Advanced Spatio-temporal Predictive Modeling: A Comparative Evaluation of Kriging Methods for Environmental Pollution Mapping.pdf
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About This Presentation
This technical report performs a rigorous comparative evaluation of advanced geostatistical methods for environmental predictive modelling, focusing on the crucial role of spatial autocorrelation and non-stationarity.
The study applies a diverse suite of Kriging algorithms (including Ordinary, Uni...
Slide Content
DSA 554 3.0 Spatio-temporal Data Analysis
Mid-Semester Evaluation Assignment
Student Name: G.D.E. Lakmali
Examination No: MSC/DSA/063
Registration No: 9152PS2023015
Mid-Semester Evaluation Assignment
Student Name: G.D.E. Lakmali
Examination No: MSC/DSA/063
Registration No: 9152PS2023015
Question1: Interpolation of Mean NO2 Concentrations in Germany,
2017
Methodology Overview:
1) Data Loading and Preprocessing:
Load the NO2 concentration data for Germany in 2017 using the ‘gstat’ package.
2) Spatial Data Handling:
Convert the data into a spatial object using the ‘sf’ package and establish a Coordinate Reference
System (CRS) to ensure accurate spatial analysis and mapping.
3) Visualization:
Visualise the spatial distribution of NO2 concentrations using maps generated with the ‘ggplot2’
package. These visualisations help in understanding the overall patterns and spatial variability of
NO2 levels across Germany.
4) Inverse Distance Weighting (IDW) Interpolation:
Utilise the Inverse Distance Weighting (IDW) interpolation method to estimate NO2
concentrations across a regular grid. IDW assigns weights to nearby observed data points based
on their distance to the prediction location, providing a simple interpolation technique.
5) Variogram Analysis:
Conduct variogram analysis to characterise the spatial dependence structure of NO2
concentrations. Variograms display the semivariance between pairs of data points as a function of
distance, helping in understanding the spatial autocorrelation and identifying an appropriate
variogram model.
6) Variogram Modeling:
Fit variogram models to the experimental variogram obtained from the data to quantify spatial
correlation. Variogram models, such as exponential or spherical models, describe the spatial
correlation of NO2 concentrations across different distances.
7) Ordinary Kriging (OK):
Implement Ordinary Kriging to interpolate NO2 concentrations considering spatial
autocorrelation. OK provides unbiased estimates of NO2 concentrations at unsampled locations
by incorporating spatial correlation from nearby observations.
8) Universal Kriging (UK):
2017
Methodology Overview:
1) Data Loading and Preprocessing:
Load the NO2 concentration data for Germany in 2017 using the ‘gstat’ package.
2) Spatial Data Handling:
Convert the data into a spatial object using the ‘sf’ package and establish a Coordinate Reference
System (CRS) to ensure accurate spatial analysis and mapping.
3) Visualization:
Visualise the spatial distribution of NO2 concentrations using maps generated with the ‘ggplot2’
package. These visualisations help in understanding the overall patterns and spatial variability of
NO2 levels across Germany.
4) Inverse Distance Weighting (IDW) Interpolation:
Utilise the Inverse Distance Weighting (IDW) interpolation method to estimate NO2
concentrations across a regular grid. IDW assigns weights to nearby observed data points based
on their distance to the prediction location, providing a simple interpolation technique.
5) Variogram Analysis:
Conduct variogram analysis to characterise the spatial dependence structure of NO2
concentrations. Variograms display the semivariance between pairs of data points as a function of
distance, helping in understanding the spatial autocorrelation and identifying an appropriate
variogram model.
6) Variogram Modeling:
Fit variogram models to the experimental variogram obtained from the data to quantify spatial
correlation. Variogram models, such as exponential or spherical models, describe the spatial
correlation of NO2 concentrations across different distances.
7) Ordinary Kriging (OK):
Implement Ordinary Kriging to interpolate NO2 concentrations considering spatial
autocorrelation. OK provides unbiased estimates of NO2 concentrations at unsampled locations
by incorporating spatial correlation from nearby observations.
8) Universal Kriging (UK):
Perform Universal Kriging by incorporating trend models (e.g., linear or polynomial trends) to
account for spatially varying means or trends in NO2 concentrations. UK combines spatial
correlation from nearby observations with information about spatial trends to improve prediction
accuracy.
9) Indicator Kriging (IK):
Apply Indicator Kriging to classify NO2 levels into categorical classes (e.g., low, moderate, high)
based on predefined threshold values. IK estimates the probability of exceeding threshold values
at unsampled locations, providing insights into spatial patterns of NO2 pollution levels.
10) Simple Kriging (SK):
Utilise Simple Kriging as a basic form of spatial interpolation that assumes a constant mean. SK
provides estimates of NO2 concentrations at unsampled locations based solely on the spatial
correlation structure derived from observed data.
11) Co-Kriging:
Implement Co-Kriging to incorporate additional covariate variables (e.g., land use, elevation) into
the spatial interpolation process, enhancing prediction accuracy. Co-Kriging accounts for the
spatial correlation between the target variable (NO2 concentrations) and auxiliary variables,
improving the accuracy of predictions in areas with sparse data.
12) Model Evaluation:
Assess the accuracy of each kriging technique through cross-validation techniques such as leave-
one-out cross-validation or spatial cross-validation. Compute Root Mean Square Error (RMSE) to
quantify the discrepancy between predicted and observed NO2 concentrations, providing a
measure of the model’s predictive performance.
4) Inverse Distance Weighting (IDW) Interpolation:
Analysis:
Data Loading and Preprocessing: The NO2 concentration data was loaded and transformed into
a spatial object. Country boundaries were loaded to provide context to the spatial data.
Initial Visualization: The country boundaries were plotted to visualise the study area.
Visualization with Observed Values: Observed NO2 concentration values were plotted on the
map along with country boundaries to visualise the distribution of NO2 concentrations.
Grid Creation: A regular grid with a cell size of 10km × 10km was created within the study area.
account for spatially varying means or trends in NO2 concentrations. UK combines spatial
correlation from nearby observations with information about spatial trends to improve prediction
accuracy.
9) Indicator Kriging (IK):
Apply Indicator Kriging to classify NO2 levels into categorical classes (e.g., low, moderate, high)
based on predefined threshold values. IK estimates the probability of exceeding threshold values
at unsampled locations, providing insights into spatial patterns of NO2 pollution levels.
10) Simple Kriging (SK):
Utilise Simple Kriging as a basic form of spatial interpolation that assumes a constant mean. SK
provides estimates of NO2 concentrations at unsampled locations based solely on the spatial
correlation structure derived from observed data.
11) Co-Kriging:
Implement Co-Kriging to incorporate additional covariate variables (e.g., land use, elevation) into
the spatial interpolation process, enhancing prediction accuracy. Co-Kriging accounts for the
spatial correlation between the target variable (NO2 concentrations) and auxiliary variables,
improving the accuracy of predictions in areas with sparse data.
12) Model Evaluation:
Assess the accuracy of each kriging technique through cross-validation techniques such as leave-
one-out cross-validation or spatial cross-validation. Compute Root Mean Square Error (RMSE) to
quantify the discrepancy between predicted and observed NO2 concentrations, providing a
measure of the model’s predictive performance.
4) Inverse Distance Weighting (IDW) Interpolation:
Analysis:
Data Loading and Preprocessing: The NO2 concentration data was loaded and transformed into
a spatial object. Country boundaries were loaded to provide context to the spatial data.
Initial Visualization: The country boundaries were plotted to visualise the study area.
Visualization with Observed Values: Observed NO2 concentration values were plotted on the
map along with country boundaries to visualise the distribution of NO2 concentrations.
Grid Creation: A regular grid with a cell size of 10km × 10km was created within the study area.
IDW Interpolation: Inverse Distance Weighting (IDW) interpolation was performed to estimate
NO2 concentrations at the grid points based on observed values.
Visualisation of Interpolated Values: The interpolated NO2 concentration values were plotted
along with observed values and country boundaries on the map.
Results and Interpretations:
Interpolated Values: The IDW interpolation provided estimates of NO2 concentrations across the
study area.
Spatial Distribution: The interpolated map showed spatial patterns of NO2 concentrations,
suggesting areas of higher and lower pollution levels.
Comparison with Observed Values: The interpolated values were compared with observed
values to assess the accuracy of the interpolation method.
Conclusions:
Limitations of IDW:
• IDW interpolation relies heavily on the assumption that nearby points have similar values,
which might not always hold true for NO2 concentration data.
• It tends to produce over smoothed surfaces and does not consider spatial autocorrelation,
which might be present in NO2 data.
Need for Alternative Methods:
• Due to the limitations of IDW and the complexity of NO2 data, alternative interpolation
methods such as kriging should be considered.
• Kriging methods take into account spatial autocorrelation and can provide more accurate
estimates, especially in areas with sparse data.
Further Analysis:
• Further analysis using kriging methods should be conducted to obtain more reliable
estimates of NO2 concentrations.
