This **polynomial regression calculator** uses **y = β₀ + β₁x + β₂x² + β₃x³ + … + βₙxⁿ + ε** formula to predict the **relationship** between variables when that relationship is **not linear**.

The calculator uses a **best-fit curve** to represent the relationship between the **independent variable** (x) and the **dependent variable** (y). This curve is described by a **polynomial equation**, which can be of various **degrees** depending on the complexity of the relationship.

**Sample conversions:**

**Linear to Quadratic:**- Linear equation:
**y = 2x + 3** - Quadratic conversion:
**y = ax² + bx + c** - Result:
**y = 0x² + 2x + 3**

- Linear equation:
**Quadratic to Cubic:**- Quadratic equation:
**y = 2x² – 3x + 1** - Cubic conversion:
**y = ax³ + bx² + cx + d** - Result:
**y = 0x³ + 2x² – 3x + 1**

- Quadratic equation:

## Polynomial Regression Calculator

Data Points | Degree | Polynomial Equation | R² Value | Conversion Equation |
---|---|---|---|---|

(1,2), (2,4), (3,8), (4,16) | 2 | y = 1 + 0.5x + 0.5x² | 0.9989 | Linear to Quadratic: y = ax² + bx + c |

(0,1), (1,3), (2,3), (3,1) | 3 | y = 1 + 3x – 1.5x² + 0.1667x³ | 1.0000 | Quadratic to Cubic: y = ax³ + bx² + cx + d |

(1,5), (2,8), (3,13), (4,20) | 1 | y = -1 + 5x | 0.9929 | Quadratic to Linear: y = mx + b |

(-2,9), (-1,4), (0,1), (1,0) | 2 | y = 1 – 2x + x² | 1.0000 | Cubic to Quadratic: y = ax² + bx + c |

(1,1), (2,8), (3,27), (4,64) | 3 | y = 1x³ | 1.0000 | Linear to Cubic: y = ax³ + bx² + cx + d |

**Data Points**represent the (x,y) coordinates used for regression.**Degree**indicates the highest power of x in the polynomial equation.**Polynomial Equation**is the**best-fit equation**derived by the calculator.**R² Value**(coefficient of determination) indicates how well the model fits the data (**1.0000**is a perfect fit).**Conversion Equation**shows how the calculator transformed the data into the given polynomial form.

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## Polynomial Regression Formula

The **general formula** for polynomial regression is:

**y = β₀ + β₁x + β₂x² + β₃x³ + … + βₙxⁿ + ε**

Where:

**y**is the**dependent variable****x**is the**independent variable****β₀, β₁, β₂, …, βₙ**are the**coefficients****n**is the**degree**of the polynomial**ε**is the**error term**

For a **second-degree polynomial** (quadratic regression), the formula to find the coefficients is:

**[β₀, β₁, β₂] = [Σx⁰, Σx¹, Σx²; Σx¹, Σx², Σx³; Σx², Σx³, Σx⁴]⁻¹ [Σy; Σxy; Σx²y]**

Where **Σ** represents the **sum**, and the superscripts indicate the **power** to which x is raised.

**Example:** Let’s say we have the following **data points**: (1, 2), (2, 5), (3, 10)

- Calculate the sums:
**Σx⁰ = 3**,**Σx¹ = 6**,**Σx² = 14**,**Σx³ = 36**,**Σx⁴ = 98****Σy = 17**,**Σxy = 41**,**Σx²y = 106** - Solve the
**matrix equation**to find**β₀, β₁,**and**β₂**. - The resulting equation might be:
**y = 1 + 0.5x + 1.5x²**

## Polynomial Regression Examples

**Weather Forecasting:**Predicting**temperature changes**throughout the day using a cubic polynomial:**Temperature = 15 + 2t – 0.5t² + 0.03t³**(Where**t**is the time in hours since midnight)**Economic Growth:**Modeling**GDP growth**over time with a quadratic function:**GDP = 1000 + 50t + 2t²**(Where**t**is the number of years since a reference point)**Drug Dosage Response:**Analyzing the**effectiveness**of a drug at different dosages:**Effect = 10 + 5d – 0.2d²**(Where**d**is the dosage in milligrams)

These examples shows how **polynomial regression** can be applied in various fields to model **complex relationships**.

## What is Polynomial Regression?

**Polynomial Regression** is a form of **regression analysis** where the relationship between the independent variable **x** and the dependent variable **y** is modeled as an **nth degree polynomial**.

It’s an extension of **linear regression** that allows for more **complex**, **curvilinear relationships** between variables.

Key features of polynomial regression include:

**Flexibility:**It can model a wide range of relationships, from simple linear to complex curved patterns.**Overfitting risk:**Higher degree polynomials can lead to**overfitting**, where the model performs well on training data but poorly on new data.**Interpretability:**Lower degree polynomials (2nd or 3rd degree) are often easier to interpret than higher degree ones.**Extrapolation caution:**Polynomial models may perform poorly when extrapolating beyond the range of the training data.

**Polynomial regression** is particularly useful when dealing with data that exhibits clear **non-linear trends** that cannot be adequately captured by a simple straight line.

## Types of Polynomial Regression

Polynomial regression can be classified based on the **degree** of the polynomial used in the model:

**Linear Regression (1st degree):****y = β₀ + β₁x**This is the simplest form and assumes a**straight-line relationship**.**Quadratic Regression (2nd degree):****y = β₀ + β₁x + β₂x²**Useful for modeling**parabolic relationships**with one turning point.**Cubic Regression (3rd degree):****y = β₀ + β₁x + β₂x² + β₃x³**Can model**S-shaped relationships**with up to two turning points.**Higher Degree Polynomials:**These can model more**complex relationships**but are prone to**overfitting**.**Fractional Polynomials:**These use**non-integer powers**of x, offering more**flexibility**in some cases.

Each type has its own **advantages** and **use cases**, depending on the nature of the data and the complexity of the relationship being modeled.