Energy Monitoring System Calibration: Linear Regression Approach for Sensor Accuracy

Introduction

In my ongoing energy monitoring project, accurate room temperature and humidity measurements are crucial for optimizing HVAC system performance and energy consumption. The low-cost DHT11 sensors used in the system required calibration against a professional-grade HTC-1 reference sensor to ensure reliable data collection.

The Calibration Challenge

The DHT11 sensor, while cost-effective, exhibited significant measurement errors compared to the HTC-1 reference:

Pre-Calibration Performance:

• Temperature MAE: 3.84°C
• Humidity MAE: 14.18%
• Temperature Bias: -3.84°C
• Maximum Error: 4.8°C (temperature), 18% (humidity)

These errors were unacceptable for energy optimization algorithms that rely on precise environmental data.

Data Collection Methodology

Data was collected simultaneously from both sensors over a 9-hour period, capturing various environmental conditions. The dataset included 19 measurement points across different times of day, with temperature readings ranging from 23.4°C to 27.7°C and humidity readings from 46.8% to 73.8%. This comprehensive sampling ensured the calibration model would be robust across typical operating conditions.

Linear Regression Model

Mathematical Foundation

The calibration uses simple linear regression to map raw sensor readings to calibrated values:

$$ y_{\text{calibrated}} = \beta_1 \cdot x_{\text{raw}} + \beta_0 $$

Where:

• $y_{\text{calibrated}}$ is the calibrated measurement
• $x_{\text{raw}}$ is the raw sensor reading
• $\beta_1$ is the slope coefficient
• $\beta_0$ is the intercept constant

Temperature Calibration Model

For temperature calibration, the regression yielded:

$$ y_{\text{temp}} = 1.343x - 7.689 $$

Model Statistics:

• R² = 0.881
• Standard Error = 0.495°C
• F-statistic = 125.95

Humidity Calibration Model

For humidity calibration:

$$ y_{\text{humidity}} = 0.914x + 5.127 $$

Model Statistics:

• R² = 0.981
• Standard Error = 0.951%
• F-statistic = 899.14

Performance Evaluation

Post-Calibration Results

The linear regression calibration significantly improved measurement accuracy:

Temperature Performance:

• MAE: 0.80°C (improved from 3.84°C)
• R²: 0.939
• Bias: +0.75°C
• Maximum Error: 1.40°C

Humidity Performance:

• MAE: 1.15% (improved from 14.18%)
• R²: 0.991
• Bias: -0.6%
• Maximum Error: 2.2%

Mean Absolute Error (MAE) Calculation

The MAE is calculated as:

$$ \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_{\text{predicted}} - y_{\text{actual}}| $$

Where $n$ is the number of samples in the validation dataset.

Implementation Code

import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_absolute_error, r2_score

class SensorCalibrator:
    def __init__(self):
        self.temp_model = LinearRegression()
        self.humidity_model = LinearRegression()

        # Calibration coefficients from regression analysis
        self.temp_coef = 1.343379812
        self.temp_intercept = -7.688931392
        self.humidity_coef = 0.914240883
        self.humidity_intercept = 5.127226491

    def calibrate_temperature(self, raw_temp):
        """Calibrate temperature reading using linear regression"""
        return self.temp_coef * raw_temp + self.temp_intercept

    def calibrate_humidity(self, raw_humidity):
        """Calibrate humidity reading using linear regression"""
        return self.humidity_coef * raw_humidity + self.humidity_intercept

    def evaluate_calibration(self, raw_readings, reference_readings):
        """Evaluate calibration performance"""
        calibrated = [self.calibrate_temperature(temp) for temp in raw_readings]

        mae = mean_absolute_error(reference_readings, calibrated)
        r2 = r2_score(reference_readings, calibrated)

        return {
            'mae': mae,
            'r2': r2,
            'calibrated_readings': calibrated
        }

# Usage example
calibrator = SensorCalibrator()

# Calibrate a new reading
raw_temp = 25.0
calibrated_temp = calibrator.calibrate_temperature(raw_temp)
print(f"Raw: {raw_temp}°C → Calibrated: {calibrated_temp:.2f}°C")

Results Visualization

The calibration significantly improved measurement accuracy across the entire operating range. The scatter plots show excellent linear correlation between calibrated DHT11 readings and HTC-1 reference values.

Key Improvements:

1. Temperature Accuracy: 79% reduction in MAE
2. Humidity Accuracy: 92% reduction in MAE
3. Measurement Consistency: High R² values indicate reliable predictions
4. Energy Optimization: More accurate data for HVAC control algorithms

Applications in Energy Monitoring

The calibrated sensors now provide reliable data for:

1. HVAC Optimization: Precise temperature control reduces energy waste
2. Occupancy Detection: Accurate environmental changes indicate room usage
3. Energy Consumption Analysis: Correlate environmental conditions with energy usage
4. Predictive Maintenance: Detect abnormal temperature/humidity patterns

Conclusion

Linear regression provides an effective method for calibrating low-cost environmental sensors. The approach:

• Reduced temperature MAE from 3.84°C to 0.80°C
• Reduced humidity MAE from 14.18% to 1.15%
• Maintained computational efficiency for embedded systems
• Provided mathematically sound calibration with interpretable coefficients

This calibration enables cost-effective deployment of environmental monitoring systems while maintaining measurement accuracy suitable for energy optimization applications.

Future Work: Exploring non-linear calibration models and automated recalibration procedures to account for sensor aging and environmental changes.