Probé con:
\documentclass{article}
\usepackage{amsmath}%amssymb,
\usepackage{bm}
\begin{document}
Outputs to machine learning models are also often represented as vectors. For instance, consider an object recognition model that takes an image as input and emits a set of numbers indicating the probabilities that the image contains a dog, human, or cat, respectively. The output of such a model is a three element vector $\vec{y} = \begin{bmatrix}y_{0}\\y_{1}\\y_{2}\\\dfrac{1}{2}\end{bmatrix}$, where the number $y_{0}$ denotes the probability that the image contains a dog, $y_{1}$ denotes the~probability that the image contains a human, and $y_{2}$ denotes the probability that the image contains a cat. Figure~\ref{fig:vec_out} shows some possible input images and corresponding output vectors.
\begin{align*}
p\left( x \right)
&= \overbrace{ \pi_{1}}^{0.33}\mathcal{N}\left( \vec{x}; \, \overbrace{ \vec{\mu}_{1} }^{\begin{bmatrix}
152\\55
\end{bmatrix}} \overbrace{ \bm{\Sigma}_{1}}^{ \begin{bmatrix}
20 &0\\0 &28
\end{bmatrix} } \right)
+ \overbrace{ \pi_{2} }^{0.33} \mathcal{N}\left(\vec{x}; \, \overbrace{ \vec{\mu}_{2} }^{ \begin{bmatrix}
175\\70
\end{bmatrix} }, \overbrace{ \bm{\Sigma}_{2}}^{ \begin{bmatrix}
35 & 39\\39 & 51
\end{bmatrix} } \right)\\
&+ \overbrace{ \pi_{3} }^{0.33} \mathcal{N}\left(\vec{x}; \, \overbrace{ \vec{\mu}_{3} }^{ \begin{bmatrix}
135\\40
\end{bmatrix} }, \overbrace{ \bm{\Sigma}_{3}}^{ \begin{bmatrix}
10 & 0\\0 & 10
\end{bmatrix} } \right)
\end{align*}
\end{document}
Salida producida como:
¿Cómo puedo hacer un espacio igual en la parte superior e inferior de la matriz? por favor avise
Además, sería más útil que alguien explicara por qué sucedió esto.
Respuesta1
No estoy seguro de si esto es lo que quieres, pero puedes incluir el contenido de cada parte con \vcenter{\hbox{$ ... $}}
.
\documentclass{article}
\usepackage{amsmath}%amssymb,
\usepackage{bm}
\begin{document}
\begin{align*}
p\left( x \right)
&= \overbrace{ \pi_{1}}^{0.33}\mathcal{N}\left(\vcenter{\hbox{$ \vec{x}; \, \overbrace{ \vec{\mu}_{1} }^{\begin{bmatrix}
152\\55
\end{bmatrix}} \overbrace{ \bm{\Sigma}_{1}}^{ \begin{bmatrix}
20 &0\\0 &28
\end{bmatrix} } $}}\right)
+ \overbrace{ \pi_{2} }^{0.33} \mathcal{N}\left(\vcenter{\hbox{$ \vec{x}; \, \overbrace{ \vec{\mu}_{2} }^{ \begin{bmatrix}
175\\70
\end{bmatrix} }, \overbrace{ \bm{\Sigma}_{2}}^{ \begin{bmatrix}
35 & 39\\39 & 51
\end{bmatrix} } $}}\right)\\
&+ \overbrace{ \pi_{3} }^{0.33} \mathcal{N}\left(\vcenter{\hbox{$ \vec{x}; \, \overbrace{ \vec{\mu}_{3} }^{ \begin{bmatrix}
135\\40
\end{bmatrix} }, \overbrace{ \bm{\Sigma}_{3}}^{ \begin{bmatrix}
10 & 0\\0 & 10
\end{bmatrix} } $}}\right)
\end{align*}
\end{document}
Respuesta2
Ha recibido una buena respuesta, pero le sugiero encarecidamente que se salga con la suya \overbrace
:
\documentclass{article}
\usepackage{amsmath}
\usepackage{bm}
\begin{document}
Outputs to machine learning models are also often represented as vectors.
