Linear and Polynomial Fitting

### Linear and Polynomial Fitting

Linear and Polynomial regressions in Origin make use of weighted least-square method to fit a linear model function to data and a polynomial model function to data, respectively.

For Linear Fit, Polynomial Fit and Multiple Linear Regression, Origin provides multiple tools with powerful features such as Weighted Fitting, Residual Analysis, Analysis Themes, Consolidated Analysis Reports and Recalculation of Analysis Results upon changes in data. Origin also offers a rich avenue of options for output, for instance, Parameter Values, Standard Errors, Coefficient of Determination (COD), Standard Deviation, Number of Points (N), P-Value, t-Value, R, R-Square, Adjusted R-Square, ANOVA table, Parameter Correlation Matrix, Covariance Matrix.

We will demonstrate the key features of Origin Linear Fit and Polynomial Fit in the following sections:

Linear Fit

The Linear Fit tool fits the simple linear model to your data:

y = β0 + β1x

where β0 is the intercept and β1 is the slope.

In Origin, linear fitting process is highly controllable and it allows graphical examination of linearity:

• Masking Outlier During Linear Fit:

• Linear Fit with Fixed Intercept or Slope:

• Ellipse Plot for Graphical Examination of Linearity:

Apparent Fit

Linear Apparent Fit performs linear fit by using apparent values for the data according to the current axis scale settings.

For some data plot, its linearity only displays when we change the axis scale settings:

Linear Fit with X Error

Linear Fit with X Error minimizes the sum of square of error on both X and Y directions, which is more practical for real experimental data where errors exist in both X and Y directions.

Linear Fit with X Error tool in Origin provides York, FV and Deming methods to minimize the parameter chi-square during fitting:

Polynomial Fit

Polynomial regression in Origin performs fit to data using the following model:

y = β0 + β1x + β2x2 + .. + βnxn

where βn are the coefficients.

In Origin, Polynomial Fit can fit data with polynomial up to 9th order and it also supports fitting with fixed intercept or slope and apparent fit:

• Up to 9th Order Polynomial Fit:

• Polynomial Fit with Fixed Intercept or Slope:

• Apparent Fit According to Current Axis Settings:

Multiple Linear Regression

Multiple Linear Regression fits multiple independent variables with the following model:

y = β0 + β1x1 + β2x2 + .. + βnxn

where βn are the coefficients.

An unique feature in Multiple Linear Regression in Origin is a Partial Leverage Plot output, which can help to study the relationship between the independent variable and a given dependent variable: