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Logit model versus fuzzy logic model for representing gap-acceptance behavior. Rossi Riccardo Gastaldi Massimiliano Gecchele Gregorio Meneguzzer Claudio. Premise. Decisional variables : - time (or space ) gap size - delay - speed of incoming vehicle - ….

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Logit model versus

fuzzy logic model

for representing

gap-acceptance behavior

Rossi Riccardo

Gastaldi Massimiliano

Gecchele Gregorio

Meneguzzer Claudio


Premise
Premise

  • Decisionalvariables:

  • - time (or space) gap size

  • - delay

  • - speedofincomingvehicle

  • - …

  • Subjectiveestimates

  • (as a functionofdriver’s

  • gender, age, drivingexperience,drivingstyles,…)

  • Decision:

  • Gap Acceptance or

  • Rejection

  • Objectivefactors:

  • - Maneuvertype (left or right turn)

  • - Controltype (stop or yeldsign)

  • - …


Premise. Goal of this work

Gap-Acceptance behavior phenomenon

Gap-Acceptance behavior models

  • Better comprehension of the phenomenon...

    • Intersection capacity evaluation

    • Application in micro simulation software

    • .....




Experimental analysis
Experimental Analysis

Some basic definitions..

  • Gap : “time interval between two successive vehicles passing a section of the street measured from the rear bumper to the front bumper”

  • Lag: “residual part of the first gap that faces the minor-stream driver”

  • Critical gap: “minimal gap size acceptable to a population of drivers” (Gattis & Low, 1999; Daganzo, 1981; Mahmassani & Sheffi, 1981)

  • Follow up time: “mean headway between queued vehicles which move through the intersection during the same gap in the major street”

1

2

3

t0

S

1

4

2

3

t0 + lag

S

2

5

3

4

t0 + lag + vl2 + gap23

S

5

3

4

S

t0 + lag + vl2 + gap23 +

+ vl3 + …


Experimental analysis gap acceptance data collection
Experimental Analysis. Gap-acceptance data collection

  • Three- leg priority intersection analysed

2

9

2

Information available for each driver decision

Type of time interval (lag or gap)

Interval time size

Secondary street vehicle total waiting time

Category of secondary street vehicle

Category of primary street vehicle closing the interval

Driver decision (interval acceptance or rejection)

9


Experimental analysis gap acceptance data collection1
Experimental Analysis. Gap-acceptance data collection

  • Drivers’ decisions

lag-acceptance

gap-acceptance


Experimental analysis gap acceptance data collection2
Experimental Analysis. Gap-acceptance data collection

  • Characteristics of the dataset

  • Analysis of Data:

    • Stratified 10-fold cross validation

    • Objective of maximize the information available


Experimental analysis gap acceptance data collection3
Experimental Analysis. Gap-acceptance data collection

  • Stratified 10-fold cross validation:

  • Dataset randomly divided in 10 folds, with approximately the same proportion of acceptance and rejection.

  • Fuzzy and Logit models calibrated and validated 10 times:

    • 9 folds used as a calibration set

    • 1 fold at each time used as a validation set to calculate the performance of the models

  • Final Calibration of the models using the full dataset

  • Global performances of the models calculated averaging the 10 values of the performances.


Experimental analysis gap acceptance data collection4
Experimental Analysis. Gap-acceptance data collection

  • Stratified 10-fold cross validation (Example):

  • Full dataset:

  • 30 drivers’ decisions (dots)

    • 10 Acceptance (red)

    • 20 Rejection (blue)


Experimental analysis gap acceptance data collection5
Experimental Analysis. Gap-acceptance data collection

  • Stratified 10-fold cross validation (Example):

  • 1. Creation of 10 folds:

    • Same proportion of acceptances-rejections


Experimental analysis gap acceptance data collection6
Experimental Analysis. Gap-acceptance data collection

  • Stratified 10-fold cross validation (Example):

Validation

  • 2. Calibration-Validation

    • 9 folds (purple) used as a calibrations set

    • 1 fold (red) used as a validation test

    • Repetition of this step 10 times, using each fold as a validation set

Calibration


Experimental analysis gap acceptance data collection7
Experimental Analysis. Gap-acceptance data collection

  • Stratified 10-fold cross validation (Example):

  • 3. Calculation of the overall performances averaging the values obtained for each fold as a validation set


Experimental analysis gap acceptance data collection8
Experimental Analysis. Gap-acceptance data collection

