My File 2

Statistical Applications in Traffic Engineering

/

7.1 Importance of Statistics in Traffic Engineering

• Traffic engineering requires collecting and analyzing large amounts of data to perform traffic studies.

• Statistics is crucial for:

  • Determining required data quantities.

  • Making confident inferences from collected data.

• Full data measurement for average characteristics is often impractical.

  • Example: Average speed of vehicles on a road cannot be directly observed for all vehicles due to logistics.

• Engineers sample from the population to estimate characteristics and the necessary degree of certainty.

Questions Addressed Through Statistical Analysis

1. How many samples are required?

2. What is the confidence level in the estimate?

3. What statistical distribution best represents the observed data?

4. Did the traffic engineering design change population characteristics?

Statistical Techniques Overview

• Not a substitute for a course in statistics; references provided for further reading.

7.1 Overview of Probability Functions and Statistics

7.1.1 Discrete vs. Continuous Functions

• Discrete Functions: Composed of discrete variables; whole numbers only (e.g., number of children in a family).

• Continuous Functions: Composed of continuous variables; any value possible within a range (e.g., height of individuals).

• Common Discrete Distributions: Bernoulli, binomial, Poisson.

• Common Continuous Distributions: Normal, exponential, chi-square.

7.1.2 Randomness and Events

• Some events are predictable (e.g., gas pedal pressure and vehicle speed), while others are random (e.g., radioactive particle emission).

• Engineer's perspective: simplify modeling based on prediction feasibility.

7.1.3 Organizing Data

• Raw traffic data can be organized into classes for clarity.

• Common distributions: normal, exponential, chi-square, Bernoulli, binomial, and Poisson.

• Creating frequency distribution tables helps simplify data analysis.

7.1.4 Common Statistical Estimators

Measures of Central Tendency

• Mean: Average of observations.

• Median: Middle value when data is ordered.

• Mode: Most frequent value. Can be bimodal (two modes).

Measures of Dispersion

• Variance: Squared deviation from the mean.

• Standard Deviation (STD): Square root of variance. Indicates data spread.

• The coefficient of variation indicates the relative spread of outcomes versus the mean.

7.2 The Normal Distribution and Its Applications

Normal Distribution Characteristics

• It has a bell-shaped curve and is defined by mean and variance.

• Probability of outcomes lies under the curve.

• Total area under the curve equals 1.

• Commonly used for speed, travel time, and delays measurements.

7.2.1 Standard Normal Distribution

• Translates an arbitrary normal distribution to a standard normal distribution (mean of 0, variance of 1) using z-scores.

• Calculating probabilities involves comparison against standard normal distribution tables.

Important Characteristics of Normal Distribution

• Approx. 68.3% of values lie within ±1 standard deviation from the mean.

• Approx. 95% lie within ±1.96 standard deviations.

• Approx. 99.7% lie within ±3 standard deviations.

7.3 Confidence Bounds

Understanding Mean Estimates

• If 70 estimates of the mean are collected, they tend to be normally distributed.

• Standard error of the mean shrinks the more samples are taken, calculated as the population standard deviation divided by the square root of sample size.

7.4 Sample Size Computations

• Older equations for computing necessary sample sizes to achieve specified tolerances and confidence.

• Example calculations demonstrate how increased confidence can lead to significantly larger sample size requirements.

7.5 Addition of Random Variables

Summing Random Variables

• The summation leads to the expected value equal to sum of means.

• Through examples, demonstrates how random variables are calculated including standard deviation and variance.

7.6 The Binomial Distribution Related to the Bernoulli and Normal Distributions

Bernoulli Distribution

• Simplest discrete distribution, two outcomes.

• Probability function defined for fixed events and independent outcomes.

• Binomial distribution derived from repeated Bernoulli trials.

7.7 The Poisson Distribution

• Used for counting occurrences within a fixed interval.

• Defined by a single parameter specifying mean and variance.

• Relevant for modeling vehicle occurrences in traffic studies.

7.8 Hypothesis Testing

Framework and Errors

• Formulate null and alternative hypotheses. Assess effectiveness with probability of Type I and Type II errors defined.

• Errors quantified, with examples illustrating potential outcomes in real studies.

Summary and Closing Comments

• Statistics is integral to traffic studies and aids decision-making under uncertainty. A range of statistical techniques beyond presented can enhance study outcomes.

• Suggestions for further reading and study are provided, alongside problems to reinforce learning.