T-shaped Partial Least Squares (TPLS)

TPLS is a multi-block method designed for T-shaped data structures that arise naturally in batch processes, formulation studies, and similar settings where information is organized in interconnected blocks.

When to Use TPLS

Use TPLS when your data has a natural multi-block structure that standard PLS cannot represent. Typical applications include:

• Pharmaceutical manufacturing - formulation recipes, raw material properties, process conditions, and tablet quality form distinct blocks.

• Chemical reaction optimization - catalyst properties, feed compositions, operating conditions, and product quality.

• Food processing - ingredient properties, recipes, process settings, and sensory or nutritional outcomes.

• Biotechnology - media composition, strain properties, fermentation trajectories, and yield/quality metrics.

If your data fits naturally into a single X matrix and a single Y matrix, standard PLS is simpler and should be preferred.

Data Structure

TPLS operates on four interconnected data blocks:

Block	Name	Description
D	Properties	Intrinsic properties of raw materials or design factors. Rows = materials, columns = properties.
F	Formulations	How materials are combined in each batch/experiment. Rows = batches, columns = materials. Column names must match D’s row index.
Z	Conditions	Process conditions or final-state measurements for each batch. Rows = batches, columns = condition variables.
Y	Quality	Response or quality variables for each batch. Rows = batches, columns = quality variables.

The “T-shape” comes from the way D and F link: D describes the materials (rows) while F describes how those same materials (columns) are used in each batch (rows). F, Z, and Y must all have the same number of rows (batches).

Basic Usage

Data blocks are organized using DataFrameDict - a dictionary of DataFrames, optionally grouped:

from process_improve.multivariate.methods import TPLS, DataFrameDict

data = DataFrameDict(
    {
        "D": {"Group_A": properties_a, "Group_B": properties_b},
        "F": {"Group_A": formulas_a, "Group_B": formulas_b},
        "Z": {"Conditions": process_conditions},
        "Y": {"Quality": quality_responses},
    }
)

model = TPLS(n_components=3, d_matrix=data["D"])
model.fit(data)

Key requirements:

• Column names in each F group must match the row index of the corresponding D group - this is how TPLS knows which material properties correspond to which formulation amounts.

• All F, Z, and Y DataFrames must have the same number of rows.

• The d_matrix parameter passed to the constructor should be the D block (material properties).