Key Terms Explained

If you are new to PCA/PLS, a bunch of abbreviations will look like "天书" (heavenly script): R², Q², VIP, T², SPE...

Don't panic. This page explains the terms that appear most often in the platform in plain language: what question each one answers, how to interpret it, and where it is easy to trip up.

💡 First, a "cheat sheet" for you:

• R² (Fit): How well the model explains the "known data".
• Q² (Prediction): How accurate the model is on "unseen data" (obtained by cross-validation).
• T² (In-model anomaly): The point is inside the model plane, but it runs too far off (a maverick).
• SPE (Out-of-model anomaly): The point doesn't follow the model plane at all (a large residual).
• VIP / Loading (Variable contribution): Which variables are actually "calling the shots".
• Orthogonal variation (OPLS): Systematic noise unrelated to Y, which can be peeled off.
• MV / DV (Optimization roles): Which variables you can actively adjust and which you can only accept passively.
• S-Plot (Markers): Variables that are both big contributors and stable gather in the two corners of the plot.
• SHAP (Individual explanation): For this one prediction, how much did each variable contribute.
• DML (Causality): Is it merely "correlated", or can you really change the result through it.
• Accuracy: How many classification questions were answered correctly (it lies when classes are imbalanced).
• F1 Score: The even-handed master who doesn't play favorites (it considers both not wronging the good and not letting the bad go).
• AUC (Discrimination): The overall strength of a classification model in telling positive from negative classes.

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✅ R² (r squared): Goodness of Fit

🧠 What question does it answer?

R² is about: how well the model explains the training data. You can think of it as "the model's ability to retell this homework".

In the platform you will commonly see:

• R2X: How much information of X (features) the model explains.
• R2Y: How much variation of Y (target) the model explains.

🧐 How to interpret the value?

• Range: Usually between 0 and 1, the larger the better.
• Intuition: R2Y = 0.90 roughly means "90% of the variation in Y can be explained by the model".
• Note: A high R² does not guarantee accurate prediction, because it is only "recalling the training set".

⚠️ Common misconceptions

• Looking only at R² and not at Q²: It is very easy to "learn by rote" (overfitting).
• Comparing across different tasks: Across regression, discriminant (classification), and different data scales, the comparability of R² is limited. It is better to compare within the same business data and the same modeling objective.

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✅ Q² (q squared): Predictive Ability

🧠 What question does it answer?

Q² is about: the model's predictive ability on unseen data. You can think of it as "the model's ability to draw inferences from one instance".

The Q² in the platform generally comes from cross-validation (see the next section), and is closer to what you really care about: "will it work well after going live".

🧐 How to interpret the value?

• Range: Usually between 0 and 1, the larger the better (in some cases a low or even negative value may appear, which means very poor prediction).
• Practical experience:
  • Low Q²: the model generalizes poorly, and you usually need to reduce components, clean outliers, or re-check the X/Y configuration.
  • Very high R² but clearly low Q²: a classic sign of overfitting.

💡 A good habit:

Treat R² as the "fit check-up" and Q² as the "prediction check-up". Only when both are healthy is the model reliable.

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✅ Cross Validation: the model's "mock exam"

🧠 What question does it answer?

The core idea of cross-validation is very plain:

• First "hide" part of the samples as exam questions;
• Train the model on the remaining samples;
• Then use the model to predict the "hidden part" and see how well it answers;
• Repeat several times to obtain a more stable evaluation.

This is one source of Q² in the platform, and also an important basis for the platform to automatically find the best number of principal components / components (see Modeling Analysis).

🧩 Common practices (you don't have to compute by hand)

• K-fold cross-validation: Split the samples into K parts and take turns using 1 part for validation.
• Leave-one-out (LOO): Leave only 1 sample out for validation each time; common when samples are few.

⚠️ Cases where cross-validation also fails

• Too few samples: The evaluation fluctuates a lot and Q² becomes unstable.
• Data leakage: For example, samples from the same batch are split into both training and validation, which makes the result look "inflated".
• Extremely imbalanced classes (classification): Q²/Accuracy alone may be optimistic; you need to combine them with metrics such as AUC.

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✅ Fit: letting the model "learn" the relationship

Fit in the platform simply means "train the model":

• PCA (exploration): Learn to summarize the main variation patterns of X with fewer dimensions (dimensionality reduction).
• PLS (prediction): Learn the relationship between X and Y, and use X to explain / predict Y.

