multivariate statistics
Multivariate statistics is the branch of statistics that analyzes several variables simultaneously, treating them not as a collection of separate measurements but as a correlated whole, and it is essential in semiconductor engineering because the quality of a wafer is described not by any single metrology reading but by a high-dimensional profile of many readings that move together. When an engineer measures film thickness, sheet resistance, defect density, and critical dimension across a wafer, those measurements are not independent, because they are all shaped by the same underlying process conditions, and analyzing them one at a time throws away the correlations that carry the most diagnostic information. The univariate methods of the earlier statistics keywords, applied to each variable separately, miss the multivariate structure: a wafer may look normal on every individual measurement yet be a clear outlier in the joint space of all of them. Multivariate statistics supplies the tools to describe the joint distribution, to reduce the many correlated measurements to a few meaningful composites, to test whether groups of wafers differ across all variables at once, and to detect faults that no single variable reveals. This document develops the covariance structure, the multivariate normal distribution, and the methods of dimension reduction and classification, and it shows how each applies to the multivariate metrology and process monitoring of a fab.
**The data of a multivariate analysis are organized in a data matrix, and the structure of that matrix is the foundation on which every multivariate method is built.** The data matrix has one row for each observational unit, such as a wafer, a lot, or a die site, and one column for each variable, such as a thickness, a resistance, or a defect count, so that the entry in the $i$-th row and the $j$-th column is the value of the $j$-th variable measured on the $i$-th unit. The analysis then characterizes the relationships among the columns, which are the variables, using the pattern of their joint variation across the rows. The two most fundamental summaries of a set of variables are their means, which locate the data, and their variances, which measure how much each variable varies, but the essential new ingredient of multivariate analysis is the covariance, which measures how two variables vary together. The means, variances, and covariances of all the variables are assembled into the mean vector and the covariance matrix, and these two objects together summarize the entire joint structure of the data under the assumption that the joint distribution is approximately multivariate normal. The data matrix is the raw material, and every multivariate method is a way of extracting structure from this matrix. An engineer who can read a data matrix and its covariance matrix can understand any multivariate analysis.
**The covariance matrix is the central object of multivariate statistics, because it captures the degree to which the variables move together, and it is the generalization of the variance of a single variable to several variables.** The covariance of two variables $x$ and $y$ is the average product of their deviations from their means, and it is positive when high values of one tend to accompany high values of the other, negative when high values of one accompany low values of the other, and near zero when they are unrelated. The covariance matrix is the square table in which the entry in the $i$-th row and $j$-th column is the covariance of the $i$-th and $j$-th variables, so that the diagonal entries are the variances of the individual variables and the off-diagonal entries are the pairwise covariances. Because the covariance of $x$ with $y$ equals the covariance of $y$ with $x$, the covariance matrix is symmetric, and because variances are always positive, it is positive semidefinite, which means that its eigenvalues are never negative. The covariance matrix carries a great deal of information, but it depends on the units of measurement, so the engineer often works instead with the correlation matrix, which rescales each variable to unit variance and so expresses the associations in unitless values between negative one and one. The covariance and correlation matrices are the lens through which all multivariate structure is first examined. The pattern of large and small entries in the correlation matrix reveals which variables form groups that move together.
**The multivariate normal distribution is the natural model for a vector of continuous measurements, and it generalizes the bell curve of a single variable to several correlated variables.** A vector of $p$ variables has a multivariate normal distribution when every linear combination of the variables is normally distributed, and the distribution is completely described by its mean vector and its covariance matrix. The density of the multivariate normal distribution is a bell-shaped surface in $p$-dimensional space, and its contours of constant density are ellipsoids that are centered at the mean and whose shape and orientation are governed by the covariance matrix. When the variables are independent, the ellipsoids are aligned with the coordinate axes and are circular in the directions of equal variance, whereas when the variables are correlated, the ellipsoids are tilted so that the direction of greatest spread is along the combination of variables that moves together. The multivariate normal distribution is the assumption behind many multivariate procedures, because under it the mean vector and the covariance matrix are sufficient to describe the entire joint distribution, and it is approximately valid by the multivariate central limit theorem when the underlying data are averages. In a fab the multivariate normal model describes the joint distribution of correlated metrology readings across a wafer, and the elliptical contours become the natural control boundaries. The multivariate normal distribution is the bridge between the raw data matrix and the inferential multivariate methods.
