# Get Prediction Intervals

A confidence interval answers: where is the true mean? A prediction interval answers: where will the next observed value fall? The prediction interval is always wider because it must account for both uncertainty in the estimated mean and the irreducible noise in each individual observation (the residual variance, σ²). Unlike confidence intervals, prediction intervals cannot shrink to zero with more data (the observation-level noise remains no matter how large the sample).


# Fit

Fit a GAM on the motorcycle helmet dataset.


``` python
import whittaker as wk
import numpy as np

data = wk.load_dataset("mcycle")
model = wk.GAM("accel ~ s(times)").fit(data)
model.summary()
```


    GAM fit summary
    ============================================================
    Formula:    accel ~ s(times)
    Family:     Gaussian(link='identity')
    Inference:  GCV
    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    663.379    < 1e-16

    Total EDF:  8.92
    Scale est:  448.520495
    Deviance:   55654.5201
    Null dev:   357878.4929
    Dev. expl:  84.4%
    GCV score:  480.746115
    AIC:        1198.44
    BIC:        1224.21


# Build a prediction grid

A 200-point grid spanning the full range of `times` is sufficient to compare both interval types.


``` python
grid = {"times": np.linspace(data["times"].min(), data["times"].max(), 200)}
```


# Confidence interval

Pass `interval="confidence"` to quantify uncertainty about the mean curve.


``` python
# Predict with confidence intervals
ci = model.predict(grid, interval="confidence")

# Compute mean CI width
ci_width = (ci.upper - ci.lower).mean().round(3)
ci_width
```


    np.float64(23.08)


This width shrinks as sample size increases (it measures how well the mean is estimated).


# Prediction interval

Pass `interval="prediction"` to quantify where individual future observations are expected to land.


``` python
# Predict with prediction intervals
pi = model.predict(grid, interval="prediction")

# Compute mean PI width
pi_width = (pi.upper - pi.lower).mean().round(3)
pi_width
```


    np.float64(87.111)


The prediction interval is noticeably wider than the confidence interval. The excess comes from the estimated residual variance, which is a property of the data-generating process and does not decrease with sample size.


# Width ratio

Quantify how much wider the prediction interval is.


``` python
round(pi_width / ci_width, 3)
```


    np.float64(3.774)


A ratio well above 1 confirms that observation-level noise dominates the uncertainty. If the model were fit on thousands of observations the confidence interval would nearly collapse, but the prediction interval would remain wide by roughly the same absolute amount.


# Compare bounds directly

Inspect the first five rows of each set of bounds to see the difference concretely.


``` python
import numpy as np

np.column_stack([
    ci.lower[:5].round(2),
    ci.upper[:5].round(2),
    pi.lower[:5].round(2),
    pi.upper[:5].round(2),
])
```


    array([[-16.62,  23.17, -43.13,  49.67],
           [-14.65,  22.98, -41.78,  50.11],
           [-12.75,  22.85, -40.49,  50.59],
           [-10.91,  22.77, -39.25,  51.1 ],
           [ -9.18,  22.75, -38.07,  51.64]])


Columns are: CI lower, CI upper, PI lower, PI upper. The prediction bounds extend substantially further from the fitted values.


# When to use each

Use `interval="confidence"` when you want to make statements about the mean response: "the average acceleration at time 20 ms is between X and Y." Use `interval="prediction"` when you need to bound where a new individual measurement will fall: "the next helmet test at 20 ms should record a value between X and Y." Reporting prediction intervals as if they were confidence intervals understates the true uncertainty for individual forecasts.