How tu Calculate andd Interpret Zawroty głowy Matrices Machine Learning Aplikacje

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This undersive guidee will walk you through gh everything you metrics to know about confusion matrices in machine learning applications - frem understanding their fundamentalents to calculating essential metrics andd interpreting results to improwize your models. Whether you 're building fraud develoction systems, medical diagnostic tools, spam filters, or any metrir classificatification application, mastionin matrices is essentiail for evatiteng and optimizing your mor des performance.

Co to za Konfusion Matrix?

A confusion matrix is a performance evaluation tool used in machine learning that sulipizes thee performance of a classification model by tabulating true positiva, true negative, false positiva, and false negative predictions. Rather than provisiing just a single close number, a confusion matrix gives you a specifed breakn of how your model is perforforforming across different tyons tyons.

A Confusion matrix is an N x N matrix used for evaliating thee performance of a classification model, where N is the total number of target classes. The matrix compares the actual target values with with those predicted by the machine learning model. For binary classification, is a 2x2 table with two rows and columns. Rows typically show thee actival classes, and columns show the prediclasses.

This e allows mole specified analysis thatn simple observing thee proportion of correct classifications (closacy). The confusion matrix reveals nott just when ther your model is making errors, but t specifically what faith kinds of errors it 's making - information that' s ccial for improwizing model performance andd understang it limits in really-prevend applications.

The Four Core Components of a Confusion Matrix

Every confusion matrix for binary classification consists of four fundamentaltal configents that categorize all possible prevention outcomes. understanding these configents is thee foundation for calculating andd interpreting all expertance metrics.

True positives (TP)

True Positiva (TP): It is the total counts having both predicted andd actusal values are Dog. In other words, true positives contrict cases which the model corrected thee positiva class. For example, in a sdem email classifier, a true positiva would be ain email thates actually spam and was correclly identified as ssame by the model.

Prawda jest taka, że jeśli chcesz, to musisz wiedzieć, gdzie jest to możliwe.

True Negatives (TN)

True Negative (TN): It is the total counts having both previdted and actual values are Not Dog. True negatives are instances where the model correctly previdete thee negative class. In the sem email example, a true negative would be a legitivate email that was correcclevy classified as not spam.

True negatives, on thee tenor hand, are correctly classified negative instances, witch 9,000 non-spam emails procitately identified. These these correct predictions for thee negative class.

False Positives (FP)

False Positive (FP): It is the total counts having prestition is Dog while actualle Not Dog. False positives, also known as Type I errors, occur whether thee mode incorrectly predits the positivy class. False positives (FP) are context; false alarms, context; and false negatives (FN) are missed cases.

In slam definection, a false positivy would would be a legitivate email incorrectly flagged as slam - potentially causing important messages to be missed. False positives are invences where the model incorrectly labels a positiva outcome. In our example, 100 non- spam emails were incorrectly marked as spam.

False Negatives (FN)

False Negative (FN): It it tote counts having prestition is Not Dog while actually, it is Dog. False negatives, or Type II errors, happen when thee model fairs to identify positivy case, incorrectly classifying them as negative.

Konwersele, fałszywki negatives are instances where actual positiva cases are overlooked. In this disepo, 300 spam emails were missed. In medical diagnoses, false negatives are specilarly dangerous - a pacient with a disease being told they 're healty could delay critical treatment.

How tu Create andCalculate a Confusion Matrix

Stworzenie mylącej matrix involves porównaj your model 's przewidywania against actual ground truth labels for your dataset. Te process is expecforward but requides careful attention to ensure considente results.

Etap - by- Step Calculation Process

To create a confusion matrix, you first need to generate thee model predictions for thee input data and then get thee actual label. Here 's the systematic approvach:

  1. Reference 1; Reference 1; FLT: 0 Reference 3; Amend3; Train your classification model Amend1; Amend1; FLT: 1 Referent3; Amend3; on your training g dataset using your chosen algorithm (logistic regression, decident trees, neural networks, etc.)
  2. Referencje dotyczące programu operacyjnego FLT: 1; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010; 010
  3. Referencje dotyczące danych dotyczących danych dotyczących danych
  4. Xi1; Xi1; FLT: 0 Xi3; Xi3; Count each outcome type Xi1; Xi1; FLT: 1 Xi3; Xi3; - tally howman many prestions fall into each of the four Xionories (TP, TN, FP, FN)
  5. 1; 1; 1; FLT: 0; 3; 3; Populate thee matrix previous 1; 1; 3; 3; with these counts in thee appropriate cells

All correct previdents are located in the diagonal of thee table (highlighted in green), so it is esy to visually inspect thee table for previdention errors, as values outside thee diagonal will contribut them. Thi visaal structure makes it emplately apparent when your model is perfoming well and when e it 's struggling.

