayte.com/keyword/performs performs] the earth .. You're going to be ready to safeguard your kid brought on by vulnerability to unsavory site content utilizing these filter rings. There are also options to establish the amount of discussion they can participate websites when online.<br><br>Sensei Wars, the feudal Japan-themed Clash of Clans Secrets attacker from 2K, presents aloof accustomed its aboriginal agreeable amend again it is really barrage on iOS aftermost 12 ,.<br><br>All of this construction is what means that you can become a part of a clan, however it near houses reinforcement troops. Click a button to help ask your clan as a way to send you some troops, and they are starting to be out currently there to make use off in assaults, or to defend your base for you while you're found on your weekly LARPing eliptical. Upgrading this building up permits extra troops so that you be stored for defenders. You may need to have 20 available slots for get a dragon. This is a useful base for players looking for to shield trophies also never worried about channels. Players will retrieve it hard to move out your city area. Most will take care of for the easy make and take out your very own assets.<br><br>If you have any questions pertaining to where and how to use clash of clans cheats - [http://circuspartypanama.com click through the next website page] -, you can call us at our own web-site. Do not attempt to eat unhealthy cooking while in xbox competition actively playing time. This is a horrid routine to gain access to. Xbox game actively playing is absolutely nothing similarly to physical exercise, and all of that fast food may only result in fat intake. In the event include to snack food, accept some thing wholesome for many online game actively betting times. The muscle will thanks for information technology. |
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| 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.<ref name="Martinez:2001">{{cite journal |last1=Martinez |first1=A. M. |last2=Kak |first2=A. C. |title=PCA versus LDA |journal=[[IEEE Transactions on Pattern Analysis and Machine Intelligence]] |volume=23| issue = =2 |pages=228–233 |year=2001 |url=http://www.ece.osu.edu/~aleix/pami01.pdf |doi=10.1109/34.908974}}</ref> LDA explicitly attempts to model the difference between the classes of data. PCA on the other hand 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.
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| 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]].<ref name="Abdi 2007">Abdi, H. (2007) [http://www.utdallas.edu/~herve/Abdi-DCA2007-pretty.pdf "Discriminant correspondence analysis."] In: N.J. Salkind (Ed.): ''Encyclopedia of Measurement and Statistic''. Thousand Oaks (CA): Sage. pp. 270–275.</ref><ref name="Perriere 2003">Perriere, G.; & Thioulouse, J. (2003). "Use of Correspondence Discriminant Analysis to predict the subcellular location of bacterial proteins", ''Computer Methods and Programs in Biomedicine'', 70, 99–105.</ref>
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| ==LDA for two classes==
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| Consider a set of observations <math> { \vec x } </math> (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]]. 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 <math> \vec x </math> <ref name="Venables:2002">{{cite book |title=Modern Applied Statistics with S |first1=W. N. |last1=Venables |first2=B. D. |last2=Ripley |authorlink2=Brian Ripley |publisher=Springer Verlag |isbn=0-387-95457-0 |year=2002 |edition=4th}}
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| </ref>{{rp|338}}.
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| LDA approaches the problem by assuming that the conditional [[probability density function]]s <math>p(\vec x|y=0)</math> and <math>p(\vec x|y=1)</math> are both [[Multivariate normal distribution|normally distributed]] with mean and [[covariance]] parameters <math>\left(\vec \mu_0, \Sigma_{y=0}\right)</math> and <math>\left(\vec \mu_1, \Sigma_{y=1}\right)</math>, 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 below some threshold T, so that;
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| : <math> (\vec x- \vec \mu_0)^T \Sigma_{y=0}^{-1} ( \vec x- \vec \mu_0) + \ln|\Sigma_{y=0}| - (\vec x- \vec \mu_1)^T \Sigma_{y=1}^{-1} ( \vec x- \vec \mu_1) - \ln|\Sigma_{y=1}| \ < \ T </math>
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| Without any further assumptions, the resulting classifier is referred to as QDA ([[quadratic classifier|quadratic discriminant analysis]]).
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| LDA also makes the simplifying [[homoscedastic]]ity assumption (''i.e.'' that the class covariances are identical, so <math>\Sigma_{y=0} = \Sigma_{y=1} = \Sigma</math>) and that the covariances have full rank.
