# Fit Your First Smooth

Whittaker GAMs are specified with a formula string, fitted in one call, and ready to interpret immediately. This recipe walks through loading data, fitting a single-smooth Gaussian GAM, reading the summary, and visualizing the smooth (all in just a few lines).


# Load data

The built-in `mcycle` dataset records head acceleration (in g) of a motorcycle rider over time (in milliseconds) during a simulated crash. It is a classic benchmark for nonlinear smoothing.


``` python
import whittaker as wk

# load the dataset and inspect its columns
data = wk.load_dataset("mcycle")
list(data.keys())
```


    ['times', 'accel']


The dataset has two columns: `times` (milliseconds post-impact) and `accel` (head acceleration in *g*). Let's confirm the sample size.


``` python
len(data["times"])
```


    133


With 133 observations, there is enough data to fit a smooth with several degrees of freedom.


# Fit a GAM

The formula `"accel ~ s(times)"` tells Whittaker to model `accel` as a smooth function of `times`. The `s()` wrapper requests a penalized spline. With no `family` argument, Whittaker defaults to `wk.Gaussian()`.


``` python
# Fit a single-smooth Gaussian GAM
model = wk.GAM("accel ~ s(times)").fit(data)
model.edf_total
```


    8.916222078330314


The total effective degrees of freedom tells us how many parameters the fitted smooth effectively consumed. A value substantially above 1 means the smooth is genuinely nonlinear. Next, check how much of the variation it accounts for.


``` python
model.deviance_explained
```


    0.8444899139289328


A value near or above 0.70 suggests the smooth captures a substantial portion of the variance in this noisy dataset.


# Inspect the fit

`model.summary()` returns a formatted text block. The two numbers to look at first are the EDF (effective degrees of freedom) for `s(times)` and the deviance explained at the bottom.


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


    GAM fit summary
    ============================================================
    Formula:    accel ~ s(times)
    Family:     Gaussian(link='identity')
    Observations: 133
    Coefficients: 10

    Parametric coefficients:
      Term                       Estimate    Std.Err    t value    p-value
      ------------------------ ---------- ---------- ---------- ----------
      (Intercept)                -45.6924     1.8364    -24.882    < 1e-16

    Approximate significance of smooth terms:
      Term                        EDF Ref.df     Chi.sq    p-value
      ------------------------ ------ ------ ---------- ----------
      s(times)                   7.92      8    663.396    < 1e-16

    Total EDF:  8.92
    Deviance:   55653.7152
    Null dev:   357878.4929
    Dev. expl:  84.4%
    GCV score:  480.746119
    Scale est:  448.517253
    AIC:        1198.44
    BIC:        1224.22


An EDF well above 1 confirms the relationship is genuinely nonlinear. Deviance explained near or above 70 % suggests a reasonable fit for this noisy dataset.


# Visualize the smooth

The [partial_effects()](../reference/partial_effects.md#whittaker.partial_effects) function renders the fitted smooth with a confidence ribbon.


``` python
wk.partial_effects(model)
```


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