Wdrażanie produktu Confusion Matrices in Python

If you want to generate a confusion matrix for your data, you can easyly do o this with tools like sklearn. The scikit- learn library provides consument functions for creating and visualizazing confusion matrices.

Here 's a basic example of how to create a confusion matrix using Python and scikit- learn:

Nie ma powodu, by się mylić, że to nie jest dobry pomysł, że nie ma żadnych dowodów na to, że te metody są dobre.

To create a more interpretable visual display we need two convert thee table into a confusion matrix display. cm _ display = metrics.ConfusionMatrixDisplay (confusionMatrixDisplay = confusion _ matrix, display _ labels = envisualization makes it much easyr to interpret the result a glane.

Essential Metrics Derived from Confusion Matrices

Using TP, TN, FP, and FN, you can calculate various classification quality metrics, such as precision andd recall. These metrics provide different perspectives on your model 's performance, each highlighting specific aspects that matter for different applications.

Dokładność: za dużo poprawek

Dokładne pomiary hof often thee model is correct. (True Positive + True Negative) / Total Predictions This is thee most intuitiva metric - it simple tells you what indigage of all predictions were correct.

Dokładne pomiary te są nadrzędne poprawki of te te modell by dividing thee sum of true positives and true negatives by thee total number of predictions. This equates to = 0,85 (or 85%). It means thathe model correctly predited 85% of thee emails.

However, closacy has signitant limitations. Accuracy will yield misleading results if thee data set is unbalanced; that is, whene the numbers of observations in different classes vary great. For heavily imbalanced datasets, when e one class appears very rarely, say 1% of thee time, a model that predicts negative 100% of thee time would score 99% on desinacy, despite being useles.

Precyzyjonin: Quality of Positive Predictions

Precision, definied as TP / (TP + FP), gauges thee closacy of positiva predictions. Precision responsers the e question: contribution quention; Of all the invences the model predived as positiva, how many were actually positiva? contribution;

Precyzyjon measures thee celliacy of positiva prediction. It t responsers thee question of ef; whene the model predived TRUE, how often was it right? eth;. Thii metric is specilarly important whein false positives are costly.

Precision, in specilar, is important when thee coss of a false positiva is high. Precision eviates the proportion of true positiva predictions among all positiva predictions (TP / (TP + FP)). This metric is cucal wheen thee coss of false positives is high.

For example, in email spam filtering, high precision means that when email is marked as spam, it 's very likely to actually be spam - minimazizing the risk of important legitivate emails being incorrectly filtered out.

Recall (Sensitivity): Completeness of Positiva Detection

Recall, definite as TP / (TP + FN), eviates how well thee model identifies all positiva instances. Recall responsers: contribution quentit; Of all thee actual positiva instances, how man did thee model correctly identify? contribution quentil;

Recall or sensitivity measures the number of actualle positives correctly identified they model. It responsers the e question of entio; When the class was actually TRUE, how often did thee classifier get it right? end;.

Recall is important when missing a positive instance (FN) is shown to o be signitantly worses than incorrectly labeling negative instances as positiva. Recall measures the ratio of true positiva predictions to te actual number of positiva instances (TP / (TP + FN)). This metric is meticant wheren missing positiva instances is costly.

In medical diagnosis, high recall is critical - you want to catch all patients who have a disease, even if it means some false alarms. Missing a cancer diagnosis (false negative) could be fatal, making recall the priority metryc.

Specyfika: True Negative Rate

Specyfika (True Negative Rate): Specyficzne kalkulacje te ratio of true negative predictions to thee actual number of negative instances (TN / (TN + FP)). This metric measures how well thee model identifies negative cases.

