Interpolation for Datasets
Interpolation is a statistical technique that estimates an unknown price or possible yield of an asset using related known variables. The unknown values can be interpolated by using other known values that are in the same order.
Interpolation is a method of estimating a function’s value for any intermediate value of the independent variable, whereas extrapolation is the process of calculating the function’s value outside of the specified range.
Interpolation is fundamentally a straightforward concept in mathematics. If a set of data points exhibits a relatively consistent trend, it is possible to estimate the value of the set at uncalculated points.
Investors and stock analysts typically produce a line chart with interpolated data points in the stock market to predict the stock. These charts are a crucial component of technical analysis to display changes in the price of stock.
The majority of the time when pre-processing data, interpolation is utilised to fill in missing values in the data frame or series.
In computer graphics discipline, interpolation technique is used for image processing so that, when an image is expanded, the pixel value can be estimated with the aid of nearby pixels.
Interpolation is the best method when average imputing missing values does not fit the data at the best.
In scientific field, when working with time-series data, interpolation is frequently used to fill in the gaps left by missing values with the preceding one or two values. E.g., temperature, wind pressure, humidity, solar radiation, etc,.
Python Pandas Syntax for interpolation:
DataFrame.interpolate(method=’linear’, axis=0, limit=None, inplace=False, limit_direction=None, limit_area=None, downcast=None, **kwargs)
Primary types of Interpolation in Data
1. Linear Interpolation
Linear interpolation refers to estimating a missing value by joining points in a straight line in ascending order. In essence, it projects the unknown value in the same ascending sequence as the ones that came before. Interpolation uses linear as its default method.
Formula:
𝑦=𝑦1+(𝑥−𝑥1)*(𝑦2−𝑦1) / (𝑥2−𝑥1)
Note: It interprets values by joining points in a straight line rather than by using the index.
a. Forward Interpolation
Mathematical Explanation
Forward Differences: The differences f1 — f0, f2 — f1, f3 — f2, ……, fn — fn–1 when denoted by △f0, △f1, △f2, ……, △fn–1 are respectively, called the first forward differences.
Formula :
h = difference between x1-x0 or x2-x1
r = (x — x0)/h
fr = f0 + r△f0 + r * (r-1)/2! * △f20 + r * (r-1) * (r-2)/3! *△f20 + …
a. Backward Interpolation
Mathematical Explanation
h = difference between x1-x0 or x2-x1
r = (x — x0)/h
fr = f0 + r ▽f0 + r* (r+1)/2! *▽f20 + r * (r+1) * (r+2)/3!*▽f20 + …
1. Polynomial Interpolation
In polynomial interpolation, an order must be specified. In other words, polynomial interpolation fills in missing values to the lowest degree possible through the data points that are still available. The polynomial interpolation curve has a form that resembles a parabola or the trigonometric sin curve.
The most precise objective of polynomial interpolation is to identify the polynomial with the lowest degree that connects the points in the dataset. The simplest form of the polynomial is the lowest degree.
There are three common methods for Polynomial Interpolation:
· Lagrange Polynomial Interpolation
· Newton Polynomial Interpolation, also called Newton’s divided differences interpolation polynomial
· Spline Interpolation and more specifically Cubic Spline Interpolation
Lagrange Polynomial Interpolation
Even though they arrive at the same conclusion via distinct calculations, the Lagrange and Newton polynomials produce the same precise outcome. The polynomial function with the shortest order that passes across the data points is produced by the Lagrange and Newton methods.
Newton Polynomial Interpolation
A Newton polynomial is an interpolation polynomial for a given set of data points in the mathematical subject of numerical analysis. It is named after its creator, Isaac Newton. Because its coefficients are determined by applying Newton’s split differences method, the Newton polynomial is occasionally referred to as the “Newton’s divided differences interpolation polynomial.”
Spline Interpolation
Spline interpolation differs slightly from other methods in that it estimates a piece-wise polynomial rather than a single polynomial. For “real” polynomial interpolation to perfectly cover all of your data points, a very complicated polynomial may be needed. The polynomial may adopt an excessively volatile curve with numerous unwelcome spikes in between data points when complexity rises to an unacceptably high level.
Conclusion — Calculating the data points between the given data points is helped by interpolation. In a nutshell, interpolation is a technique used in data analytics and wrangling to identify unknown values that lie between known data points.
