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Questions and Answers
Polynomials are never used for curve fitting.
Polynomials are never used for curve fitting.
False
In exact polynomial fitting, there are always more data points than the degree of the polynomial.
In exact polynomial fitting, there are always more data points than the degree of the polynomial.
False
Regression analysis maximizes the norm of the error vector.
Regression analysis maximizes the norm of the error vector.
False
Spline curve fitting involves passing a single polynomial through all the data points.
Spline curve fitting involves passing a single polynomial through all the data points.
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The coefficients of a polynomial can be determined by solving a non-linear system of equations.
The coefficients of a polynomial can be determined by solving a non-linear system of equations.
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In the given system of linear equations, the element aij is given by (xi)j-1.
In the given system of linear equations, the element aij is given by (xi)j-1.
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The polynomial y = 1 - (10/3)x + 3x2 - (2/3)x3 passes through the points (0, 1), (1, 0), (2, 1), and (3, 0).
The polynomial y = 1 - (10/3)x + 3x2 - (2/3)x3 passes through the points (0, 1), (1, 0), (2, 1), and (3, 0).
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The standard deviation of a sample is given by S = sqrt((y_i - y)^2 / (n - 1)).
The standard deviation of a sample is given by S = sqrt((y_i - y)^2 / (n - 1)).
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The arithmetic mean is calculated by dividing the sum of the individual data points by the number of points minus one.
The arithmetic mean is calculated by dividing the sum of the individual data points by the number of points minus one.
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The descriptive statistics are used to represent the shape of the distribution of the data.
The descriptive statistics are used to represent the shape of the distribution of the data.
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Study Notes
Curve Fitting
- Curve fitting involves finding a functional relationship between one or more independent variables and a dependent variable for a given set of data points.
- The most commonly used functions for curve fitting are polynomials.
Types of Curve Fitting
- Exact polynomial fitting: The number of data points is exactly equal to (n+1), where n is the degree of the polynomial.
- Inexact polynomial fitting: The number of data points is greater than (n+1), and a single polynomial of degree n cannot pass through all the data points exactly.
Exact Polynomial Fitting
- Given (n+1) points, there is a unique polynomial of degree n that passes through these points.
- The coefficients of the polynomial can be determined by solving a linear system of equations.
Linear System of Equations
- The linear system of equations can be written in matrix form as [A]{B} = {Y}.
- The elements of [A] are given by aij = (xi-1)j-1, where i = 1, 2, ..., n+1, and j = 1, 2, ..., n+1.
Example of Exact Polynomial Fitting
- Example 2.1: Determine the 3rd degree polynomial that passes through the points (0, 1), (1, 0), (2, 1), and (3, 0).
- Solution: The coefficients of the polynomial are b0 = 1.0, b1 = -3.3333, b2 = 3.0, and b3 = -0.66666.
Mathematical Background
Statistics
- Descriptive statistics provide a summary of the data and convey information about specific characteristics of the data set.
- Common descriptive statistics include:
- Arithmetic mean: The sum of individual data points divided by the number of points.
- Standard deviation: A measure of the spread of the data.
Arithmetic Mean
- The arithmetic mean is calculated as Σ yi / n, where yi is the individual data point and n is the number of points.
Standard Deviation
- The standard deviation is calculated as S y = √[Σ (yi - y)^2 / (n - 1)], where yi is the individual data point, y is the arithmetic mean, and n is the number of points.
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Description
Determine the coefficients of a functional relationship between independent and dependent variables using polynomial curve fitting. Learn how to apply polynomial equations to fit a given set of data points.