For instance, consider an object recognition model that takes an image as
input and emits a set of numbers indicating the probabilities that the
image contains a dog, human, or cat, respectively. The output of such
a model is a three element vector
$\vec{y} = [\begin{matrix}y_{0} & y_{1} & y_{2} & \frac{1}{2}\end{matrix}]^T$,
where the number $y_{0}$ denotes the probability that the image contains a dog,
$y_{1}$ denotes the~probability that the image contains a human, and $y_{2}$
denotes the probability that the image contains a cat. Figure~\ref{fig:vec_out}
shows some possible input images and corresponding output vectors.
\begin{gather*}
p(x) = \pi_{1} \mathcal{N} ( \vec{x}; \, \vec{\mu}_{1}, \bm{\Sigma}_{1})
+ \pi_{2} \mathcal{N} ( \vec{x}; \, \vec{\mu}_{2}, \bm{\Sigma}_{2})
+ \pi_{3} \mathcal{N} ( \vec{x}; \, \vec{\mu}_{3}, \bm{\Sigma}_{3})
\\[1ex]
\begin{aligned}
\pi_1&=0.33 & \pi_2&=0.33 & \pi_3&=0.33
\\
\vec{\mu}_{1}&=\begin{bmatrix} 152 \\ 55 \end{bmatrix}, &
\vec{\mu}_{2}&=\begin{bmatrix} 175 \\ 70 \end{bmatrix}, &
\vec{\mu}_{3}&=\begin{bmatrix} 135 \\ 40 \end{bmatrix}
\\
\bm{\Sigma}_{1}&=\begin{bmatrix} 20 & 0 \\ 0 & 28 \end{bmatrix}, &
\bm{\Sigma}_{2}&=\begin{bmatrix} 35 & 39 \\ 39 & 51 \end{bmatrix}, &
\bm{\Sigma}_{3}&=\begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}
\end{aligned}
\end{gather*}
\end{document}
Tenga en cuenta el vector columna escrito como la transposición de un vector fila, lo que evita espacios entre líneas.
Con una alineación alternativa
\documentclass{article}
\usepackage{amsmath}
\usepackage{bm}
\begin{document}
Outputs to machine learning models are also often represented as vectors.
For instance, consider an object recognition model that takes an image as
input and emits a set of numbers indicating the probabilities that the
image contains a dog, human, or cat, respectively. The output of such
a model is a three element vector
$\vec{y} = [\begin{matrix}y_{0} & y_{1} & y_{2} & \frac{1}{2}\end{matrix}]^T$,
where the number $y_{0}$ denotes the probability that the image contains a dog,
$y_{1}$ denotes the~probability that the image contains a human, and $y_{2}$
denotes the probability that the image contains a cat. Figure~\ref{fig:vec_out}
shows some possible input images and corresponding output vectors.
\begin{alignat*}{3}
p(x) = \pi_{1} &\mathcal{N} ( \vec{x}; \, \vec{\mu}_{1}, \bm{\Sigma}_{1})
&{}+ \pi_{2} &\mathcal{N} ( \vec{x}; \, \vec{\mu}_{2}, \bm{\Sigma}_{2})
&{}+ \pi_{3} &\mathcal{N} ( \vec{x}; \, \vec{\mu}_{3}, \bm{\Sigma}_{3})
\\[1ex]
\pi_1&=0.33 & \pi_2&=0.33 & \pi_3&=0.33
\\
\vec{\mu}_{1}&=\begin{bmatrix} 152 \\ 55 \end{bmatrix}, &
\vec{\mu}_{2}&=\begin{bmatrix} 175 \\ 70 \end{bmatrix}, &
\vec{\mu}_{3}&=\begin{bmatrix} 135 \\ 40 \end{bmatrix}
\\
\bm{\Sigma}_{1}&=\begin{bmatrix} 20 & 0 \\ 0 & 28 \end{bmatrix}, &
\bm{\Sigma}_{2}&=\begin{bmatrix} 35 & 39 \\ 39 & 51 \end{bmatrix}, &
\bm{\Sigma}_{3}&=\begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}
\end{alignat*}
\end{document}
Para completar, así es como puede realizar la tarea propuesta. Dejé el gran vector de columna en línea para mostrar por qué es realmente malo.
Comparando las salidas no tengo ninguna duda.
\documentclass{article}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{delarray}
\begin{document}
Outputs to machine learning models are also often represented as vectors.