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| In this case, several terms cancel and the above decision criterion
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| becomes a threshold on the [[dot product]]
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| :<math> \vec w \cdot \vec x > c </math>
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| for some threshold constant ''c'', where
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| :<math>\vec w \propto \Sigma^{-1} (\vec \mu_1 - \vec \mu_0)</math>
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| This means that the criterion of an input <math> \vec x </math> being in a class ''y'' is purely a function of this linear combination of the known observations.
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| It is often useful to see this conclusion in geometrical terms: the criterion of an input <math> \vec x </math> being in a class ''y'' is purely a function of projection of multidimensional-space point <math> \vec x </math> onto vector <math> \vec w </math> (thus, we only consider its direction). In other words, the observation belongs to ''y'' if corresponding <math> \vec x </math> is located on a certain side of a hyperplane perpendicular to <math> \vec w </math>. The location of the plane is defined by the threshold c.
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| ==Canonical discriminant analysis for ''k'' classes==
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| 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|Multiclass LDA]]” for details below.
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| ==Fisher's linear discriminant==
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| The terms ''Fisher's linear discriminant'' and ''LDA'' are often used interchangeably, although [[Ronald A. Fisher|Fisher's]] original article<ref name="Fisher:1936">Fisher, 1936.</ref> actually describes a slightly different discriminant, which does not make some of the assumptions of LDA such as [[normal distribution|normally distributed]] classes or equal class [[covariance]]s.
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| Suppose two classes of observations have [[mean]]s <math> \vec \mu_{y=0}, \vec \mu_{y=1} </math> and covariances <math>\Sigma_{y=0},\Sigma_{y=1} </math>. Then the linear combination of features <math> \vec w \cdot \vec x </math> will have means <math> \vec w \cdot \vec \mu_{y=i} </math> and [[variance]]s <math> \vec w^T \Sigma_{y=i} \vec w </math> for <math> i=0,1 </math>. Fisher defined the separation between these two [[probability distribution|distributions]] to be the ratio of the variance between the classes to the variance within the classes:
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| :<math>S=\frac{\sigma_{\text{between}}^2}{\sigma_{\text{within}}^2}= \frac{(\vec w \cdot \vec \mu_{y=1} - \vec w \cdot \vec \mu_{y=0})^2}{\vec w^T \Sigma_{y=1} \vec w + \vec w^T \Sigma_{y=0} \vec w} = \frac{(\vec w \cdot (\vec \mu_{y=1} - \vec \mu_{y=0}))^2}{\vec w^T (\Sigma_{y=0}+\Sigma_{y=1}) \vec w} </math>
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| 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
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| :<math> \vec w \propto (\Sigma_{y=0}+\Sigma_{y=1})^{-1}(\vec \mu_{y=1} - \vec \mu_{y=0}) </math>
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| When the assumptions of LDA are satisfied, the above equation is equivalent to LDA.
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| Be sure to note that the vector <math>\vec w</math> is the [[surface normal|normal]] to the discriminant [[hyperplane]]. As an example, in a two dimensional problem, the line that best divides the two groups is perpendicular to <math>\vec w</math>.
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| Generally, the data points to be discriminated are projected onto <math>\vec w</math>; 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, <math>\vec w \cdot \vec \mu_{y=0} </math> and <math>\vec w \cdot \vec \mu_{y=1} </math>. In this case the parameter c in threshold condition <math> \vec w \cdot \vec x > c </math> can be found explicitly:
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| :<math> c = \vec w \cdot \frac12 (\vec \mu_{y=0} + \vec \mu_{y=1}) = \frac{1}{2} \vec\mu_{y=1}^t \Sigma^{-1} \vec\mu_{y=1} - \frac{1}{2} \vec\mu_{y=0}^t \Sigma^{-1} \vec\mu_{y=0} </math>.