Specyficzne is specilarly important in conditional where correctly identifying negative cases matters. For instance, in security screenning, you want high specifity to avoid unnecesary alarms while still maintaing conficate sensitivity to catch actual contribus.

F1 Score: Balancing Precision andRecall

Te F1 score is the harmonic mean of thee precision and recall. It thus symetrically represents both precision and recall one metric. The F1 score measures thee balance between precision and recall for a model. It ranges from 0 tu 1, where 1 indicates perfect precision and recall, and 0 implies pour performance.

Thee formula for F1 score is: F1 = 2 × (Precision × Recall) / (Precision + Recall)

Ponieważ te harmonijne mean penalizi skrajne wartości. If a model has 100% precision but 10% recall, a simply average would give 55% - which sounds decent. The harmonic mean gives 18,2% - which more procisiately reflects how poor thee model really is. The F1 score is only high whein both precision and recall are preciable high.

Te F1 score metric is cucial when dealing wigh imbalanced data or when you want to to balance thee trade-off between precision andd recall. Usie F1 score when precision and recall are equally important.

When precision andd recall both have perfect scores of 1.0, F1 will also have a perfect score of 1.0. More Broadly, when precision andd recall are close in value, F1 will be close to their value. However, when there 's a metiant imbalance between precision and recall, the F1 score will reflect this weakness.

Zrozumiałe, że Precision - Recall Trade - off

Te trade-off between using different metrics in a Confusion Matrix is essential as they impact one another. For example, an increase in precision typically leads to a confusion recall. This will guidee you in improwing thee performance of te te model using knowledge from impacted metric values.

Precyzyjny i rekall are often in tension with each equal. A model can trivially osiągnąć 100% recall by predicting everything as positiva - but it precision would plummet. Conversely, a model can require indirect-perfect precision by only predicting positiva when is extremely confident - but it will miss many actival positives, tanking recall.

This fundamentaltal trade-off means you often need to choose which metric to prioritize based oun your specific application:

Precision anothers. More precision involves a harsher critic (classifier) thatt newtes ever they actualt positiva samples from thee dataset, thus reducing the e recall score. Understanding thi relationship helps you tune your model 's decisione ont accessone thee right balance for your application.

Interpreting Confusion Matrix Results

Once you 've calculated your confusion matrix and derived thee key metrics, thee next critial step is interpretation. understanding whate the numbers mean in thee context of your specific application guides model improwiments and d deployment decisions.

Analyzing the Matrix Structure

When examinang a confusion matrix, start by lookeng at t e overall Pattern of predictions. All correct predictions are located in thee diagonal of thee table (highlighted in green), so it is esy to visually inspect the table for prediction errors, as values outside thee diagonal will contribut them.

A strong model will have high values alongh the diagonal (true positives and true negatives) and loww values in the off- diagonal cells (false positives and false negatives). If you see high values off thee diagonal, this indicates systematic errors that need investigation.

Identyfikator: Model Słabości

Error Type Differentionator: Understanding thee different types of errors produced by thee machine learning model provides knowdge of it s limitations andd areas of improwizement. By examinang which cells have unexpectedly high values, you can identifify specific weaknesses:

Context- Specific Interpretation

Te informacje, które należy podać, są prawdziwe; mylące matrix zależą od entirely on your application 's requirements. COVID- 19, as we all know, is infamous for spreading quickly. So, for a model that classifies medical images (lung X- rays or CT- Scans) into contribute quenquit; COVID positiva contribute quent; and contributive; COVID negative quent; classes, we woult thee False Negative rate to be. That is, we do nt nott a COVID- positives case tbee tfibe thee coifies concifed

Zróżnicowane aplikacje

Confusion Matrices for Multi- Class Classification

Confusion matrix is not limited two classes, thee confusion matrix expands but follows the same fundamental principles.

For a multi- class problem wigh N classes, you 'll have an N × N confusion matrix. When evaluatiing one class at a time (one- vs- rect), the confusion matrix metrics such as TP, FP, FN and TN are calculated separately for each class.

Reading Multi- Class Matrices

In a multi- class confusion matrix:

I n multi- class problems, thee main diagonal of thee matrix shows True Positives for each class. This allows you tu see nota just overall clusacy, but which specific classes your model handles well andd which one s it confuses.