For instance, consider an object recognition model that takes an image as
input and emits a set of numbers indicating the probabilities that the
image contains a dog, human, or cat, respectively. The output of such
a model is a three element vector
$\vec{y} = \begin{bmatrix}y_{0} \\ y_{1} \\ y_{2} \\ \dfrac{1}{2}\end{bmatrix}$,
where the number $y_{0}$ denotes the probability that the image contains a dog,
$y_{1}$ denotes the~probability that the image contains a human, and $y_{2}$
denotes the probability that the image contains a cat. Figure~\ref{fig:vec_out}
shows some possible input images and corresponding output vectors.
\begin{equation*}
\begin{aligned}
p(x)
&= \overbrace{ \pi_{1}}^{0.33}\mathcal{N}
\begin{array}[b]({c})
\vec{x}; \, \overbrace{ \vec{\mu}_{1} }^{\begin{bmatrix}
152\\55
\end{bmatrix}} \overbrace{ \bm{\Sigma}_{1}}^{ \begin{bmatrix}
20 &0\\0 &28
\end{bmatrix} }
\end{array}
+ \overbrace{ \pi_{2} }^{0.33} \mathcal{N}
\begin{array}[b]({c})
\vec{x}; \, \overbrace{ \vec{\mu}_{2} }^{ \begin{bmatrix}
175\\70
\end{bmatrix} }, \overbrace{ \bm{\Sigma}_{2}}^{ \begin{bmatrix}
35 & 39\\39 & 51
\end{bmatrix} }
\end{array}\\
&+ \overbrace{ \pi_{3} }^{0.33} \mathcal{N}
\begin{array}[b]({c})\vec{x}; \, \overbrace{ \vec{\mu}_{3} }^{ \begin{bmatrix}
135\\40
\end{bmatrix} }, \overbrace{ \bm{\Sigma}_{3}}^{ \begin{bmatrix}
10 & 0\\0 & 10
\end{bmatrix} }
\end{array}
\end{aligned}
\end{equation*}
\end{document}
Respuesta3
Basta con sustituir las tres instancias de \left( ... \right)
por \bigl( ... \bigr)
. Obsérvese que, dado que los segundos argumentos de los \overbrace
constructos más amplios son explicativos más que definitorios, no es necesario encerrarlos entre paréntesis (que ahora ya no son muy altos).
Ah, y a menos que quieras llamar mucho la atención sobre la definición de \vec{y}
en el párrafo que precede al align*
entorno, lo escribiría como un vector de fila en lugar de como un vector de columna.
\documentclass{article}
\usepackage{amsmath,amssymb,bm}
\begin{document}
Outputs to machine learning models are also often represented as vectors. For instance, consider an object recognition model that takes an image as input and emits a set of numbers indicating the probabilities that the image contains a dog, human, or cat, respectively. The output of such a model is a three element vector
$\vec{y} = \begin{bmatrix} y_{0} & y_{1} & y_{2} \end{bmatrix}'$,
where the number $y_{0}$ denotes the probability that the image contains a dog, $y_{1}$ denotes the~probability that the image contains a human, and $y_{2}$ denotes the probability that the image contains a cat. Figure~\ref{fig:vec_out} shows some possible input images and corresponding output vectors.
\begin{align*}
p(x)
&=\overbrace{ \pi_{1}\mathstrut}^{0.33}\mathcal{N}
\bigl( \vec{x};
\overbrace{ \vec{\mu}_{1} }^{
\begin{bmatrix} 152\\55 \end{bmatrix}} ,
\overbrace{ \bm{\Sigma}_{1}}^{
\begin{bmatrix} 20 &0\\0 &28 \end{bmatrix} }
\bigr)
+\overbrace{ \pi_{2}\mathstrut}^{0.33} \mathcal{N}
\bigl(\vec{x};
\overbrace{ \vec{\mu}_{2} }^{
\begin{bmatrix} 175\\70 \end{bmatrix} },
\overbrace{ \bm{\Sigma}_{2}}^{
\begin{bmatrix} 35 & 39\\39 & 51 \end{bmatrix} }
\bigr)
\\[2\jot] % insert a bit more vertical whitespace
&\quad+\overbrace{ \pi_{3}\mathstrut}^{0.33} \mathcal{N}
\bigl(\vec{x};
\overbrace{ \vec{\mu}_{3} }^{
\begin{bmatrix} 135\\40 \end{bmatrix} },
\overbrace{ \bm{\Sigma}_{3}}^{
\begin{bmatrix} 10 & 0\\0 & 10 \end{bmatrix} }
\bigr)
\end{align*}
\end{document}