5.M Focus on Modeling (Chapter Supplement)
Ideas in this lesson
- Read full datasets and choose a model family
- Estimate a sinusoid from observed features
- Fit all observations by nonlinear sine regression
- Normalize and compare hand and fitted phases
- Solve every contextual threshold interval
Read full datasets and choose a model family · Worked example
The complete original rapid positive-offset sensor data use output units and seconds.
Construct a scatterplot of every pair with labeled scales. Determine whether sustained exponential growth is credible. Calculate one positive and one negative secant slope and locate observed turning points; explain why a positive baseline alone is not exponential growth. Do not calibrate parameters or run a regression in this task. Show the requested calculations and interpretation. Written reasoning and graphs require teacher review; selecting a summary does not verify them. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
| seconds | output units |
|---|---|
| \(0.1\) | \(19.92\) |
| \(0.2\) | \(22.93\) |
| \(0.3\) | \(22.93\) |
| \(0.4\) | \(21.29\) |
| \(0.5\) | \(18.36\) |
| \(0.6\) | \(13.67\) |
| \(0.7\) | \(9.1\) |
| \(0.8\) | \(6.44\) |
| \(0.9\) | \(6.2\) |
| \(1\) | \(8.85\) |
| \(1.1\) | \(12.9\) |
| \(1.2\) | \(16.73\) |
| \(1.3\) | \(20.66\) |
| \(1.4\) | \(23.49\) |
| \(1.5\) | \(23.21\) |
Answer: Sinusoidal trial; interior increases and decreases support oscillation; exact periodicity remains unproved.
The full input horizon is \([0.1,1.5]\); the output range of the observations is \([6.2,23.49]\) in output units. All 15 observations are plotted.
On \([1,1.1]\), the secant slope is \(\dfrac{12.9-8.85}{0.1}=40.5\). On \([0.5,0.6]\), it is \(\dfrac{13.67-18.36}{0.1}=-46.9\). Thus the record contains both genuine increases and decreases.
Count each consecutive run of tied readings as one sampled extremum when the neighboring readings are lower (peak) or higher (trough). Listed run endpoints are sample times, not a claim that the underlying signal is constant between them. Use the first sample in each run as its representative time. Interior sampled peak times or runs: [0.2, 0.3] (tied sampled readings), 1.4. Interior sampled trough times or runs: 0.9. Gaps between the declared representative peak times are 1.2. These are estimates influenced by sampling and noise.
The endpoint change is \(23.21-19.92=3.29\), but the full excursion is \(23.49-6.2=17.29\). An endpoint secant hides the interior reversals and does not justify a line.
A sinusoidal trial is reasonable because the record falls and rises around a bounded central level rather than sustaining one direction. A line has constant slope; a positive-base exponential with fixed nonzero coefficient is monotone; a usual saturating trend approaches a level without repeated reversals. A single U-turn is compatible with a quadratic and does not alone establish a repeating cycle. Here the full sampled turning pattern supports testing a sinusoid, not declaring it uniquely proved.
Finite, noisy samples do not prove exact periodicity, identify a unique unrestricted frequency, or establish future repeats. A translated origin changes horizontal labels, not the order or spacing of observations. Every endpoint remains part of the evidence; choosing a shorter favorable window would discard relevant data.
There are not two interior sampled trough runs, so a repeated trough-to-trough spacing cannot be measured from this record alone.
Written-work review:
- Plot all observed pairs with their actual time labels and units.
- Use the actual requested numerical evidence to compare model families; distinguish a modeling suggestion from proof.
- Accept equivalent forms at the stated precision. Written reasoning and graphs require teacher review.