🎛️ Why are there "components / latent variables"?

You can understand the number of components as the number of patterns the model allows itself to "memorize":

• Too few components: the model is too simple and doesn't learn enough patterns (underfitting).
• Too many components: the model is too complex and even memorizes the noise (overfitting).

The C+1/C-1 in the platform is exactly this "complexity knob", and Q² (cross-validation) helps you choose a more stable complexity (see Modeling Analysis).

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✅ Discriminant (Classification) and Regression

These are the two "tracks" you must be clear about before modeling. Your Y (target variable) decides which track you are on.

🧠 Discriminant (classification task)

• Goal: Predict a class (discrete value).
• Examples: Deciding whether a product is "qualified" or "unqualified"; deciding whether a patient is "positive" or "negative".
• Which metrics to watch: Accuracy, F1 Score, AUC.
• In the platform: If your Y column is a text label (such as 0/1), the platform automatically uses PLS-DA (Partial Least Squares Discriminant Analysis) to handle it.

🧠 Regression (continuous prediction task)

• Goal: Predict a specific value (continuous value).
• Examples: Predicting whether tomorrow's temperature is 25.3℃ or 26.1℃; predicting whether a product's purity is 98.5% or 99.2%.
• Which metrics to watch: R², Q², RMSE (root mean square error).
• In the platform: If your Y column is a continuous number, the platform automatically uses standard PLS to handle it.

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✅ Accuracy: how many classification questions you "answered correctly"

🧠 What question does it answer?

Accuracy = number of correctly predicted samples / total number of samples. For example, if 92 out of 100 samples are predicted correctly, Accuracy = 0.92.

⚠️ When does Accuracy lie?

When classes are extremely imbalanced, Accuracy may look "good" but be meaningless.

For example: out of 100 samples, 95 are "qualified" and 5 are "unqualified".

• If you always predict "qualified", Accuracy is still 95%;
• But this model completely fails to catch "unqualified", which is actually the most dangerous outcome for the business.

✅ More reliable combinations

• Also look at the confusion matrix (TP/FP/TN/FN), Recall, and Precision.
• Then combine them with the F1 Score or ROC/AUC in the next sections to see the model's overall performance under different conditions.

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✅ F1 Score: the even-handed master who doesn't play favorites

🧠 What question does it answer?

When you find that Accuracy lies under "class imbalance" (for example, 95 qualified and 5 unqualified), you need to bring out the F1 Score. It is computed from two metrics:

• Precision: Among the "unqualified" items you predicted, how many are really unqualified? (Don't wrong the good ones)
• Recall: Among the "unqualified" items that truly exist, how many did you successfully catch? (Don't let the bad ones go)

🧐 How to interpret it?

The F1 Score is the "harmonic mean" of Precision and Recall (it demands a good score in both subjects — no playing favorites).

• Range: 0 to 1, the closer to 1 the stronger the model.
• Intuition: If the model always guesses randomly, or only cares about catching one class, its F1 Score will be dragged down very low by that "weak subject". It forces the model to be an "even-handed master".

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✅ AUC Curve (ROC/AUC): the classification model's "threshold check-up"

🧠 What is the ROC curve?

A classification model usually outputs a score (or probability) of "how much it looks like the positive class". Choosing different thresholds gives you different:

• True positive rate TPR (recall): The proportion of the positive class that is caught.
• False positive rate FPR (false alarm): The proportion of the negative class that is misjudged as positive.

The ROC curve simply connects the (FPR, TPR) pairs at different thresholds, to see whether the model overall is "close to the top-left corner (good)" or "close to the diagonal (mediocre)".

🧮 What is AUC?

AUC is the area under the ROC curve. Intuitive interpretation:

• AUC = 0.5: Close to random guessing.
• The closer AUC is to 1: The stronger the discrimination between positive and negative classes.

✅ Why is AUC more reliable than Accuracy in many cases?

• AUC does not depend on one fixed threshold, so it better reflects the model's overall discrimination ability.
• Under class imbalance, AUC is usually more stable than Accuracy and less likely to be "good-looking but useless".

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✅ Hotelling T²: in-model anomaly

🧠 What question does it answer?

T² measures how far a sample deviates within the model's "principal space (score space)":

• The sample point still varies along the directions described by the model, but it deviates too far;
• It is commonly seen when "some variables are particularly extreme", and is an in-model maverick.

🧐 How to read it?

• There is usually a Limit (confidence limit) acting as the warning line.
• Beyond the Limit: Anomaly candidates that deserve close attention.