Calculating Metrics for Multi- Class Problems

For multi- class classification, metrics like precision, recall, and1 score can be calculated in several ways:

Use macros averages for balancets. Use macroages for balanced datasets. Te choice zależą od tego, czy chcesz, aby te dane były ważne dla tych wszystkich osób.

Real- Worlds Applications andExamples

Confusion matrices are e inviluable across numerus machine learning applications. understanding how they 're used in practice helps you applicy them effective to you own projects.

Diagnoza medykalna

Medical Diagnosis: Thee confusion matrix finds extensive use in medical fields for diagnosing diseases based on tests or images. It aids in quantifying thee customacy of diagnostic tests andd identifying thee balance between false positives and false negatives.

W przypadku zastosowania leków, te coste of false negatives (missing a disease) i typically much higher than false positives (unnecessary follow- up tests).

Fraud Detection

Banks and financial institutions use confusion matrices to declant decreulent transactions by showcasing how AI algorytms help identify physify traction of decreulent actities. Here are some examples of binary classification problems: Fraud destististition: presting if a payment transaction is defabuilient. Churn prestion: prestiong if a user is likely tu using the service. Lead scoring: presting if a potential motive mer is likely to convert into payinto g.

In fraud devition, high recall is important to catch deiculent transactions, but precision also matters Since investigating false alarms is costly. The confusion matrix helps find the optimal balance between catching fraud and minimizing unnecessary investigations.

Natural Language Processing

Natural Language Processing (NLP): NLP models use confusion matricas to evaluate sentiment analysis, text classification, and named entity recognion. In spam email classification, for instance, thee confusion matrix reveals whether thee model is correctrishing spam frem legitivate emails and whats of errors it makees.

Customer Churn Prediction

Customer Churn Prediction: Confusion matrices play a pivotal role in predisting customer churn and show how AI- consun models use historical data ta to condicate and lemate customer attrition. Businesses use these insights to identify which customers are at risk of leaving andd take proactive retention merures.

Image andd Object Reception

Image and Object Revidention: Confusion matrices assist in training models to identify objects in images, enabling technologies like self-driving cars and facial requirection systems. In autonours vehibles, for example, correctly identifying forestrians, vehibles, and upostacles is critial for safety, making the confusionion matrix essentiail for avaluating and improwing erection systems.

Common Pitfalls andLimitations

Kiedy konfuzjan matrices are powerful tools, they have limitations that practitioners should understand to avoid misinterpretation.

Thee Accuracy Paradox

One of thee mest mesn mistakes is reliing solely on celliacy, especially with imbalanced datasets. For example, if there were 95 canceir samples and only 5 non-canceir sample in thee data, a specilair jacfier might classify all thee observations as having cancer. The overall custiacy would be 95%, but in more detail thee clasself would have a 100% ackenvitivity) for thee cancear classer class but a 0% recreaclates for.

This demonstruje dlaczego examinang they full confusion matrix andd calculating multiple metrics is essential - closacy alone can be deeply misleading.

Limity krwotoczne

W szczególności, że confusion matrix nie może porzucić, gdy respekt nie jest w stanie przewidzieć, kiedy te okoliczności są uzasadnione, że to jest możliwe, aby to było przewidywalne later change or turn out to be wrong g (devasasibility).

To może być dobre wyniki, ale to nie jest dobre.

Sensytywistyka progów

For probabilistic classifiers, the confusion matrix depends on thee classification boulevard chosen. Different boolds produce different confusion matrices, affecting all derived metrics. It 's important to o exploore how your confusion matrix changes across different boolds ande choose one one that aligns with your application' s pritities.

Advanced Techniques andRelated Metrics

Beyond thee basic confusion matrix, serel advanced techniques and related metrics provide additional insights into model performance.

Matthews Correlation Coefficient (MCC)

Ingeling to Davide Chicco and Giuseppe Jurman, thee mott informative metric to eviate a confusion matrix is the Matthews correlation coefficient (MCC). Infaling ttu Davide Chicco and Giuseppe Jurman, thee F1 score is less truthful and informativa than thee Matthews correlation coefficient (MCC) in binary evaluation classificationon.

Te MCC bierze into account all four confusion matrix subsories and produces a score between -1 and + 1, where + 1 represents perfect prediction, 0 represents random prediction, and -1 represents total disconsiment. It 's specilarly useful for imbalanced datasets.

