dorsal/arxiv
View SchemaDetrending Moving Average variance: a derivation of the scaling law
| Authors | Sergio Arianos, Anna Carbone |
|---|---|
| Categories | |
| ArXiv ID | physics/0608313 |
| URL | https://arxiv.org/abs/physics/0608313 |
| DOI | 10.1016/j.physa.2007.02.074 |
| Journal | Physica A: Statistical Mechanics and its Applications Volume 382, (2007) Pages 9-15 |
Abstract
The Hurst exponent $H$ of long range correlated series can be estimated by means of the Detrending Moving Average (DMA) method. A computational tool defined within the algorithm is the generalized variance $ \sigma_{DMA}^2={1}/{(N-n)}\sum_i [y(i)-\widetilde{y}_n(i)]^2\:$, with $\widetilde{y}_n(i)= {1}/{n}\sum_{k}y(i-k)$ the moving average, $n$ the moving average window and $N$ the dimension of the stochastic series $y(i)$. This ability relies on the property of $\sigma_{DMA}^2$ to scale as $n^{2H}$. Here, we analytically show that $\sigma_{DMA}^2$ is equivalent to $C_H n^{2H}$ for $n\gg 1$ and provide an explicit expression for $C_H$.
{
"annotation_id": "880f7205-a877-4456-ab5b-7543ee9be122",
"date_created": "2026-03-02T18:01:11.157000Z",
"date_modified": "2026-03-02T18:01:11.157000Z",
"file_hash": "22a34110eb3efec1d2288fd91fded7b50eb59efb7654f2ae2bb14d200b606eea",
"private": false,
"record": {
"abstract": "The Hurst exponent $H$ of long range correlated series can be estimated by\nmeans of the Detrending Moving Average (DMA) method. A computational tool\ndefined within the algorithm is the generalized variance $\n\\sigma_{DMA}^2={1}/{(N-n)}\\sum_i [y(i)-\\widetilde{y}_n(i)]^2\\:$, with\n$\\widetilde{y}_n(i)= {1}/{n}\\sum_{k}y(i-k)$ the moving average, $n$ the moving\naverage window and $N$ the dimension of the stochastic series $y(i)$. This\nability relies on the property of $\\sigma_{DMA}^2$ to scale as $n^{2H}$. Here,\nwe analytically show that $\\sigma_{DMA}^2$ is equivalent to $C_H n^{2H}$ for\n$n\\gg 1$ and provide an explicit expression for $C_H$.",
"arxiv_id": "physics/0608313",
"authors": [
"Sergio Arianos",
"Anna Carbone"
],
"categories": [
"physics.data-an",
"q-fin.ST"
],
"doi": "10.1016/j.physa.2007.02.074",
"journal_ref": "Physica A: Statistical Mechanics and its Applications Volume 382,\n (2007) Pages 9-15",
"title": "Detrending Moving Average variance: a derivation of the scaling law",
"url": "https://arxiv.org/abs/physics/0608313"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "72b40ff1-5070-4db8-932f-68e5ade43ec6",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}