Extrapolation

Extrapolation is a mathematical method used to estimate unknown values. It works by looking at the relationship between variables within a known range. Once that relationship is understood, researchers project it beyond the original observations. This process is distinct from interpolation. Interpolation estimates values that fall between known data points. Extrapolation, however, attempts to predict what happens in the unknown territory outside the data set. Because it relies on assumptions about the future or the unseen, it carries a high risk of uncertainty. If the assumptions are wrong, the results can become entirely meaningless.

To choose a successful method, experts must understand the process that created the data. This is known as a priori knowledge. Scientists must ask specific questions about the data's nature. They check if the data is continuous, smooth, or perhaps periodic, meaning it repeats in a cycle. Some experts even suggest evaluating extrapolation through the study of causal forces. These are the underlying reasons why the data moves the way it does. Without understanding these forces, a mathematical model might fail to capture the true behavior of the system.

One of the most basic techniques is linear extrapolation. This method creates a tangent line at the very end of the known data points. It then extends that straight line into the unknown. This works well if the function is approximately linear or if the prediction does not go too far. If you have two points, $(x_1, y_1)$ and $(x_2, y_2)$, the linear formula is identical to linear interpolation. You can also use more points to improve the guess. By using regression-like techniques, you can average the slope of several points. This approach is very similar to what is called linear prediction.

More complex shapes can be used for harder problems, such as polynomial extrapolation. This method uses curves to fit the data. You can create a curve through the entire data set or just near the end. A linear approach uses two points, while a quadratic approach uses three points. Mathematicians often use Lagrange interpolation or Newton's method of finite differences to create these curves. However, high-order polynomials must be used with extreme care. If the degree is too high, the error grows quickly. This problem is related to a concept known as Runge's phenomenon.

Other methods include conic section extrapolation and French curve extrapolation. A conic section can be built using five points near the end of a data set. If the resulting shape is an ellipse or a circle, the extrapolation will eventually loop back to rejoin itself. If the shape is a parabola or a hyperbola, it will not rejoin. French curve extrapolation is specifically useful for distributions that show exponential tendencies. This method has been used for important health forecasts. For example, it helped project the growth of HIV/AIDS in the UK starting in 1987. It was also used to track variant CJD in the UK for several years.

Even when the math is correct, extrapolation can diverge from reality. A classic example involves the power series for a sine function, or $\sin(x)$. Near the point where $x = 0$, a simple linear estimate works perfectly. However, as you move away from that center, the estimate flies away from the axis. While the real sine wave stays between $-1$ and $1$, the extrapolated line moves toward infinity. This is called divergence. The error increases without bound because the mathematical tool does not perfectly match the true nature of the function. This is a specific property of extrapolation that is very difficult to avoid.

Extrapolation also appears in complex analysis and human reasoning. In the complex plane, a change of variable can sometimes turn an extrapolation problem into an interpolation problem. This involves mapping the area outside a unit circle to the area inside it. In biology, scientists use extrapolation arguments to move from animal studies to human studies. We also use it to trust technology. We believe light microscopes work because they extend the vision we have with magnifying glasses. We trust electron microscopes because they extend what we see with light microscopes. We are constantly using known patterns to navigate the unknown.