**The eigenvalues and eigenvectors of the covariance matrix are the key to the structure of a multivariate dataset, because they reveal the directions in which the data vary most.** An eigenvector of the covariance matrix is a direction such that multiplying the covariance matrix by that vector is the same as scaling the vector by a factor, and that factor is the corresponding eigenvalue, which measures the amount of variance in that direction. The eigenvectors of the covariance matrix are the directions of the principal axes of the data ellipsoid, and they are mutually orthogonal, meaning that they point along the independent directions of variation in the data. The eigenvector with the largest eigenvalue points in the direction of greatest variance, the eigenvector with the second-largest eigenvalue points in the direction of the next greatest variance that is orthogonal to the first, and so on, so that the eigenvectors order the independent directions of variation from most to least variable. The sum of all the eigenvalues equals the total variance of all the variables, and each eigenvalue's share of that total is the proportion of variance explained by the corresponding direction. The eigenvalues and eigenvectors therefore decompose the covariance matrix into its independent components of variation, and this decomposition is the engine of principal component analysis. An engineer who computes the eigenvalues and eigenvectors of a covariance matrix has found the natural axes of the data.
**The principal component analysis is the most important method of dimension reduction in multivariate statistics, and it reduces a large number of correlated variables to a few uncorrelated composites that capture most of the variation.** Each principal component is a weighted combination of the original variables, and the components are chosen so that the first component has the largest possible variance, the second has the largest variance among those orthogonal to the first, and so on, and it turns out that the principal components are exactly the eigenvectors of the covariance matrix weighted by their eigenvalues. The first few principal components typically capture most of the total variance of the data, so that the engineer can project the high-dimensional data onto the small set of components and lose very little information. The coefficients that define each component are called its loadings, and they show which of the original variables the component is most associated with, so that a component can often be interpreted as a meaningful combination, such as an overall thickness or an overall profile shape. The value of a unit on a component is called its score, and the scatterplot of the first two scores is the standard two-dimensional view of a high-dimensional dataset. In a fab the principal component analysis reduces many correlated metrology readings to a few summary scores, and it is the foundation of multivariate process monitoring. The reduction from many variables to a few components is the heart of the method.
**The proportion of variance explained by the principal components is the guide to how many components to keep, and the scree plot and the eigenvalue threshold are the standard tools for that decision.** Because the eigenvalues sum to the total variance, the fraction of the total variance explained by the first $k$ components is the sum of their eigenvalues divided by the total, and this fraction is the primary measure of how much information is retained. The scree plot graphs the eigenvalues in decreasing order, and it typically falls steeply and then levels off, and the number of components to keep is chosen at the elbow of the plot, where the drop in eigenvalue becomes gradual. A common rule keeps the components whose eigenvalues exceed one, on the grounds that such a component explains more variance than a single standardized variable, while a more principled rule keeps enough components to explain a target fraction of the variance, such as ninety percent. The choice involves a trade-off between the simplicity of a low-dimensional model and the completeness of a higher-dimensional one, and the engineer balances the two against the needs of the analysis. In a fab the explained-variance analysis determines how many summary scores to monitor in the multivariate control chart. The disciplined choice of the number of components keeps the model simple without losing the important structure.
**The factor analysis is a method related to principal component analysis that seeks to explain the correlations among the observed variables by a smaller number of unobserved latent factors, and it is used when the engineer believes that the measured variables are driven by a few underlying constructs.** In factor analysis the model writes each observed variable as a weighted combination of a few common factors plus a unique error, and the weights, called factor loadings, measure how strongly each variable reflects each factor. The common factors account for the correlations among the variables, so that once the factors are held fixed, the variables are conditionally independent, and the remaining variation of each variable is its unique variance. Factor analysis differs from principal component analysis in that it models the covariance structure through latent variables rather than merely transforming the observed variables, and it requires the engineer to choose the number of factors and often to rotate the solution to make the loadings more interpretable. In a fab factor analysis might reveal that many correlated metrology readings are driven by a small number of underlying process factors, such as an overall temperature gradient or a pressure uniformity. The latent-factor structure gives the engineer a simplified causal picture of the data. Factor analysis and principal component analysis together form the classical toolkit of dimension reduction.
**The multivariate regression extends ordinary regression to the setting in which the engineer wishes to predict several correlated response variables from a set of predictor variables, and it is the natural multivariate generalization of the models in the inference statistics keyword.** In multivariate regression, each response variable is regressed on the same set of predictors, and the responses are modeled jointly, so that the correlations among the responses are captured in the covariance matrix of the errors. The least squares estimates of the regression coefficients are the same as those obtained by regressing each response separately, but the multivariate model provides a coherent covariance structure and enables joint tests of whether a set of predictors affects the vector of responses as a whole. The multivariate regression is the foundation of many engineering models that predict a profile of outputs from a set of process inputs, and it is closely related to the analysis of the multivariate analysis of variance. In a fab the multivariate regression predicts the full vector of wafer metrology from the process settings, so that the engineer can anticipate how a change in a single input shifts the entire quality profile. The joint modeling of the responses is the advantage of the multivariate regression over a set of univariate regressions.
**The multivariate analysis of variance, abbreviated MANOVA, extends the analysis of variance to several correlated response variables at once, and it tests whether the mean vectors of several groups differ rather than testing each response separately.** Where the univariate analysis of variance asks whether the means of a single response differ across groups, MANOVA asks whether the entire vector of response means differs, and it does so by comparing the covariance structure within the groups with the covariance structure between the groups. The MANOVA test statistics, such as Wilks lambda, Hotelling trace, and Pillai trace, are functions of the eigenvalues of a certain matrix, and they reduce to the univariate F statistic when there is only a single response. The advantage of MANOVA is that it respects the correlations among the responses, so that it can detect a difference in the joint mean vector that no single response would reveal, and it protects against the inflation of the error rate that would come from testing each response separately. In a fab MANOVA compares the full metrology profiles of wafers produced under different process recipes, testing whether the recipes differ across all the measured characteristics at once. The joint test of the response vectors is the contribution of MANOVA. The engineer who uses MANOVA tests the whole profile rather than a single measurement.
**The discriminant analysis is a method for classifying observations into groups and for identifying which variables best separate the groups, and it is the classification tool that grows out of the multivariate normal model.** In linear discriminant analysis, developed by Ronald Fisher, each group is modeled as a multivariate normal distribution with a common covariance matrix, and a new observation is classified into the group whose mean is nearest in the Mahalanobis distance, which is the covariance-adjusted distance that respects the correlations of the variables. The discriminant function is a weighted combination of the variables that maximizes the separation between the groups relative to the within-group variation, and its coefficients indicate which variables contribute most to the separation. The performance of a discriminant rule is assessed by how well it classifies the observations, often by leaving out one observation at a time and predicting its group, and the misclassification rate measures the quality of the rule. In a fab the discriminant analysis classifies wafers as good or defective, or assigns a wafer to the process condition that produced it, based on its full metrology profile. The classification of new observations into known groups is the practical use of discriminant analysis. The discriminant rule turns the multivariate profile into a group assignment.
**The cluster analysis is an unsupervised method that groups the observations into clusters of similar units without any pre-existing labels, and it is used to discover structure in the data rather than to test a hypothesis.** In cluster analysis the observations are grouped so that units within the same cluster are similar to one another and units in different clusters are dissimilar, where similarity is measured by a distance between the multivariate profiles of the units. The hierarchical clustering builds a tree of nested groups by successively merging the closest clusters, and it is displayed as a dendrogram that shows the hierarchy of similarity, while the k-means method partitions the observations into a specified number of clusters by iteratively assigning each unit to the nearest cluster center and recomputing the centers. The number of clusters is chosen by examining how the within-cluster variation decreases as the number of clusters increases, and by the interpretability of the resulting groups. In a fab the cluster analysis groups wafers or lots by their metrology profiles, revealing distinct populations that may correspond to different process conditions or failure modes. The discovery of natural groupings in unlabeled data is the purpose of cluster analysis. The cluster structure often points the engineer toward the process factors that produced the distinct groups.
