Gaussians

Section 21.1 Gaussians

A Gaussian is a function of the form

\begin{equation} f(x)=N e^{-\frac{(x-x_0)^2}{2\sigma^2}}.\tag{21.1.1} \end{equation}

The interactive graph below shows the role of the parameters \(N\text{,}\) \(x_0\text{,}\) and \(\sigma\) on the shape of the graph.

Figure 21.1. The graph of a Gaussian with parameters \(N\text{,}\) \(x_0\text{,}\) and \(\sigma\text{.}\)

Sensemaking 21.1. The Relationship between Gaussian Parameters and the Shape of the Graph.

The interactive graph above allows you to explore the relationship between the parameters \(N\text{,}\) \(x_0\text{,}\) and \(\sigma\) in the formula for the Gaussian function and the shape of the graph. EXPLAIN these relationships.

Answer.

\(N\) is an overall multiplicative factor, so increasing (or decreasing) \(N\) increases (or decreases) the overall amplitude (height) of the function.
\(x_0\) appears in the function in the form \(x-x_0\text{,}\) i.e. it appears as a shift in the value of the independent variable, so increasing (or decreasing) \(x_0\) results in a shift of the graph to the right (or left).
\(\sigma\) occurs in the denominator of a fraction, with the shifted independent variable \(x-x_0\) in the numerator. When \(\sigma\) increases (or decreases), the value of the fraction decreases (or increases). This fraction squared appears in a negative exponent, so as the value of the fraction decreases (or increases), the value of the exponential increases (or decreases) which makes the Gaussian wider (or narrower).

Sensemaking 21.2. Derivatives of the Gaussian.

What do the first and second derivatives of the Gaussian function tell you about the shape of the graph?

Hint.

The sign of the first derivative tells you whether the function is increasing (positive derivative) or decreasing (negative derivative). The sign of the second derivative tells you whether the function is concave up (positive second derivative) or concave down (negative second derivative).