Logistic Regression for Ultrasonic Flow Meter Health Classification

Created on January 10, 2025

2025   ·   machine-learning   classification   optimization   ·   sample-posts

Introduction

Condition monitoring and diagnostics play a critical role in ensuring the reliability of industrial measurement systems. In ultrasonic flow metering, installation effects can significantly distort measurements, leading to inaccurate flow estimation and potential operational risks.

This work focuses on developing a binary classification model based on logistic regression to determine whether an 8-path liquid ultrasonic flow meter (USM) is operating in a healthy state or is affected by installation effects. Using diagnostic parameters derived from experimental observations, we formulate the problem as a supervised learning task and analyze both optimization behavior and classification performance.

Problem Formulation

Given a dataset
\(\mathcal{D} = \{(x_i, y_i)\}_{i=1}^{N},\) where $x_i \in \mathbb{R}^{36}$ represents diagnostic features and

\[y_i \in \{0,1\}\]

denotes the health state of the flow meter, the objective is to learn a parameter vector $\beta$ that models the probability

\[P(y = 1 \mid x) = \sigma(x^\top \beta),\]

where $\sigma(\cdot)$ is the sigmoid function.

About the Dataset

The dataset, referred to as Meter A, consists of 87 experimental observations collected from an 8-path liquid ultrasonic flow meter .

Key characteristics:

  • 87 total samples
  • 36 continuous input features
  • Binary target variable:
    • 1 → Healthy condition
    • 0 → Installation effect

The relatively small dataset size motivated careful preprocessing and evaluation to avoid biased conclusions.

Data Preprocessing

Feature Normalization

Initial exploration revealed that the predictors varied significantly in scale and dispersion. To ensure numerical stability during optimization, all features were standardized using Z-score normalization:

\[z = \frac{x - \mu}{\sigma},\]

where $\mu$ and $\sigma$ denote the mean and standard deviation of each feature.

No missing values or categorical variables were detected in the dataset.

Train–Test Splitting Strategy

Due to the sequential ordering of healthy and faulty observations, the dataset was shuffled prior to splitting to ensure representative samples in both subsets.

Multiple split ratios were evaluated: 60:40, 75:25, 80:20.

This allowed us to analyze how training data size affects convergence, generalization, and classification accuracy.

Logistic Regression Model

Sigmoid Function

Logistic regression maps linear combinations of features to probabilities using the sigmoid function:

\[\sigma(x) = \frac{1}{1 + e^{-x}}.\]

This formulation enables direct probabilistic interpretation of model outputs.

Cost Function and Optimization

The model parameters were estimated by minimizing the negative log-likelihood:

\[\mathcal{L}(\beta) = - \sum_{i=1}^{n} \left[ y_i \log(\sigma(X_i \beta)) + (1 - y_i)\log(1 - \sigma(X_i \beta)) \right].\]

Optimization Method

  • Initial weights: zero initialization
  • Bias term embedded directly into the feature matrix
  • Optimization performed using MATLAB’s fminsearch
  • No explicit stopping criteria imposed

A custom output function tracked:

  • loss value per iteration
  • total number of iterations until convergence

This setup allowed us to study the natural convergence behavior of the optimizer.

Optimization Behavior

Key observations:

  • All splits show monotonic decrease in loss
  • Larger training sets require more optimization iterations
  • Absence of stopping criteria increases computation cost for larger datasets

Model Evaluation Metrics

Model performance was assessed using:

  • Accuracy
  • Confusion Matrix
  • Precision
  • Recall
\[\text{Precision} = \frac{TP}{TP + FP}, \quad \text{Recall} = \frac{TP}{TP + FN}\]

Classification Accuracy Results

pretty_table: true

| split ratio | Tr. accuracy | Te. accuuracy |
| :---------- | :----------: | ------------: |
| 60:40       |    80.86     |         82.86 |
| 75:25       |    83.08     |        100.00 |
| 80:20       |     90.0     |         76.47 |

The 75:25 split achieved perfect test accuracy, though this should be interpreted cautiously due to the small sample size.

Feature Importance Analysis

Across all splits, the 25th feature (gain at the fifth end of each of the eight paths) consistently exhibited the largest coefficient magnitude.

  • Strong influence regardless of split ratio
  • Indicates a dominant diagnostic role in identifying installation effects

Confusion Matrix Analysis

Discussion

This study demonstrates that logistic regression, when combined with appropriate normalization and numerical optimization, can effectively classify ultrasonic flow meter health states—even with limited data.

However:

  • Small datasets increase variance in performance metrics
  • Absence of stopping criteria impacts computational efficiency
  • Feature dominance suggests opportunities for dimensionality reduction

Final Remarks

Logistic regression remains a powerful and interpretable baseline for industrial diagnostic problems. Beyond classification accuracy, analyzing optimization dynamics, feature influence, and data splitting strategies provides deeper insight into model behavior and reliability.

Future work may include:

  • regularization
  • alternative optimizers
  • cross-validation
  • nonlinear extensions

Code and experiments related to this project are available in the accompanying repository.



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