Editing Linear discriminant analysis (section)

==Applications==
In addition to the examples given below, LDA is applied in [[positioning (marketing)|positioning]] and [[product management]].

===Bankruptcy prediction===
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]]<ref >{{Cite journal
  |last=Altman  |first=Edward I.  |author-link=Edward I. Altman 
  |title=Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy
  |journal=[[The Journal of Finance]]
  |volume=23  |issue=4 
  |year=1968
  |pages=589–609
  |doi=10.2307/2978933
|jstor=2978933 }}</ref> is still a leading model in practical applications.<ref >{{Cite web
  |last1=Agarwal  |first1=Vineet
  |last2=Taffler  |first2=Richard
  |year=2005
  |title=Twenty-five years of z-scores in the UK: do they really work?
  |url=https://efmaefm.org/0efmameetings/EFMA%20ANNUAL%20MEETINGS/2006-Madrid/papers/932609_full.pdf
}}</ref><ref >{{Cite journal
  |last1=Agarwal  |first1=Vineet
  |last2=Taffler  |first2=Richard
  |year=2007
  |title=Twenty-Five Years of the Taffler Z-Score Model: Does It Really Have Predictive Ability?
  |journal=Accounting and Business Research
  |volume=37
 |issue=4
  |pages=285–300
  |doi=10.1080/00014788.2007.9663313
}}</ref><ref >{{Cite journal
  |first1=Patrick  |last1=Bimpong 
  |first2=Ishmael  |last2=Arhin
  |first3=Thomas hezkeal Khela  |last3=Nan
  |first4=Edward  |last4=Danso
  |first5=Pious  |last5=Opoku
  |first6=Arthur  |last6=Benedict
  |first7=Grace |last7=Tettey
  |display-authors=1
  |title=Assessing Predictive Power and Earnings Manipulations. Applied Study on Listed Consumer Goods and Service Companies in Ghana Using 3 Z-Score Models
  |journal=Expert Journal of Finance
  |volume=8  |issue=1
  |pages=1–26
  |year=2020
  |url=https://finance.expertjournals.com/23597712-801/
}}</ref>

===Face recognition===
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]]''.

===Marketing===
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:
#Formulate the problem and gather data—Identify the [[Social 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).
#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 regression|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''.
#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]].

===Biomedical studies===

The main application of discriminant analysis in medicine is the assessment of severity state of a patient and prognosis of disease outcome. For example, during retrospective analysis, patients are divided into groups according to severity of disease – mild, moderate, and severe form. Then results of clinical and laboratory analyses are studied to reveal statistically different variables in these groups. Using these variables, discriminant functions are built to classify disease severity in future patients. Additionally, Linear Discriminant Analysis (LDA) can help select more discriminative samples for data augmentation, improving classification performance.<ref>{{cite journal | last1 = Moradi | first1 = M | last2 = Demirel | first2 = H | year = 2024 | title = Alzheimer's disease classification using 3D conditional progressive GAN-and LDA-based data selection | journal = Signal, Image and Video Processing | volume = 18 | issue = 2 | pages = 1847–1861 | doi = 10.1007/s11760-023-02878-4 }}</ref>

In biology, similar principles are used in order to classify and define groups of different biological objects, for example, to define phage types of Salmonella enteritidis based on Fourier transform infrared spectra,<ref>{{cite journal | last1 = Preisner | first1 = O | last2 = Guiomar | first2 = R | last3 = Machado | first3 = J | last4 = Menezes | first4 = JC | last5 = Lopes | first5 = JA | year = 2010 | title = Application of Fourier transform infrared spectroscopy and chemometrics for differentiation of Salmonella enterica serovar Enteritidis phage types | journal = Appl Environ Microbiol | volume = 76 | issue = 11| pages = 3538–3544 | doi=10.1128/aem.01589-09| pmid = 20363777 | pmc = 2876429 | bibcode = 2010ApEnM..76.3538P }}</ref> to detect animal source of ''Escherichia coli'' studying its virulence factors<ref>{{cite journal | last1 = David | first1 = DE | last2 = Lynne | first2 = AM | last3 = Han | first3 = J | last4 = Foley | first4 = SL | year = 2010 | title = Evaluation of virulence factor profiling in the characterization of veterinary Escherichia coli isolates | journal = Appl Environ Microbiol | volume = 76 | issue = 22| pages = 7509–7513 | doi=10.1128/aem.00726-10| pmid = 20889790 | pmc = 2976202 | bibcode = 2010ApEnM..76.7509D }}</ref> etc.

===Earth science===
This method can be used to {{clarify|date=May 2021 |reason=separate what, where?|text=separate the alteration zones}}. For example, when different data from various zones are available, discriminant analysis can find the pattern within the data and classify it effectively.<ref>{{cite journal | last1 = Tahmasebi | first1 = P. | last2 = Hezarkhani | first2 = A. | last3 = Mortazavi | first3 = M. | year = 2010 | title = Application of discriminant analysis for alteration separation; sungun copper deposit, East Azerbaijan, Iran. Australian | url = http://ajbasweb.com/old/ajbas/2010/564-576.pdf | journal = Journal of Basic and Applied Sciences | volume = 6 | issue = 4| pages = 564–576 }}</ref>