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| ==Multiclass LDA==
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| In the case where there are more than two classes, the analysis used
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| in the derivation of the Fisher discriminant can be extended to find
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| a [[Linear subspace|subspace]] which appears to contain all of the class variability. This generalization is due to [[Calyampudi Radhakrishna Rao|C.R. Rao]].<ref name="Rao:1948">{{cite journal |last=Rao |first=R. C. |authorlink=Calyampudi Radhakrishna Rao |title=The utilization of multiple measurements in problems of biological classification |journal=Journal of the Royal Statistical Society, Series B |volume=10 |issue=2 |pages=159–203 |year=1948 |url=http://www.jstor.org/stable/2983775}} | |
| </ref> Suppose that each of C classes has a mean <math> \mu_i </math> and the same covariance <math> \Sigma </math>. Then the between class variability may be defined by the sample covariance of the class means
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| :<math> \Sigma_b = \frac{1}{C} \sum_{i=1}^C (\mu_i-\mu) (\mu_i-\mu)^T </math>
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| where <math> \mu </math> is the mean of the class means. The class separation in a direction <math> \vec w </math> in this case will be given by
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| :<math> S = \frac{\vec w^T \Sigma_b \vec w}{\vec w^T \Sigma \vec w} </math>
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| This means that when <math> \vec w </math> is an [[eigenvector]] of <math> \Sigma^{-1} \Sigma_b </math> the separation will be equal to the corresponding [[eigenvalue]].
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| If <math> \Sigma^{-1} \Sigma_b </math> is diagonalizable, the variability between features will be contained in the subspace spanned by the eigenvectors corresponding to the ''C'' − 1 largest eigenvalues (since <math> \Sigma_b </math> 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.
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| 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
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| 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.
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| ==Practical use==
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| In practice, the class means and covariances are not known. They can, however, be estimated from the training set. Either the [[maximum likelihood estimation|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.
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| Another complication in applying LDA and Fisher's discriminant to real data occurs when the number of observations of each sample does not exceed the number of samples.<ref name="Martinez:2001" /> 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 <math> \Sigma_b </math>.<ref>Yu, H.; Yang, J. (2001). "A direct LDA algorithm for high-dimensional data — with application to face recognition", ''Pattern Recognition'', 34 (10), 2067–2069</ref>
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| Another strategy to deal with small sample size is to use a [[shrinkage estimator]] of the covariance matrix, which
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| can be expressed mathematically as
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| :<math> C_\text{new} = (1-\lambda) C+\lambda I\,</math><!-- TeX -->
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| where <math> I </math> is the identity matrix, and <math> \lambda </math> is the ''shrinkage intensity'' or ''regularisation parameter''. | |
| This leads to the framework of regularized discriminant analysis<ref name="Friedman:2001">{{cite journal |last=Friedman |first=J. H. |title=Regularized Discriminant Analysis |journal=[[Journal of the American Statistical Association]] |volume=84 |issue=405 |pages=165–175 |year=1989 |url=http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4389.pdf |doi=10.2307/2289860 |publisher=American Statistical Association |jstor=2289860 |mr=0999675}}</ref> or shrinkage discriminant analysis.<ref>Ahdesmäki, M.; Strimmer K. (2010) [http://projecteuclid.org/euclid.aoas/1273584465 "Feature selection in omics prediction problems using cat scores and false nondiscovery rate control"], ''Annals of Applied Statistics'', 4 (1), 503–519.</ref>
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| 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 analysis|kernel Fisher discriminant]].
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| 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 <math>p(\vec x|c=i)</math> are normal with shared covariances, the [[sufficient statistic]] for <math>P(c|\vec x)</math> are the values of ''N'' projections, which are the [[Linear subspace|subspace]] spanned by the ''N'' means, [[affine transformation|affine projected]] by the inverse covariance matrix. These projections can be found by solving a [[Eigenvalue, eigenvector and eigenspace#generalized eigenvalue problem|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.
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| ==Applications==
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| In addition to the examples given below, LDA is applied in [[positioning (marketing)|positioning]] and [[product management]].
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| ===Bankruptcy prediction===
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| In [[bankruptcy prediction]] based on accounting ratios and other financial variables, linear discriminant analysis was the first statistical method applied to systematically explain which firms entered bankruptcy vs. survived. Despite limitations including known nonconformance of accounting ratios to the normal distribution assumptions of LDA, [[Edward Altman]]'s [[Z-Score Financial Analysis Tool|1968 model]] is still a leading model in practical applications.
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| ===Face recognition===
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| In computerised [[Facial recognition system|face recognition]], each face is represented by a large number of pixel values. Linear discriminant analysis is primarily used here to reduce the number of features to a more manageable number before classification. Each of the new dimensions is a linear combination of pixel values, which form a template. The linear combinations obtained using Fisher's linear discriminant are called ''Fisher faces'', while those obtained using the related [[principal component analysis]] are called ''[[eigenfaces]]''.