• Additional factors such as temporal variability and sources of pollution should also be
considered in future analyses.
7) Ordinary Kriging
NO2 concentrations at the grid points based on observed values.
Visualisation of Interpolated Values: The interpolated NO2 concentration values were plotted
along with observed values and country boundaries on the map.
Results and Interpretations:
Interpolated Values: The IDW interpolation provided estimates of NO2 concentrations across the
study area.
Spatial Distribution: The interpolated map showed spatial patterns of NO2 concentrations,
suggesting areas of higher and lower pollution levels.
Comparison with Observed Values: The interpolated values were compared with observed
values to assess the accuracy of the interpolation method.
Conclusions:
Limitations of IDW:
• IDW interpolation relies heavily on the assumption that nearby points have similar values,
which might not always hold true for NO2 concentration data.
• It tends to produce over smoothed surfaces and does not consider spatial autocorrelation,
which might be present in NO2 data.
Need for Alternative Methods:
• Due to the limitations of IDW and the complexity of NO2 data, alternative interpolation
methods such as kriging should be considered.
• Kriging methods take into account spatial autocorrelation and can provide more accurate
estimates, especially in areas with sparse data.
Further Analysis:
• Further analysis using kriging methods should be conducted to obtain more reliable
estimates of NO2 concentrations.
• Additional factors such as temporal variability and sources of pollution should also be
considered in future analyses.
7) Ordinary Kriging
Analysis and Interpretation
Ordinary Kriging Results: The Ordinary Kriging model was applied to predict NO2
concentrations across the study area. The resulting map illustrates the spatial distribution of
predicted NO2 concentrations based on the observed data and spatial autocorrelation.
Visual Interpretation: From the visualisation, areas with higher predicted NO2 concentrations
are depicted in darker shades, while lighter shades represent lower concentrations. Spatial patterns
and gradients in NO2 concentrations can be observed across the study area.
Model Evaluation: The Root Mean Square Error (RMSE) was calculated to evaluate the
predictive performance of the Ordinary Kriging model. The RMSE value obtained was 3.8703,
indicating the average discrepancy between observed and predicted NO2 concentrations.
Conclusions:
The Ordinary Kriging model provides valuable insights into the spatial distribution of NO2
concentrations in the study area. The RMSE value of 3.8703 suggests moderate prediction
accuracy, indicating that while the model captures overall spatial trends, there may be variability
or local deviations not adequately accounted for. Additionally, alternative interpolation methods
could be explored to address the limitations of the Ordinary Kriging approach.
8) Universal Kriging
Analysis and Interpretation
Universal Kriging Results: Universal Kriging was performed to predict NO2 concentrations
across the study area, incorporating both a spatial trend model and the variogram model for
residuals.
Visual Interpretation: The map illustrates the spatial distribution of predicted NO2
concentrations obtained through Universal Kriging. Darker shades represent areas with higher
predicted concentrations, while lighter shades indicate lower concentrations. Spatial patterns and
gradients in NO2 concentrations can be observed across the study area.
Model Evaluation: Cross-validation was conducted to assess the predictive performance of the
Universal Kriging model. The Root Mean Square Error (RMSE) was calculated as 3.8904,
providing a measure of the average discrepancy between observed and predicted NO2
concentrations.
Ordinary Kriging Results: The Ordinary Kriging model was applied to predict NO2
concentrations across the study area. The resulting map illustrates the spatial distribution of
predicted NO2 concentrations based on the observed data and spatial autocorrelation.
Visual Interpretation: From the visualisation, areas with higher predicted NO2 concentrations
are depicted in darker shades, while lighter shades represent lower concentrations. Spatial patterns
and gradients in NO2 concentrations can be observed across the study area.
Model Evaluation: The Root Mean Square Error (RMSE) was calculated to evaluate the
predictive performance of the Ordinary Kriging model. The RMSE value obtained was 3.8703,
indicating the average discrepancy between observed and predicted NO2 concentrations.
Conclusions:
The Ordinary Kriging model provides valuable insights into the spatial distribution of NO2
concentrations in the study area. The RMSE value of 3.8703 suggests moderate prediction
accuracy, indicating that while the model captures overall spatial trends, there may be variability
or local deviations not adequately accounted for. Additionally, alternative interpolation methods
could be explored to address the limitations of the Ordinary Kriging approach.
8) Universal Kriging
Analysis and Interpretation
Universal Kriging Results: Universal Kriging was performed to predict NO2 concentrations
across the study area, incorporating both a spatial trend model and the variogram model for
residuals.
Visual Interpretation: The map illustrates the spatial distribution of predicted NO2
concentrations obtained through Universal Kriging. Darker shades represent areas with higher
predicted concentrations, while lighter shades indicate lower concentrations. Spatial patterns and
gradients in NO2 concentrations can be observed across the study area.
Model Evaluation: Cross-validation was conducted to assess the predictive performance of the
Universal Kriging model. The Root Mean Square Error (RMSE) was calculated as 3.8904,
providing a measure of the average discrepancy between observed and predicted NO2
concentrations.
Conclusions:
Universal Kriging offers a sophisticated approach to spatial interpolation, incorporating both a
trend model and variogram modeling to account for spatial autocorrelation and trend components
in the data. The RMSE value of 3.8904 suggests moderate prediction accuracy, indicating that the
model captures overall spatial trends in NO2 concentrations. However, the model may not
adequately capture local variability or deviations. Further exploration of alternative interpolation
techniques may be required to improve predictive accuracy and enhance the reliability of NO2
concentration estimates. Additionally, considering alternative trend models could provide valuable
insights into the spatial distribution of NO2 concentrations and improve the performance of the
Universal Kriging model.
9) Indicator Kriging
Threshold Definition and Indicator Creation: Three threshold values were defined to categorise
NO2 levels into low, moderate, and high categories:
• Low: <= 10
• Moderate: > 10 and <= 20
• High: > 20 and <= 30
Indicator variables were created based on these thresholds, where each observation is assigned a
value of 1 if it falls within the corresponding category and 0 otherwise.
Variogram Analysis: Variogram analysis was performed for each indicator variable (low,
moderate, and high NO2 levels) to characterize the spatial autocorrelation of each category.
Indicator Kriging: Indicator Kriging was applied separately for each category of NO2 levels
using the corresponding variogram model. The resulting Indicator Kriging predictions provide
spatial estimates of the probability of each NO2 level category across the study area.
Visualisation: Indicator Kriging predictions were visualized for each NO2 level category using
thematic maps. These maps illustrate the spatial distribution of low, moderate, and high NO2 levels
across the study area.
Cross-Validation: Cross-validation was conducted to assess the predictive performance of the
Indicator Kriging models.
RMSE Interpretation: Root Mean Square Error (RMSE) was calculated for each NO2 level
category to quantify the accuracy of the predictions. The RMSE values provide insight into the
accuracy of the Indicator Kriging predictions for each NO2 level category:
• Low NO2 Levels: RMSE = 2.7392
Universal Kriging offers a sophisticated approach to spatial interpolation, incorporating both a
trend model and variogram modeling to account for spatial autocorrelation and trend components
in the data. The RMSE value of 3.8904 suggests moderate prediction accuracy, indicating that the
model captures overall spatial trends in NO2 concentrations. However, the model may not
adequately capture local variability or deviations. Further exploration of alternative interpolation
techniques may be required to improve predictive accuracy and enhance the reliability of NO2
concentration estimates. Additionally, considering alternative trend models could provide valuable
insights into the spatial distribution of NO2 concentrations and improve the performance of the
Universal Kriging model.
9) Indicator Kriging
Threshold Definition and Indicator Creation: Three threshold values were defined to categorise
NO2 levels into low, moderate, and high categories:
• Low: <= 10
• Moderate: > 10 and <= 20
• High: > 20 and <= 30
Indicator variables were created based on these thresholds, where each observation is assigned a
value of 1 if it falls within the corresponding category and 0 otherwise.
Variogram Analysis: Variogram analysis was performed for each indicator variable (low,
moderate, and high NO2 levels) to characterize the spatial autocorrelation of each category.
Indicator Kriging: Indicator Kriging was applied separately for each category of NO2 levels
using the corresponding variogram model. The resulting Indicator Kriging predictions provide
spatial estimates of the probability of each NO2 level category across the study area.
Visualisation: Indicator Kriging predictions were visualized for each NO2 level category using
thematic maps. These maps illustrate the spatial distribution of low, moderate, and high NO2 levels
across the study area.
Cross-Validation: Cross-validation was conducted to assess the predictive performance of the
Indicator Kriging models.
RMSE Interpretation: Root Mean Square Error (RMSE) was calculated for each NO2 level
category to quantify the accuracy of the predictions. The RMSE values provide insight into the
accuracy of the Indicator Kriging predictions for each NO2 level category:
• Low NO2 Levels: RMSE = 2.7392
• Moderate NO2 Levels: RMSE = 5.5857
• High NO2 Levels: RMSE = 13.4751
Lower RMSE values indicate better predictive performance, while higher values suggest greater
prediction errors.
