Lesson 1 - Meet Lantern & Vine

Welcome to Meet Lantern & Vine#

Every module in this course, from decomposition through backtesting, was built and tested on Cyclepath, a single monthly bike-share series. That consistency made each new idea easier to isolate, but it also means you have never had to apply this course’s tools to a series you were not already familiar with. This capstone fixes that. Meet Lantern & Vine, a fictional home-goods shop, and its weekly unit sales over four years, a new series with its own structure, explored here for the very first time.

By the end of this lesson, you will be able to:

• Generate and explore a new seeded series the same way Module 1 explored Cyclepath
• Read summary statistics, yearly totals, and a rolling mean for a series you have not seen before
• Identify three structural differences from Cyclepath that this capstone will need to handle
• State the seven-stage pipeline this module will run from scratch

Let’s meet the series.

Generating Lantern & Vine#

import numpy as np
import pandas as pd

def lantern_vine():
    idx = pd.date_range("2020-01-06", periods=208, freq="W-MON")
    t = np.arange(208)
    rng = np.random.default_rng(7)
    growth_rate = np.where(t < 104, 0.008, 0.003)
    log_level = np.cumsum(growth_rate)
    level = 500 * np.exp(log_level)
    seasonal_factor = 1 + 0.35 * np.sin(2 * np.pi * (t - 35) / 52)
    noise_factor = rng.normal(1, 0.04, 208)
    y = level * seasonal_factor * noise_factor
    return pd.Series(np.round(y).astype(int), index=idx, name="units_sold")

y = lantern_vine()
print(len(y), y.index[0].date(), y.index[-1].date())
print(y.iloc[:5].tolist())
print(y.iloc[-5:].tolist())

208 2020-01-06 2023-12-25
[660, 662, 639, 613, 613]
[2163, 2258, 2227, 2001, 1943]

208 weeks, exactly four years, from the first Monday of January 2020 through the last Monday of December 2023. The series is built from a level that grows over time, multiplied by a seasonal factor that swings around 1.0, multiplied by a small amount of random noise also centered near 1.0. That multiplication, rather than addition, is deliberate, and Lesson 2 will confirm exactly what it implies. For now, treat this the way Module 1 treated Cyclepath on first contact: a series to explore, not yet to theorize about.

A First Look#

print(y.describe())

count     208.000000
mean     1070.826923
std       453.250577
min       362.000000
25%       714.250000
50%       992.000000
75%      1381.250000
max      2258.000000
Name: units_sold, dtype: float64

Weekly unit sales range from 362 to 2,258, averaging 1,070.8 across the full four years, with a standard deviation (453.3) that is unusually large relative to the mean, over 40%, a first hint that this series’ swings are large relative to its level, exactly the kind of relationship Module 2 taught you to check for directly rather than assume.

yearly = y.resample("YE").sum()
print(yearly.values.tolist())

[32464, 49558, 64549, 76161]

Yearly totals climb every year, 32,464 in 2020 up to 76,161 in 2023, more than doubling over four years. But look at the year-over-year growth rate, not just the totals: 2020 to 2021 grew by 52.6%, 2021 to 2022 by 30.3%, and 2022 to 2023 by only 18.0%. Cyclepath’s yearly growth was remarkably steady across all eight of its years. Here, growth is clearly slowing down, a second hint this lesson will not yet resolve, and Lesson 2 will chase down directly.

ma = y.rolling(52).mean()
print(round(ma.dropna().iloc[0], 1), ma.dropna().index[0].date())
print(round(ma.dropna().iloc[-1], 1), ma.dropna().index[-1].date())

624.3 2020-12-28
1464.6 2023-12-25

A 52-week rolling mean, the weekly equivalent of Cyclepath’s 12-month rolling mean, smooths away the seasonal swing and confirms the same rising trend the yearly totals already showed, climbing from about 624 to about 1,465 units a week.

Three Differences From Cyclepath, Already Visible#

Before running a single formal test, three structural differences are already showing:

• Weekly, not monthly. Cyclepath’s seasonal period was 12 (months in a year). Lantern & Vine’s is 52 (weeks in a year), a much longer cycle relative to the data available, only four repeats of it across the whole series instead of Cyclepath’s eight.
• A standard deviation unusually large relative to the mean, an early hint (not yet a confirmed diagnosis) that the seasonal swing might scale with the level rather than stay fixed, the multiplicative structure Module 2’s swing-to-level test was built to check.
• Decelerating year-over-year growth (52.6%, then 30.3%, then 18.0%), unlike Cyclepath’s steady climb. Something about this series’ trend is not constant, and finding out exactly what changed, and when, is Lesson 2’s first job.

Practice Exercises#

Exercise 1: Why check standard deviation relative to the mean?#

Lantern & Vine’s standard deviation (453.3) is about 42% of its mean (1,070.8). Why is that ratio worth noticing before running any formal test?

Exercise 2: Why does four cycles matter for period 52?#

Cyclepath had eight full yearly cycles (96 months, period 12). Lantern & Vine has four full yearly cycles (208 weeks, period 52). Why might having fewer cycles make some of this course’s later tools harder to apply?

Exercise 3: Predict what caused the growth deceleration#

Without running any more code, and using only the yearly growth rates already computed (52.6%, 30.3%, 18.0%), does this look more like a single smooth slowdown or a sudden change at one specific point?

Summary#

Lantern & Vine is a new, seeded, 208-week series, four years of a fictional home-goods shop’s weekly unit sales, ranging from 362 to 2,258 and averaging 1,070.8. Yearly totals climb from 32,464 to 76,161, but the year-over-year growth rate itself decelerates, 52.6%, then 30.3%, then 18.0%, unlike Cyclepath’s steady pace. A first exploration already surfaces three structural differences from Cyclepath that this capstone will have to handle: a weekly frequency with a 52-week seasonal period instead of 12, an unusually large standard deviation relative to the mean hinting at multiplicative structure, and a decelerating trend rather than a constant one.

Key Concepts#

• A new series, the same first steps — describe, resample, and a rolling mean, exactly Module 1’s approach, applied fresh.
• Three real differences from Cyclepath — weekly frequency (period 52), a possible multiplicative structure, and a decelerating rather than constant trend.
• Fewer seasonal cycles — four repeats of the yearly pattern instead of eight, which will make some later diagnostics noisier.
• The seven-stage pipeline — explore, decompose, stationarize, identify, fit, backtest, report, the structure this entire module follows.

Why This Matters#

Every technique this course has built was learned on a series designed to teach that technique cleanly. A real forecasting project rarely offers that convenience, and the true test of whether you have learned a method is whether you can apply it to a series that does not match the one you learned it on. Lantern & Vine’s three visible differences, weekly data, a possibly multiplicative structure, and a changing growth rate, are exactly the kind of complications a real dataset introduces, and this module will confront each one directly rather than assume Cyclepath’s answers still apply. Next, Lesson 2 chases down the growth-rate deceleration precisely, and settles the additive-versus-multiplicative question with real evidence.

Next Steps#

Continue Building Your Skills#

You have met Lantern & Vine and already spotted three ways it differs from Cyclepath: a weekly cadence, a hint of multiplicative structure, and a decelerating trend. Next, you will pin down exactly when that deceleration happened and confirm, with the same real-evidence discipline Module 2 taught, whether this series is genuinely multiplicative rather than additive.