**The Hotelling T-squared statistic is the multivariate generalization of the t statistic, and it is the fundamental tool for detecting outliers and monitoring a multivariate process.** For a single new observation, the Hotelling T-squared is the squared Mahalanobis distance of the observation from the center of the reference data, and it measures how far the observation is from the typical profile in units of the covariance structure. The Hotelling T-squared distribution provides control limits for the statistic, so that a new wafer whose T-squared exceeds the limit is flagged as an outlier, and this is the multivariate analogue of the control limits of a univariate control chart. Unlike a univariate chart, which monitors each variable separately, the Hotelling chart monitors the joint profile, so that it can detect a wafer that is normal on every individual measurement but unusual in the combination of its measurements. The Mahalanobis distance, developed by Prasanta Chandra Mahalanobis, is the covariance-adjusted distance that underlies the T-squared statistic, and it reduces to the ordinary Euclidean distance when the variables are uncorrelated. The sampling distribution of the T-squared statistic is related to the F distribution, and in large samples it approaches a chi-square distribution with degrees of freedom equal to the number of variables, so that the control limit is read from the chi-square tables. In a fab the Hotelling T-squared chart is the standard multivariate control chart for the correlated metrology profile of a wafer, and it flags wafers whose joint profile has drifted. The monitoring of the joint profile is the core of multivariate statistical process control.
**The Mahalanobis distance is the natural measure of distance between a point and a distribution in multivariate space, and it corrects the ordinary Euclidean distance for the scale and correlation of the variables.** The Mahalanobis distance of an observation from the mean divides the deviation in each direction by the standard deviation in that direction, and it further accounts for the correlation, so that a deviation in the direction where the data are spread widely counts less than an equal deviation in a direction where the data are tightly packed. When the variables are uncorrelated and have unit variance, the Mahalanobis distance reduces to the Euclidean distance, but in general it is the only measure of distance that treats the elliptical contours of the data as the natural notion of closeness. The Mahalanobis distance is the basis of the Hotelling T-squared statistic, of the linear discriminant analysis, and of many outlier-detection methods, because it measures how unusual an observation is relative to the covariance structure of the reference population. In a fab the Mahalanobis distance flags wafers that lie far from the normal process region in the joint metrology space, even when no single measurement is out of spec. The covariance-adjusted distance is the key to all multivariate outlier detection. An engineer who thinks in terms of the Mahalanobis distance thinks correctly about multivariate proximity.
**The curse of dimensionality is the phenomenon that the behavior of high-dimensional spaces differs profoundly from the intuition developed in one or two dimensions, and it is the central caution of multivariate analysis.** In high-dimensional space, the volume of a region grows exponentially with the dimension, so that the data become sparse, and the distances between points become more similar, making the nearest neighbors less informative and the estimates of the covariance matrix less stable. The number of observations needed to estimate a covariance matrix grows with the square of the number of variables, so that with many variables and few wafers the covariance matrix becomes ill-conditioned or singular and the multivariate methods become unreliable. This is why dimension reduction is so important: reducing the many correlated measurements to a few principal components concentrates the information and stabilizes the estimation. The engineer must also be alert to the danger of overfitting, in which a model with too many parameters describes the noise of the training data and fails on new data, and the confirmation of a multivariate model on new data is the guard against it. In a fab, where the number of wafers in a study is often small while the number of metrology readings is large, the curse of dimensionality is a real constraint. The discipline of keeping the model dimension small relative to the sample size is the practical lesson.
**The mean vector and the generalized variance are the multivariate analogues of the mean and the variance of a single variable, and they summarize the location and the spread of the data in a single pair of numbers. The mean vector stacks the means of all the variables into a vector that locates the center of the data, while the covariance matrix measures the spread, and the generalized variance is the determinant of the covariance matrix, which is a scalar that measures the overall volume of the data cloud. The generalized variance is large when the variables are spread out and small when they are tightly concentrated, and because it equals the product of the eigenvalues of the covariance matrix, it is also a measure of how much total variation the data contain. The trace of the covariance matrix, which is the sum of its diagonal entries and also the sum of its eigenvalues, measures the total variance without regard to the correlations, while the determinant captures the dependence through the geometry of the data. In a fab the engineer uses the mean vector and the generalized variance to compare the overall location and spread of a metrology profile across process conditions. The two numbers give a compact summary of a high-dimensional dataset.
