Chapter 4.6 The lycanthrope’s log

 

The lycanthrope’s log

  

 

🔍: Here the auther continues the discussion of objects in JavaScript  with an application to a problem of correlating events with transformations into a squirrel. 

👩‍🎓: Breaking Down the JavaScript Code and the Concept of Correlation:

1. Journal Array and addEntry Function:

   • let journal = []; initializes an empty array that will store the journal entries.
   • addEntry(events, squirrel) is a function that creates a new entry in the journal. Each entry is an object with properties events and squirrel.
   • The shorthand {events, squirrel} in the function is ES6 property shorthand for {events: events, squirrel: squirrel}, where the property name is the same as the variable name.

2. Data Recording:

   • Jacques logs his daily events and whether he turned into a squirrel using the addEntry function.

3. Correlation and the Phi Coefficient (ϕ):

   • Correlation measures the relationship between two variables. It can range from -1 (perfect inverse correlation), through 0 (no correlation), to 1 (perfect correlation).
   • The Phi coefficient is a specific measure used for binary variables (true/false, yes/no) to determine the degree of association between them.
   • To compute ϕ, a frequency table is used to count the occurrences of the variables' combinations.

📚: Understanding the Frequency Table and Phi Coefficient Calculation:

1. Frequency Table:

   • The table counts occurrences of four combinations: when there is neither a squirrel transformation nor pizza (n00), when there's pizza but no squirrel (n01), when there's a squirrel but no pizza (n10), and when both a squirrel transformation and pizza occur (n11).

    

   Phi Coefficient Formula:

   • The formula for ϕ given in the book is: $\varphi =\frac{n_{11}{n}_{00}-n_{10}{n}_{01}}{\sqrt{n_{1\cdot }{n}_{0\cdot }{n}_{\cdot 1}{n}_{\cdot 0}}}$
   • ​
   • This formula calculates the correlation between two binary variables.

2. Application to the Pizza Table:

   • Using the frequency table, we calculate the dividend as $1\times 76-4\times 9$, which equals 40.
   • The divisor is the square root of $5\times 85\times 10\times 80$, which equals the square root of 340000.
   • The resulting ϕ is approximately 0.069, indicating a very weak correlation between eating pizza and transforming into a squirrel.

🎨: Interpreting the Results:

• The value of ϕ, in this case, suggests that eating pizza has little to no effect on Jacques' transformations. This type of analysis is essential in statistics to understand the strength of the relationship between variables.

🔄: Real-World Application:

• While this example is whimsical, the approach is used in real-world scenarios. Data scientists use similar techniques to find patterns and correlations in data, although with more complex variables and often using software that can handle large datasets.

🔬: In Conclusion:

• The code and concepts presented illustrate how programming can intersect with statistics to analyze data. The Phi coefficient is a practical application of this intersection, allowing us to quantify the relationship between two binary variables in a dataset.

I need comprehensive explanation of Phi Coefficient Formula used in this example

🔍: Let's delve deeper into the Phi Coefficient (ϕ) formula used in the example from "Eloquent JavaScript" and provide a comprehensive explanation.

👩‍🎓: The Phi Coefficient Formula: The Phi Coefficient (ϕ) is a measure of correlation specifically designed for binary variables. In the context of the weresquirrel example, these binary variables are whether Jacques turned into a squirrel, and whether he ate pizza.

The general formula for ϕ is:

$$\varphi =\frac{n_{11}{n}_{00}-n_{10}{n}_{01}}{\sqrt{n_{1\cdot }{n}_{0\cdot }{n}_{\cdot 1}{n}_{\cdot 0}}}$$

where:

• $n_{\mathrm{11 }}$ is the frequency of both events occurring together (squirrel and pizza).
• $n_{\mathrm{00 \; \; }}$ is the frequency of neither event occurring.
• $n_{\mathrm{10 }}$ is the frequency of the first event occurring without the second (squirrel, no pizza).
• $n_{\mathrm{01 \; }}$ is the frequency of the second event occurring without the first (pizza, no squirrel).
• $n_{1\cdot }$ is the total number of times the first event occurred (squirrel).
• $n_{0\cdot }$is the total number of times the first event did not occur.
• $n_{\cdot \mathrm{1 }}$ is the total number of times the second event occurred (pizza).
• $n_{\cdot 0}$ is the total number of times the second event did not occur.

Breaking Down the Formula:

1. Numerator: The numerator $n_{11}{n}_{00}-n_{10}{n}_{\mathrm{01\; }}$ calculates the product of the frequency of both variables occurring together and the frequency of neither occurring, and then subtracts the product of the frequencies of each occurring alone. This part of the formula assesses the degree of association between the two events.

2. Denominator: The denominator is the square root of the product of the totals of each event occurring and not occurring. This part normalizes the numerator, accounting for the different frequencies of each event, and ensures that the Phi Coefficient stays within the -1 to 1 range.

Calculating ϕ for the Weresquirrel Example:

Let's assign the frequencies based on the table you've provided:

• $n_{11}\mathrm{}$ (Squirrel and pizza)
• $n_{00}$ (No squirrel, no pizza)
• $n_{10}\mathrm{}$ (Squirrel, no pizza)
• $n_{01}\mathrm{}$ (No squirrel, pizza)
• $n_{1\cdot }=n_{11}+n_{10}=\mathrm{1\; +\; 4 }$ (Total squirrel)
• $n_{0\cdot }=n_{00}+n_{01}=\mathrm{76+9}$ (Total no squirrel)
• $n_{\cdot 1}=n_{11}+n_{01}=\mathrm{1\; +9}$ (Total pizza)
• $n_{\cdot 0}=n_{00}+n_{10}=\mathrm{76\; +4}$ (Total no pizza)

Plugging these into the formula gives us:

$$\varphi =\frac{1\times 76-4\times 9}{\sqrt{5\times 85\times 10\times 80}}$$

​

Calculating the numerator:

$$1\times 76-4\times 9=76-36=40$$

Calculating the denominator:

$$\sqrt{5\times 85\times 10\times 80}=\sqrt{340000}$$

​

Finally, we find ϕ by dividing the numerator by the denominator:

$$\varphi \approx \frac{40}{\sqrt{340000}}\approx 0.069$$

ϕ $$\approx 0.069$$

🔬: Interpreting ϕ:

• A ϕ of 0.069 indicates a very weak positive correlation between eating pizza and turning into a squirrel. In practical terms, it suggests that the occurrence of one does not reliably predict the occurrence of the other.

💡: Conclusion:

• The Phi Coefficient is a concise way to represent the relationship between two binary variables. While the formula might look intimidating, it's essentially comparing the occurrence of events to what would be expected if there were no relationship at all. The closer the value of ϕ is to 0, the weaker the relationship. Values near -1 or 1 indicate strong relationships, with -1 being a perfect inverse relationship and 1 being a perfect direct relationship.

 

Link to the chapter of the Book Eloquent Javascript

https://eloquentjavascript.net/04_data.html