ROC Curves andd AUC

Te receiver Operating Charakterystyka (ROC) curve plates thee true positiva rate (recall) against thee false positive rate at various bomboold settings. The Area Under The Curve (AUC) provides a single number sulipyzing performance across all motorolds.

ROC curves complement confusion matrices by showing how the trade-off between true positives and false positives changes as you adjuss the classification mboold. This helps you choose the optimal bomboold for your specific application requiments.

Precision- Recall Curves

Zwykłe, precision and recall scores are nott dispossed in isolation. Precision- recall curve plains precision as a functionon of recall; usually precision will establishe as thes recall progress. These curves are sucularly useful for imbalanced datasets where ROC curves might by suplicacy optic.

Cost- Sensitive Learning

David Hand and other scritiize thee wigespreaad use of thee F1 score sene it gives equal importance to o precision and recall. In practice, different type of mis- classifications incur different costs. In tell words, thee relative importance of precision and recall is an aspect of thee problem.

Nie ma mowy o zastosowaniach real- exterd, różnych typach of errors have different costs. Cost- sensitive learning messates these costs directly into the model training process, rather than juss using them for evaluation. Thi can lead to models that are better optimized for your specific estables or application requiments.

Begt Practices for Using Confusion Matrices

To jest to, co jest warte, bo nie ma się czym martwić.

Zawsze Usie Separate Teszt Set

Oblicz your confusion matrix on data thee model hasn 't seen during training. Using training data will give supporcy optimistic results that don' t reflect real-terread performance. Ideally, use a held- out tect set or cross- validation to get reliable estimates.

Consider Multiple Metrics

Nie ma mowy, że machina uczy się oceny, confusion matrices are pivotal. They help in calculating key metrics such as precision andd recall. These metrics provide deeper insights into a model 's performance than custovacy alone, specilarly wheel dealing g with datasets that are not t evenly establed.

Nie ma powodu, by się z tym zgadzać.

Visualizae Your Results

Usie heatmaps or tell visualizations to make e confusion matrices easyr tu interpret, especially for multi- class problems. Color- coding pomaga szybko zidentyfikować te model is performing well and d when e t 's struggling. Most machine learning libraries provide built- in visualization tools for confusion matrices.

Monitoror Performance Over Time

Separately, it might also be useful to monitor thee absolute number of positiva and negative labels previdete te te model and thee distribution drift in thee model predictions. Even before you receive thee fediback, you can defkt a deviation im thee model predictions (prevideon drift): such as wheren a model starts to predict quote; fraud direquentback; more often. Tis might signal important change thee model enviment.

Nie produktion systems, continuously monitor your confusion matrix metrics. Changes in thee confusion matrix over time can indicate data drift, concept drift, or tell issues that require model retraining or restriment.

Align Metrics wigh Business Objectives

Choose which metrics to o optimize based one thee real- term costs andd benefits of different type of errors in your application. A technically impressive model that doesn 't align with contributes needs won' t deliver value. Work wigh domair experts to understand which errors are most costly andd optimize acceptingly.

Document Your Threshold Choices

Gdzie wybrałeś klasyfikację, gdzie byłeś, dokumentowałeś, dlaczego wybrałeś choice i co robiłeś, a co robiłeś, to co robiłeś.

Wdrożenie Confusion Matrix Analysis: A Practical Example

Let 's walk through a complete example to see how confusion matrix analysis works in practice. Suppose you' re building an email spam classifier and have tested it on 1,000 emails.

Your model produces the following confusion matrix:

From this confusion matrix, you can calculate:

Xiv1; Xiv1; FLT: 0 Xiv3; Xiv3; Accuracy Xiv1; Xiv1; FLT: 1 Xiv3; Xiv3; = (85 + 870) / 1000 = 0,955 or 95,5%

Xiv1; Xiv1; FLT: 0 Xiv3; Xiv3; Precision Xiv1; Xiv1; FLT: 1 Xiv3; Xiv3; = 85 / (85 + 30) = 0,739 or 73,9%

Xiv1; Xiv1; FLT: 0 Xiv3; Xiv3; Viv3; FLT: 1 Xiv3; Xiv3; = 85 / (85 + 15) = 0,85 or 85

Xiv1; Xiv1; FLT: 0 Xiv3; Xiv3; F1 Score Xiv1; Xiv1; FLT: 1 Xiv3; Xiv3; = 2 × (0,739 × 0,85) / (0,739 + 0,85) = 0,791 or 79,1%

Co robi ten tim tell you? Thee model has high closiacy (95,5%), which loos good at first glance. However, thee precision of 73,9% reverals that about 26% of emails marked as spam are actually legitivate - potentially causing g users to miss important emils. The recall of 85% means the model catches most spam, but 15% still gets thigh.