Conclusions:
• Indicator Kriging effectively models the spatial variability of NO2 levels across the study
area by categorising observations into distinct levels based on predefined thresholds.
• The maps generated from Indicator Kriging predictions offer valuable insights into the
spatial distribution of different NO2 level categories, aiding in environmental monitoring
and management efforts.
• The RMSE values indicate reasonable accuracy in estimating NO2 levels using Indicator
Kriging, especially for low and moderate levels. However, higher RMSE for high NO2
levels suggests challenges in accurately predicting areas with elevated pollution levels,
which may require further investigation or model refinement.
• Overall, Indicator Kriging provides a valuable tool for spatial analysis of NO2 pollution
levels and can assist policymakers and environmental agencies in making informed
decisions to mitigate air quality issues.
10) Simple Kriging (SK)
Analysis Codes:
Perform Simple Kriging: Implement Simple Kriging to predict NO2 levels at unsampled
locations using the variogram model previously fitted (‘v.m’).
Visualise Simple Kriging Predictions: Create a plot to visualize the predicted NO2 levels
obtained from Simple Kriging.
Perform Cross-Validation and calculate RMSE: Conduct cross-validation to assess the
predictive performance of Simple Kriging and compute the Root Mean Squared Error (RMSE).
Results and Interpretations:
RMSE for Simple Kriging: The RMSE value obtained for Simple Kriging is approximately
3.8703.
Conclusions:
• High NO2 Levels: RMSE = 13.4751
Lower RMSE values indicate better predictive performance, while higher values suggest greater
prediction errors.
Conclusions:
• Indicator Kriging effectively models the spatial variability of NO2 levels across the study
area by categorising observations into distinct levels based on predefined thresholds.
• The maps generated from Indicator Kriging predictions offer valuable insights into the
spatial distribution of different NO2 level categories, aiding in environmental monitoring
and management efforts.
• The RMSE values indicate reasonable accuracy in estimating NO2 levels using Indicator
Kriging, especially for low and moderate levels. However, higher RMSE for high NO2
levels suggests challenges in accurately predicting areas with elevated pollution levels,
which may require further investigation or model refinement.
• Overall, Indicator Kriging provides a valuable tool for spatial analysis of NO2 pollution
levels and can assist policymakers and environmental agencies in making informed
decisions to mitigate air quality issues.
10) Simple Kriging (SK)
Analysis Codes:
Perform Simple Kriging: Implement Simple Kriging to predict NO2 levels at unsampled
locations using the variogram model previously fitted (‘v.m’).
Visualise Simple Kriging Predictions: Create a plot to visualize the predicted NO2 levels
obtained from Simple Kriging.
Perform Cross-Validation and calculate RMSE: Conduct cross-validation to assess the
predictive performance of Simple Kriging and compute the Root Mean Squared Error (RMSE).
Results and Interpretations:
RMSE for Simple Kriging: The RMSE value obtained for Simple Kriging is approximately
3.8703.
Conclusions:
• Simple Kriging provides predictions of NO2 levels at unsampled locations based on spatial
autocorrelation captured by the fitted variogram model.
• The RMSE value indicates the average prediction error of Simple Kriging across the study
area.
• Further analysis, including comparison with other kriging methods and consideration of
spatial patterns, can provide additional insights into the performance and suitability of
Simple Kriging for predicting NO2 levels in the study area.
Analysis of Kriging Models for Predicting NO2 Concentrations
Introduction
In this analysis, four different kriging models were evaluated for predicting NO2 concentrations:
Ordinary Kriging, Universal Kriging, Indicator Kriging, and Simple Kriging. The root mean
square error (RMSE) was used as the metric to assess the predictive performance of each model.
Methodology
Ordinary Kriging: Assumes stationarity and estimates NO2 concentrations at unsampled
locations based on the spatial autocorrelation of the observed data.
Universal Kriging: Similar to Ordinary Kriging but incorporates a trend model to account for
non-stationarity.
Indicator Kriging: Predicts the probability that NO2 concentrations exceed a certain threshold,
providing robust predictions, especially for extreme values.
Simple Kriging: Assumes a constant mean and accounts for the spatial autocorrelation of the data.
Results
The table below summarises the RMSE values obtained for each kriging model:
Kriging Model RMSE
Ordinary Kriging 3.8703
Universal Kriging 3.8904
Indicator Kriging (Low) 2.7392
Indicator Kriging (Moderate) 5.5857
Indicator Kriging (High) 13.4751
autocorrelation captured by the fitted variogram model.
• The RMSE value indicates the average prediction error of Simple Kriging across the study
area.
• Further analysis, including comparison with other kriging methods and consideration of
spatial patterns, can provide additional insights into the performance and suitability of
Simple Kriging for predicting NO2 levels in the study area.
Analysis of Kriging Models for Predicting NO2 Concentrations
Introduction
In this analysis, four different kriging models were evaluated for predicting NO2 concentrations:
Ordinary Kriging, Universal Kriging, Indicator Kriging, and Simple Kriging. The root mean
square error (RMSE) was used as the metric to assess the predictive performance of each model.
Methodology
Ordinary Kriging: Assumes stationarity and estimates NO2 concentrations at unsampled
locations based on the spatial autocorrelation of the observed data.
Universal Kriging: Similar to Ordinary Kriging but incorporates a trend model to account for
non-stationarity.
Indicator Kriging: Predicts the probability that NO2 concentrations exceed a certain threshold,
providing robust predictions, especially for extreme values.
Simple Kriging: Assumes a constant mean and accounts for the spatial autocorrelation of the data.
Results
The table below summarises the RMSE values obtained for each kriging model:
Kriging Model RMSE
Ordinary Kriging 3.8703
Universal Kriging 3.8904
Indicator Kriging (Low) 2.7392
Indicator Kriging (Moderate) 5.5857
Indicator Kriging (High) 13.4751
Simple Kriging 3.8703
Analysis
Ordinary vs. Universal Kriging: Both models incorporate spatial autocorrelation, but Universal
Kriging, with its ability to account for trend, does not significantly outperform Ordinary Kriging
in this case.
Indicator Kriging: This model shows competitive performance for low and moderate NO2 levels,
providing robust predictions by considering the probability of exceeding a threshold. However, for
high NO2 levels, the RMSE is significantly higher, indicating limitations in prediction accuracy
for extreme values.
Simple Kriging: While Simple Kriging yields comparable results to Indicator Kriging for overall
NO2 concentrations, it assumes a constant mean and may not fully capture the spatial variability
of NO2 concentrations.
Conclusion
Based on the RMSE values and analysis, Indicator Kriging (Low) emerges as the best-performing
model for predicting NO2 concentrations. It provides robust predictions, especially for low NO2
levels, with the lowest RMSE among all models evaluated.
Recommendation
Given the performance of Indicator Kriging (Low), further investigation into its application and
potential improvements is required. Additionally, exploring alternative modelling approaches or
incorporating additional variables may further enhance predictive accuracy.
Limitations and Future Work
This analysis focused solely on RMSE values as the metric for evaluating predictive performance.
Future work could consider additional metrics such as mean absolute error (MAE). Further
investigation could also examine the impact of different variogram models and parameters on
model performance.
Analysis
Ordinary vs. Universal Kriging: Both models incorporate spatial autocorrelation, but Universal
Kriging, with its ability to account for trend, does not significantly outperform Ordinary Kriging
in this case.
Indicator Kriging: This model shows competitive performance for low and moderate NO2 levels,
providing robust predictions by considering the probability of exceeding a threshold. However, for
high NO2 levels, the RMSE is significantly higher, indicating limitations in prediction accuracy
for extreme values.
Simple Kriging: While Simple Kriging yields comparable results to Indicator Kriging for overall
NO2 concentrations, it assumes a constant mean and may not fully capture the spatial variability
of NO2 concentrations.
Conclusion
Based on the RMSE values and analysis, Indicator Kriging (Low) emerges as the best-performing
model for predicting NO2 concentrations. It provides robust predictions, especially for low NO2
levels, with the lowest RMSE among all models evaluated.
Recommendation
Given the performance of Indicator Kriging (Low), further investigation into its application and
potential improvements is required. Additionally, exploring alternative modelling approaches or
incorporating additional variables may further enhance predictive accuracy.
Limitations and Future Work
This analysis focused solely on RMSE values as the metric for evaluating predictive performance.
Future work could consider additional metrics such as mean absolute error (MAE). Further
investigation could also examine the impact of different variogram models and parameters on
model performance.
Question 2: Kriging Model Selection for Zinc Concentration
Interpolation in the Meuse Dataset
Methodology Overview
The methodology section outlines the step-by-step approach used to analyze the Meuse dataset,
which contains information on heavy metal concentrations, particularly zinc. The analysis involves
various data preprocessing steps, spatial data handling techniques, interpolation methods,
variogram analysis, and geostatistical modeling.
The following steps outline the approach used in detail.
1) Data Loading and Preprocessing:
• Load the Meuse dataset containing information on zinc concentrations and associated
spatial attributes.
