> ## Documentation Index
> Fetch the complete documentation index at: https://nixtlaverse.nixtla.io/llms.txt
> Use this file to discover all available pages before exploring further.

# INAR (integer counts)

INAR — INteger-valued AutoRegression — generates *count* series:
non-negative integers with autocorrelation, for demand, arrivals, and
case counts where a Gaussian model makes no sense. It carries dependence
forward through binomial thinning rather than a linear lag, so the
output stays integer-valued.

> **The model**
>
> $X_t = \alpha_1 \circ X_{t-1} + \dots + \alpha_p \circ X_{t-p} + \varepsilon_t, \qquad \alpha \circ X = \sum_{i=1}^{X} \mathrm{Bernoulli}(\alpha)$
>
> Each step keeps a binomially-thinned fraction of the previous count
> and adds a fresh integer innovation. The thinning probability sets the
> persistence; the innovation distribution (`poisson` or
> `negative_binomial`) sets the marginal spread — negative binomial for
> overdispersion.

```python theme={null}
import polars as pl
import matplotlib.pyplot as plt

from synforecast.generators import INARGenerator
```

## 1. Innovation distribution

The innovation distribution sets the marginal spread. With the thinning
parameter and innovation mean held fixed, negative binomial innovations
produce overdispersion — variance above the mean — while Poisson
innovations keep the two equal.

```python theme={null}
poisson_df = INARGenerator(
    engine="polars",
    min_length=200,
    max_length=200,
    freq="D",
    p=1,
    alpha=[0.5],
    innovation_type="poisson",
    innovation_mean=3.0,
    seed=42,
).generate(n_series=1)

negative_binomial_df = INARGenerator(
    engine="polars",
    min_length=200,
    max_length=200,
    freq="D",
    p=1,
    alpha=[0.5],
    innovation_type="negative_binomial",
    innovation_mean=3.0,
    innovation_dispersion=2.0,
    seed=42,
).generate(n_series=1)

fig, axes = plt.subplots(2, 1, figsize=(12, 6), sharex=True)
panels = [("poisson", poisson_df), ("negative binomial", negative_binomial_df)]
for ax, (label, df) in zip(axes, panels):
    counts = df["y"].to_list()
    mean = sum(counts) / len(counts)
    variance = sum((count - mean) ** 2 for count in counts) / len(counts)
    ax.step(df["ds"].to_list(), counts, where="mid", alpha=0.85)
    ax.set(
        ylabel="Count",
        title=f"{label} innovations (mean {mean:.2f}, variance {variance:.2f})",
    )
axes[-1].set_xlabel("Timestamp")
plt.tight_layout()
plt.show()

```

<img src="https://mintcdn.com/nixtla/kY7DsWHSCHfeJfFk/synforecast/docs/generators/statistical/inar_files/figure-markdown_strict/cell-3-output-1.png?fit=max&auto=format&n=kY7DsWHSCHfeJfFk&q=85&s=9f5b2a338981253aafc666aa0e5a9bbe" alt="" width="1189" height="590" data-path="synforecast/docs/generators/statistical/inar_files/figure-markdown_strict/cell-3-output-1.png" />

## 2. Higher-order dependence

INAR(2) models have memory over two lags, producing smoother count
dynamics.

```python theme={null}
inar2_gen = INARGenerator(engine="polars", 
    min_length=200,
    max_length=200,
    freq="D",
    p=2,
    alpha=[0.3, 0.2],
    innovation_type="poisson",
    innovation_mean=2.0,
    seed=42,
)
inar2_df = inar2_gen.generate(n_series=1)

fig, ax = plt.subplots(figsize=(12, 4))
ax.step(inar2_df["ds"].to_list(), inar2_df["y"].to_list(), where="mid", alpha=0.8)
ax.set_xlabel("Timestamp")
ax.set_ylabel("Count")
ax.set_title("INAR(2) with two-lag dependence")
plt.tight_layout()
plt.show()
```

<img src="https://mintcdn.com/nixtla/kY7DsWHSCHfeJfFk/synforecast/docs/generators/statistical/inar_files/figure-markdown_strict/cell-4-output-1.png?fit=max&auto=format&n=kY7DsWHSCHfeJfFk&q=85&s=ba0c73a31dd43264d066df853d7ef1c6" alt="" width="1189" height="390" data-path="synforecast/docs/generators/statistical/inar_files/figure-markdown_strict/cell-4-output-1.png" />

## 3. Multiple series

```python theme={null}
multi_gen = INARGenerator(engine="polars", 
    min_length=150, max_length=150, freq="D",
    p=1, alpha=[0.5], innovation_mean=3.0, seed=42,
)
multi_df = multi_gen.generate(n_series=3)

fig, ax = plt.subplots(figsize=(12, 4))
for uid in multi_df["unique_id"].unique().to_list():
    series = multi_df.filter(pl.col("unique_id") == uid)
    ax.step(series["ds"].to_list(), series["y"].to_list(), where="mid", label=uid, alpha=0.8)
ax.set_xlabel("Timestamp")
ax.set_ylabel("Count")
ax.set_title("Multiple INAR(1) series")
ax.legend()
plt.tight_layout()
plt.show()
```

<img src="https://mintcdn.com/nixtla/kY7DsWHSCHfeJfFk/synforecast/docs/generators/statistical/inar_files/figure-markdown_strict/cell-5-output-1.png?fit=max&auto=format&n=kY7DsWHSCHfeJfFk&q=85&s=1d7b1f8203c5b211d0c2efee5b6279df" alt="" width="1186" height="390" data-path="synforecast/docs/generators/statistical/inar_files/figure-markdown_strict/cell-5-output-1.png" />

> **Related generators**
>
> * [Intermittent demand](../domain/intermittent_demand) — sparse
>   counts with many zeros.
> * [Poisson process](../stochastic/poisson_process) — event arrivals
>   in continuous time.
>
> Innovation and thinning parameters are in the [generator
> reference](https://github.com/Nixtla/synforecast/blob/main/GENERATORS.md).