Ordinary linear or parabolic trend, as met in Applied Statistics, extrapolates the same growth pattern forever. Real-world series (sales of a new product, adoption of technology, population of a city, bacterial cultures) typically exhibit an S-shaped (sigmoid) growth pattern:
Three classical curves capture this behaviour: modified exponential, logistic and Gompertz.
The simplest growth curve. It approaches an upper asymptote \(a\) at a rate proportional to the remaining gap.
Here \(a\) = upper asymptote (the saturation level); \(b\) sets the initial gap; \(c\) is the rate parameter.
As \(t \to \infty\), \(c^t \to 0\) and \(Y_t \to a\). The curve is monotone increasing and concave (it does not have the typical sigmoid bend; it just rises and levels off).
Choose three equidistant time points \(t_1, t_2, t_3\) with values \(Y_1, Y_2, Y_3\). Then:
Divide the series of \(n\) equal-spaced observations into three equal groups of \(k\) observations each (so \(n = 3k\)), and number the years \(t = 0, 1, \dots, 3k-1\) from the first observation. Sum each group:
Why these work. Summing \(a + b c^t\) over a group of \(k\) consecutive years gives \(S_1 = k a + b\,\dfrac{c^k - 1}{c - 1}\), and \(S_2\), \(S_3\) are the same with the \(b\) term multiplied by \(c^k\) and \(c^{2k}\). Subtracting consecutive sums removes \(a\): \(S_2 - S_1 = b\,\dfrac{(c^k - 1)^2}{c - 1}\) and \(S_3 - S_2 = c^k (S_2 - S_1)\). Their ratio gives \(c^k\); the first difference then gives \(b\); and \(S_1\) gives \(a\). Because every observation enters a sum, the fit is less sensitive to one irregular year than the three-selected-points fit.
Sales data for years 2018–2026 (9 years). Pick \(t = 2018, 2022, 2026\) with values 50, 80, 95.
\(c^4 = (95 - 80)/(80 - 50) = 15/30 = 0.5\); \(c = 0.5^{1/4} = 0.841\).
\(a = (50 \cdot 95 - 80^2)/(50 + 95 - 2 \cdot 80) = (4750 - 6400)/(145 - 160) = -1650/-15 = 110\).
\(Y_t = 110 + b\, c^t\); using point 2018: \(50 = 110 + b\, 0.841^{2018}\). Plug appropriate normalization (shifting time origin to 2018 so \(t = 0\)): \(b = 50 - 110 = -60\).
Fitted: \(Y_t = 110 - 60(0.841)^t\). Saturation = 110.
Values: 10, 15, 22, 30, 38, 45, 51, 56, 60. Group sums: \(S_1 = 47,\; S_2 = 113,\; S_3 = 167\).
\(c^3 = (167 - 113)/(113 - 47) = 54/66 = 0.8182\); \(c = 0.8182^{1/3} = 0.9353\).
\(b = 66\,(0.9353 - 1)/(0.8182 - 1)^2 = 66\,(-0.0647)/0.03306 = -129.2\).
\(a = \tfrac13\left[47 - 66/(0.8182 - 1)\right] = \tfrac13\,(47 + 363.0) = 136.7\).
Fitted (with \(t = 0\) at the first value): \(Y_t = 136.7 - 129.2\,(0.9353)^t\). The fitted values are 7.5, 15.8, 23.7, 31.0, 37.8, 44.2, 50.2, 55.8, 61.0 against the data 10, 15, 22, 30, 38, 45, 51, 56, 60. The estimated saturation level is \(a \approx 137\), more than twice the last observation: an asymptote this far beyond the data is an extrapolation and should be quoted with that caution.
The classical S-shaped growth model, used for population biology, market saturation, technology adoption, epidemic spread.
\(k\) = upper asymptote (carrying capacity); \(c\) = intrinsic growth rate; \(b\) sets the inflection time.
Properties:
Take reciprocals and rearrange:
Letting \(Z_t = 1/Y_t\) reduces the logistic to a modified exponential in \(Z_t\). Apply either of the methods from Section 2 to \(Z_t\) data.
Compute \(Z_t = 1/Y_t\), form three group sums of \(Z\), then proceed exactly as in modified-exponential partial-sums fitting.