  • Stratified 10-fold cross validation (Example):

  • 4. Final calibration of the models on the full dataset. They can be used to predict gap-acceptance behavior on further datasets



Model formulation and parameters estimation logit gap acceptance m odel
Model formulation and parameters estimation. Logit Gap-Acceptance Model

  • Logit gap-acceptance model estimation (GAL Model)

  • Explanatory Variables:

  • Logarithm of the Size of the Time Interval

  • Driver’s total delay on the minor approach

  • Type of Interval

    • Dummy variable: 1 for lag, 0 for gap

  • Response Variable:

  • Dummy Variable: 1 for acceptance, 0 for rejection of a certain interval

  • Note:

  • The introduction of natural logarithm of Time Interval Size in the utility function gives an acceptance probability that tends to zero for time interval size close to zero


Model formulation and parameters estimation logit gap acceptance m odel1
Model formulation and parameters estimation. Logit Gap-Acceptance Model

  • Logit gap-acceptance model estimation (GAL Model)

  • where:

  • = Logarithm of the Size of the Time Interval (seconds)

  • = Driver’s total delay on the minor approach (seconds)

  • =Type of Interval

    • 1 if interval type = lag

    • 0 if interval type = gap


Model formulation and parameters estimation logit gap acceptance m odel2
Model formulation and parameters estimation. Logit Gap-Acceptance Model

  • Logit gap-acceptance model estimation (GAL Model)


Model formulation and parameters estimation logit gap acceptance m odel3
Model formulation and parameters estimation. Logit Gap-Acceptance Model

  • Logit gap-acceptance model estimation (GAL Model)

Gap-acceptanceprobability

Lag-acceptanceprobability

Total Delay[s]

Total Delay[s]

Time IntervalSize [s]

Time IntervalSize [s]

  • Lag-acceptance surface

  • Gap-acceptance surface


Model formulation and parameters estimation logit gap acceptance m odel4
Model formulation and parameters estimation. Logit Gap-Acceptance Model

  • Logit gap-acceptance model estimation (GAL Model)

Total delay = 15 [s]

LAG-acceptance probability

GAP-acceptance probability


Model formulation and parameters estimation fuzzy gap acceptance m odel
Model formulation and parameters estimation. Fuzzy Gap-Acceptance Model

  • Fuzzy gap-acceptance model identification (GAF Model)

Gap-acceptance

behavior

Fuzzy systems theory:

subjective and qualitative

evaluation of input variables

Gap-acceptance

fuzzymodel


Model formulation and parameters estimation fuzzy gap acceptance m odel1
Model formulation and parameters estimation. Fuzzy Gap-Acceptance Model

  • Fuzzy gap-acceptance model identification(GAF Model)

Software used:

  • FisPro (Fuzzy Inference System Professional)http://www.inra.fr/bia/M/fispro

    • HierarchicalFuzzyPartitioning and Fast PrototypeAlgorithm

      References:

  • Guillaume, S. and Charnomordic, B. (2011) “Learning interpretable Fuzzy Inference Systems with FisPro”, Information Sciences, 181: 4409-4427.

  • Guillaume, S. (2011) “Designing fuzzy inference systems from data: an interpretability-oriented review”, IEEE Transactions on Fuzzy Systems, 9 (3): 426-443, June 2001.

  • Guillaume, S. and Charnomordic, B. (2003) “A new method for inducing a set of interpretable fuzzy partitions and fuzzy inference systems from data”, Studies in Fuzziness and Soft Computing, 128: 148-175, Springer, 2003.

  • Guillaume, S. and Charnomordic, B. (2004) “Generating an interpretable family of fuzzypartitions”, IEEE Transactions on Fuzzy Systems, 12 (3): 324-335, June 2004.


Model formulation and parameters estimation fuzzy gap acceptance m odel2
Model formulation and parameters estimation. Fuzzy Gap-Acceptance Model

  • Fuzzy gap-acceptance model identification(GAF Model)

IntervalSize

FUZZY

Total Delay

FUZZY

#15

Rules

Acceptance

FUZZY

Intervaltype

CRISP


Model formulation and parameters estimation fuzzy gap acceptance m odel3
Model formulation and parameters estimation. Fuzzy Gap-Acceptance Model

  • Fuzzy gap-acceptance model identification (GAF Model)

#15

Rules

  • 3 Non-compensatory rules:

  • Ifinterval size is “very small” then the interval is rejected

  • Ifinterval size is “large” then the interval is accepted

  • If interval size is “very large” then the interval is accepted

  • 12 Compensatory rules:

  • ...