• Conduct initial data preprocessing to ensure data integrity and compatibility with
spatial analysis techniques.
2) Spatial Data Handling:
• Convert the Meuse dataset into a SpatialPointsDataFrame, enabling efficient spatial
data handling.
• Define an appropriate Coordinate Reference System (CRS) to maintain spatial
consistency and accuracy.
3) Visualization:
• Visualize the spatial distribution of zinc concentrations using maps, scatter plots, and other
visualization techniques.
• Explore different visualization methods to identify spatial patterns and variations in zinc
concentrations across the study area.
4) Inverse Distance Weighted (IDW) Interpolation:
• Apply Inverse Distance Weighting (IDW) interpolation estimate zinc concentrations across
a regular grid.
5) Variogram Analysis:
• Conduct a Variogram analysis to characterise the spatial dependence structure of zinc
concentrations.
Interpolation in the Meuse Dataset
Methodology Overview
The methodology section outlines the step-by-step approach used to analyze the Meuse dataset,
which contains information on heavy metal concentrations, particularly zinc. The analysis involves
various data preprocessing steps, spatial data handling techniques, interpolation methods,
variogram analysis, and geostatistical modeling.
The following steps outline the approach used in detail.
1) Data Loading and Preprocessing:
• Load the Meuse dataset containing information on zinc concentrations and associated
spatial attributes.
• Conduct initial data preprocessing to ensure data integrity and compatibility with
spatial analysis techniques.
2) Spatial Data Handling:
• Convert the Meuse dataset into a SpatialPointsDataFrame, enabling efficient spatial
data handling.
• Define an appropriate Coordinate Reference System (CRS) to maintain spatial
consistency and accuracy.
3) Visualization:
• Visualize the spatial distribution of zinc concentrations using maps, scatter plots, and other
visualization techniques.
• Explore different visualization methods to identify spatial patterns and variations in zinc
concentrations across the study area.
4) Inverse Distance Weighted (IDW) Interpolation:
• Apply Inverse Distance Weighting (IDW) interpolation estimate zinc concentrations across
a regular grid.
5) Variogram Analysis:
• Conduct a Variogram analysis to characterise the spatial dependence structure of zinc
concentrations.
• Compute experimental variograms to examine the relationship between spatial separation
and variability in zinc concentrations.
6) Variogram Modeling:
• Fit Variogram models, such as spherical, exponential, Gaussian, or linear models to the
experimental variogram data.
• Detremine the Optimal variogram model parameters, including nugget, sill, and range to
effectively capture spatial correlation.
• Fit theoretical variogram models, such as spherical, exponential, or Gaussian models, to
the experimental variogram data to capture spatial correlation effectively.
7) Ordinary Kriging:
• Implement Ordinary Kriging using the fitted variogram model to interpolate zinc
concentrations while considering spatial autocorrelation.
• Assess the accuracy of the kriging model through the calculation of Root Mean Square
Error (RMSE), a common metric for evaluating interpolation performance.
• Compare predicted zinc concentrations with observed values to interpret the RMSE and
understand the predictive performance of the kriging model.
8) Directional Variograms:
• Analyze directional variograms to investigate potential directional trends or anisotropy in
the spatial dependence structure of zinc concentrations.
• Explore directional variograms to understand spatial variability and anisotropic behaviour
across different directions.
9) Regression Kriging:
• Incorporate auxiliary variables, such as distance to the river, using regression kriging to
improve the accuracy of spatial predictions.
• Fit regression models to the data and incorporate residuals into the kriging predictions to
account for additional spatial variability.
10) Blocking Kriging:
• Implement blocking kriging to address large spatial variability or focus on specific spatial
patterns at different scales.
• Partition the study area into blocks and perform kriging within each block to capture local
variations effectively.
11) Indicator Kriging:
and variability in zinc concentrations.
6) Variogram Modeling:
• Fit Variogram models, such as spherical, exponential, Gaussian, or linear models to the
experimental variogram data.
• Detremine the Optimal variogram model parameters, including nugget, sill, and range to
effectively capture spatial correlation.
• Fit theoretical variogram models, such as spherical, exponential, or Gaussian models, to
the experimental variogram data to capture spatial correlation effectively.
7) Ordinary Kriging:
• Implement Ordinary Kriging using the fitted variogram model to interpolate zinc
concentrations while considering spatial autocorrelation.
• Assess the accuracy of the kriging model through the calculation of Root Mean Square
Error (RMSE), a common metric for evaluating interpolation performance.
• Compare predicted zinc concentrations with observed values to interpret the RMSE and
understand the predictive performance of the kriging model.
8) Directional Variograms:
• Analyze directional variograms to investigate potential directional trends or anisotropy in
the spatial dependence structure of zinc concentrations.
• Explore directional variograms to understand spatial variability and anisotropic behaviour
across different directions.
9) Regression Kriging:
• Incorporate auxiliary variables, such as distance to the river, using regression kriging to
improve the accuracy of spatial predictions.
• Fit regression models to the data and incorporate residuals into the kriging predictions to
account for additional spatial variability.
10) Blocking Kriging:
• Implement blocking kriging to address large spatial variability or focus on specific spatial
patterns at different scales.
• Partition the study area into blocks and perform kriging within each block to capture local
variations effectively.
11) Indicator Kriging:
• Utilize indicator kriging to model spatial discontinuities or threshold effects in zinc
concentrations, which is particularly useful for identifying areas with specific
characteristics or categorical variables.
12) Co-Kriging:
• Apply Co-kriging to model the spatial dependence between multiple variables
simultaneously, leveraging their cross-correlation structure.
• This technique improves the accuracy of spatial predictions by incorporating additional
information from correlated variables.
13) Indicator Co-Kriging:
• Extend co-kriging to indicator co-kriging to incorporate indicator variables, useful for
dealing with multiple correlated variables with different measurement scales or
distributions.
14) Universal Kriging:
• Perform universal kriging to interpolate zinc concentrations while considering spatial trend
information that may not be captured by the variogram alone.
• Enhance the accuracy of predictions by incorporating additional spatial trend information
into the kriging model.
4) Inverse Distance Weighted (IDW) Interpolation
Analysis:
Data: The dataset under analysis comprises observed zinc concentrations at specific locations,
denoted as the original data. Additionally, zinc concentrations were estimated at unsampled
locations using the Inverse Distance Weighting (IDW) interpolation method, resulting in the
interpolated data.
Method: The IDW interpolation method was applied to estimate zinc concentrations at unsampled
locations based on observed values at sampled locations.
Evaluation: The evaluation of the IDW interpolation method was conducted through a visual
comparison of the original zinc concentration map with the IDW-interpolated map.
Results:
concentrations, which is particularly useful for identifying areas with specific
characteristics or categorical variables.
12) Co-Kriging:
• Apply Co-kriging to model the spatial dependence between multiple variables
simultaneously, leveraging their cross-correlation structure.
• This technique improves the accuracy of spatial predictions by incorporating additional
information from correlated variables.
13) Indicator Co-Kriging:
• Extend co-kriging to indicator co-kriging to incorporate indicator variables, useful for
dealing with multiple correlated variables with different measurement scales or
distributions.
14) Universal Kriging:
• Perform universal kriging to interpolate zinc concentrations while considering spatial trend
information that may not be captured by the variogram alone.
• Enhance the accuracy of predictions by incorporating additional spatial trend information
into the kriging model.
4) Inverse Distance Weighted (IDW) Interpolation
Analysis:
Data: The dataset under analysis comprises observed zinc concentrations at specific locations,
denoted as the original data. Additionally, zinc concentrations were estimated at unsampled
locations using the Inverse Distance Weighting (IDW) interpolation method, resulting in the
interpolated data.
Method: The IDW interpolation method was applied to estimate zinc concentrations at unsampled
locations based on observed values at sampled locations.
Evaluation: The evaluation of the IDW interpolation method was conducted through a visual
comparison of the original zinc concentration map with the IDW-interpolated map.
Results:
Visual Comparison: Upon visual inspection, notable disparities were observed between the
original zinc concentration map and the IDW-interpolated map. The spatial patterns and
concentration values exhibited significant differences, suggesting potential limitations in the IDW
interpolation method.
Interpretations:
Visual Inspection: The visual analysis underscored the inadequacy of the IDW interpolation
method in accurately capturing the spatial distribution and concentration levels of zinc across the
study area. Discrepancies between the original and IDW-interpolated maps were evident, raising
concerns about the reliability of the interpolation results.
Conclusions:
In conclusion, based on the visual analyses, it is evident that the IDW interpolation method
employed in this study is not suitable for accurately estimating zinc concentrations in the study
area. The disparities observed between the original and IDW-interpolated maps underscore the
limitations of the IDW method for this application. Further exploration of alternative interpolation
techniques or refinement of parameters may be warranted to improve the accuracy of zinc
concentration estimates.