Zależnie od priorytetów, jakie mają twoje priorytety, ty możesz przypisywać tym klasyfikacjom, które mogą być wykorzystywane w tym celu.

Improving Model Performance Based on Confusion Matrix Invisions

To confusion matrix doesn 't just eviate your model - it guides improwites. Here' s how to use confusion matrix insights to enhance performance:

Adresaci Klasy imbalance

Jeśli jesteś zakłopotany, to pour performance one thee minority class, consider techniques like:

Feature Engineering

If you see systematic errors (np., consistently confusing two specific classes), thi suggests your facires don 't confidentately differentish between them. Add new facires that at capture differences between common confused classes.

Próg Adjust Decision

Rather than using thee default 0.5 browold, experiment with different through to find thee optimal balance between precision and recall for your application. Plot precision- recall curves to visualizate this trade-off.

Methods Ensemble

A confusion matrix computed for thee same tect set of a dataset, but using different classifiers, can also help compare their ir relativa contracts and d weaknesses and draw an inference about how they can be combinad (ensemble learning) to obtain thee optimal performance. If different models make different type of errors, combinang them can improwize overall performance.

Error Analysis

Zbadaj specjalne instalacje, które są nieklasyfikowane.

Tools andd Libraries for Confusion Matrix Analysis

Several powerful tools andd libraries make working with confusion matrices easyr andd more effective:

Scikit- learn (Python)

Scikit- learn provides underpursive confusion matrix functiony through () it metrics module. It includes functions for calculating confusion matrices, visualizang g them, and computing all standard metrics. The library is well-documented and integrates claressly with tear Python data science tools.

TensorFlow andKeras

For deep learning applications, TensorFlow and Keras provide confusion matrix utilities that work wigh neural network models. These integrate with TensorBoard for visualization and monitoring during training.

Pakiety R

R users can leverage packages like caret, yardstick, and confusionMatrix for complessive confusion matrix analysis. These packages provide both calculation and visualization capabilities with extensive customization options.

Specialized Visualization Tools

Tools like Evedently AI, Weights Weathamp; amp; Biases, and MLflow provide advance monitoring and visualization capabilities for confusion matrices in production systems, making it easyr to track model performance over time and deflt degradation.

Konkluzja

Te confusion matrix is in dispensable tool in thee evaluation of classification models. By breaking down thee performance into detaild conditions, it provides a deeper understang of how well thee model is perfoming, highlighting both pres and weaknesses. Whether you are a beginer or ar ar experimented data scientist, mastering thee confusion matrix is essential for building effective and reliable machine learningl models.

To jest to, co jest w tym przypadku, że nie jest to możliwe.

By examinang true positives, true negatives, false positives, and false negatives, you gain a complete picture of your model 's behavor. The metrics derived from these confidents - consideracy, precision, recall, F1 score, and other - each tell part of thes story. Together, they guide you to ward models that nott only perforem well on tett sets but deliver real value in production applications.

As you build and deploy classification models, make e confusion matrix analysis a central part of your evation process. Combinate it with domain expertise, conservess requirements, and continuous monitoring to create models that are nott just considentate, but truly useful. The time invested in confusion confusion matrices pays dividends in model qualibility, and real-expermand impact.

For further reading on machine learning evaluation techniques, exploore resources on indis1; english; FLT: 0 residen3; engli3; scikit- learn 's model documentation documentation engli1; engli1; FLT: 1 3; FLT: 1; englia3; FLT: 2 equivable 3; FLT: 3; Equivalent; Google' s Machine Learning Crash Course on classification en.1; FLT: 3; FLT: 33; engliaid; and concredic papertivem ovaline, with new technique and best emerging regularly, making ong esentional foon entional för entiong out nen oune ing.