**The partial correlation extends the concept of correlation to the relationship between two variables after removing the influence of the other variables, and it is a more precise measure of association in a multivariate setting.** The partial correlation of two variables given a set of others is the correlation of the residuals after each variable has been regressed on the others, and it measures the association that remains once the common influences have been removed. Two variables may have a strong ordinary correlation that is entirely due to a third variable, and the partial correlation exposes such spurious associations by holding the third variable fixed. The partial correlation is closely related to the inverse of the covariance matrix, because the entries of the inverse covariance matrix are the partial covariances of the variables, and this connection makes the inverse covariance matrix a tool for understanding the conditional structure of the data. In a fab the partial correlation can reveal whether two metrology readings are genuinely related or merely share a common process cause. The partial correlation gives the engineer a sharper picture of which associations are real.
**The maximum likelihood estimation of the multivariate normal parameters is the principled way to estimate the mean vector and the covariance matrix from the data, and it is the foundation of the inference methods of the subject.** Under the multivariate normal model, the maximum likelihood estimate of the mean vector is the sample mean vector, and the maximum likelihood estimate of the covariance matrix is the sample covariance matrix divided by the sample size, and these estimates are consistent as the sample grows. The log-likelihood of a multivariate normal sample is a function of the determinant and the inverse of the covariance matrix, and maximizing it balances the fit of the mean with the fit of the spread, and the resulting estimates have the familiar asymptotic properties of maximum likelihood. The likelihood ratio is also the basis of the multivariate tests, because a comparison of the maximized likelihood under different models yields the test statistics for the mean vector and the covariance matrix. In a fab the engineer uses maximum likelihood to estimate the parameters of the joint metrology distribution that serves as the baseline for monitoring. The likelihood framework unifies the estimation and the testing of multivariate models.
**The tests of the mean vector generalize the single-sample and two-sample tests to the multivariate setting, and they ask whether the mean vector of the population equals a specified value or whether two populations share a common mean vector.** The one-sample Hotelling test generalizes the Student t test by comparing the sample mean vector with a hypothesized vector using the Mahalanobis distance and the Hotelling T-squared statistic, and it rejects the hypothesis when the mean vector lies far from the hypothesized value in the units of the covariance structure. The two-sample Hotelling test compares the mean vectors of two groups, and it reduces to the pooled t test when there is a single variable. These tests require more data than their univariate counterparts, because they estimate the full covariance matrix, and they illustrate the general rule that multivariate inference needs samples that grow with the dimension of the data. In a fab the one-sample test checks whether the current metrology profile matches the target profile, and the two-sample test compares the profiles of two tools or two recipes. The multivariate mean tests are the inferential engine of the subject.
**The visual display of multivariate data is a distinct challenge, because the human eye cannot see beyond a few dimensions, and several graphical methods have been developed to show high-dimensional structure.** The scatterplot matrix is an array of pairwise scatterplots that shows all the two-dimensional projections of the data at once, and it reveals the pairwise correlations and the outliers. The biplot displays both the observations and the variables of a principal component analysis on a single plot, with the observations as points and the variables as arrows, so that the engineer can see which variables drive the separation of the observations. The parallel coordinates plot draws each observation as a broken line across parallel vertical axes, one for each variable, so that the engineer can see the profile of each observation and the patterns of the correlated variables. The star plot and the Andrews plot are further devices for representing many variables in two dimensions, and each trades some fidelity for the ability to see the whole profile. In a fab the scatterplot matrix and the biplot are standard first views of a new metrology dataset. The graphics of multivariate data are the first step toward understanding them.
**The factor rotation is the device that makes the factor loadings interpretable, and it transforms the factor solution to concentrate the loadings on the individual factors.** The initial factor solution is not unique, because any rotation of the factors reproduces the same covariance structure, and the rotation is chosen to make each variable load strongly on as few factors as possible. The varimax rotation maximizes the variance of the squared loadings within each factor, which tends to produce factors with a few large loadings and many near-zero ones, and the oblique rotations such as promax allow the factors themselves to be correlated. The interpretation of a rotated factor is the pattern of variables that load strongly on it, and the engineer names the factor by the common theme of those variables, such as an overall thickness factor or a uniformity factor. The choice between orthogonal and oblique rotation depends on whether the underlying factors are believed to be independent, and the interpretation is judged by the clarity of the loadings. In a fab the rotated factors often correspond to distinct physical drivers of the metrology profile. The rotation turns a mathematical factor solution into an interpretable scientific finding.
