The estimation and approximation errors are related concepts often used in the fields of mathematics, statistics, and machine learning. However, they relate to somewhat distinct parts of the same fundamental concept in the context of measurement and numerical analysis. In this Answer, we'll learn about the estimation and approximation errors and the difference between them.
The estimation error refers to the difference between the true value of a quantity and its estimated or predicted value based on a model, method, or measurement. In other words, it indicates how far off our estimate is from the actual value. The estimation error can arise due to various factors, such as the inherent variability in the data, the assumptions made by a model, or the limitations of the measurement instruments.
Let’s say we want to estimate the average height of people in a city based on a sample of heights. The estimation error is the difference between our estimated average and the actual average for the entire population of a city.
Let’s have a look at the coding example:
import numpy as np# Simulated sample heights of people (in inches)sample_heights = np.array([65, 68, 70, 62, 72, 67])# Calculate the estimated average heightestimated_average = np.mean(sample_heights)# Actual average height for the entire population (known for this example)actual_average = 68.5# Calculate the estimation errorestimation_error = actual_average - estimated_averageprint("Estimated Average Height:", estimated_average)print("Actual Average Height:", actual_average)print("Estimation Error:", estimation_error)
Let's have a look at the code explanation:
Line 4: We create a NumPy array, sample_heights, which contains six people’s heights.
Line 7: We calculate the estimated_average height by taking the mean of the heights in the sample_heights array.
Line 10: We define the actual_average height for the entire population. In this example, we use it as 68.5.
Line 13: We calculate the estimation_error by subtracting the estimated_average height from the actual_average height.
Lines 15–17: Finally, we'll print the values of estimated_average, actual_average, and estimation_error on the console.
The approximation error, on the other hand, is specifically related to representing a complex or continuous function or value by a simpler or discrete one. It’s the discrepancy between the true value of a function and the value obtained by using an approximation method, such as truncating a Taylor series or using a numerical method to solve an equation.
The approximation error can arise when we simplify a complex mathematical expression or function to make it more manageable for computation or analysis. For example, suppose we’re approximating the value of
Let's have a look at the coding example:
import numpy as np# Number of terms in the series for approximationnum_terms = 5# Calculate pi using the first 'num_terms' terms of the Leibniz seriesapproximation = 0for i in range(num_terms):approximation += (-1) ** i / (2 * i + 1)approximation *= 4# Actual value of piactual_pi = np.pi# Calculate the approximation errorapproximation_error = actual_pi - approximationprint("Approximated value of pi:", approximation)print("Actual of pi:", actual_pi)print("Approximation Error:", approximation_error)
Let's have a look at the code explanation:
Lines 6–9: The for loop calculates the approximation of num_terms of the Leibniz series. The Leibniz series alternates between the addition and subtraction of fractions based on the index i.
Line 11: The Leibniz series approximates 4 gives the approximation of
Line 14: We store the actual value of numpy library using np.pi.
Line 17: Then, we calculate the approximation_error by subtracting the calculated approximation from the actual value of actual_pi.
Lines 19–20: Finally, we'll print the values of approximation, actual_pi, and approximation_error on the console.
Let's summarize the main differences between estimation error and approximation error in the given table:
Characteristic | Estimation Error | Approximation Error |
Context | Usually, this occurs when we are involved in data analysis, and our goal is to estimate a parameter or value based on a sample. | In analysis and mathematics, it is common to utilize techniques, such as numerical integration, finite element analysis, or series approximations. These methods help simplify functions and solve equations effectively. |
Calculation basis | The estimation error is determined using metrics like squared error (MSE), absolute error, or relative error. This typically entails comparing the estimated value to the actual value. | The approximation error is determined using mathematical techniques that depend on the nature of the approximation, such as Taylor series expansion or numerical methods like finite differences. |
Goals | Decrease the value as close to the true value, especially in predictive modeling, hypothesis testing, and regression analysis. | Minimize it to achieve a balance between precision and computational efficiency, often in numerical simulations, differential equation solving, or function approximation. |
Examples | Predicting future sales, estimating the mean of a population, or forecasting stock prices. | Numerical solutions to differential equations, polynomial interpolation, or when approximating functions by simpler functions like piecewise linear approximations. |
In essence, while estimation and approximation errors involve the difference between a calculated and true value, the distinction lies in their context and the reasons for the variations. The estimation error deals with the difference in making predictions or conclusions from data, while the approximation error focuses on simplifying complex values or functions for computation or analysis. These concepts are critical in the domains of decision-making, model-building, and data analysis.
Free Resources