Find Parabola of the Form y = a + bx + cx² Using Least Square Method

Find Parabola of the Form y = a + bx + cx² Using Least Square Method

Introduction:

The least squares method is a useful technique for using mathematics to model real-world occurrences. This methodology provides a dependable method for determining the best-fit line or curve, whether it is utilized for data trend analysis or fitting curves to experimental data. This article explains how to find a parabola of the form y = a + bx + cx² using the least squares approach. It also discusses its importance and usefulness in other fields.

Understanding the Least Square Method:

The least square method is a mathematical approach to determining the best-fit curve for a set of data points. It operates by reducing the sum of the squares of the vertical deviations between each point and the curve. To put it another way, the goal is to minimize the total error and identify the curve that “best” fits the data.
The least squares method is based on the idea of creating an objective function that reflects the sum of squared disparities between observed data points and the associated model-predicted values. Minimizing this objective function allows us to identify the model parameters that best suit the data.

Understanding the Objective:

Before delving into how to use the least square method to find a parabola, it’s critical to grasp its function. When working with a set of data points (x, y), the goal is to identify the parameters a, b, and c that define the parabolic curve y = a + bx + cx² and best fit the presented data. To do this, the overall error between observed y-values and predicted values by the parabolic model must be minimized.

Finding a Parabola of the Form y = a + bx + cx²:

Now, let’s focus on finding a parabola of the form y = a + bx + cx² using the least square method. This form of a parabola includes three parameters: a (the y-intercept), b (the coefficient of x), and c (the coefficient of x²).

To apply the least square method, we need a set of data points (x, y). Once we have the data, we can proceed with the following steps:

1. Minimizing the Objective Function:

To minimize the objective function, find the values of a, b, and c that minimize the sum of squared discrepancies between the actual y-values and the parabolic model’s predictions. This is frequently performed using calculus techniques, specifically by computing and setting to zero the partial derivatives of the objective function with respect to each parameter (a, b, and c). Solving these equations simultaneously yields the optimal values for a, b, and c.

2. Implementation Using the Least Square Method:

Several steps are required to perform the least square approach for determining the parabolic curve, which can be facilitated by computational tools such as Python, NumPy, SciPy, and MATLAB. Here’s a summary of the implementation process:

• Data Collection: Begin by gathering the data points (x, y) for which you intend to fit the parabolic curve.
• Formulate Objective Function: Using the collected data, create the objective function outlined before.

• Minimize Objective Function: Use optimization techniques supplied by computational libraries to reduce the objective function and find the best values for parameters a, b, and c.
• To fit a parabolic curve: Use the ideal parameters (y = a + bx + cx²).

• Evaluate Fit: Evaluate the quality of the fit by examining the residuals (the differences between observed and predicted y-values) and assessing the overall goodness of fit metrics.

Applications and Significance:

The ability to find a parabola of the form y = a + bx + cx² using the least square method holds significant applications across various domains:

Data Analysis and Prediction: The least square method is used in domains such as finance, economics, and social sciences to examine data trends and create predictions using fitted curves.

Engineering and Physics: Engineers and physicists use the least square method to model physical phenomena and generate equations that represent experimental data, which aids in design and analytical procedures.

Optimization and Control Systems: Fitting curves to data is a common aspect of optimization issues, and the least square method is a reliable way to accomplish this. Accurate modeling of system dynamics is critical in control systems, and the least square method makes this easier by fitting curves to experimental data.

Conclusion:

A parabola (y = a + bx + cx²) that best fits a set of data points is produced by the least square method. This approach provides a stable solution for curve fitting and data analysis by minimizing the sum of squared differences between actual and expected values. Because of its adaptability and use, it is an essential tool for a variety of industries, helping analysts, engineers, and researchers to get insightful information from data and form reliable conclusions.

Parabolic Equation y = a+bx+cx2 :

This equation represents a general quadratic function, where:

Overall, the equation y=a+bx+cx2 provides a comprehensive framework for describing and studying quadratic functions, which have a wide range of applications in domains like as physics, engineering, and economics.

For Example:

Question:

Find the parabola of the form y = a+bx+cx2 which fits most closely with the following observations by using least square method.

x

-3

-2

-1

0

1

2

3

y

4.63

2.11

0.67

0.09

0.63

2.15

4.58

Find the parabola of the form y = a+bx+cx2 which fits most closely with the following observations by using least square method.

x

y

-3

4.63

-2

2.11

-1

0.67

0

0.09

1

0.63

2

2.15

3

4.58

Solution:

Let y = a+bx+cx2 be the parabola                (Eq 1)

Constructing the table as shown below;

x

y

xy

x2

x3

x4

x2y

-3

4.63

-13.89

9

-27

81

41.67

-2

2.11

-4.22

4

-8

16

8.44

-1

0.67

-0.67

1

-1

1

0.67

0

0.09

0

0

0

0

0

1

0.63

0.63

1

1

1

0.63

2

2.15

4.3

4

8

16

8.6

3

4.58

13.74

9

27

81

41.22

∑x = 0

∑y = 14.86

∑xy = -0.11

∑x2 = 28

∑x3 = 0

∑x4 = 196

∑x2y = 101.23

The normal equations are;

∑y = na + b∑x + c∑x2                   (Eq 2)

∑xy = a∑x +b∑x2 +c∑x3              (Eq 3)

∑x2y = a∑x2 + b∑x3 + c∑x4           (Eq 4)

From  (Eq 2)

14.86 = 7a + 28c              (Eq 5)

From  (Eq 3)

-0.11 = a (0) + b (28) +c (0)

-0.11 = 28b

b = -0.004

From  (Eq 4)

101.23 = 28a + 196c               (Eq 5)

Eq(5) – 4(Eq 4)

101.23 = 28a + 196c

59.44 = 28a + 112c

41.79 = 84c

c = 0.4975

Put value of “c” in eq 5

14.86 = 7a + 28 (0.4975)

7a = 14.86 – 13.93

7a = 0.93

a = 0.1328

Put values of “a,b and c” in  (Eq 1)

y = a +bx +cx2

y = 0.1328 – 0.004x + 0.4975x2

 

Picturized Solution: