Predicting Cumulative Totals From Sampled Rates

From Rate Measurements to Cumulative Change

A rate describes how quickly a quantity changes. If the rate is known continuously over a time interval, accumulated change is represented by the integral of that rate. In practice, measurements are often available only at selected times. The rate between those times is not directly recorded, so the total change must be estimated.

The basic strategy is to approximate the rate between observations by a simple shape. A rectangle treats the rate as constant over an interval; a trapezoid treats it as changing linearly between the measured endpoint rates. The estimated accumulated change is then the area of the chosen shapes.

If the quantity is a stock \(C(t)\), its estimated value at the final time is the initial stock plus the estimated change. The rate and time units must combine to give the units of the accumulated quantity: for example, litres per minute multiplied by minutes gives litres.

Rectangle and Trapezoid Formulas

Write \(h_i=t_{i+1}-t_i\) for the length of interval \(i\). The left-rectangle method uses the rate measured at the interval's left endpoint; the right-rectangle method uses the right endpoint. The trapezoidal method uses the average of the two endpoint rates.

Here \(L\), \(R\), and \(T\) denote estimates of accumulated change, not rates. If rates can be negative, these formulas estimate net change: positive and negative contributions are added with their signs. If all rates are nonnegative, the estimates represent nonnegative accumulated change.

For equally spaced measurements, every interval has the same length \(h\), so the formulas simplify. In particular, the trapezoidal estimate can be written as \(h\) times half the first rate, all interior rates, and half the last rate:

When intervals are unequal, use each interval's own length \(h_i\). A single average interval length cannot generally replace those individual widths: a rate measured across a longer interval contributes over more time.

Why the Trapezoidal Formula Works

The trapezoidal method corresponds to a specific assumption about the unobserved rate: on each interval, the rate varies linearly between its endpoint measurements. For an interval \([a,b]\), with endpoint rates \(r(a)\) and \(r(b)\), the line joining those values has average height \(\bigl(r(a)+r(b)\bigr)/2\). Its area is therefore the average height times the width.

This derivation explains both the formula and its limitation. A trapezoid is exact for a rate that is linear on the interval, but a real rate may curve or vary irregularly between observations. The measured endpoints alone do not reveal precisely what happened in between.

How the Sampling Interval Affects an Estimate

Shorter intervals generally give the approximation more opportunities to track changes in the rate. They do not, by themselves, guarantee that a particular estimate moves closer to the true total: the shape of the rate between observations matters. Under a smoothness assumption, however, the effect can be quantified for trapezoids.

To justify the bound on one interval \([a,b]\), let \(p\) be the line through the endpoint values. The interpolation remainder for a twice-differentiable function gives, at each interior \(t\), \(r(t)-p(t)=\frac{r''(\xi)}{2}(t-a)(t-b)\) for some \(\xi\) between \(a\) and \(b\). Consequently, if \(\lvert r''\rvert\le M\), then \(\lvert r(t)-p(t)\rvert\le\frac{M}{2}(t-a)(b-t)\). Integrating this inequality over the interval gives

The integral of \(p\) is the trapezoidal contribution. Applying the one-interval bound to every interval and adding the bounds proves the stated composite estimate. With equal spacing, there are \(n=(t_n-t_0)/h\) intervals; thus \(nMh^3/12=M(t_n-t_0)h^2/12\). The bound shows why reducing the spacing can reduce the possible trapezoidal error when the curvature stays bounded. It does not apply without the stated smoothness and derivative bound.

Choosing and Reporting an Estimate

A rectangle estimate can be informative when there is a meaningful reason to treat the rate as constant over each interval, or when a one-sided estimate is specifically useful. If the rate is increasing, left rectangles fall below the accumulated change and right rectangles rise above it; if the rate is decreasing, the roles reverse. These comparisons require monotonicity throughout the intervals, not just a visual trend between a few measurements.

The trapezoidal estimate uses both endpoint measurements and often gives a reasonable approximation when the rate changes gradually. Yet it is not automatically more accurate for every set of data. In particular, knowing that endpoint values differ does not show whether the rate between them is nearly linear, sharply curved, or variable in some other way. More frequent measurements reduce the width over which such unobserved behavior can occur, but no method can recover details that the observations and assumptions do not determine.

When reporting a cumulative prediction, state the method and the sampling intervals, include units, and distinguish estimated change from the final cumulative total. For instance, if a stock begins at \(C(t_0)\), its estimated final value is \(C(t_0)+T\) when \(T\) is the trapezoidal estimate of net change. Do not report the estimate with unjustified precision: the arithmetic may be exact while the approximation of the unmeasured rate remains uncertain.

A Common Calculation Error

A frequent mistake is to average endpoint rates for a trapezoid but forget to multiply by the time interval. In the first example, the three average rates are \(4\), \(4.5\), and \(3\) L/min. Adding them gives \(11.5\), but this is still a sum of rates, not a volume. Each interval is two minutes long, so the correct calculation is

Always multiply a representative rate by its own interval length before adding interval contributions. This also prevents a second error: using one common width when the measurement times are unevenly spaced.