# Model Location and Scale Together (GAMLSS)

A standard GAM models only the mean of the response. Generalized Additive Models for Location, Scale, and Shape (GAMLSS) go further: every distributional parameter gets its own linear predictor. This is essential when the variance itself depends on covariates. Ignoring that structure produces inefficient estimates and misleading confidence intervals.

The `climate` dataset records monthly temperatures across weather stations that vary widely in altitude and latitude. Mountainous stations and high-latitude stations tend to be not just colder on average but also more variable (a pattern that a standard model cannot capture).


# Fit

Load the dataset and fit a [GAMLSS](../reference/GAMLSS.md#whittaker.GAMLSS) with a [GaussianLS()](../reference/GaussianLS.md#whittaker.GaussianLS) family. The `mu` formula captures the smooth mean trend. The `sigma` formula captures how spread changes with geography.


``` python
import whittaker as wk

data = wk.load_dataset("climate")

# Fit GAMLSS with separate mu and sigma formulas
model = wk.GAMLSS(
    formulas={
        "mu":    "temperature ~ s(altitude) + s(latitude) + month",
        "sigma": "temperature ~ altitude + latitude",
    },
    family=wk.GaussianLS(),
).fit(data, method="GCV")
```


# Summarize

Inspect the fitted model. The summary reports effective degrees of freedom for each smooth in `mu`, coefficient estimates for `sigma`, and overall fit statistics.


``` python
model.summary()
```


    'GAMLSS fit summary\n========================================\nFamily: GaussianLS(mu=identity, sigma=log)\nN obs: 600\nGlobal deviance: 4614.5107\nAIC: 4646.2817\nBIC: 4716.1292\nLog-likelihood: -2307.2554\nConverged: True (27 iterations)\n\n--- mu ---\n  EDF total: 12.89\n  Smooth 1: edf = 7.90\n  Smooth 2: edf = 2.99\n\n--- sigma ---\n  EDF total: 3.00\n'


Check `model.converged` and `model.n_iter` to confirm the alternating estimation algorithm reached a stable solution.


``` python
model.converged, model.n_iter
```


    (True, 27)


# Predict across an altitude gradient

Build a prediction grid that holds latitude and month fixed while sweeping altitude. `model.predict()` returns a [GAMLSSPrediction](../reference/GAMLSSPrediction.md#whittaker.GAMLSSPrediction) object whose `.values` dict contains one array per parameter.


``` python
import numpy as np

# Build prediction grid across altitude gradient
new_data = {
    "altitude": np.linspace(0, 2000, 50),
    "latitude": np.full(50, 50.0),
    "month":    np.full(50, 7),
}

# Predict and inspect mean temperature values
pred = model.predict(new_data)
pred.values["mu"][:5]
```


    array([17.00599998, 17.5139282 , 18.00578257, 18.44388959, 18.77893675])


# Inspect predicted scale

The `sigma` predictions show how temperature variability grows with altitude, independent of the mean.


``` python
pred.values["sigma"][:5]
```


    array([5.03063254, 5.13064434, 5.23264444, 5.33667235, 5.4427684 ])


# Interpret

Higher altitude drives `mu` downward (temperatures are colder) while `sigma` tends to increase, reflecting greater day-to-day variability at elevation. A standard Gaussian GAM would fix the residual variance at a single global value, masking this pattern entirely. By modeling both parameters simultaneously, GAMLSS produces better-calibrated prediction intervals at every altitude level.