By scaling the error back to the data and taking the square root of it, we get sqrt(0.8)=0.89, so on average, the predictions differ by 0.89 from the real value. It takes into account the sum of squared errors instead of the errors as they are because sometimes they can be negative or positive and they could sum up to a nearly null value.įor example, if your real values are 2, 3, 5, 2, and 4 and your predicted values are 3, 2, 5, 1, 5, then the total error would be (3-2)+(2-3)+(5-5)+(1-2)+(5-4)=1-1+0-1+1=0 and the average error would be 0/5=0, which could lead to false conclusions. One may ask themselves why we choose to minimize the sum of squared errors instead of the sum of errors directly. Intuitively speaking, the aim of the ordinary least squares method is to minimize the prediction error, between the predicted and real values. Where p* is the number of explanatory variables to which we add 1 if the intercept is not fixed, wi is the weight of the ith observation, W is the sum of the wi weights, y the * vector of the observed values and y* the vector of predicted values What is the intuitive explanation of the least squares method? We can even the variance σ² of the random error ε by the following formula : The vector of the predicted values can be written as follows: With X the matrix of the explanatory variables preceded by a vector of 1s, D is a matrix with the wi weights on its diagonal, and y the vector of the n observed values of the dependent variable That way, the vector β of the coefficients can be estimated by the following formula The OLS method aims to minimize the sum of square differences between the observed and predicted values. How do ordinary least squares (OLS) work? If you want to find out more about the calculations, read the next paragraph. Moreover, before predicting, our method has to find the β coefficients: we just start out by inputting a table containing the heights of several plants along with the number of days they have spent in the sun. Of course, it is not always exact, which is why we must take into account the random error ε. β1 is 0.1 because it is the coefficient multiplied by the number of days.Ī plant being exposed 5 days to the sun has therefore an estimated height of Y = 30 + 0.1*5 = 30.5 cm.β0 is 30 because it is the value of Y when X is 0.X is the number of days spent in the sun.A plant grows 1 mm (0.1 cm) after being exposed to the sun for a day. In the case where there are n observations, the estimation of the predicted value of the dependent variable Y for the i th observation is given by:Įxample: We want to predict the height of plants depending on the number of days they have spent in the sun. Where Y is the dependent variable, β 0, is the intercept of the model, X j corresponds to the j th explanatory variable of the model (j= 1 to p), and e is the random error with expectation 0 and variance σ². In the case of a model with p explanatory variables, the OLS regression model writes: economy, if you need to predict a company’s turnover based on the amount of sales.Ī bit of theory: Equations for the Ordinary Least Squares regression The ordinary least squares formula: what is the equation of the model?.biology, if you need to predict the number of remaining individuals in a species depending on the number of predators or life resources.meteorology, if you need to predict temperature or rainfall based on external factors.In practice, you can use linear regression in many fields: Maximum likelihood and Generalized method of moments estimator are alternative approaches to OLS. Least squares stand for the minimum squares error (SSE). Ordinary Least Squares regression ( OLS) is a common technique for estimating coefficients of linear regression equations which describe the relationship between one or more independent quantitative variables and a dependent variable (simple or multiple linear regression).
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