# Administrative Information - **Class Meeting**: Friday, September 25, 2026, 1:00–1:50pm — **Coolbaugh 212** - Previous meeting: [[MATH310F26(Day 12) - Spectrograms, and a yodel]] - 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]] > [!info]- Admin. Notes > Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes. > > - 🎯 **[LR] [Day 12 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121396)** — 4 questions drawn from Wednesday's material, unlimited attempts, highest kept — due **Friday 9/25, 11:59pm** > - 🎯 **[LR] [Day 12 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121397)** — one free response, no wrong answers — due **Friday 9/25, 11:59pm** > - 🎯 **[MP] [One Question, and the First Arrow](https://elearning.mines.edu/courses/81632/quizzes/121327)** — due **Friday 9/25, 11:59pm** > - 🎯 **[WB] [Workbook 10](https://elearning.mines.edu/courses/81632/assignments/587683)** · [[MATH310F26_WB10.pdf|blank]] · [[MATH310F26(Workbook 10 Companion) - Reading a sound's fingerprint|companion]] — due **Wednesday 9/30, 11:59pm** > - **Workbook check-in: Workbooks 1–10 due Wednesday 9/30** (soft); no more workbooks publish before the check-in; in-person checks begin soon > - **Presentations repackaged**: the TA's Connections story and the instructor's mock presentation move to next week, together > [!example]- 🗃️ COMAP Corner — three from the archive > *TK — three picks (one stat, one computational, one weird) from the catalog land here.* > [!warning] Skeleton — this is the pre-class page > Day 13 has not met yet (title provisional). The plan from Wednesday: the instructor breaks down a model built by two students in a last-semester independent study — **whale communication, background noise, and the traveling of pods** — as a live model-decoding exercise, possibly alongside a class walk-through of [[MATH310F26_WB10.pdf|Workbook 10]]'s pictures. Carried forward for Monday: the **discrete Fourier transform meets sea-level data**, and the unlabeled time series on the [[MATH310F26(Day 12) - Spectrograms, and a yodel|Day 12 page]] gets its axes back. The transcript and any captures fill in after Friday's class. --- # One record, three pictures ![[day13_mystery_week.png]] ![[day13_mystery_spectrum.png]] ![[day13_mystery_spectrogram.png]] > [!question] Look first, then guess > Three views of one record, with the labels taken off. The top picture is a stretch of the record. The middle one is its amplitude spectrum. The bottom one is a spectrogram of the whole record, with time running left to right, frequency running up, and brighter meaning stronger. > > 1. What repeats in the top picture? How many bumps does one repeat have, and is every repeat the same? > 2. The spectrum is a row of evenly spaced teeth. Count them. One tooth is nearly missing. Which one, and what about the shape of a single repeat could make exactly that tooth vanish? > 3. Small bumps sit beside each tooth. What slower rhythm would put them there? > 4. Partway across the spectrogram something changes. Which bands survive and which fade? Is there anything that looks like a one-off? > 5. What is being counted, and what happened? > [!success]- The reveal (open after the guesses) > **The data.** Hourly counts of bicycles crossing the Fremont Bridge in Seattle, from the city's automatic counter, October 2012 to August 2023. The top picture is three weeks starting Monday 29 April 2019. The spectrum is April and May 2019. The spectrogram is the whole record in 56-day windows, each column scaled to its own brightest cell. > > ![[day13_mystery_week_ticks.png]] > ![[day13_mystery_spectrum_ticks.png]] > ![[day13_mystery_spectrogram_ticks.png]] > > **Two rush hours make a comb.** A weekday has two sharp peaks, near 8 am and 5 pm. That shape repeats every 24 hours but looks nothing like a sine, so the transform needs a whole family of harmonics to build it, with teeth at 1, 2, 3, 4, 5 and 6 cycles per day (periods of 24, 12, 8, 6, 4.8 and 4 hours). A weekend day is one broad midday hump, which is nearly all fundamental. > > **Why the fourth tooth vanishes.** Model a weekday as two equal spikes $d$ hours apart. The height of harmonic $n$ is then proportional to > $2\left|\cos\left(\frac{\pi n d}{24}\right)\right|.$ > With $d = 9$ hours this gives 0.77, 1.41, 1.85, **0**, 1.85, 1.41 for $n = 1, \dots, 6$. The fourth harmonic has a 6-hour period, and 9 hours is one and a half of those periods, so the evening spike lands on a trough of the wave the morning spike lands on a crest of, and the two cancel. The measured weekday amplitudes for April and May 2019, in bikes per hour, are 123, 155, 135, **22**, 89, 35. The upper teeth come in lower than the model says because a real rush hour lasts an hour or two rather than an instant, and a wider bump carries less high-frequency content. > > **The side bumps are the week.** Weekdays and weekends differ, so the daily pattern is switched up and down with a 7-day period. That modulation puts small peaks at $k \pm 1/7$ cycles per day beside each tooth $k$. > > **What the spectrogram shows.** Before March 2020 the bands at 1, 2, 3 and 5 cycles per day are bright and the band at 4 is dark. In spring 2020 the upper bands fade and the band at 1 survives. Weekday riding roughly halved (4,147 to 2,161 bikes per day), but not evenly. Harmonic 1 fell 17 percent, while harmonics 2, 3 and 5 fell 68, 77 and 83 percent. The morning spike disappeared, and in spring 2020 weekends outdrew weekdays. > > **How to read the loss.** The fundamental measures daylight, since people ride more by day than by night whatever the reason. The higher harmonics measure how tightly riders are locked to a shared clock, the workday. Their share of the power above 1 cycle per day was 77 percent in 2019, 27 percent in 2020 and 55 percent in 2022, so the commute came partway back. The vertical stripe in February 2019 is a week of snow. > > **One to carry.** What would have to change about when people ride for the 6-hour band to light up? --- # How much work is a transform ![[day13_growth_0_to_2.png]] ![[day13_growth_1_to_1e5.png]] ![[day13_growth_1_to_1e5_loglog.png]] > [!question] Four curves, two windows > The four curves are $N^2$, $N \log_2 N$, $N$ and $N^{1/2}$. Computing the DFT straight from its definition takes about $N^2$ multiplications. The FFT takes about $N \log_2 N$. > > 1. In the first window, $N$ between 0 and 2, where do the curves cross, and which is on top on each side of $N = 1$? Why does $N \log_2 N$ dip below zero there? > 2. In the second window, $N$ up to 100,000, why do three of the curves look like they lie flat on the axis? > 3. The third picture puts both axes on a log scale. What does a power of $N$ become there, and where does $N \log_2 N$ sit? > 4. The sea-level record has $N = 1168$ points. How many times less work is the FFT? What about a year of hourly bike counts? > [!success]- Numbers to check against > At $N = 2$ the curves read 4, 2, 2 and 1.41, so $N \log_2 N$ and $N$ meet there. Below $N = 1$ the logarithm is negative, and $N \log_2 N$ bottoms out near $-0.53$ at $N = 1/e$. For the sea-level record, $N^2 = 1{,}364{,}224$ against $N \log_2 N \approx 11{,}904$, a ratio of about 115. A year of hourly counts has $N = 8760$ and a ratio $N/\log_2 N$ of about 670. At $N = 100{,}000$ the ratio is about 6,000.