注:部分希腊字母在数学公式中常以变量形式出现,例如 ϵ epsilon ϵ在数学中一般写法为 ε varepsilon ε, ϕ phi ϕ在数学中通常写作 φ varphi φ
符号语法符号语法符号语法 α alpha αalpha β eta βeta γ gamma γgamma θ heta θ heta ε varepsilon εvarepsilon δ delta δdelta μ mu μmu ν u ν u η eta ηeta ζ zeta ζzeta λ lambda λlambda ψ psi ψpsi σ sigma σsigma ξ xi ξxi τ au τ au ϕ phi ϕphi φ varphi φvarphi ρ ho ρ ho χ chi χchi ω omega ωomega π pi πpi 大写希腊字母大写希腊字母通常是小写希腊字母的LATEX语法第一个字母改为大写,见下表
符号语法符号语法符号语法 Σ Sigma ΣSigma Π Pi ΠPi Δ Delta ΔDelta Γ Gamma ΓGamma Ψ Psi ΨPsi Θ Theta ΘTheta Λ Lambda ΛLambda Ω Omega ΩOmega Φ Phi ΦPhi Ξ Xi ΞXi 常用字体默认的字体为 A B C d e f ABCdef ABCdef,也就是mathnormal{ABCdef}(当然,打公式的时候不需要加上这个mathnormal,直接打字母就是这个效果)
字体语法字体语法 A B C d e f mathrm{ABCdef} ABCdefmathrm{ABCdef} A B C d e f mathbf{ABCdef} ABCdefmathbf{ABCdef} A B C d e f mathit{ABCdef} ABCdefmathit{ABCdef} A B C d e f pmb{ABCdef} ABCdefpmb{ABCdef} A B C d e f mathscr{ABCdef} ABCdefmathscr{ABCdef} A B C d e f mathcal{ABCdef} ABCdefmathcal{ABCdef} A B C d e f mathfrak{ABCdef} ABCdefmathfrak{ABCdef} A B C d e f mathbb{ABCdef} ABCdefmathbb{ABCdef} 常见运算符 运算符语法运算符语法运算符语法 + + ++ − - −- × imes × imes ± pm ±pm ⋅ cdot ⋅cdot ∗ ast ∗ast ∪ cup ∪cup ∩ cap ∩cap ∘ circ ∘circ ∨ lor ∨lor或vee ∧ land ∧land或wedge ¬ lnot ¬lnot ⊕ oplus ⊕oplus ⊖ ominus ⊖ominus ⊗ otimes ⊗otimes ⊙ odot ⊙odot ⊘ oslash ⊘oslash ∙ ullet ∙ullet x sqrt{x} x sqrt{x} x n sqrt[n]{x} nx sqrt[n]{x} 大尺寸运算符 运算符语法运算符语法运算符语法 ∑ sum ∑sum ∏ prod ∏prod ∫ int ∫int ⋃ igcup ⋃igcup ⋂ igcap ⋂igcap ∮ oint ∮oint ⋁ igvee ⋁igvee ⋀ igwedge ⋀igwedge ∬ iint ∬iint ∐ coprod ∐coprod ⨆ igsqcup ⨆igsqcup ∯ oiint ∬ oiint 常见关系符号 符号语法符号语法符号语法 < < >> = = == ≤ leq ≤leq ≥ geq ≥geq ≠ eq = eq ≪ ll ≪ll ≫ gg ≫gg ≡ equiv ≡equiv ⊂ subset ⊂subset ⊃ supset ⊃supset ≈ approx ≈approx ⊆ subseteq ⊆subseteq ⊇ supseteq ⊇supseteq ∼ sim ∼sim ∈ in ∈in ∋ i ∋ i ∝ propto ∝propto ⊢ vdash ⊢vdash ⊣ dashv ⊣dashv ⊨ models ⊨models ∣ mid ∣mid ∥ parallel ∥parallel ⊥ perp ⊥perp ∉ otin ∈/ otin ⋈ Join ⋈Join ≁ sim ≁ sim ⊊ subsetneq ⊊subsetneq ⊋ supsetneq ⊋supsetneq 数学模式重音符 符号语法符号语法符号语法 a ^ hat{a} a^hat{a} a ˉ ar{a} aˉar{a} a ~ ilde{a} a~ ilde{a} a ⃗ vec{a} a vec{a} a ˙ dot{a} a˙dot{a} a ¨ ddot{a} a¨ddot{a} a b c ^ widehat{abc} abc widehat{abc} a b c ~ widetilde{abc} abc widetilde{abc} a b c ‾ overline{abc} abcoverline{abc} 箭头如果需要长箭头,只需要在语法前面加上long,例如longleftarrow即为 ⟵ longleftarrow ⟵,如果加上Long则变为双线长箭头,例如Longleftarrow即为 ⟸ Longleftarrow ⟸
符号语法符号语法符号语法 ← leftarrow ←leftarrow → ightarrow → ightarrow ↔ leftrightarrow ↔leftrightarrow ⇐ Leftarrow ⇐Leftarrow ⇒ Rightarrow ⇒Rightarrow ⇔ Leftrightarrow ⇔Leftrightarrow ↑ uparrow ↑uparrow ↓ downarrow ↓downarrow ↕ updownarrow ↕updownarrow ⇑ Uparrow ⇑Uparrow ⇓ Downarrow ⇓Downarrow ⇕ Updownarrow ⇕Updownarrow ↼ leftharpoonup ↼leftharpoonup ↽ leftharpoondown ↽leftharpoondown ⇀ ightharpoonup ⇀ ightharpoonup ⇁ ightharpoondown ⇁ ightharpoondown ⇌ ightleftharpoons ⇌ ightleftharpoons ⇋ leftrightharpoons ⇋leftrightharpoons ⟺ iff ⟺iff ↦ mapsto ↦mapsto 括号 括号语法括号语法括号语法 ( ) () ()() [ ] [] [][] { } {} {}{} ⌊ ⌋ lfloor floor ⌊⌋lfloor floor ⌈ ⌉ lceil ceil ⌈⌉lceil ceil ⟨ ⟩ langle angle ⟨⟩langle angle 大尺寸括号 括号语法括号语法 ( ) left( ight) ()left( ight) [ ] left[ ight] []left[ ight] x 1 x 2 … x n ⏞ n overbrace{x_1x_2ldots x_n}^{n} x1x2…xn noverbrace{x_1x_2ldots x_n}^{n} x 1 x 2 … x n ⏟ n underbrace{x_1x_2ldots x_n}_{n} n x1x2…xnunderbrace{x_1x_2ldots x_n}_{n}注:大尺寸的()和[]是可以根据公式的高度自动调节的,例如
