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spatial_extrapolation

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Spatial Extrapolation

Purpose

To fill in gaps in the City of Toronto's count program in order to provide a complete picture of volumes across the entire city.

Methodology

Several methods are tested for this purpose and are detailed below.

  1. Average of Nearest Neighbours (confidence code = 3)
    Nearest neighbours: 5 (or less) segments of the same road class that are maximum 300m (nearest point to point distance) away from the target segment

  2. Linear Regression (Based on Proximity only) (confidence code = 4)
    Take the volumes of the nearest 5 segments of the same road class as dependent variables (ordered by proximity).

  3. Linear Regression (Directional) (confidence code = 2)
    Take the volumes of the nearest 2 parallel segments and 2 perpendicular segments as dependent variables.

  4. Kriging
    Implemented using the Gaussian Process model from scikit-learn
    Input: (4-dimensional) from_x, from_y, to_x, to_y (coordinate information from the start and end of the segment)
    Output: volume
    Covariance matrix is constructed based on the coordinate information of the segments in order to find the spatial correlation of volumes.

Methodology Evaluation

Regression

Major Arterials

- Linear Regression (proximity only) Direction Linear Regression Average of Nearest Neighbours
Scatter plot major_arterials_proximity_regr major_arterials_directional_regr major_arterials_neighbour_avg
Root Mean Squared Error 4374 4232 4554
Coef. of Det. 0.480 0.542 0.492

major_arterials_proximity_regr_scores

Minor Arterials

- Linear Regression (proximity only) Direction Linear Regression Average of Nearest Neighbours
Scatter plot minor_arterials_proximity_regr minor_arterial_directional_regr minor_arterial_neighbour_avg
Root Mean Squared Error 2285 2143 2067
Coef. of Det. 0.345 0.461 0.341

minor_arterials_proximity_regr_scores

Collectors

- Linear Regression (proximity only) Direction Linear Regression Average of Nearest Neighbours
Scatter plot collectors_proximity_regr collectors_directional_regr collectors_neighbour_avg
Root Mean Squared Error 1349 1263 1233
Coef. of Det. 0.312 0.268 0.364

collectors_proximity_regr_scores

Locals

- Linear Regression (proximity only) Direction Linear Regression Average of Nearest Neighbours
Scatter Plot locals_proximity_regr locals_directional_regr locals_neighbour_avg
Root Mean Squared Error 736 732 718
Coef. of Det. 0.230 0.046 0.213

locals_proximity_regr_scores

Directional Regression Coefficients

Road Class Perpendicular Segs Coef Parallel Segs Coef
Major Arterials 0.0077 -0.0013 0.4404 0.4340
Minor Arterials -0.0132 0.0429 0.4129 0.2954
Collectors 0.0104 0.0249 0.3937 0.1681
Locals 0.0037 0.0129 0.1779 0.2441

The coefficients indicate a strong correlation between upstream and downstream segments and a week if existent relationship between perpendicular segments. As we move from major arterials to locals, the relationship gets messier.

Kriging

Road Class Semivariogram
Major Arterial major_arterials_semivariogram
Minor Arterial minor_arterials_semivariogram
Collector collectors_semivariogram

The relationship between distance and volume relationship is weak. The variance does not fit any model very well. A Gaussian Process Kriging model was fitted to each road class anyway and the results are inferior than regression. Therefore kriging is not used in actual implementation.

Implementation

Road Class Method
Major Arterials Directional Linear Regression
Minor Arterials Directional Linear Regression
Collectors Average of Neighbours
Locals Average of Neighbours

Note that expressways are not included. However, there are uncounted expressways that need to be included in the future.