So gleichen Sie mithilfe einer Matrix den oberen und unteren Raum an

Question

Ich habe es versucht mit:

\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}

Ausgabe wie folgt:

Wie kann ich oben und unten in der Matrix den gleichen Abstand schaffen? Bitte beraten Sie

Außerdem wäre es hilfreicher, wenn jemand erklären würde, warum das passiert ist.

Answer 1

Ich bin nicht sicher, ob das das ist, was Sie möchten, aber Sie könnten den Inhalt jedes Teils mit einschließen \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}

Answer 2

Sie haben eine nette Antwort erhalten, aber ich schlage dringend vor, damit davonzukommen \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}

Beachten Sie, dass der Spaltenvektor als Transponierte eines Zeilenvektors geschrieben ist, wodurch Lücken zwischen den Zeilen vermieden werden.

Mit einer alternativen Ausrichtung

\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}

Der Vollständigkeit halber finden Sie hier, wie Sie die vorgeschlagene Aufgabe erledigen können. Ich habe den großen Inline-Spaltenvektor gelassen, um zu zeigen, warum er wirklich schlecht ist.

Wenn ich die Ergebnisse vergleiche, habe ich keinerlei Zweifel.

\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}

Answer 3

\left( ... \right)Es genügt, die drei Instanzen von durch zu ersetzen \bigl( ... \bigr). Beachten Sie, dass die zweiten Argumente der größeren \overbraceKonstrukte erklärenden und nicht definierenden Charakter haben und daher nicht in die (jetzt nicht mehr sehr hohen) Klammern eingeschlossen werden müssen.

Oh, und wenn Sie nicht zu viel Aufmerksamkeit auf die Definition \vec{y}im Absatz vor der align*Umgebung lenken möchten, würde ich es als Zeilenvektor und nicht als Spaltenvektor schreiben.

\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}