**The classification assessment of a discriminant rule is the measure of how well it will perform on new data, and it guards against the optimism of evaluating a rule on the data that built it.** The apparent error rate of a rule is the fraction of the training observations it misclassifies, but this rate is optimistic, because the rule has been fit to those very observations, and the honest assessment requires a separate evaluation. The cross-validation evaluates the rule by splitting the data into a training part that builds the rule and a test part that measures its error, and the leave-one-out method repeats this by holding out one observation at a time and classifying it with the rule built on the rest. The confusion matrix displays the counts of the true and predicted groups, and the misclassification rate and the sensitivity and specificity are derived from it. In a fab the engineer uses cross-validation to confirm that a wafer-classification rule will generalize to future wafers, and the confusion matrix shows which failure modes are confusable. The honest assessment of a classifier is as important as the classifier itself.
**The comparison of the multivariate process monitoring with the univariate control charts shows the advantage of the joint monitoring, and it also shows the cost of ignoring the correlations.** A univariate chart monitors each variable with its own control limits, and it treats a wafer as out of control when any single variable exceeds its limits, but this approach inflates the false-alarm rate as the number of variables grows, because the probability that at least one variable crosses its limit rises with the dimension. The multivariate chart such as the Hotelling T-squared chart monitors the joint profile with a single statistic, so that it controls the false-alarm rate of the whole profile, and it detects the faults that change the correlation structure without moving any single mean. The multivariate charts also provide the interpretation of a signal through the contribution plot, which shows which variables contributed most to the out-of-control statistic. In a fab the choice between many univariate charts and one multivariate chart is the choice between many false alarms and a single honest monitor of the profile. The multivariate chart is the disciplined alternative to the proliferation of univariate charts.
**The application of multivariate statistics to the fault detection and classification of a semiconductor fab is the most direct industrial use of the subject, and it ties all the methods to the core task of yield management.** In the fault detection step, the Hotelling T-squared statistic monitors the multivariate profile of the tool states and the metrology, and it flags the wafers or the lots that drift from the normal region, while the contribution plot identifies the variables responsible for the drift. In the fault classification step, the discriminant analysis and the cluster analysis assign the flagged lots to known or newly discovered fault modes, and the principal component analysis reduces the high-dimensional signature of a fault to a recognizable pattern. The engineering baseline for the monitoring is built from a period of known-good production, and the model is updated as the process shifts. In a fab the multivariate methods turn the flood of tool and metrology data into a small set of actionable alarms, each tied to a diagnosis. The subject is the mathematical core of modern fab intelligence.
**The history of multivariate statistics is the story of a few pioneering statisticians who developed the field in the first half of the twentieth century, and their names mark the principal results.** Karl Pearson developed the ideas of correlation and the method of principal components around the turn of the twentieth century, while Prasanta Chandra Mahalanobis introduced the distance that bears his name and the tests based on it, and Harold Hotelling developed the principal component analysis as it is used today and the T-squared statistic and the trace test that carry his name. Samuel Wilks introduced the likelihood-ratio statistic for the multivariate analysis of variance that carries his name, and Ronald Fisher developed the discriminant analysis, while Charles Roy developed the largest-root criterion and the approach that bears his name. The later developments by William Krasker and others consolidated the theory, and the books by Richard Johnson and Dean Wichern and by T. W. Anderson became the standard references that shaped the teaching of the field. The subject grew from the insight that the correlations among measurements are themselves the signal. The history shows that the tools of multivariate statistics were built to serve exactly the kind of many-measurement data a fab produces.