7) Ordinary Kriging
Analysis:
Variogram Analysis: The variogram analysis was conducted to assess the spatial dependence
structure of zinc concentrations in the study area. The number of point-pairs was calculated to
determine the total number of pairwise combinations, indicating the sample size for variogram
analysis. Additionally, the distance and semivariance between the first two points in the dataset
were computed to provide insight into the spatial autocorrelation at a local scale. The experimental
variogram of log-transformed zinc concentrations was plotted to visualize the relationship between
spatial separation and semivariance.
Model Fitting: A spherical variogram model was fitted to the experimental variogram to
characterize the spatial correlation of zinc concentrations. Initially, the model fit was assessed
visually, revealing discrepancies between the original estimate and the empirical variogram.
Subsequently, the variogram model was adjusted using the gstat automatic fit function to improve
the fit to the experimental variogram.
original zinc concentration map and the IDW-interpolated map. The spatial patterns and
concentration values exhibited significant differences, suggesting potential limitations in the IDW
interpolation method.
Interpretations:
Visual Inspection: The visual analysis underscored the inadequacy of the IDW interpolation
method in accurately capturing the spatial distribution and concentration levels of zinc across the
study area. Discrepancies between the original and IDW-interpolated maps were evident, raising
concerns about the reliability of the interpolation results.
Conclusions:
In conclusion, based on the visual analyses, it is evident that the IDW interpolation method
employed in this study is not suitable for accurately estimating zinc concentrations in the study
area. The disparities observed between the original and IDW-interpolated maps underscore the
limitations of the IDW method for this application. Further exploration of alternative interpolation
techniques or refinement of parameters may be warranted to improve the accuracy of zinc
concentration estimates.
7) Ordinary Kriging
Analysis:
Variogram Analysis: The variogram analysis was conducted to assess the spatial dependence
structure of zinc concentrations in the study area. The number of point-pairs was calculated to
determine the total number of pairwise combinations, indicating the sample size for variogram
analysis. Additionally, the distance and semivariance between the first two points in the dataset
were computed to provide insight into the spatial autocorrelation at a local scale. The experimental
variogram of log-transformed zinc concentrations was plotted to visualize the relationship between
spatial separation and semivariance.
Model Fitting: A spherical variogram model was fitted to the experimental variogram to
characterize the spatial correlation of zinc concentrations. Initially, the model fit was assessed
visually, revealing discrepancies between the original estimate and the empirical variogram.
Subsequently, the variogram model was adjusted using the gstat automatic fit function to improve
the fit to the experimental variogram.
Ordinary Kriging: Ordinary Kriging was performed using the fitted variogram model to predict
zinc concentrations at unsampled locations across the study area. The kriging predictions were
visualized on a map to illustrate the spatial distribution of predicted zinc concentrations.
Model Evaluation: Cross-validation was conducted to evaluate the performance of the Ordinary
Kriging model. The Root Mean Square Error (RMSE) was calculated as a measure of prediction
accuracy, indicating the average difference between observed and predicted zinc concentrations.
Results:
Variogram Analysis: The variogram analysis revealed a spatial dependence structure in zinc
concentrations, with semivariance increasing with spatial separation up to a certain range.
Model Fitting: The spherical variogram model provided a reasonable fit to the experimental
variogram after adjustment, capturing the spatial correlation of zinc concentrations across the study
area.
Ordinary Kriging: The Ordinary Kriging predictions depicted spatial patterns in zinc
concentrations, with higher concentrations observed in certain areas and lower concentrations in
others.
Model Evaluation: The RMSE for Ordinary Kriging was computed as 0.397, indicating moderate
prediction accuracy relative to the range of zinc concentrations in the dataset.
Conclusions:
In conclusion, the variogram analysis, model fitting, and Ordinary Kriging provided valuable
insights into the spatial distribution of zinc concentrations in the study area. The fitted variogram
model adequately captured the spatial correlation, enabling reliable predictions through Ordinary
Kriging. However, the moderate RMSE suggests some level of prediction uncertainty,
emphasizing the need for cautious interpretation of the kriging predictions. Overall, Ordinary
Kriging serves as a valuable tool for spatial interpolation of zinc concentrations, providing
valuable information for environmental monitoring and management efforts.
8) Directional Variogram
Results:
The analysis of the directional variogram model reveals the following:
zinc concentrations at unsampled locations across the study area. The kriging predictions were
visualized on a map to illustrate the spatial distribution of predicted zinc concentrations.
Model Evaluation: Cross-validation was conducted to evaluate the performance of the Ordinary
Kriging model. The Root Mean Square Error (RMSE) was calculated as a measure of prediction
accuracy, indicating the average difference between observed and predicted zinc concentrations.
Results:
Variogram Analysis: The variogram analysis revealed a spatial dependence structure in zinc
concentrations, with semivariance increasing with spatial separation up to a certain range.
Model Fitting: The spherical variogram model provided a reasonable fit to the experimental
variogram after adjustment, capturing the spatial correlation of zinc concentrations across the study
area.
Ordinary Kriging: The Ordinary Kriging predictions depicted spatial patterns in zinc
concentrations, with higher concentrations observed in certain areas and lower concentrations in
others.
Model Evaluation: The RMSE for Ordinary Kriging was computed as 0.397, indicating moderate
prediction accuracy relative to the range of zinc concentrations in the dataset.
Conclusions:
In conclusion, the variogram analysis, model fitting, and Ordinary Kriging provided valuable
insights into the spatial distribution of zinc concentrations in the study area. The fitted variogram
model adequately captured the spatial correlation, enabling reliable predictions through Ordinary
Kriging. However, the moderate RMSE suggests some level of prediction uncertainty,
emphasizing the need for cautious interpretation of the kriging predictions. Overall, Ordinary
Kriging serves as a valuable tool for spatial interpolation of zinc concentrations, providing
valuable information for environmental monitoring and management efforts.
8) Directional Variogram
Results:
The analysis of the directional variogram model reveals the following:
Nugget Effect (Nug):
• Partial sill: 0.08305825
• Range: 0.000
• Angle: 0
• Anisotropy: 1.0
Spherical Component (Sph):
• Partial sill: 0.20023979
• Range: 1452.826
• Angle: 45
• Anisotropy: 0.4
Interpretations:
The directional variogram model indicates the following:
• The nugget effect suggests a moderate level of variability at very short distances, likely
due to measurement error or small-scale variability.
• The spherical component represents the main spatial structure, with higher variability along
the direction of 45 degrees.
• Anisotropic behavior is observed, with greater variability along the direction of 45 degrees
compared to other directions.
Conclusions:
Based on the analysis:
• The dataset exhibits both isotropic and anisotropic spatial variability, as indicated by the
directional variogram model.
• It is important to consider directional trends in spatial analysis for accurate interpolation
and prediction.
• It is required to consider anisotropic interpolation methods, such as Regression Kriging
with Anisotropy, Co-Kriging and Blocking Kriging to account for the observed directional
variability and improve spatial predictions.
9) Regression Kriging:
Regression kriging is a spatial interpolation technique that combines regression analysis with
kriging to improve the accuracy of predictions by incorporating auxiliary variables. In this
analysis, regression kriging was attempted to predict zinc concentrations in the Meuse dataset
using distance to the river as an auxiliary variable.
• Partial sill: 0.08305825
• Range: 0.000
• Angle: 0
• Anisotropy: 1.0
Spherical Component (Sph):
• Partial sill: 0.20023979
• Range: 1452.826
• Angle: 45
• Anisotropy: 0.4
Interpretations:
The directional variogram model indicates the following:
• The nugget effect suggests a moderate level of variability at very short distances, likely
due to measurement error or small-scale variability.
• The spherical component represents the main spatial structure, with higher variability along
the direction of 45 degrees.
• Anisotropic behavior is observed, with greater variability along the direction of 45 degrees
compared to other directions.
Conclusions:
Based on the analysis:
• The dataset exhibits both isotropic and anisotropic spatial variability, as indicated by the
directional variogram model.
• It is important to consider directional trends in spatial analysis for accurate interpolation
and prediction.
• It is required to consider anisotropic interpolation methods, such as Regression Kriging
with Anisotropy, Co-Kriging and Blocking Kriging to account for the observed directional
variability and improve spatial predictions.
9) Regression Kriging:
Regression kriging is a spatial interpolation technique that combines regression analysis with
kriging to improve the accuracy of predictions by incorporating auxiliary variables. In this
analysis, regression kriging was attempted to predict zinc concentrations in the Meuse dataset
using distance to the river as an auxiliary variable.
Methodology:
Fit a Regression Model: The first step was fitting a regression model using the ‘lm’ function in
R. The regression model aimed to predict the natural logarithm of zinc concentration (‘log(zinc)’)
based on the square root of the distance to the river (‘sqrt(dist)’).
Compute Residuals: After fitting the regression model, residuals of the regression model were
computed by subtracting the observed values from the predicted values.
Perform Ordinary Kriging on Residuals: Following the computation of residuals, ordinary
kriging was performed on the residuals obtained from the regression model. Ordinary kriging
estimates the spatial dependence of the residuals using a variogram model and provides predictions
at unsampled locations.
