import numpy as np
import pandas as pd
from sklearn.cross_decomposition import PLSRegression
from sklearn.model_selection import KFold, cross_val_score
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import mean_squared_error, mean_absolute_error, r2_score

df = pd.read_csv("dataset.csv")

predictors = [
    "G1", "G2", "studytime", "failures", "absences",
    "age", "Medu", "Fedu", "school", "sex", "address", "famsize"
]
X = pd.get_dummies(df[predictors], drop_first=True, dtype=float)
y = df["G3"].to_numpy(dtype=float)

# Component selection on a standardized outcome.
y_scaled = StandardScaler().fit_transform(y.reshape(-1, 1)).ravel()
cv = KFold(n_splits=5, shuffle=True, random_state=42)

cv_rmse = {}
for n_components in range(1, 6):
    model = PLSRegression(n_components=n_components, scale=True)
    scores = cross_val_score(
        model, X, y_scaled,
        cv=cv,
        scoring="neg_root_mean_squared_error"
    )
    cv_rmse[n_components] = -scores.mean()

best_components = min(cv_rmse, key=cv_rmse.get)

final_model = PLSRegression(
    n_components=best_components,
    scale=True
).fit(X, y)

predicted = final_model.predict(X).ravel()
print("R2:", r2_score(y, predicted))
print("RMSE:", mean_squared_error(y, predicted) ** 0.5)
print("MAE:", mean_absolute_error(y, predicted))

library(pls)

df <- read.csv("dataset.csv")

fit <- plsr(
  G3 ~ G1 + G2 + studytime + failures + absences +
    age + Medu + Fedu + school + sex + address + famsize,
  data = df,
  ncomp = 5,
  scale = TRUE,
  validation = "CV"
)

summary(fit)
validationplot(fit, val.type = "RMSEP")
predicted <- drop(predict(fit, ncomp = 5))
scores(fit)
loadings(fit)
coef(fit, ncomp = 5)

Install the PLS extension supported by the available SPSS version, then specify:

• Dependent variable: G3
• Scale predictors: G1, G2, studytime, failures, absences, age, Medu, Fedu
• Categorical predictors: school, sex, address, famsize
• Maximum components: at least 5
• Selection: cross-validation error
• Output: scores, loadings, coefficients, predictions, residuals, and VIP values when supported

Record the installed extension version and default scaling rules because SPSS PLS syntax and available output can vary by installation.

Prediction from reconstructed coefficients:
=Intercept
 + G1_Input*G1_Coefficient
 + G2_Input*G2_Coefficient
 + Studytime_Input*Studytime_Coefficient
 + Failures_Input*Failures_Coefficient
 + ...

Residual:
=Observed_G3-Predicted_G3

Absolute residual:
=ABS(Residual)

Squared residual:
=Residual^2

RMSE:
=SQRT(AVERAGE(Squared_Residual_Range))

MAE:
=AVERAGE(Absolute_Residual_Range)

PLS regression replaces a crowded predictor set with a few outcome-focused patterns. Those patterns summarize shared predictor information and retain the parts most useful for predicting the response.

A complete Partial Least Squares Regression tutorial should cover data preparation, dummy coding, scaling, component tuning, model fitting, coefficient reconstruction, VIP calculation, residual review, and validation on unseen data.

In this PLS regression example, five components explain 85.15% of G3 variation. G2 and G1 dominate the latent prediction structure, while failures is close to the conventional VIP threshold.

Principal components regression selects directions that explain X variance. Partial Least Squares Regression selects directions with high X–Y covariance. The response therefore influences the PLS components but not the initial PCR components.

Multiple linear regression estimates original-variable effects directly. PLS regression uses latent components to stabilize prediction when original predictors overlap strongly or the design is high dimensional.

OLS minimizes residual sum of squares in the original predictor space. Partial Least Squares Regression first reduces X to supervised latent dimensions, then performs prediction through that reduced representation.

Scaling prevents predictors with large units from dominating component construction. The scaling operation must be estimated inside each training fold to prevent information leakage.

Partial Least Squares Regression coefficients reconstruct the latent model in the original design space. Their signs and magnitudes describe prediction contributions, but they do not automatically include classical inferential uncertainty.

VIP scores combine a variable’s contribution across all selected components, weighted by outcome explanation. Values near or above 1 are often highlighted, but the threshold is conventional rather than a hypothesis test.

Scores locate observations in latent space; loadings describe how the original variables relate to that space. Both are useful for interpretation, clustering, and identifying unusual observations.

R² = 0.8515 summarizes fitted-sample outcome variation. Adjusted R-Squared using five components is 0.8503, but external or cross-validated prediction remains more important than training fit.

Categorical predictors enter Partial Least Squares Regression through dummy or contrast coding. Their coefficients depend on the selected reference category and the scaling used for the design matrix.

Prediction with PLS regression does not require every predictor to be normally distributed. Severe skew, outliers, and nonlinear response patterns can still destabilize components and should be investigated.

The selected components provide a lower-dimensional supervised representation of the predictor matrix. Five components replace twelve design columns in this worked model.

Component truncation acts as a form of regularization because weak directions are excluded. Unlike Lasso or Elastic Net, the method regularizes through latent projection rather than direct coefficient penalties.

Sparse Partial Least Squares Regression adds variable selection to component construction. It can improve interpretability in very wide datasets but introduces additional tuning decisions.

Robust PLS regression reduces the influence of outliers or heavy-tailed observations during component estimation. It is useful when standard PLS scores change substantially after unusual cases are removed.

Kernel Partial Least Squares Regression represents nonlinear relationships through a kernel function. Kernel choice and tuning can improve prediction but make interpretation less direct.

Orthogonal PLS separates predictive variation from structured X variation unrelated to Y. The separation can aid visualization, but validation remains necessary because added model flexibility can overfit.

SIMPLS is an alternative algorithm for extracting Partial Least Squares Regression components. Different algorithms can reach equivalent prediction spaces while using different computational procedures.

Machine learning workflows treat component count as a hyperparameter and evaluate PLS regression through resampling, pipelines, and an untouched test set.

Scikit-learn provides PLSRegression for Partial Least Squares Regression Python projects. Preprocessing, component tuning, and metric calculation should be placed in a reproducible pipeline.

R implementations include pls, mixOmics, and model interfaces in broader resampling frameworks. Defaults for scaling, validation, and coefficient extraction should be checked.

Partial Least Squares Regression SPSS output should identify the number of latent factors, fit metrics, predictor weights, loadings, coefficients, and validation method. Available tables depend on the installed extension.

Excel can reproduce a fitted prediction equation and summarize results, but deriving stable latent components, resampling, and VIP scores is safer in Python, R, SPSS extensions, MATLAB, Minitab, or specialist software.

Report the outcome, design columns, scaling, selected components, cross-validation method, R², RMSE, MAE, coefficients, VIP scores, and residual findings. Use Confidence Interval language only when uncertainty has actually been estimated.