Question

Face recognition systems pick faces out of crowds at airports to see if any matches occur with law enforcement databases. Performance of the systems can be affected by lighting, gender and age of the target, and age of the database. In Problems \(24-27,\) the tables give identification rates for faces under various conditions. Decide whether the rate of recognition is independent of the given factor. Table 17.24 compares face recognition for different ages.$$ \begin{array}{c|c|c} \hline \text { Age } & \text { Did recognize } & \text { Did not recognize } \\\ \hline 18-22 & 248 & 152 \\ \hline 38-42 & 222 & 78 \\ \hline \end{array} $$

Short answer

Answer: No, the rate of face recognition is not independent of age; age has a significant impact on face recognition.

Step-by-step solution

Set up the contingency table

We are given the following table: $$ \begin{array}{c|c|c} \hline \text { Age } & \text { Did recognize } & \text { Did not recognize } \\\ \hline 18-22 & 248 & 152 \\\ \hline 38-42 & 222 & 78 \\\ \hline \end{array} $$ We will add an additional row and column to this table to get the sum of each row and column.

Calculate the row and column totals

We need to determine the totals for each row and column in the table. $$ \begin{array}{c|c|c|c} \hline \text { Age } & \text { Did recognize } & \text { Did not recognize } & \text{Total} \\\ \hline 18-22 & 248 & 152 & 400 \\ \hline 38-42 & 222 & 78 & 300 \\ \hline & 470 & 230 & 700 \\ \hline \end{array} $$

Calculate the expected values

To calculate the expected values, we will use the following formula for each cell: $$Expected = \frac{Row Total \times Column Total}{Overall Total}$$ Calculate the expected values for each cell: 18-22, Did recognize: $$Expected = \frac{400 \times 470}{700} = 268.57$$ 18-22, Did not recognize: $$Expected = \frac{400 \times 230}{700} = 131.43$$ 38-42, Did recognize: $$Expected = \frac{300 \times 470}{700} = 201.43$$ 38-42, Did not recognize: $$Expected = \frac{300 \times 230}{700} = 98.57$$

Calculate the Chi-square value

Calculate the Chi-square value using the following formula: $$\chi^2 = \sum \frac{(Observed - Expected)^2}{Expected}$$ Chi-square value for the given table: $$\chi^2 = \frac{(248-268.57)^2}{268.57} + \frac{(152-131.43)^2}{131.43} + \frac{(222-201.43)^2}{201.43} + \frac{(78-98.57)^2}{98.57} = 19.03$$

Compare the Chi-square value to the critical value

We will use a significance level of 0.05 and 1 degree of freedom for this problem (since there are two age groups). The critical value for the Chi-square test with 1 df and significance level of 0.05 is 3.84. Our calculated Chi-square value is 19.03, which is greater than the critical value of 3.84.

Conclusion

Since our calculated Chi-square value (19.03) is greater than the critical value (3.84), we can reject the null hypothesis. This means that the rate of face recognition is not independent of age; age has a significant impact on face recognition.

Key concepts

Contingency Table Analysis

Contingency table analysis is a method used to examine the relationship between two categorical variables. It's an easy way to visualize and analyze how often different combinations of factors occur in a dataset. This is especially useful in situations where you need to determine if there is any association or dependency between these variables.

In the context of our face recognition exercise, the contingency table helps to analyze the relationship between age and the rate at which faces are recognized. By organizing data into a table format, it's possible to see at a glance which age groups performed better in face recognition tasks. From our exercise, you note the arrangement of age groups (18-22, 38-42) against recognition outcomes ("Did recognize," "Did not recognize").

Using a chi-square test for independence on this contingency table allows us to statistically assess whether there is a significant relationship between these age groups and recognition success. This involves calculating expected frequencies for each category combination, the difference between observed frequencies (actual data collected) and these expected frequencies, and ultimately deriving a chi-square statistic to test for independence.

Face Recognition

Face recognition technology is designed to identify or verify a person from a digital image or video frame. The algorithms rely on unique facial features from different angles and lighting conditions and compare these features with a database to find a match.

One of the main applications of face recognition systems is in security, such as in airports to identify persons of interest. However, factors like lighting, pose, makeup, and most importantly, age can significantly affect system performance.

In our exercise, the focus is on age as a factor. When a system tries to recognize faces under various conditions, the algorithms must compensate for age-related changes in appearance. This can be a challenge because most systems are trained on a defined set of face images, usually differing in age from the new faces encountered in real-world scenarios. This highlights the importance of updating and training systems with diverse datasets to improve accuracy.

Age Impact on Recognition

Age can significantly influence the performance of face recognition systems. As people age, their facial features can change, which creates a challenge for recognition algorithms that rely on consistency in these features.

The exercise presented a clear example of age impact on recognition systems, where data was organized in a contingency table showing different age groups' recognition rates. Analyzing this table with a chi-square test demonstrated that different age groups were recognized at different rates, indicating a dependence on age.

Reasons for this impact include:

• Variability in age-related facial changes such as wrinkles, facial structure, and skin tone.
• Potential bias if the recognition system was primarily trained on images from age groups not included in the test set.
• Different environmental and situational factors faced by different age groups.

By understanding these impacts, developers can improve face recognition systems to be more inclusive and accurate across all age groups.