Combine Kriging Predictions with Regression Predictions: Finally, the kriging predictions of
the residuals were combined with the predictions from the regression model to obtain the final
regression kriging predictions.
Interpretation:
Regression kriging leverages both the spatial autocorrelation captured by kriging and the
relationships between predictor variables and the target variable captured by regression modeling.
By incorporating the distance to the river as an auxiliary variable, an attempt was made to enhance
the spatial predictions of zinc concentrations in the Meuse dataset. The combination of regression
and kriging methods allows leveraging both the global trend captured by the regression model and
the local spatial variability captured by kriging.
Conclusions:
Regression kriging holds promise for improving spatial predictions in environmental studies where
auxiliary information is available. In this analysis, while the methodology was successfully
implemented and predictions were obtained, further validation and refinement may be necessary
to assess the accuracy and reliability of the regression kriging predictions. Additionally, exploring
alternative auxiliary variables and variogram models could provide valuable insights into the
spatial distribution of zinc concentrations in the study area.
10) Blocking Kriging:
Analysis:
Fit a Regression Model: The first step was fitting a regression model using the ‘lm’ function in
R. The regression model aimed to predict the natural logarithm of zinc concentration (‘log(zinc)’)
based on the square root of the distance to the river (‘sqrt(dist)’).
Compute Residuals: After fitting the regression model, residuals of the regression model were
computed by subtracting the observed values from the predicted values.
Perform Ordinary Kriging on Residuals: Following the computation of residuals, ordinary
kriging was performed on the residuals obtained from the regression model. Ordinary kriging
estimates the spatial dependence of the residuals using a variogram model and provides predictions
at unsampled locations.
Combine Kriging Predictions with Regression Predictions: Finally, the kriging predictions of
the residuals were combined with the predictions from the regression model to obtain the final
regression kriging predictions.
Interpretation:
Regression kriging leverages both the spatial autocorrelation captured by kriging and the
relationships between predictor variables and the target variable captured by regression modeling.
By incorporating the distance to the river as an auxiliary variable, an attempt was made to enhance
the spatial predictions of zinc concentrations in the Meuse dataset. The combination of regression
and kriging methods allows leveraging both the global trend captured by the regression model and
the local spatial variability captured by kriging.
Conclusions:
Regression kriging holds promise for improving spatial predictions in environmental studies where
auxiliary information is available. In this analysis, while the methodology was successfully
implemented and predictions were obtained, further validation and refinement may be necessary
to assess the accuracy and reliability of the regression kriging predictions. Additionally, exploring
alternative auxiliary variables and variogram models could provide valuable insights into the
spatial distribution of zinc concentrations in the study area.
10) Blocking Kriging:
Analysis:
Variogram Analysis: The variogram analysis was conducted to assess the spatial dependence
structure of the logarithm of zinc concentration in the Meuse dataset. The empirical variogram was
calculated to quantify the relationship between spatial separation and semivariance. A theoretical
variogram model was then fitted to the empirical variogram, with parameters optimized to capture
the spatial correlation of zinc concentrations. The fitted model, represented by a spherical
variogram with parameters psill = 1, range = 900, and nugget = 1, adequately described the spatial
dependence of the data.
Ordinary Kriging: Before performing blocking kriging, ordinary kriging was applied to
interpolate the logarithm of zinc concentration across the study area. Ordinary kriging predictions
provided estimates of zinc concentrations at unsampled locations based on nearby sampled points.
The visualization of ordinary kriging predictions depicted the spatial distribution of predicted zinc
concentrations, highlighting areas of high and low concentrations.
Blocking Kriging: Blocking kriging was then performed using a block size of 500 by 500 units.
This method incorporates spatially correlated blocks of data rather than individual point
observations, potentially capturing spatial patterns that ordinary kriging might overlook. The
resulting predictions from blocking kriging were evaluated using cross-validation to assess the
model’s predictive performance.
Results:
Model Evaluation: Cross-validation was conducted to evaluate the accuracy of the blocking
kriging model. The Root Mean Square Error (RMSE) was calculated as a measure of prediction
error, providing insight into the average discrepancy between observed and predicted zinc
concentrations. The calculated RMSE for blocking kriging was found to be 0.3918, indicating
the level of prediction uncertainty associated with the model.
Conclusions:
The variogram analysis, ordinary kriging, and blocking kriging provided valuable insights into the
spatial distribution of zinc concentrations in the study area. The fitted variogram model adequately
captured the spatial correlation, enabling reliable predictions through both ordinary and blocking
kriging methods. However, the moderate RMSE value for blocking kriging suggests some level of
prediction uncertainty, emphasizing the need for cautious interpretation of the kriging predictions.
Overall, blocking kriging serves as a valuable tool for spatial interpolation of zinc concentrations,
complementing ordinary kriging and providing additional insights for environmental monitoring
and management efforts.
structure of the logarithm of zinc concentration in the Meuse dataset. The empirical variogram was
calculated to quantify the relationship between spatial separation and semivariance. A theoretical
variogram model was then fitted to the empirical variogram, with parameters optimized to capture
the spatial correlation of zinc concentrations. The fitted model, represented by a spherical
variogram with parameters psill = 1, range = 900, and nugget = 1, adequately described the spatial
dependence of the data.
Ordinary Kriging: Before performing blocking kriging, ordinary kriging was applied to
interpolate the logarithm of zinc concentration across the study area. Ordinary kriging predictions
provided estimates of zinc concentrations at unsampled locations based on nearby sampled points.
The visualization of ordinary kriging predictions depicted the spatial distribution of predicted zinc
concentrations, highlighting areas of high and low concentrations.
Blocking Kriging: Blocking kriging was then performed using a block size of 500 by 500 units.
This method incorporates spatially correlated blocks of data rather than individual point
observations, potentially capturing spatial patterns that ordinary kriging might overlook. The
resulting predictions from blocking kriging were evaluated using cross-validation to assess the
model’s predictive performance.
Results:
Model Evaluation: Cross-validation was conducted to evaluate the accuracy of the blocking
kriging model. The Root Mean Square Error (RMSE) was calculated as a measure of prediction
error, providing insight into the average discrepancy between observed and predicted zinc
concentrations. The calculated RMSE for blocking kriging was found to be 0.3918, indicating
the level of prediction uncertainty associated with the model.
Conclusions:
The variogram analysis, ordinary kriging, and blocking kriging provided valuable insights into the
spatial distribution of zinc concentrations in the study area. The fitted variogram model adequately
captured the spatial correlation, enabling reliable predictions through both ordinary and blocking
kriging methods. However, the moderate RMSE value for blocking kriging suggests some level of
prediction uncertainty, emphasizing the need for cautious interpretation of the kriging predictions.
Overall, blocking kriging serves as a valuable tool for spatial interpolation of zinc concentrations,
complementing ordinary kriging and providing additional insights for environmental monitoring
and management efforts.
11) Indicator Kriging
Analysis:
Variogram Analysis: The variogram analysis was conducted to explore the spatial dependence
structure of zinc concentrations in the study area. The empirical variogram was developed based
on the indicator variable representing whether the logarithm of zinc concentrations exceeds 6. This
analysis provided insights into the spatial autocorrelation and variability of high zinc
concentrations across different distances.
Model Fitting: A spherical variogram model was fitted to the empirical variogram to characterize
the spatial correlation of high zinc concentrations. The fitted model included parameters such as
partial sill, range, and nugget effect, which were estimated to capture the underlying spatial
variability of zinc concentrations exceeding the threshold value of 6.
Indicator Kriging: Indicator Kriging was employed to predict the probability that the logarithm
of zinc concentrations exceeds 6.21 at unsampled locations within the study area. By incorporating
the fitted variogram model, Indicator Kriging generated predictions of the likelihood of elevated
zinc concentrations, offering valuable insights into areas with potentially high pollution levels.
Model Evaluation: Cross-validation was performed to assess the performance of the Indicator
Kriging model. The Root Mean Square Error (RMSE) was computed as a measure of prediction
accuracy. The obtained RMSE value of approximately 0.327 indicates the average discrepancy
between observed and predicted probabilities of high zinc concentrations, providing an assessment
of the model’s reliability.
Results:
Variogram Analysis: The variogram analysis highlighted the spatial dependence structure of zinc
concentrations exceeding the threshold, indicating the presence of spatial autocorrelation.
Model Fitting: The spherical variogram model effectively characterized the spatial correlation of
high zinc concentrations, providing a basis for accurate predictions through Indicator Kriging.
Indicator Kriging: Predictions from Indicator Kriging depicted the spatial distribution of the
probability of elevated zinc concentrations, enabling the identification of areas at higher risk of
contamination.
Analysis:
Variogram Analysis: The variogram analysis was conducted to explore the spatial dependence
structure of zinc concentrations in the study area. The empirical variogram was developed based
on the indicator variable representing whether the logarithm of zinc concentrations exceeds 6. This
analysis provided insights into the spatial autocorrelation and variability of high zinc
concentrations across different distances.