  • Ifinterval size is “small” andinterval type is “lag” and total delay is “small” then the interval is rejected

  • ...

  • If interval size is “medium” and interval type is “gap” then the interval is accepted


Comparison between models roc curve analysis
Comparison between models. ROC Curve Analysis

  • Fuzzy gap-acceptance model identification (GAF Model)

  • Acceptance Index

    • Centroid value of output fuzzy set

    • 0.5 Threshold for gap/lag acceptance or rejection


Model formulation and parameters estimation fuzzy gap acceptance m odel4
Model formulation and parameters estimation. Fuzzy Gap-Acceptance Model

  • Fuzzy gap-acceptance model identification (GAF Model)

Gap-acceptanceprobability

Lag-acceptanceprobability

Total Delay[s]

Total Delay[s]

Time IntervalSize [s]

Time IntervalSize [s]

  • Lag-acceptance surface

  • Gap-acceptance surface


Model formulation and parameters estimation fuzzy gap acceptance m odel5
Model formulation and parameters estimation. Fuzzy Gap-Acceptance Model

  • Fuzzy gap-acceptance model identification(GAF Model)

Total delay = 15 sec

Total delay = 15 sec

  • Lag-acceptance index

  • Gap-acceptance index



Comparison between models roc curve analysis1
Comparison between models. ROC Curve Analysis

  • Introduction to ROC Curve Analysis (Fawcett, 2006)

True Positive TP: the model predicts an Acceptance and the driver accepted the gap/lag

False Positive FP: the model predicts an Acceptance but the driver rejected the gap/lag

True Negative TN: the model predicts a Rejection and driver rejected the gap/lag

False Negative FN: the model predicts a Rejection but the driver accepted the gap/lag


Comparison between models roc curve analysis2
Comparison between models. ROC Curve Analysis

  • Introduction to ROC Curve Analysis (Fawcett, 2006)

  • TPR (True Positive Rate) or Sensitivity = number of TP / number of P

  • TNR (True Negative Rate) or Specificity = number of TN / number of N

  • Precision= number of TP/number of MP

  • F-measure= 2/(1/Precision + 1/TPR)

  • Percent right = [(number of TP + number of TN)/(number of P + N)]*100

  • Youden Index = TPR+TNR-1


Comparison between models roc curve analysis3
Comparison between models. ROC Curve Analysis

  • Introduction to ROC Curve Analysis (Fawcett, 2006)

AUC (Area Under Curve) = measured in the plane (1-TNR,TPR)

  • Analysis of different cut-off values

  • Its value increases with the accuracy of model predictions:

    • 1,0 = Perfects Forecasts

    • 0,5 = Random Forecasts

  • Equivalence with:

    • Gini coefficient=2*AUC-1

    • Mann–Whitney–Wilcoxon two-independent sample test statistic

  • Quick comparison between models


Comparison between models roc curve analysis4
Comparison between models. ROC Curve Analysis

  • Introduction to ROC Curve Analysis (Fawcett, 2006)

High values of all metrics

Good Model

TPR (True Positive Rate) = number of TP / number of P

TNR (True Negative Rate) = number of TN / number of N

Precision = number of TP/number of MP

F-measure = 2/(1/Precision + 1/TPR)

Percent right = [(number of TP + number of TN)/(number of P + N)]*100

Youden Index = TPR+TNR-1

AUC


Comparison between models roc curve analysis5
Comparison between models. ROC Curve Analysis

  • ROC Curve Analysis of the models

  • Average ROC curves

    built from cross-validated data

    (bars represent 95% confidence region):

    • GALmodel = solid line

    • GAFmodel = dashed line

  • Comparison of AUC values:

    • GALmodel = 0.978

    • GAFmodel = 0.958

  • GALslightly outperforms GAF

  • The two lines are indistinguishable

    in correspondence of 0.5 threshold (which maximize performances)


Comparison between models roc curve analysis6
Comparison between models. ROC Curve Analysis

  • ROC Curve Analysis of the models

  • Comparison Between Models: Performance Metrics

    • Cross-validated data (Average of 10 validation sets)

    • Cut-off value of 0.5

  • Paired t-tests conducted on the performances of GAL and GAF

    • The differences between the performances of GALand GAF are not significant at the 95% confidence level.

    • The two models are equivalent in terms of accuracy




Thankyou

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