Forecasting with time series

Every January, Lakeview Health's clinics are overwhelmed. Respiratory infections peak, the emergency department fills, and patients who cannot get a clinic appointment go to the hospital instead. Every summer the clinics are quiet and staff take leave. Hana Kowalski sets the clinic nursing schedule six months ahead, and last winter she under-staffed badly: visits ran about eight percent above the budget, which had been built by adding three percent to the previous year. Agency nurses filled the gaps at nearly twice the cost of staff nurses. Owen faces a similar problem at Prairie Ridge, where gas demand peaks in cold weather and wind output varies with the season. Dr. Zhou devotes the tenth session to forecasting, which she describes as the art of being usefully wrong.

What a time series contains

A time series is a sequence of measurements taken at regular intervals: monthly visits, weekly sales, daily electricity demand. Most business series can be thought of as a combination of four components. The trend is the long-run direction, such as Lakeview's gradual growth in visits as the population of its catchment area rises. Seasonality is a pattern that repeats over a fixed period, such as the winter peak in respiratory illness or the December peak in retail sales. Cycles are longer, irregular swings, often tied to the economy. And the remainder is the irregular noise that no model can predict.

Hana's four years of monthly clinic visits show all of this clearly. Visits peak in December and January at around 6,500 to 6,800 a month and fall to about 4,500 to 5,000 in mid-summer. A twelve-month moving average, which averages each run of twelve consecutive months and so removes the seasonal pattern, rises steadily from about 5,300 to about 5,800 over the period. The first step in any forecast is to plot the series and identify these components; a forecast that ignores a strong seasonal pattern will be badly wrong at exactly the times that matter most.

Simple benchmarks

Before building anything sophisticated, a forecaster should establish simple benchmarks that any proposed method must beat. The naive forecast says that next period will equal this period. The seasonal naive forecast says that next January will equal last January, and so on for each month. The mean forecast uses the average of all past values. These sound crude, but they are surprisingly hard to beat for many series, and a method that cannot beat them on past data has no business being used.

The evidence from forecasting competitions supports this caution. The M competitions, organized by Spyros Makridakis and colleagues since 1982, have asked researchers and practitioners to forecast thousands of real time series, with results scored on data the forecasters had not seen. A recurring finding is that simple methods perform well relative to complex ones, and that combining the forecasts of several methods usually beats the individual methods. The M4 competition in 2018, with 100,000 series, found that most of the best performers were combinations, often mixing statistical and machine-learning methods. The M5 competition in 2020, using Walmart sales data, found that machine-learning methods could do well on large collections of related series, but the advantage came from careful engineering rather than from any single clever algorithm.

Moving averages and exponential smoothing

A moving average forecast predicts the next value as the average of the last k values. It smooths out noise, and a longer window smooths more but responds more slowly to real changes. Exponential smoothing refines this idea by giving the most recent observation the greatest weight and older observations geometrically decreasing weights. The forecast is updated each period by moving it part of the way toward the latest actual value.

The smoothing constant alpha controls how quickly the forecast reacts. With alpha near one, the forecast chases every movement; with alpha near zero, it barely moves. Simple exponential smoothing assumes no trend or seasonality, so on Hana's data it produces a flat forecast near recent levels and misses the winter peak entirely. Extensions add components: Holt's method adds a smoothed trend, and the Holt-Winters method adds smoothed seasonal factors as well. These methods, developed in the late 1950s and early 1960s for inventory control, remain among the most widely used forecasting tools in business, built into spreadsheets and planning software.

Seasonal indices and decomposition

A transparent way to handle seasonality is to compute a seasonal index for each month: the average value for that month divided by the overall average. An index of 1.17 for January means January visits typically run 17 percent above the average month; an index of 0.85 for July means July runs 15 percent below. Dividing each observation by its month's index gives a deseasonalized series, which shows the underlying trend more clearly. A trend line can be fitted to the deseasonalized data, projected forward and then multiplied by the seasonal index for each future month to produce the forecast.

Seasonal indices are also useful in their own right. Hana can use them to plan staffing for each month of the year in proportion to expected demand, and Grace's colleagues use them to judge whether a month's sales are good once the normal seasonal pattern has been taken into account. A report that December sales rose 30 percent over November says nothing if December always runs 35 percent above November; the deseasonalized change shows whether the business actually improved.

Measuring forecast accuracy

Forecast accuracy should be measured on data that were not used to build the forecast. A common approach is to hold back the most recent period, fit each method on the earlier data, forecast the held-back period and compare. Several measures are useful. The mean absolute error is the average size of the errors in the original units, such as visits. The mean absolute percentage error, or MAPE, expresses the average error as a percentage of the actual values, which allows comparison across series of different sizes, though it misbehaves when actual values are near zero. The root mean squared error penalizes large errors more heavily. And the bias, the average signed error, shows whether a method tends to forecast too high or too low; a biased forecast is a sign that something systematic has been missed.