Model Fitting: A spherical variogram model was fitted to the empirical variogram to characterize
the spatial correlation of high zinc concentrations. The fitted model included parameters such as
partial sill, range, and nugget effect, which were estimated to capture the underlying spatial
variability of zinc concentrations exceeding the threshold value of 6.
Indicator Kriging: Indicator Kriging was employed to predict the probability that the logarithm
of zinc concentrations exceeds 6.21 at unsampled locations within the study area. By incorporating
the fitted variogram model, Indicator Kriging generated predictions of the likelihood of elevated
zinc concentrations, offering valuable insights into areas with potentially high pollution levels.
Model Evaluation: Cross-validation was performed to assess the performance of the Indicator
Kriging model. The Root Mean Square Error (RMSE) was computed as a measure of prediction
accuracy. The obtained RMSE value of approximately 0.327 indicates the average discrepancy
between observed and predicted probabilities of high zinc concentrations, providing an assessment
of the model’s reliability.
Results:
Variogram Analysis: The variogram analysis highlighted the spatial dependence structure of zinc
concentrations exceeding the threshold, indicating the presence of spatial autocorrelation.
Model Fitting: The spherical variogram model effectively characterized the spatial correlation of
high zinc concentrations, providing a basis for accurate predictions through Indicator Kriging.
Indicator Kriging: Predictions from Indicator Kriging depicted the spatial distribution of the
probability of elevated zinc concentrations, enabling the identification of areas at higher risk of
contamination.
Model Evaluation: The computed RMSE value indicates moderate prediction accuracy,
suggesting that while the model provides valuable insights, caution should be exercised in
interpreting the predictions.
Conclusions:
In conclusion, Indicator Kriging proved to be a valuable technique for assessing the likelihood of
high zinc concentrations in the study area. The analysis, supported by variogram modeling and
cross-validation, provided meaningful insights into spatial patterns of zinc pollution, aiding in
environmental risk assessment and management strategies. Despite its moderate prediction
accuracy, Indicator Kriging serves as a useful tool for identifying areas requiring further
investigation and remediation efforts.
12) Co-Kriging:
Data:
The dataset comprises observed zinc concentrations at specific locations in the Meuse region,
along with the distance of each location from the river, serving as a covariate. These observed data
points form the basis for the Co-Kriging interpolation method.
Method:
Linear Model Fitting: Initially, a linear model was fitted to the logarithm of zinc concentrations
using the square root of the distance from the river as a covariate. This model serves as the basis
for Co-Kriging.
Residual Analysis: Residuals were computed to assess the goodness of fit of the linear model.
Visualizations of fitted values and residuals were plotted to understand the relationship between
the covariate and zinc concentrations.
Variogram Modeling: A variogram model was fitted to the semivariance of zinc concentrations
as a function of the square root of the distance from the river. This model characterizes the spatial
correlation between zinc concentrations and distance from the river.
Co-Kriging Interpolation: Co-Kriging interpolation was performed using the fitted variogram
model, incorporating information from both zinc concentrations and distance from the river to
estimate zinc concentrations at unsampled locations.
Results:
suggesting that while the model provides valuable insights, caution should be exercised in
interpreting the predictions.
Conclusions:
In conclusion, Indicator Kriging proved to be a valuable technique for assessing the likelihood of
high zinc concentrations in the study area. The analysis, supported by variogram modeling and
cross-validation, provided meaningful insights into spatial patterns of zinc pollution, aiding in
environmental risk assessment and management strategies. Despite its moderate prediction
accuracy, Indicator Kriging serves as a useful tool for identifying areas requiring further
investigation and remediation efforts.
12) Co-Kriging:
Data:
The dataset comprises observed zinc concentrations at specific locations in the Meuse region,
along with the distance of each location from the river, serving as a covariate. These observed data
points form the basis for the Co-Kriging interpolation method.
Method:
Linear Model Fitting: Initially, a linear model was fitted to the logarithm of zinc concentrations
using the square root of the distance from the river as a covariate. This model serves as the basis
for Co-Kriging.
Residual Analysis: Residuals were computed to assess the goodness of fit of the linear model.
Visualizations of fitted values and residuals were plotted to understand the relationship between
the covariate and zinc concentrations.
Variogram Modeling: A variogram model was fitted to the semivariance of zinc concentrations
as a function of the square root of the distance from the river. This model characterizes the spatial
correlation between zinc concentrations and distance from the river.
Co-Kriging Interpolation: Co-Kriging interpolation was performed using the fitted variogram
model, incorporating information from both zinc concentrations and distance from the river to
estimate zinc concentrations at unsampled locations.
Results:
Variogram Analysis: The variogram analysis revealed a spatial dependence structure between
zinc concentrations and distance from the river, with semivariance increasing with spatial
separation up to a certain range. The spherical variogram model provided a reasonable fit to the
experimental variogram, capturing the spatial correlation effectively.
Interpolated Surface Visualization: The Co-Kriging predictions were visualized on a map,
illustrating the spatial distribution of estimated zinc concentrations across the study area.
Model Evaluation: Cross-validation was conducted to evaluate the performance of the Co-
Kriging model. The Root Mean Square Error (RMSE) was calculated as a measure of prediction
accuracy, indicating the average difference between observed and predicted zinc concentrations.
Interpretations:
Residual Analysis: The residuals from the linear model fitting were examined to assess the
adequacy of the model in capturing the relationship between zinc concentrations and distance from
the river. The visualisations provided insights into any patterns or trends in the model residuals.
Interpolation Accuracy: The Co-Kriging interpolation method demonstrated promising results,
as evidenced by the visualisation of predicted zinc concentrations and the relatively low RMSE
value of 0.375. These findings suggest that Co-Kriging effectively incorporates spatial information
from the covariate (distance from the river) to improve the accuracy of zinc concentration
estimates.
Conclusions:
In conclusion, Co-Kriging emerges as a valuable approach for interpolating zinc concentrations in
the Meuse region, leveraging both spatial and covariate information to enhance prediction
accuracy. The methodological steps, including linear model fitting, variogram modeling, and Co-
Kriging interpolation, collectively contribute to a robust spatial analysis framework. The relatively
low RMSE value underscores the efficacy of Co-Kriging in capturing the spatial variability of zinc
concentrations and highlights its potential for informing environmental monitoring and
management initiatives. Further refinement and validation of the Co-Kriging parameters may offer
opportunities to enhance prediction accuracy and address spatial interpolation uncertainties.
13) Indicator-Co-Kriging
Analysis of Codes:
Quartile Computation: Quartiles of zinc concentration in the Meuse dataset were computed.
zinc concentrations and distance from the river, with semivariance increasing with spatial
separation up to a certain range. The spherical variogram model provided a reasonable fit to the
experimental variogram, capturing the spatial correlation effectively.
Interpolated Surface Visualization: The Co-Kriging predictions were visualized on a map,
illustrating the spatial distribution of estimated zinc concentrations across the study area.
Model Evaluation: Cross-validation was conducted to evaluate the performance of the Co-
Kriging model. The Root Mean Square Error (RMSE) was calculated as a measure of prediction
accuracy, indicating the average difference between observed and predicted zinc concentrations.
Interpretations:
Residual Analysis: The residuals from the linear model fitting were examined to assess the
adequacy of the model in capturing the relationship between zinc concentrations and distance from
the river. The visualisations provided insights into any patterns or trends in the model residuals.
Interpolation Accuracy: The Co-Kriging interpolation method demonstrated promising results,
as evidenced by the visualisation of predicted zinc concentrations and the relatively low RMSE
value of 0.375. These findings suggest that Co-Kriging effectively incorporates spatial information
from the covariate (distance from the river) to improve the accuracy of zinc concentration
estimates.
Conclusions:
In conclusion, Co-Kriging emerges as a valuable approach for interpolating zinc concentrations in
the Meuse region, leveraging both spatial and covariate information to enhance prediction
accuracy. The methodological steps, including linear model fitting, variogram modeling, and Co-
Kriging interpolation, collectively contribute to a robust spatial analysis framework. The relatively
low RMSE value underscores the efficacy of Co-Kriging in capturing the spatial variability of zinc
concentrations and highlights its potential for informing environmental monitoring and
management initiatives. Further refinement and validation of the Co-Kriging parameters may offer
opportunities to enhance prediction accuracy and address spatial interpolation uncertainties.
13) Indicator-Co-Kriging
Analysis of Codes:
Quartile Computation: Quartiles of zinc concentration in the Meuse dataset were computed.
Indicator Dataset Creation: Three indicator datasets were created based on quartiles of zinc
concentration.
Variogram Modeling: An initial semivariogram model was fitted to the indicator datasets, and
empirical variograms were estimated.
Model Fitting: A single semivariogram model was fit to all empirical variograms using a linear
model of coregionalization (LMC).
Co-Kriging Prediction: Indicator co-kriging was performed using the fitted variogram model to
predict zinc concentration at unmeasured locations.
RMSE Calculation: Root mean squared error (RMSE) was calculated as a measure of prediction
accuracy.
