Interpolation

Interpolation is a method used to estimate unknown values between known data points. In data analysis, it is often applied when you have missing values in a dataset or want to generate smoother curves for visualization.

Formally, if you have data points $(x_1,y_1),(x_2,y_2),...,(x_n,y_n)$, interpolation constructs a function $f(x)$ such that $f(x_i)=y_i$ for all $i$, and then you can use $f(x)$ to estimate $y$ at values of $x$ that are not in the original data.

Interpolation is useful for filling missing data, smoothing curves, and resampling datasets, but one should avoid extrapolating too far beyond the known points, as it can be unreliable.

Common types of interpolation:

1. Linear interpolation: Connects two adjacent points with a straight line. Simple and fast. $y=y_1+\frac{(x-x_1)(y_2-y_1)}{x_2-x_1}$

2. Polynomial interpolation: Fits a single polynomial through all points. Can be accurate for few points but unstable for many points.

3. Spline interpolation: Fits piecewise polynomials (usually cubic) between points, ensuring smoothness at joins. Often used in time series and graphics.

4. Nearest-neighbor interpolation: Assigns the value of the nearest known point. Simple but can produce jumps.

5. Time-series specific methods: Forward-fill, backward-fill, or more advanced methods like linear or spline interpolation along the time axis.