Linear regression in Origin uses weighted leastsquare method to fit a linear model function to data. The leastsquare method minimize the sum of the squares of the deviations between the theoretical curve and the experimental points for a range of independent variables. Origin provides multiple tools for Simple Linear Fit, Polynomial Fit and Multiple Linear Regression. A Quick Fit Gadget tool is also available for interactive fitting of data in a graph.
All fitting tools support powerful features such as weighted fitting, Residual Analysis, Analysis Themes, Consolidated Analysis Reports, and Recalculation of Analysis Results on data change. The output provides various options including Parameter Values, Standard Errors, Coefficient of Determination (COD), Standard Deviation, Number of Points (N), PValue, tValue, R, Rsquare, adjusted Rsquare, ANOVA table, parameter Correlation matrix, and Covariance matrix. Beside these general features in curve fitting, Origin also provides unique features for different tools:
Linear Fit
The Linear Fit tool fits the simple linear model to your data:

y = β_{0} + β_{1}x
where β_{0} is the intercept and β_{1} is the slope. With the Linear Fit tool, you have additional options to:
 Fix intercept of slope during fitting.
 Perform Apparent Fit. This option will use apparent values for the data, according to the current axis scale settings. This option is useful for example to perform a linear fit to data that appears to be linear when the x axis scale is set to log
 Ellipse Plot to allow graphical examination of correlation between two variables.
Besides the GUI tool, you can also perform linear fitting from LabTalk script using the fitlr XFunction. 
Linear Apparent Fit with Confidence and Prediction Bands 
Linear fit with X Error
Starting with OriginPro 8.1, we provide the option to perform simple linear fit with support for on both X and Y direction. The traditional leastsquare fitting minimizes the sum of square of error on Y direction only. However, with practical data, measurement errors may exist in both X and Y directions. The new Fit Linear with X Error tool provides two methods (York and FV) to minimize the chisquare in such cases.
Polynomial Fit
Polynomial regression in Origin performs fit to data using the following model:

y = β_{0} + β_{1}x + β_{2}x + .. + β_{n}x
where β_{n} are the coefficients.
Similar to simple linear fit, Polynomial Fit provides option for fixing intercept to a specific value, and also option for performing apparent fit. Polynomials up to order 9 can be used to fit data. However, it is worth noting that the higher order terms in polynomial equation have the greatest effect on the dependent variable. Consequently, models with high order terms (higher than 4) tend to be overparameterized and extremely sensitive to the precision of coefficient values, where small differences in the coefficient values can result in a larges differences in the computed y value.
The fitpoly XFunction can be used to perform polynomial fitting from LabTalk script.

Different polynomial orders to fit the data 
Multiple Linear Regression
Multiple Linear Regression fits multiple independent variables with the following model:

y = β_{0} + β_{1}x_{1} + β_{2}x_{2} + .. + β_{n}x_{n}
A 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.
