Statistics

Statistics provides tools to analyze data and evaluate machine learning (ML) models. This section introduces descriptive statistics, hypothesis testing, and confidence intervals, with a Rust lab using the statrs crate.

Descriptive Statistics

Descriptive statistics summarize data through measures like:

• Mean: Average value, $\mu =\frac{1}{n}\sum _{i=1}^n{x}_i$.
• Variance: Spread of data, ${\sigma }^2=\frac{1}{n}\sum _{i=1}^n(x_i-\mu {)}^2$.
• Standard Deviation: $\sigma =\sqrt{{\sigma }^2}$.

In ML, these describe datasets and model performance.

Hypothesis Testing

Hypothesis testing assesses if observed data supports a hypothesis. A t-test compares means of two groups to determine if they differ significantly.

Example: Test if a model’s predictions have a different mean error than a baseline. The t-statistic is:

$$t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$$

where ${\overline{x}}_i$, $s_i^2$, $n_i$ are the mean, variance, and size of group $i$.

Confidence Intervals

A confidence interval estimates a parameter’s range (e.g., mean) with a confidence level (e.g., 95%). For a mean:

$$\overline{x}\pm z\frac{s}{\sqrt{n}}$$

where $z$ is the z-score (e.g., 1.96 for 95%), $s$ is the standard deviation, and $n$ is the sample size.

Lab: T-Test with statrs

You’ll perform a t-test on two synthetic datasets using statrs to compare their means.

1. Edit src/main.rs in your rust_ml_tutorial project:

   use statrs::statistics::{Data, Distribution};
   use statrs::distribution::StudentsT;
   
   fn main() {
       // Synthetic datasets
       let data1 = vec![2.1, 2.3, 2.0, 2.2, 2.4]; // Group 1
       let data2 = vec![2.5, 2.7, 2.4, 2.6, 2.8]; // Group 2
   
       // Compute means
       let mean1 = Data::new(data1.clone()).mean().unwrap();
       let mean2 = Data::new(data2.clone()).mean().unwrap();
       println!("Mean1: {}, Mean2: {}", mean1, mean2);
   
       // Perform t-test (assuming equal variances)
       let t_stat = t_test(&data1, &data2);
       println!("T-statistic: {}", t_stat);
   
       // Check p-value (two-tailed, df=8)
       let t_dist = StudentsT::new(0.0, 1.0, 8.0).unwrap();
       let p_value = 2.0 * (1.0 - t_dist.cdf(t_stat.abs()));
       println!("P-value: {}", p_value);
   }
   
   fn t_test(data1: &[f64], data2: &[f64]) -> f64 {
       let n1 = data1.len() as f64;
       let n2 = data2.len() as f64;
       let mean1 = data1.iter().sum::<f64>() / n1;
       let mean2 = data2.iter().sum::<f64>() / n2;
   
       let var1 = data1.iter().map(|x| (x - mean1).powi(2)).sum::<f64>() / (n1 - 1.0);
       let var2 = data2.iter().map(|x| (x - mean2).powi(2)).sum::<f64>() / (n2 - 1.0);
   
       let se = ((var1 / n1) + (var2 / n2)).sqrt();
       (mean1 - mean2) / se
   }

2. Ensure Dependencies:

   • Verify Cargo.toml includes:

     [dependencies]
     statrs = "0.16.0"

   • Run cargo build.

3. Run the Program:

   cargo run

   Expected Output (approximate):

   Mean1: 2.2, Mean2: 2.6
   T-statistic: -3.46
   P-value: 0.008

   A low p-value (< 0.05) suggests the means differ significantly.

Understanding the Results

• Datasets: Two groups with slightly different means (~2.2 vs. ~2.6).
• T-Test: The t-statistic and p-value indicate a significant difference, relevant for ML model evaluation.
• ML Relevance: Hypothesis testing validates model performance (e.g., comparing error rates).

This lab prepares you for statistical ML methods.

Learning from Official Resources

Deepen Rust skills with:

• The Rust Programming Language (The Book): Free at doc.rust-lang.org/book.
• Programming Rust: By Blandy, Orendorff, and Tindall.

Next Steps

Proceed to Core Machine Learning for regression techniques, or revisit Probability.

Further Reading

• An Introduction to Statistical Learning by James et al. (Chapter 5)
• Andrew Ng’s Machine Learning Specialization (Course 1, Week 3)
• statrs Documentation: docs.rs/statrs