The choice among the multivariate methods is governed by the goal of the analysis, and the following table organizes the principal methods by their purpose, the nature of the response, and the question they answer.** The table makes it easy to select the appropriate multivariate tool for a given engineering question, and it shows how the methods divide into the descriptive, the predictive, and the inferential. The engineer reads the table by matching the goal to a method, and then applies the tool with the data matrix in mind.
| Method | Purpose | Response type | Typical question |
|---|---|---|---|
| Covariance / correlation matrix | describe joint structure | many variables | which variables move together? |
| Principal component analysis | dimension reduction | many variables | reduce to a few components? |
| Factor analysis | latent structure | many variables | what drives the correlations? |
| Multivariate regression | predict a profile | several responses | predict outputs from inputs? |
| MANOVA | compare groups | several responses | do group profiles differ? |
| Discriminant analysis | classify | categorical group | which group does a wafer belong to? |
| Cluster analysis | discover groups | unlabeled units | what natural groups exist? |
| Hotelling T² | monitor / detect outliers | many variables | is a wafer out of control? |
| Mahalanobis distance | outlier / distance | many variables | how unusual is this wafer? |
**The selection of a multivariate method is guided by a decision tree based on whether the analysis is descriptive, predictive, or inferential, and on whether the observations carry labels, and the following flowchart routes an analysis to the appropriate method.** The first question is whether the goal is to reduce dimension, to classify, to compare groups, or to discover structure, and the second is whether the response is a set of variables, a group label, or none at all. Working through these questions selects a multivariate method that matches the goal.
```flowchart
A([Multivariate question]) --> B{Goal?}
B -- reduce many variables --> C[Principal component analysis]
B -- explain latent drivers --> D[Factor analysis]
B -- predict a response profile --> E{Response type?}
E -- continuous profile --> F[Multivariate regression]
E -- group label --> G[Discriminant analysis]
B -- compare group mean vectors --> H[MANOVA]
B -- discover natural groups --> I[Cluster analysis]
B -- monitor / detect outliers --> J[Hotelling T² / Mahalanobis]
B -- describe joint structure --> K[Covariance / correlation matrix]
```
**The connection between multivariate statistics and the other keywords in the series is direct, and it completes the advanced statistics toolkit that the series has been building.** The probability stats keyword supplies the multivariate normal distribution and the concepts of covariance and correlation, while the statistics basics keyword supplies the descriptive measures that the multivariate methods generalize. The inference statistics keyword supplies the analysis of variance and the F test that MANOVA extends to the multivariate setting, and the experimental design keyword supplies the designed experiments whose responses the multivariate methods analyze. The stochastic processes keyword supplies the time-ordered structure into which multivariate monitoring is embedded, and the nonparametric statistics keyword supplies the rank-based alternatives that handle multivariate data that are not normal. Multivariate statistics, in turn, supplies the machinery that makes the high-dimensional metrology of a modern fab interpretable, reducing many correlated readings to the few that matter. The engineer who adds multivariate methods to the univariate and experimental toolkit can handle the full richness of wafer data.
**A concrete example ties the tools together and shows how a multivariate analysis is actually carried out, and the example of monitoring the quality of an etch process illustrates the complete workflow.** The engineer measures several correlated metrology readings, including the etch rate, the uniformity, and the profile angle, on each wafer, and assembles them into a data matrix whose columns are the variables and whose rows are the wafers. The engineer computes the correlation matrix and finds that the variables are strongly correlated, then performs a principal component analysis that reduces the three readings to a single dominant component that captures the overall quality profile. The engineer plots the Hotelling T-squared statistic of the new wafers on the control chart and flags the wafers whose joint profile has drifted, and a discriminant analysis classifies the flagged wafers by the process condition that most likely produced them. The example shows that multivariate statistics is not a single test but a connected set of tools that describe the joint structure, reduce its dimension, and monitor and classify it. This single example shows how a fab turns a high-dimensional metrology profile into a clear, monitored, and classified picture of process health.
**The closing lens for multivariate statistics is that it is the discipline of seeing the joint structure that univariate methods miss, and the value of the subject is not any single test but the recognition that the correlations among measurements are themselves information.** With this lens the engineer sees the covariance matrix not as a table of numbers but as the map of how the variables move together, sees the eigenvalues and eigenvectors as the natural axes along which the data vary, sees the principal components as the few combinations that capture the whole profile, and sees the Hotelling T-squared and the Mahalanobis distance as the honest way to judge how unusual a wafer is in the joint space. The mastery of multivariate statistics is the mastery of monitoring and understanding a whole profile of measurements at once, which is precisely the situation that the high-dimensional metrology of a modern fab presents every day. Read multivariate statistics through a joint-structure lens rather than a variable-bundle lens.