# Trend, Seasonality, Cycles, and Remainder

`D00-F10-A06` · Financial Mathematics, Statistics, and Data Foundations → Financial Time-Series Foundations · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/financial-time-series-foundations/trend-seasonality-cycles-and-remainder/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

```ts
import { trendSeasonalityCyclesAndRemainder } from "fintech-algorithms/foundations/financial-time-series-foundations/trend-seasonality-cycles-and-remainder";
```

## Signature

```ts
trendSeasonalityCyclesAndRemainder(input)
```

Separates a series into a rolling-mean trend, a seasonal difference taken `period` steps back, and the remainder left once the trend is removed.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | Reads `values`, a non-empty list of finite numbers, `timestamps` of the same length, `window` for the trend average, and `period` for the seasonal difference. |

## Returns

`D00Output`

`trend` is the rolling mean, `null` at positions with fewer than `window` observations behind them. `seasonalDifference` subtracts the value `period` positions earlier and is `null` for the first `period` entries. `remainder` is the value minus the trend, `null` wherever the trend is. `period` is echoed back.

## Errors

- When `values` is absent, empty, or holds a non-finite number — throws RangeError
- When `timestamps` and `values` have different lengths — throws RangeError
- When the series holds fewer than two observations — throws RangeError
- When `window` is not an integer between one and the observation count — throws RangeError
- When `period` is not an integer between one and the observation count minus one — throws RangeError

## Complexity

Time `O(n^2)`, space `O(n)`.

## Worked example

This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

### Input

`input`:

```json
{
  "timestamps": [
    "2025-01-01T00:00:00Z",
    "2025-01-02T00:00:00Z",
    "2025-01-03T00:00:00Z",
    "2025-01-04T00:00:00Z",
    "2025-01-05T00:00:00Z",
    "2025-01-06T00:00:00Z"
  ],
  "values": [100, 102, 101, 104, 106, 105],
  "lag": 1,
  "window": 3,
  "resampleSize": 2,
  "period": 3,
  "stationarityTolerance": 3,
  "alpha": 0.4,
  "splitIndex": 4
}
```

### Call

```ts
trendSeasonalityCyclesAndRemainder(input)
```

### Returns

object with 2 fields: trend, seasonalDifference

```json
{
  "trend": [null, null, 101, 102.33333333333333, 103.66666666666667, 105],
  "seasonalDifference": [null, null, null, 4, 4, 4]
}
```

## Verification and provenance

Tier: **verified** (via input-expected).

The worked example below is the figure published in this algorithm's article, replayed and asserted by the test suite on every build. The arithmetic cannot drift without the build failing.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.0.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/foundations/financial-time-series-foundations/trend-seasonality-cycles-and-remainder/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/financial-time-series-foundations/trend-seasonality-cycles-and-remainder/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt