# Distance

We define the Euclidean distance between two vectors $$a$$ and $$b$$ as the norm of their difference: $\mathbf{dist}(a, b) = ||a - b||$ For one, two, and three dimensions, this distance is exactly the usual distance between points with coordinates $$a$$ and $$b$$. However, the Euclidean distance is defined for vectors of any dimension. import numpy as np u = np.array([1.8, 2.0, -3.7, 4.7]) v = np.array([0.6, 2.1, 1.9, -1. …

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# Workshops Are A Bad Idea

If you work in an agency, especially one that prided itself on being a “digital agency” back in the early 2000s, you might encounter people who hold workshops as a way of fostering client commitment. The idea is to get all client-side stakeholders in a room and hold exercises and activities to cement mutual understanding about shared goals. However, workshops are a terrible idea from a psychological standpoint. A workshop enables a phenomenon called diffusion of responsibility which, if you ever did group projects in middle school, you should already be aware of. …

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# Norm

The Euclidean norm (or magnitude) of an $$n$$-vector $$x$$, denoted by $$||x||$$, is the square root of the sum of the squares of its elements, $||x|| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}$ or the inner product of the vector with itself, $$||x|| = \sqrt{x^Tx}$$. When $$x$$ is a scalar the norm is the same as the absolute value of $$x$$. import numpy as np x = np.array([2, -1, 2]) np. …

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# Regression Model

The regression model is a very commonly used affine function, especially when the $$n$$-vector $$x$$ represents a feature vector. The model is $\hat{y} = x^T \beta + v$ where $$\beta$$ is an $$n$$-vector (of features, i.e., variables) and $$v$$ is scalar. In this case, the entries of $$x$$ are called the regressors, and $$\hat{y}$$ is called the prediction, since the regression model is typically an approximation or prediction of some true value of $$y$$, which is called the dependent variable. …

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# Taylor Approximation

In many cases, you can approximate the relation between functions of $$n$$ variables with linear or affine functions. In these applications we refer to the linear or affine function relating the variables and the scalar variable as a model. Suppose that $$f : \mathbf{R}^n \to \mathbf{R}$$ is differentiable (meaning it has derivatives and partial derivatives). Let $$z$$ be an $$n$$-vector. The (first-order) Taylor approximation of $$f$$ near (or at) the point $$z$$ is the function $$\hat{f}(x)$$ defined as …

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