Average

Beyond simple arithmetic mean, Openmadness provides sophisticated averaging techniques for statistical analysis, signal processing, and weighted data aggregation. Below is a comprehensive guide to these advanced methods.

What is the Average?

The average (or mean) is the sum of all elements divided by the number of elements. It represents the central value of a dataset.

Core Average Methods

  1. Basic Arithmetic Mean
const data = omArray([2, 4, 6, 8]);
data.mean(); // 5
  • Axis Support for ND Arrays:
const matrix = omArray([
  [1, 5],
  [3, 7],
]);
matrix.mean(0); // [2, 6] (column-wise)
matrix.mean(1); // [3, 5] (row-wise)
  1. Weighted Average
  • Apply weights to values:
const values = omArray([10, 20, 30]);
const weights = omArray([0.2, 0.3, 0.5]);

values.weightedAverage(weights); // (10*0.2 + 20*0.3 + 30*0.5) = 23
  1. Moving Average (Rolling Mean)
  • Smooth time-series data with window size:
const stockPrices = omArray([102, 104, 103, 106, 105]);
stockPrices.movingAverage(3); // [null, 103, 104.33, 104.67, 105]

Options:

  • padding: 'wrap'|'reflect'|'clamp'
  • align: 'left'|'center'|'right'

Specialized Averages

  1. Exponential Moving Average (EMA)
  • Prioritize recent data with decay factor (α):
const sensorData = omArray([22, 24, 23, 25]);
sensorData.ema(0.5); // [22, 23, 23, 24]

Formula: EMAₜ = α * current + (1-α) * EMAₜ₋₁

  1. Mode & Median
omArray([1, 3, 3, 6]).median(); // 3
omArray([1, 2, 2, 3]).mode(); // [2] (multi-modal returns array)
  1. Truncated Mean
  • Exclude outliers:
const examScores = omArray([40, 55, 80, 85, 90, 95]);
examScores.trimmedMean(0.1); // Discard lowest/highest 10% → [55,80,85,90] → 77.5

Multi-Dimensional Averaging

  1. Layer-wise Averages (3D Arrays)
const tensor = omArray([
  [
    [1, 2],
    [3, 4],
  ], // Layer 0
  [
    [5, 6],
    [7, 8],
  ], // Layer 1
]);

tensor.mean(0); // Per-voxel across layers: [[3,4],[5,6]]
  1. Pooling (Image/Data Reduction)
const image = omArray([
  [10, 20, 30, 40],
  [50, 60, 70, 80],
  [90, 100, 110, 120],
]);

// 2x2 max pooling
image.pool(2, 2, "max"); // [[60, 80], [110, 120]]
  • Pool Types: 'mean', 'min', 'max'
  1. Segmented Averages
  • Group data via labels:
const values = omArray([10, 20, 30, 40]);
const labels = omArray(["A", "B", "A", "B"]);

values.groupAverage(labels);
// { A: (10+30)/2=20, B: (20+40)/2=30 }

Advanced Use Cases

  1. Time-Series Smoothing
  • Combine EMA and moving averages:
const rawData = omArray([...]); // Noisy sensor readings
const smoothed = rawData
  .movingAverage(5, { padding: 'reflect' })
  .ema(0.3);
  1. Weighted Spatial Averaging
  • Blur image with kernel weights:
const kernel = omArray([
  [0.05, 0.1, 0.05],
  [0.1, 0.4, 0.1],
  [0.05, 0.1, 0.05],
]);

image.convolve(kernel); // Weighted average per pixel
  1. Statistical Rollups
  • Analyze 3D climate data:
const temperatureCube = omArray([...]); // [days×lat×lon]

// Daily global averages
const dailyMeans = temperatureCube.mean([1,2]);

// Monthly zonal averages (axis1=latitude)
const monthlyZones = temperatureCube
  .reshape(12, 30, 180, 360) // Group into months
  .mean([2,3]);

Performance & Best Practices

  1. Lazy Chaining:
largeTensor.slice(...).flatten().mean(); // Minimal intermediates
  1. Precomputed Weights: Cache weight arrays for repeated calculations.
  2. Typed Arrays: Use Float32Array for large datasets.

Edge Cases

  1. Empty Inputs: .mean() returns NaN for empty arrays.
  2. All-NaN Segments: Omitted from calculations with warning.
  3. Uniform Weights: weightedAverage() defaults to arithmetic mean if weights sum to 0.