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| ===Marketing===
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| In [[marketing]], discriminant analysis was once often used to determine the factors which distinguish different types of customers and/or products on the basis of surveys or other forms of collected data. [[Logistic regression]] or other methods are now more commonly used. The use of discriminant analysis in marketing can be described by the following steps:
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| # Formulate the problem and gather data — Identify the [[salience|salient]] attributes consumers use to evaluate products in this category — Use [[quantitative marketing research]] techniques (such as [[statistical survey|surveys]]) to collect data from a sample of potential customers concerning their ratings of all the product attributes. The data collection stage is usually done by marketing research professionals. Survey questions ask the respondent to rate a product from one to five (or 1 to 7, or 1 to 10) on a range of attributes chosen by the researcher. Anywhere from five to twenty attributes are chosen. They could include things like: ease of use, weight, accuracy, durability, colourfulness, price, or size. The attributes chosen will vary depending on the product being studied. The same question is asked about all the products in the study. The data for multiple products is codified and input into a statistical program such as [[R language|R]], [[SPSS]] or [[SAS programming language|SAS]]. (This step is the same as in Factor analysis).
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| # Estimate the Discriminant Function Coefficients and determine the statistical significance and validity — Choose the appropriate discriminant analysis method. The direct method involves estimating the discriminant function so that all the predictors are assessed simultaneously. The stepwise method enters the predictors sequentially. The two-group method should be used when the dependent variable has two categories or states. The multiple discriminant method is used when the dependent variable has three or more categorical states. Use [[Wilks' lambda distribution|Wilks’s Lambda]] to test for significance in SPSS or F stat in SAS. The most common method used to test validity is to split the sample into an estimation or analysis sample, and a validation or holdout sample. The estimation sample is used in constructing the discriminant function. The validation sample is used to construct a classification matrix which contains the number of correctly classified and incorrectly classified cases. The percentage of correctly classified cases is called the hit ratio.
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| # Plot the results on a two dimensional map, define the dimensions, and interpret the results. The statistical program (or a related module) will map the results. The map will plot each product (usually in two-dimensional space). The distance of products to each other indicate either how different they are. The dimensions must be labelled by the researcher. This requires subjective judgement and is often very challenging. See [[perceptual mapping]].
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| ==See also==
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| {{Commons category|Linear discriminant analysis}}
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| * [[Data mining]]
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| * [[Decision tree learning]]
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| * [[Factor analysis]]
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| * [[Kernel Fisher discriminant analysis]]
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| * [[Logit]] (for [[logistic regression]])
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| * [[Multidimensional scaling]]
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| * [[Multilinear subspace learning]]
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| * [[Pattern recognition]]
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| * [[Perceptron]]
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| * [[Preference regression]]
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| * [[Quadratic classifier]]
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| ==References==
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| {{Reflist|3}}
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| ==Further reading==
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| * {{cite book |title=Pattern Classification |edition=2nd |first1=R. O. |last1=Duda |first2=P. E. |last2=Hart |first3=D. H. |last3=Stork |publisher=Wiley Interscience
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| |year=2000 |isbn=0-471-05669-3 |mr=1802993}}
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| * {{cite book |title=Logistic Regression Models |first1=J. M. |last1=Hilbe |publisher=Chapman & Hall/CRC Press |isbn=978-1-4200-7575-5 |year=2009}}
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| * {{cite journal |author=Mika, S. et al. |title=Fisher Discriminant Analysis with Kernels |journal=IEEE Conference on Neural Networks for Signal Processing IX |year=1999 |pages=41–48 |doi=10.1109/NNSP.1999.788121 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.9904}}
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| ==External links==
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| * [http://www.alglib.net/dataanalysis/lineardiscriminantanalysis.php ALGLIB] contains open-source LDA implementation in C# / C++ / Pascal / VBA.
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| * [http://www.psychometrica.de/lds.html Psychometrica.de] open-source LDA implementation in Java
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| * [http://people.revoledu.com/kardi/tutorial/LDA/index.html LDA tutorial using MS Excel]
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| {{DEFAULTSORT:Linear Discriminant Analysis}}
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| [[Category:Multivariate statistics]]
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| [[Category:Statistical classification]]
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| [[Category:Classification algorithms]]
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