Results:
Quartile Computation: Quartiles of zinc concentration were computed: 25th percentile =
‘quartzinc[2]’, 50th percentile = ‘quartzinc[3]’, 75th percentile = ‘quartzinc[4]’.
Variogram Modeling: Initial semivariogram models were fitted to the indicator datasets, and
empirical variograms were visualized.
Model Fitting: A single semivariogram model was successfully fitted to all empirical variograms
using LMC.
Co-Kriging Prediction: Indicator co-kriging predictions and variances for each quartile of zinc
concentration were generated and visualized.
RMSE Calculation: RMSE for the indicator co-kriging model was computed.
Interpretations:
Variogram Analysis: The variogram analysis indicated spatial correlation between zinc
concentrations, with the semivariogram model capturing the spatial dependence adequately.
Model Fitting: The LMC model effectively captured the spatial correlation between different
indicators, allowing for accurate co-kriging predictions.
concentration.
Variogram Modeling: An initial semivariogram model was fitted to the indicator datasets, and
empirical variograms were estimated.
Model Fitting: A single semivariogram model was fit to all empirical variograms using a linear
model of coregionalization (LMC).
Co-Kriging Prediction: Indicator co-kriging was performed using the fitted variogram model to
predict zinc concentration at unmeasured locations.
RMSE Calculation: Root mean squared error (RMSE) was calculated as a measure of prediction
accuracy.
Results:
Quartile Computation: Quartiles of zinc concentration were computed: 25th percentile =
‘quartzinc[2]’, 50th percentile = ‘quartzinc[3]’, 75th percentile = ‘quartzinc[4]’.
Variogram Modeling: Initial semivariogram models were fitted to the indicator datasets, and
empirical variograms were visualized.
Model Fitting: A single semivariogram model was successfully fitted to all empirical variograms
using LMC.
Co-Kriging Prediction: Indicator co-kriging predictions and variances for each quartile of zinc
concentration were generated and visualized.
RMSE Calculation: RMSE for the indicator co-kriging model was computed.
Interpretations:
Variogram Analysis: The variogram analysis indicated spatial correlation between zinc
concentrations, with the semivariogram model capturing the spatial dependence adequately.
Model Fitting: The LMC model effectively captured the spatial correlation between different
indicators, allowing for accurate co-kriging predictions.
Prediction Visualization: Co-kriging predictions provided spatial estimates of zinc concentration,
while variance maps indicated prediction uncertainties.
RMSE Evaluation: The RMSE provided a quantitative measure of prediction accuracy, indicating
the average difference between predicted and observed zinc concentrations.
Conclusions:
Spatial Distribution: Zinc concentration in the Meuse dataset exhibits spatial dependence, which
can be effectively modeled using indicator co-kriging.
Prediction Accuracy: The indicator co-kriging model provides spatially explicit predictions of
zinc concentration, with quantified uncertainties.
Analysis of Kriging Models for Predicting Zinc Concentrations in the Meuse
Dataset
Introduction
In this analysis, four different kriging models were evaluated for predicting zinc concentrations in
the Meuse dataset. The models considered are Ordinary Kriging, Blocking Kriging, Indicator
Kriging, and Co-Kriging. The root mean square error (RMSE) is used as the metric to assess the
predictive performance of each model.
Methodology
Ordinary Kriging: This model assumes stationarity and estimates the zinc concentrations at
unsampled locations based on the spatial autocorrelation of the observed data.
Blocking Kriging: Similar to Ordinary Kriging, but divides the study area into blocks and
estimates values within each block separately.
Indicator Kriging: This model is useful for handling categorical data or data with extreme values.
It predicts the probability that a variable exceeds a certain threshold and provides more robust
predictions.
Co-Kriging: Incorporates auxiliary variables to improve prediction accuracy. It exploits the
correlation between the primary variable (zinc concentration) and auxiliary variables.
while variance maps indicated prediction uncertainties.
RMSE Evaluation: The RMSE provided a quantitative measure of prediction accuracy, indicating
the average difference between predicted and observed zinc concentrations.
Conclusions:
Spatial Distribution: Zinc concentration in the Meuse dataset exhibits spatial dependence, which
can be effectively modeled using indicator co-kriging.
Prediction Accuracy: The indicator co-kriging model provides spatially explicit predictions of
zinc concentration, with quantified uncertainties.
Analysis of Kriging Models for Predicting Zinc Concentrations in the Meuse
Dataset
Introduction
In this analysis, four different kriging models were evaluated for predicting zinc concentrations in
the Meuse dataset. The models considered are Ordinary Kriging, Blocking Kriging, Indicator
Kriging, and Co-Kriging. The root mean square error (RMSE) is used as the metric to assess the
predictive performance of each model.
Methodology
Ordinary Kriging: This model assumes stationarity and estimates the zinc concentrations at
unsampled locations based on the spatial autocorrelation of the observed data.
Blocking Kriging: Similar to Ordinary Kriging, but divides the study area into blocks and
estimates values within each block separately.
Indicator Kriging: This model is useful for handling categorical data or data with extreme values.
It predicts the probability that a variable exceeds a certain threshold and provides more robust
predictions.
Co-Kriging: Incorporates auxiliary variables to improve prediction accuracy. It exploits the
correlation between the primary variable (zinc concentration) and auxiliary variables.
Results
The table below summarises the RMSE values obtained for each kriging model:
Kriging Model RMSE
Ordinary Kriging 0.3973
Blocking Kriging 0.3918
Indicator Kriging 0.3268
Co-Kriging 0.3753
Analysis
Ordinary vs. Blocking Kriging: Both models utilize the spatial autocorrelation of the data but
differ in how they handle spatial variability. Blocking Kriging partitions the study area into blocks,
resulting in slightly better predictive performance compared to Ordinary Kriging.
Indicator Kriging: This model outperforms Ordinary and Blocking Kriging significantly. By
considering the probability of exceeding a threshold, Indicator Kriging provides more robust
predictions, especially for extreme values of zinc concentrations.
Co-Kriging: While Co-Kriging shows competitive performance, it does not outperform Indicator
Kriging. However, it offers the advantage of incorporating auxiliary variables, which may be
useful in certain applications where additional data is available.
Conclusion
Best Performing Model: Indicator Kriging emerges as the best-performing model with the lowest
RMSE value (0.3268). It effectively captures the spatial variability of zinc concentrations in the
Meuse dataset and provides robust predictions.
Recommendation: Based on the RMSE values and analysis, it can be recommend utilizing
Indicator Kriging for predicting zinc concentrations in the Meuse dataset. However, it’s essential
to consider other factors such as computational complexity and data requirements before finalizing
the choice of model.
Limitations and Future Work
This analysis focused solely on RMSE values as the metric for evaluating predictive performance.
Other metrics such as mean absolute error (MAE) or cross-validation techniques could provide
additional insights.
The table below summarises the RMSE values obtained for each kriging model:
Kriging Model RMSE
Ordinary Kriging 0.3973
Blocking Kriging 0.3918
Indicator Kriging 0.3268
Co-Kriging 0.3753
Analysis
Ordinary vs. Blocking Kriging: Both models utilize the spatial autocorrelation of the data but
differ in how they handle spatial variability. Blocking Kriging partitions the study area into blocks,
resulting in slightly better predictive performance compared to Ordinary Kriging.
Indicator Kriging: This model outperforms Ordinary and Blocking Kriging significantly. By
considering the probability of exceeding a threshold, Indicator Kriging provides more robust
predictions, especially for extreme values of zinc concentrations.
Co-Kriging: While Co-Kriging shows competitive performance, it does not outperform Indicator
Kriging. However, it offers the advantage of incorporating auxiliary variables, which may be
useful in certain applications where additional data is available.
Conclusion
Best Performing Model: Indicator Kriging emerges as the best-performing model with the lowest
RMSE value (0.3268). It effectively captures the spatial variability of zinc concentrations in the
Meuse dataset and provides robust predictions.
Recommendation: Based on the RMSE values and analysis, it can be recommend utilizing
Indicator Kriging for predicting zinc concentrations in the Meuse dataset. However, it’s essential
to consider other factors such as computational complexity and data requirements before finalizing
the choice of model.
Limitations and Future Work
This analysis focused solely on RMSE values as the metric for evaluating predictive performance.
Other metrics such as mean absolute error (MAE) or cross-validation techniques could provide
additional insights.
Future work could explore the impact of different variogram models and parameters on the
performance of kriging models. Additionally, incorporating spatial covariates or machine learning
techniques may further enhance predictive accuracy.
performance of kriging models. Additionally, incorporating spatial covariates or machine learning
techniques may further enhance predictive accuracy.
Tags
spatio-temporal data
geostatistics
kriging
indicator kriging
data science
geospatial analysis
environmental modeling
spatial prediction
variogram
ordinary kriging
co-kriging
universal kriging
predictive modeling
pollution mapping
rmse evaluation
advanced analytics
non-stationarity
no2 prediction
zinc concentration prediction
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