City population (lakhs) for 9 successive years: 5.0, 7.1, 9.9, 13.5, 17.7, 22.5, 27.5, 32.3, 36.6.
Reciprocals \(Z_t = 1/Y_t\), grouped in threes: \(S_1 = 0.44186\), \(S_2 = 0.17502\), \(S_3 = 0.09465\).
For \(Z_t = A + B\,C^t\) (with \(A = 1/k\), \(B = b/k\), \(C = e^{-c}\)): \(C^3 = (0.09465 - 0.17502)/(0.17502 - 0.44186) = 0.3012\), so \(C = 0.6703\); \(B = (-0.26684)(0.6703 - 1)/(0.3012 - 1)^2 = 0.1801\); \(A = \tfrac13\left[0.44186 + 0.26684/(0.3012 - 1)\right] = 0.02000\).
So \(k = 1/A = 50.0\) lakh, \(b = B/A = 9.0\) and \(c = -\ln C = 0.40\): \(Y_t = 50/(1 + 9e^{-0.4t})\), which reproduces all nine values to one decimal. The asymptote \(k = 50\) lakh is read as the city's carrying capacity, and the inflection (\(Y = 25\) lakh) falls at \(t = \ln 9/0.4 \approx 5.5\).
Mobile-phone penetration in a country fits a logistic with \(k \approx 1.05\) (proportion of population) — even higher than 1 because of multiple SIMs per person.
The Gompertz curve is also S-shaped but asymmetric — the inflection point is at \(Y_t = k/e \approx 0.37 k\), not at \(k/2\). Suited to biological growth, mortality and product life cycles where the deceleration phase is longer than the acceleration phase.
Equivalently: \(\log Y_t = \log k + b^t \log a\), so \(\log Y_t - \log k = (\log a)\, b^t\).
Take logs: \(\log Y_t = \log k + (\log a)\, b^t\). This is a modified-exponential in \(\log Y_t\). Apply three-selected-points or partial-sums on \(\log Y_t\).
Three selected points (with \(t_1, t_2, t_3\) equispaced):
Then derive \(\log a\) from any single point.
Population (10 yrs): 20, 35, 56, 80, 105, 125, 142, 153, 161, 165. Ten values do not split into three equal groups, so drop the first year and use the nine values from 35 onwards (\(t = 0, \dots, 8\)).
Sums of \(\log_{10} Y_t\) in threes: \(S_1 = 5.1953\), \(S_2 = 6.2704\), \(S_3 = 6.6090\).
\(b^3 = (6.6090 - 6.2704)/(6.2704 - 5.1953) = 0.3150\), so \(b = 0.680\); \(\log a = (1.0751)(0.680 - 1)/(0.3150 - 1)^2 = -0.732\); \(\log k = \tfrac13\left[5.1953 - 1.0751/(0.3150 - 1)\right] = 2.2549\), so \(k \approx 180\).
Fitted: \(\log_{10} Y_t = 2.2549 - 0.732\,(0.680)^t\), giving 33, 57, 82, 106, 125, 141, 152, 160, 166 against the data 35 to 165. The saturation level is about 180.
| Curve | Equation | Inflection at | Best for |
|---|---|---|---|
| Modified exponential | \(a + b c^t\) | None (no S-shape) | Approach to asymptote, no slow start |
| Logistic | \(k / (1 + b e^{-ct})\) | \(k/2\) | Symmetric S — biological populations |
| Gompertz | \(k\, a^{b^t}\) | \(k/e \approx 0.37k\) | Asymmetric S — mortality, mature products |
Detrending means removing the trend (T) component from a time series so the seasonal (S), cyclical (C) and irregular (I) components can be studied separately.
Sales (₹ lakhs) and fitted trend:
| Year | Sales \(Y\) | Trend \(T\) | Detrended \(Y/T\) |
|---|---|---|---|
| 2020 | 120 | 110 | 1.091 |
| 2021 | 140 | 130 | 1.077 |
| 2022 | 155 | 150 | 1.033 |
| 2023 | 165 | 170 | 0.971 |
| 2024 | 180 | 190 | 0.947 |
The ratio Y/T fluctuates around 1, revealing residual S/C/I components.
For series \(Y_t = 2 + 3 t + \epsilon_t\): \(\Delta Y_t = 3 + (\epsilon_t - \epsilon_{t-1})\). The deterministic linear trend disappears; only noise remains.