Track 12 · Time Series and Forecasting · Lesson 1 of 11

Time-indexed data

Recognise what makes data a time series, prepare a single regular series from raw records, and describe change over time correctly.

25 minAdvanced7 graded exercisesFree

A distributor must decide how many ceiling fans to order for next summer. Last year's orders, month by month, are the most useful evidence it has — but only if they are kept in time order, at a sensible frequency, with gaps noticed and the right months compared with each other.

A time series is a sequence of measurements of the same quantity taken at successive, usually equally spaced, points in time: monthly units ordered, weekly sales, daily electricity use, hourly calls.

Two things set it apart from the data in earlier tracks:

Order carries information. Shuffle the rows of a survey and nothing is lost. Shuffle a time series and the trend, the seasons and the link between this month and last month all disappear.
Observations are not independent. A hot week is usually followed by another warm week; a busy Sunday at a contact centre is followed by the next busy Sunday. Methods that assume independent observations (most of Track 8) do not apply directly.

y_1, y_2, \ldots, y_T \qquad \hat{y}_{T+h\,|\,T}

Notation used throughout this track

$y_t$: the observation at time $t$ (for example the units ordered in month $t$ )
$T$: the number of observations available, so $y_T$ is the most recent one
$h$: the forecast horizon: how many periods ahead we forecast
$\hat{y}_{T+h\,|\,T}$: the forecast of $y_{T+h}$ made using data up to time $T$ only

Dataset

Monthly product demand

288 rows · one row = one product-month · synthetic data

About this dataset Open in the lab

Month	Month index	Product	Units demanded (units)
2017-01-01	1	Ceiling fan	1232
2017-01-01	1	Water heater	2430
2017-01-01	1	School backpack	1080
2017-02-01	2	Ceiling fan	1481
2017-02-01	2	Water heater	2264
2017-02-01	2	School backpack	429

Long format: one row per product-month. Three products share the same month, so you must filter to one product before treating the column units as a series.

Ceiling-fan orders, January 2017 (month 1) to December 2024 (month 96).

Show the data as a table

Month index	Units demanded
1	1232
2	1481
3	1948
4	2427
5	3360
6	4211
7	4240
8	4524
9	2822
10	2242
11	1584
12	1224
13	1440
14	1385
15	1761
16	2551
17	3545
18	4415
19	4039
20	4351
21	3335
22	2247
23	1472
24	1233
25	1301
26	1481
27	1957
28	2813
29	3940
30	4544
31	4511
32	4159
33	3133
34	2171
35	1532
36	1641
37	1299
38	1412
39	2004
40	2991
41	3951
42	4086
43	4887
44	4262
45	3731
46	2230
47	1735
48	1514
49	1362
50	1891
51	2167
52	3136
53	4405
54	4744
55	5290
56	4815
57	3244
58	2415
59	1690
60	1232
61	1535
62	1768
63	2208
64	2791
65	4240
66	5441
67	4976
68	4866
69	3403
70	2631
71	1961
72	1310
73	1627
74	1985
75	2508
76	3386
77	3874
78	5516
79	5929
80	4521
81	4182
82	2768
83	2024
84	1399
85	1561
86	1766
87	2401
88	2751
89	4458
90	5186
91	5489
92	5730
93	4069
94	2937
95	1881
96	1731

In this (synthetic) dataset the yearly ceiling-fan total grows from 31,295 units in 2017 to 39,960 in 2024, and every year has a summer peak. A line chart in time order shows both features at a glance; a histogram of the same 96 numbers would hide them completely.

Data recorded	Repeating pattern	Period $m$
Monthly	Calendar year	12
Quarterly	Calendar year	4
Weekly	Calendar year	52 (approximately)
Daily	Week	7 (or 5 for working days only)
Hourly, open 24 h	Day	24

Frequency and the natural seasonal period

Frequency is a choice. The contact-centre data are recorded per hour (13 opening hours a day), but staffing rotas are planned per day. Summing the hours of each date gives a daily series of 90 values with a mean of 278 calls. Choose the frequency at which the decision is made, and aggregate with the right operation: sum for flows (calls, units, kWh), mean or last value for levels (temperature, price, stock on hand).

Worked example

Comparing like with like

Ceiling-fan orders were 31,295 units in 2017 and 39,960 in 2024. A manager also notices that December 2024 is far below July 2024 and asks whether demand is collapsing. How should the change be described?

Seasonal data must be compared with the same season. December is always a low month for fans, so December vs July mostly measures the season, not a change in demand.
Year-over-year change compares a period with the same period one cycle earlier: $y_t - y_{t-12}$ for monthly data.
Annual totals remove the season altogether. From 2017 to 2024 the total rose by $39{,}960 - 31{,}295 = 8{,}665$ units, a growth of $8{,}665 / 31{,}295 = 27.7\%$ over seven years.

Demand is not collapsing: the December dip is seasonal. Over the whole period demand grew by about 28%.

Quick check

You randomly split 96 months of orders into 80% training rows and 20% test rows to evaluate a forecasting method. What is wrong?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

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Exercise 1 · Guided, difficulty 1 of 3

Year-over-year growth from monthly records

Not started

Using the monthly_demand data for Ceiling fan only, add up the units for 2023 (months 73–84) and for 2024 (months 85–96). Then give the year-over-year percentage change from 2023 to 2024, to 2 decimal places.

Data: monthly_demand (product = Ceiling fan) · Open in the lab to explore and calculate.

Exercise 2 · Practice, difficulty 1 of 3

Match the frequency to its seasonal period

Not started

Match each series to the natural seasonal period $m$ (the number of observations in one repeating cycle).

Exercise 3 · Application, difficulty 2 of 3

From hourly records to a daily series

Not started

The call_center data have one row per opening hour (13 per day). Staff rotas are planned per day, so build the daily series: the total calls_offered on each date.

Give the total for the first date (2024-01-07) and the mean daily total over all 90 days (2 decimal places).

Data: call_center · Open in the lab to explore and calculate.

Exercise 4 · Practice, difficulty 2 of 3

Compare the right quarters

Not started

Quarterly sales of a product (units) for three years, oldest first (Q1, Q2, Q3, Q4 of year 1, then year 2, then year 3):

176, 214, 277, 214, 192, 251, 306, 227, 218, 261, 334, 250

The sales director wants to know how the latest quarter (Q4 of year 3) compares with a year earlier. Give the change in units and the percentage change (2 decimal places).

Exercise 5 · Interpretation, difficulty 2 of 3

Missing, or simply not part of the calendar?

Not started

For each situation, decide whether the absent values are missing data that must be handled, or an absence by design that defines the calendar of the series.

Exercise 6 · Practice, difficulty 1 of 3

Prepare a series for forecasting

Not started

Put these preparation steps for forecasting water-heater demand from the long-format monthly_demand table in a sensible order.

Exercise 7 · Interpretation, difficulty 2 of 3

A mean that hides the week

Not started

The mean daily total of calls is 278.0. A planner proposes staffing every day for that number. Which statement best describes the problem?

Data: call_center · Open in the lab to explore and calculate.

What you can now do

Explain why order and dependence make time series different from independent observations
Build a single regular series from long-format or hourly records, aggregating flows by summing
Choose the seasonal period that matches the frequency
Compare seasonal data with the same season a cycle earlier, or with annual totals
Distinguish missing values from absences that define the calendar