# Backtest three forecasting methods on Lakeview's monthly clinic visits
import statistics
visits = [6022, 5941, 5417, 5046, 4922, 4432, 4504, 4604, 4940, 5419, 5767, 6507,
          6370, 6189, 5770, 5122, 4918, 4852, 4629, 4692, 5137, 5441, 6086, 6558,
          6847, 6348, 5815, 5387, 5209, 4935, 4928, 4921, 5195, 5966, 6070, 6740,
          6690, 6593, 5819, 5456, 5347, 5152, 4950, 5123, 5304, 5951, 6230, 6671]
train, test = visits[:36], visits[36:]          # fit on years 1 to 3, test on year 4

# 1. Seasonal naive: next January = last January, and so on
f_naive = train[-12:]

# 2. Simple exponential smoothing (no trend or season), alpha = 0.3
level = train[0]
for v in train[1:]:
    level = 0.3 * v + 0.7 * level
f_ses = [level] * 12

# 3. Seasonal index times linear trend
avg = statistics.mean(train)
index = [statistics.mean(train[m::12]) / avg for m in range(12)]
deseason = [v / index[i % 12] for i, v in enumerate(train)]
trend = statistics.linear_regression(range(36), deseason)
f_trend = [(trend.intercept + trend.slope * t) * index[t % 12] for t in range(36, 48)]

def scores(f):
    err = [a - b for a, b in zip(test, f)]
    mae = statistics.mean(abs(e) for e in err)
    mape = statistics.mean(abs(e) / a for e, a in zip(err, test))
    bias = statistics.mean(err)
    return mae, mape, bias
f_combo = [(a + b) / 2 for a, b in zip(f_naive, f_trend)]       # simple average of methods 1 and 3
for name, f in [('Seasonal naive', f_naive), ('Exponential smoothing', f_ses), ('Trend x seasonal index', f_trend), ('Average of 1 and 3', f_combo)]:
    mae, mape, bias = scores(f)
    print(f"{name:24s} MAE {mae:5.0f}   MAPE {mape:5.1%}   bias {bias:+5.0f}")
print("Seasonal index by month:", " ".join(f"{i:.2f}" for i in index))

Seasonal naive           MAE   117   MAPE  2.0%   bias   +77
Exponential smoothing    MAE   574   MAPE 10.2%   bias  -178
Trend x seasonal index   MAE   125   MAPE  2.1%   bias   -76
Average of 1 and 3       MAE    93   MAPE  1.6%   bias    +0
Seasonal index by month: 1.17 1.12 1.03 0.94 0.91 0.86 0.85 0.86 0.93 1.02 1.09 1.20

The backtest fits each method on the first three years and forecasts the fourth. Simple exponential smoothing, which ignores the seasonal pattern, has an average error of about 574 visits a month, or about ten percent. The seasonal naive forecast and the trend-times-seasonal-index forecast both do far better, at about two percent. The seasonal naive method tends to forecast slightly too low, because it ignores growth, while the trend method forecasts slightly too high. Their simple average cancels the biases and cuts the error to about 1.6 percent, a small-scale echo of the M competitions' finding that combinations usually win.

Forecast intervals and scenarios

A point forecast is almost certain to be wrong; the question is by how much. A forecast interval gives a range likely to contain the actual value, based on the size of past forecast errors. A useful rule of thumb is to look at the distribution of errors in the backtest and use, for example, the range covering 80 or 95 percent of them. Intervals should widen for forecasts further into the future, because errors accumulate. They should also be checked: if the 80 percent intervals from past forecasts contained the actual value only half the time, the intervals are too narrow, a common failing because forecasters tend to be overconfident.

Some uncertainties are better handled with scenarios than with intervals. An unusually severe respiratory season, a new clinic opening nearby, or a change in provincial or state funding could shift Hana's demand by more than any past pattern suggests. Planning for a base case, a high case and a low case, and deciding in advance what she would do in each, gives her a staffing plan that is robust rather than merely accurate on average. For Owen, the equivalent is to plan gas supply for a normal winter while holding contracts that can cover a cold-snap scenario.

Judgment and forecasting

Most organizations adjust statistical forecasts with human judgment, and the evidence on whether this helps is mixed. Studies of supply chain forecasting, including work by Robert Fildes and Paul Goodwin with several companies, found that large adjustments based on specific information, such as a planned promotion or a known store closure, often improved accuracy, while frequent small adjustments, especially upward ones driven by optimism, tended to make forecasts worse. The practical lessons are to adjust only when there is information the model does not have, to record the reason for each adjustment and to track whether adjustments improve accuracy over time.

Forecasts can also be distorted by incentives. Sales teams may lower forecasts so that targets are easier to beat; product managers may raise them to secure resources. Separating the forecast, an honest estimate of what will happen, from the target, what the organization wants to happen, and from the plan, what it will do about the difference, is one of the most useful disciplines a manager can introduce. Hana's budget had confused all three: the three percent growth figure was partly a forecast and partly a hope that visits would grow no faster than funding.

Forecasting demand, not sales

A final trap is that recorded sales or visits may not reflect true demand. When a store runs out of stock, sales stop even though shoppers still want the product; when a clinic is fully booked, patients who are turned away do not appear in the visit count. Forecasting from such censored data underestimates demand exactly when it is highest, and the forecast then justifies the same capacity that caused the shortfall. Hana adds a count of patients turned away or redirected to the emergency department to her data, so that next year's forecast reflects demand rather than the clinics' past capacity.