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In Bishop book, page 4, section 1.1, there's a notation I don't seem to understand what's meant by it. The whole paragraph, with which the section begins, is:

We begin by introducing a simple regression problem, which we shall use as a running example throughout this chapter to motivate a number of key concepts. Suppose we observe a real-valued input variable x and we wish to use this observation to predict the value of a real-valued target variable t. For the present purposes, it is instructive to consider an artificial example using synthetically generated data because we then know the precise process that generated the data for comparison against any learned model. The data for this example is generated from the function sin(2πx) with random noise included in the target values, as described in detail in Appendix A. Now suppose that we are given a training set comprising N observations of x, written $x \equiv (x_1, . . . , x_N)^T$, together with corresponding observations of the values of t, denoted $t \equiv (t_1, . . . , t_N)^T$. Figure 1.2 shows a plot of a training set comprising N = 10 data points. The input data set x in Figure 1.2 was generated by choosing values of xn, for n = 1, . . . , N, spaced uniformly in range [0, 1], and the target data set t was obtained by first computing the corresponding values of the function 1.1. Example: Polynomial Curve Fitting 5 sin(2πx) and then adding a small level of random noise having a Gaussian distribution (the Gaussian distribution is discussed in Section 1.2.4) to each such point in order to obtain the corresponding value tn. By generating data in this way, we are capturing a property of many real data sets, namely that they possess an underlying regularity, which we wish to learn, but that individual observations are corrupted by random noise. This noise might arise from intrinsically stochastic (i.e. random) processes such as radioactive decay but more typically is due to there being sources of variability that are themselves unobserved.

I'm not clear what the notation $x \equiv (x_1, . . . , x_N)^T$ might mean here, especially the so called variable T. I guess it means it's the set of input vectors together with its corresponding set of target vectors (t). If I am right, so is this an example of supervised learning? Still, what's T?

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$x$ is a vector, composed of the individual elements $x_1$, $x_2$, etc. The "T" is for transpose. It is not a variable.

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  • $\begingroup$ Thank you, I should have guessed since x was a vector. But why x^T and not x? $\endgroup$ – Gigili Sep 26 '12 at 15:05
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    $\begingroup$ In textbooks vectors are pretty much always set up as row vectors simply because that is much more compact space-wise in a book. Sometimes you need them to be column vectors, though, when you multiply them with another vector or matrix. Look at how $x$ is used, that should tell you why it needs to be a column vector. $\endgroup$ – Jim Clay Sep 26 '12 at 15:07
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    $\begingroup$ Another bit of notation that you might see sometimes is $x \equiv (x_1, \ldots , x_N)^H$, which is similar. In this case, the $()^H$ indicates the Hermitian (or conjugate) transpose of the vector. $\endgroup$ – Jason R Sep 26 '12 at 15:39
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    $\begingroup$ @JasonR Do Hermitian transposes of vectors occur all that frequently in this context? $n\times n$ complex-valued matrices, yes, but vectors? $\endgroup$ – Dilip Sarwate Sep 26 '12 at 17:46
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    $\begingroup$ @Dilip: You're right, it's more commonly used with matrices, but I have seen it before. One example: the definition of a dot product between two vectors $x$ and $y$ in the Euclidean vector space is $y^Hx$. I've seen that notation used in books before to express the action of a vector of FIR filter coefficients on a window of input samples. $\endgroup$ – Jason R Sep 26 '12 at 18:45

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