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If I execute the commands

my_reg = LinearRegression()
lin.reg.fit(X,Y)

I train my model. To my understanding training a model is calculating coefficient estimators.

I do not really understand the difference between this and e.g.

scipy.stats.linregress(X,Y)

calculating a 'normal' regression that also gives me the coefficient estimators and all the other statistics connected with it.

Could anyone tell me what is the difference here?

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They both solve the exact same objective, which is minimizing the mean squared error. However the second method can answer "how confident it is that slope is not zero, i.e. $Y$ is correlated with $X$?" via p-value.

In detail

Lets denote the data as $(X, Y) = \{(x_n, y_n)|x_n \in \mathbb{R}^D, y_n \in \mathbb{R}\}$. And the regression as $\hat{y} = Ax+B$.

Extra quantities returned by scipy.stats.linregress(X,Y) are: rvalue ($r$), and pvalue ($p$).

In statistics, $r^2$ (known as r-squared) measures the "goodness-of-fit" . That is, as regression $\hat{y}=Ax+B$ gets closer to observation $y$, $r^2$ gets closer to $1$. Since it is a function of $y$ and $\hat{y}$, it can be calculated for the first method too. So no difference here.

However, $p$ is specific to second method. scipy.stats.linregress(X,Y) adds a normality assumption to noise, i.e. assumes $\epsilon \sim N(0, \sigma^2)$ where $$\epsilon = y - \overbrace{Ax+B}^{\hat{y}}$$ On the basis of this assumption, it can answer an additional question: "how confident it is that the slope is not zero?". The first method cannot answer this question.

For example, suppose the estimated slope is $2.1$ for both methods, we still cannot tell whether this slope is significant or $Y$ is actually independent of $X$. Unless we look at the value of $p$. For example, for $p < 0.01$ we are confident (at significance level $0.01$) that $Y$ is correlated with $X$, but for $p > 0.1$ we cannot be confident, i.e. slope $2.1$ could be due to chance and $Y$ might be independent of $X$.

This link gives more details on how p-value is actually calculated in second method.

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There is no difference in the conceptual sense - both methods calculate linear regression coefficients. The difference lies in the interface - while through scipy.stats you gain the coefficients directly (and it is up to you to put them into an equation to calculate the predictions), scikit-learn wraps them into a model object so that you can use it in a similar fashion to other ML models such as decision trees, for example. (Actually, you can obtain the regression coefficients from the fitted scikit-learn model using my_reg.coef_.)

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