提升模拟程序循环速度（apply方法无改善效果）

Hey there, let's figure out how to speed up that slow simulation component of yours. You're checking how sample mean estimation gets more precise as n goes from 1 to 500 (sampling from a fixed normal distribution with set μ and σ)—I've got some practical, battle-tested tweaks to cut down runtime big time:

1. Ditch Loops for Vectorized Operations #

If you're using Python (with NumPy) or R, stop iterating over individual samples one by one. Vectorized operations are optimized at the low level (C under the hood for NumPy) and will slash your runtime drastically.

For example, in Python with NumPy:

import numpy as np

# Fixed distribution parameters
true_mu = 10
true_sigma = 2
max_n = 500
num_simulations = 10000  # Adjust based on your needs

# Generate ALL samples in one go (shape: [max_n, num_simulations])
rng = np.random.default_rng()  # Faster, modern RNG
all_samples = rng.normal(true_mu, true_sigma, size=(max_n, num_simulations))

# Calculate cumulative means for every n in one step
cumulative_sums = np.cumsum(all_samples, axis=0)
sample_means = cumulative_sums / np.arange(1, max_n + 1)[:, np.newaxis]

This way, you don't re-sample for each n—the largest dataset (n=500) already includes all samples needed for smaller n values.

2. Increment Precision Metrics (Don't Recalculate From Scratch) #

Instead of recomputing precision metrics like mean squared error (MSE) or standard error for each n from scratch, update them incrementally. For example, tracking MSE relative to the true mean:

• For n=1: MSE = (sample_mean_1 - true_mu)²
• For n>1: MSE = ((n-1)*previous_mse + (new_sample_mean - true_mu)²) / n

This avoids reprocessing all past samples every time you increase n, saving tons of unnecessary computation.

3. Use a Faster Random Number Generator (RNG) #

Default RNGs are general-purpose, but not always the fastest for large-scale sampling. Swap to a optimized generator:

• In Python: Use np.random.default_rng() (uses PCG64, faster and more statistically robust than the legacy np.random module)
• In R: Try the rfast package's optimized normal sampling functions, or use base::rnorm with vectorized calls instead of loops

4. Parallelize Simulation Runs #

If you need thousands of independent simulations, split the work across your CPU cores. In Python, joblib makes this dead simple:

from joblib import Parallel, delayed

def run_single_simulation(max_n, mu, sigma):
    samples = rng.normal(mu, sigma, size=max_n)
    return np.cumsum(samples) / np.arange(1, max_n + 1)

# Run all simulations in parallel (uses all available cores)
simulation_results = Parallel(n_jobs=-1)(
    delayed(run_single_simulation)(max_n, true_mu, true_sigma) 
    for _ in range(num_simulations)
)

# Convert results to a numpy array for easy analysis
results_array = np.array(simulation_results).T

5. Cut Redundant Work #

Double-check if you're running more simulations than you actually need. For example, if you're seeing stable precision trends after 5,000 runs, you don't need 100,000. You can even add a early-stopping check: stop simulating once the precision metric's change between iterations falls below a small threshold (like 0.001).

内容的提问来源于stack exchange，提问作者JimGrange