# Zero-Lag Exponential Moving Average (ZLEMA)

`D07-F01-A15` · Technical Indicators → Trend Smoothing · archetype `series-transform` · difficulty 2/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/technical-indicators/trend-smoothing/zero-lag-exponential-moving-average-zlema/
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 { zeroLagExponentialMovingAverageZlema } from "fintech-algorithms/technical-indicators/trend-smoothing/zero-lag-exponential-moving-average-zlema";
```

## Signature

```ts
zeroLagExponentialMovingAverageZlema(input)
```

Removes most of the EMA's lag by first de-lagging close into `2 * close[t] - close[t - lag]`, where `lag` is `floor((period - 1) / 2)`, and then running an ordinary EMA of `period` over that adjusted series.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `TopicInput` | yes | `bars` is a non-empty array of OHLCV records with strictly increasing `timestamp`, finite `open`, `high`, `low`, `close` and non-negative `volume`, all sharing one adjustment basis. From `parameters` this topic reads only `period`, an integer of at least 2, default 14, which sets both the de-lag distance and the EMA smoothing constant. |

## Returns

`TopicResult`

`series` and `latest` carry a single key, `value`. The warm-up stacks the de-lag offset on top of the EMA seed, so with the default period `ready_at` is 19, not 13.

## Warm-up

The first `floor((period - 1) / 2) + period - 1 bars (19 at the default period of 14)` positions are `null`. The de-lagged series is null for its first `floor((period - 1) / 2)` bars, and the EMA then needs a further `period` non-null values to seed from their mean. `ready_at` is the index of the first non-null `value`.

## Errors

- When `parameters.period` is not an integer of at least 2 — throws Error
- When `bars` is empty, or a bar has a non-finite price, a negative volume, or a timestamp that does not increase — throws Error

## Complexity

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

## Worked example

Captured by running this function on the input its own test provides. Real output of real code — but not asserted against a published figure.

### Input

`input`:

```json
{
  "bars": [
    {
      "timestamp": "2024-01-02",
      "basis": "synthetic-unadjusted",
      "open": 100,
      "high": 101.45,
      "low": 98.695,
      "close": 100,
      "volume": 750000,
      "benchmark": 200
    },
    {
      "timestamp": "2024-01-03",
      "basis": "synthetic-unadjusted",
      "open": 101.49111452,
      "high": 103.38381693,
      "low": 100.05480022,
      "close": 101.78791214,
      "volume": 795117,
      "benchmark": 200.56326135
    },
    {
      "timestamp": "2024-01-04",
      "basis": "synthetic-unadjusted",
      "open": 102.45519048,
      "high": 104.6701838,
      "low": 100.91147007,
      "close": 102.9549389,
      "volume": 840234,
      "benchmark": 201.11020913
    }
  ],
  "parameters": {}
}
```

### Call

```ts
zeroLagExponentialMovingAverageZlema(input)
```

### Returns

object with 9 fields: topic_id, title, state, ready, ready_at, series, latest, parameters, …

```json
{
  "topic_id": "D07-F01-A15",
  "title": "Zero-Lag Exponential Moving Average (ZLEMA)",
  "state": "calculated",
  "ready": true,
  "ready_at": 19,
  "series": {
    "value": [null, null, null, null, null, null]
  },
  "latest": {
    "value": 101.8372551421456
  },
  "parameters": {},
  "diagnostics": {
    "causal": true,
    "input_count": 96
  }
}
```

## Verification and provenance

Tier: **verified** (via E).

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/technical-indicators/trend-smoothing/zero-lag-exponential-moving-average-zlema/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/technical-indicators/trend-smoothing/zero-lag-exponential-moving-average-zlema/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/technical-indicators/llms.txt