For a more intuitive chart explanation see: Hotelling T² Plot

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✅ SPE (Squared Prediction Error): out-of-model anomaly

🧠 What question does it answer?

SPE is also often called DModX. It measures the "perpendicular distance" (residual) from a sample to the model plane:

• Large SPE: the variation pattern of this sample cannot be explained by the model;
• It is commonly seen when "the operating conditions changed, a new pattern appeared, or data quality is abnormal", and is an out-of-model floating sample.

🧩 How do T² and SPE work together?

• High T², not-so-high SPE: The model can explain it, but the point is extreme (in-model anomaly).
• Not-so-high T², high SPE: The direction is not extreme, but the model cannot explain it (out-of-model anomaly).
• Both high: Both extreme and unexplainable — usually the first to investigate.

For a more intuitive chart explanation see: SPE Plot

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✅ VIP (Variable Importance in Projection): the variable importance ranking

🧠 What question does it answer?

VIP is used for PLS: it tells you which X variables are most critical for explaining / predicting Y.

You can treat VIP as a leaderboard of "who contributes the most to the result".

🧐 How to interpret it?

• A commonly used empirical line: VIP = 1 as the reference threshold for an "important variable".
• The higher it is: the more likely this variable is a key control point.

For a more intuitive chart explanation see: VIP Variable Contribution

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✅ Loading: the "standing position" of variables in the model

🧠 What question does it answer?

Loading describes: the direction and magnitude of each variable's contribution to a given principal component / latent variable.

In one sentence: the score plot looks at samples (rows), the loading plot looks at variables (columns).

🧐 How to read a loading plot?

• Farther from the origin: The greater this variable's influence on that component.
• Two variables pointing the same way and close together: They often carry similar information (positively correlated).
• Two variables pointing opposite ways, on opposite sides: They often represent a trade-off (negatively correlated).

💡 A small reminder:

The interpretation of loadings / scores depends heavily on whether the data was standardized and how many components you selected. It is recommended to read the conclusions together with the Model Summary and business common sense.

For a more intuitive chart explanation see: Loading Plot

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✅ OPLS / OPLS-DA (Orthogonal Partial Least Squares): "showing the noise out"

🧠 What question does it answer?

Ordinary PLS faces an awkward situation: the data often contains a lot of systematic fluctuation unrelated to Y — ambient temperature drift, equipment aging, batch baseline drift.

These fluctuations turn the score plot into "a tangled mess": the model predicts decently, yet you cannot explain it clearly.

OPLS's approach is to split the variation of X in two:

• Predictive variation: related to Y, kept
• Orthogonal variation: unrelated to Y, peeled off

Then it runs PLS on the clean data.

🧐 How to interpret it?

• There is only 1 predictive component: This is the biggest characteristic of OPLS — the score plot has only one predictive direction, and the separation between groups is obvious at a glance
• ⚠️ The meaning of the component count is different: Clicking C+1 on OPLS / OPLS-DA does not add "predictive power", but rather the number of orthogonal noise layers peeled off

💡 When to use it: When the data has obvious systematic noise, or when you need to explain clearly to others "which variable is actually at work". Note that the predictive ability of OPLS is usually not better than PLS; its benefit lies mainly in interpretability.

For a more intuitive chart explanation see: S-Plot Marker Plot

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✅ S-Plot (marker plot): both important and reliable

🧠 What question does it answer?

VIP only looks at "how big the contribution is", but a variable with a big contribution is not necessarily stable.

The S-Plot draws the two dimensions together:

• Horizontal axis: Covariance — how big the contribution is
• Vertical axis: Correlation coefficient — how reliable it is

🧐 How to read it?

• Top-right / bottom-left corners ⭐: Big contribution and stable correlation → marker candidate
• Middle of the right side: Big contribution but unstable correlation → suspicious result
• Close to the origin: Small contribution → can be considered for removal

💡 One-line memory aid: The closer to the corner, the more trustworthy.

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✅ SHAP (individual contribution breakdown): who did the work in this prediction?

🧠 What question does it answer?

VIP and regression coefficients both give global average conclusions. But in reality we often see this:

A certain variable makes a positive contribution on some samples, yet is a negative drag on other samples.

SHAP is based on the Shapley value from game theory, and fairly splits each prediction among the individual variables.

🧐 How to read it?

• Importance bar chart: Sorted by mean absolute SHAP value; the longer, the more important
• Beeswarm plot: One point per sample
  • Point position: Whether this variable pushes the prediction up or down on that sample
  • Point color: The actual value of that variable (usually dark means a high value)
  • Points mixed on the left and right, not clearly separated → This variable is non-monotonic and has an optimal interval

💡 "Mixed left and right" often means an optimization opportunity: Too large doesn't work and too small doesn't work, which means there is a best value waiting for you to find.

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✅ DML (Double Machine Learning): correlation ≠ causation

🧠 What question does it answer?

All the previous metrics (VIP, SHAP, correlation coefficients) tell you "who varies together with whom", not "changing whom will make the result better".

A classic trap: In summer, ice cream sales and the number of drownings are highly correlated. But banning ice cream sales will not reduce drownings — the real common cause is "air temperature".

DML uses double machine learning to strip out confounding factors and give an effect estimate closer to causality.

🧐 How to use it?

In [Analysis Exploration → Causal Inference (DML)], specify three roles:

Role	Meaning
Outcome Variable (Y - Outcome)	The metric you want to improve
Treatment Variable (T - Treatment)	The variable you plan to "move"
Confounder Library (W - Confounders)	The other variables that need to be controlled

💡 Practical value: The biggest fear in process optimization is "adjusting an accompanying variable". DML helps you confirm whether this parameter is really worth adjusting.

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✅ Anomaly detection algorithms: Isolation Forest and LOF

Besides the classic T² / SPE, the platform also has two built-in unsupervised anomaly detection algorithms:

Algorithm	Intuition	Good at
Isolation Forest	Randomly splits the data; the easier a point is "isolated", the more anomalous it is	Global outliers
LOF (Local Outlier Factor)	Compares the density of a point's neighbors; a clearly lower density means an anomaly	Samples in locally sparse regions

💡 It is normal for the two to disagree, because they focus on different types of "anomaly". Samples flagged by both are the most worth investigating first.

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✅ MV / DV (variable roles in optimization)

Before doing parameter optimization, you must first tell the algorithm which variables you can move:

Mark	Full name	Meaning
MV	Manipulated Variable	Manipulated variable — one you can actively adjust (such as a temperature setpoint)
DV	Disturbance Variable	Disturbance variable — uncontrollable but affecting the result (such as ambient humidity)

⚠️ The consequence of marking it wrong: If you mark a DV as an MV, the algorithm may give advice such as "go adjust the ambient humidity", which simply cannot be executed.

For a more complete method of variable classification see Factory Scenario Data Modeling Guide.

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✅ Lag: causal relationships often have a delay

🧠 What question does it answer?

You adjust the feed rate today, and the effect may only show up in product quality two hours later.

Lag analysis finds out: how many steps must pass before a change in the source variable shows up in the target variable?

🧐 How to read it?

• The peak appears at Lag = 0 → the two variables change in sync
• The peak appears at Lag = k → the source variable leads the target by k steps, so there is a transmission delay

⚠️ Prerequisite for use: Lag analysis relies on time order, so the model must choose a "non-random" data set split. See Lag Impact Analysis.

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📌 One last cheat sheet: what each of them "catches"

• R²: How well the overall explanation / fit is (training set).
• Q²: How accurate the overall prediction is (cross-validation).
• Accuracy: How many predictions were correct (beware of being fooled).
• F1 Score: The master of balance between precision and recall.
• AUC: The hard power of the model in distinguishing positive from negative classes.
• T²: In-model deviation (running too far along the model direction).
• SPE: Out-of-model deviation (large residual, unexplainable by the model).
• VIP: The ranking of X variables that contribute most to Y (PLS).
• Loading: The contribution and correlation structure of variables on components (PCA/PLS).
• Orthogonal variation: The systematic noise "unrelated to Y" that OPLS peels off.
• S-Plot: Two-dimensional screening by contribution + reliability; the closer to the corner, the more trustworthy.
• SHAP: Splitting the credit of a single prediction among the individual variables.
• DML: Distinguishing "correlation" from "causation".
• Isolation Forest / LOF: Two complementary anomaly detection algorithms.
• MV / DV: Which variables are adjustable and which are not during optimization.
• Lag: The time delay in a causal relationship.

If you run into situations such as "high R² but low Q²" or "both T² and SPE out of limits" in your actual analysis, it is usually not because you did something wrong, but because the data is giving you a hint: points that need cleaning, variables that need reconfiguring, complexity that needs reducing — none of them can be skipped.