argmin_{ heta}left[ -sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight] ight]arg min θ [ − ∑ i = 1 n [ y ( i ) ln ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) ] ] argmin_{ heta} left[ -sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight] ight] argθmin[−i=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]]
可以看出,括号高度可以框住整个公式
因此在这种大型的公式中,使用大尺寸括号视觉效果更美观
其他常见符号 符号语法符号语法符号语法 ∀ forall ∀forall ∃ exist ∃exist ∠ angle ∠angle ∅ emptyset ∅emptyset ∂ partial ∂partial ∞ infty ∞infty … ldots …ldots ⋯ cdots ⋯cdots … dots …dots ⋮ vdots ⋮vdots ⋱ ddots ⋱ddots ′ prime ′prime ∵ ecause ∵ecause ∴ herefore ∴ herefore □ Box □Box △ riangle △ riangle § S §S 数学公式写法 上下标 ^:上标_:下标例如:
sum_{i=1}^{n}X_n表示 ∑ i = 1 n X n sum_{i=1}^{n}X_n ∑i=1nXnint_{0}^{infty}x^2dx表示 ∫ 0 ∞ x 2 d x int_{0}^{infty}x^2dx ∫0∞x2dxprod_{i=1}^{n}X_n表示 ∏ i = 1 n X n prod_{i=1}^{n}X_n ∏i=1nXn 分数使用frac{}{}即可,例如frac{a}{b}表示 a b frac{a}{b} ba
插入文字使用 ext,例如 ext{hello,world!}表示 hello,world! ext{hello,world!} hello,world!
常见函数 函数语法函数语法函数语法 log ( ) log() log()log() ln ( ) ln() ln()ln() lg ( ) lg() lg()lg() max max maxmax min min minmin lim x → ∞ lim_{x o infty} limx→∞lim_{x o infty} arg max c ∈ C argmax_{c in C} argmaxc∈Cargmax_{c in C} arg min c ∈ C argmin_{c in C} argminc∈Cargmin_{c in C} exp exp expexp 矩阵、行列式&表示分隔元素,\表示换行
A=egin{pmatrix}a_{11} & a_{12} \a_{21} & a_{22}end{pmatrix}A = ( a 11 a 12 a 21 a 22 ) A= egin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{pmatrix} A=(a11a21a12a22)
A=egin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}end{bmatrix}A = [ a 11 a 12 a 21 a 22 ] A= egin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{bmatrix} A=[a11a21a12a22]
A=egin{Bmatrix}a_{11} & a_{12} \a_{21} & a_{22}end{Bmatrix}A = { a 11 a 12 a 21 a 22 } A= egin{Bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{Bmatrix} A={a11a21a12a22}
A=egin{vmatrix}a_{11} & a_{12} \a_{21} & a_{22}end{vmatrix}A = ∣ a 11 a 12 a 21 a 22 ∣ A= egin{vmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{vmatrix} A= a11a21a12a22
A=egin{Vmatrix}a_{11} & a_{12} \a_{21} & a_{22}end{Vmatrix}A = ∥ a 11 a 12 a 21 a 22 ∥ A= egin{Vmatrix} a_{11} & a_{12} \ a_{21} & a_{22} end{Vmatrix} A= a11a21a12a22
A=egin{matrix}a_{11} & a_{12} \a_{21} & a_{22}end{matrix}A = a 11 a 12 a 21 a 22 A= egin{matrix} a_{11} & a_{12} \ a_{21} & a_{22} end{matrix} A=a11a21a12a22
多行公式对齐使用egin{split} end{split},在需要对齐的地方添加&符号,注意需要用\来换行。
例如:
egin{split}L( heta)&=argmax_{ heta}ln(P_{All})\&=argmax_{ heta}lnprod_{i=1}^{n} left[ (h_{ heta}(mathbf{x}^{(i)}))^{mathbf{y}^{(i)}}cdot (1-h_{ heta}(mathbf{x}^{(i)}))^{1-mathbf{y}^{(i)}} ight]\&=argmax_{ heta}sum_{i=1}^{n}left[mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) +(1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight]\&=argmin_{ heta}left[ -sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight] ight]\&=argmin_{ heta}mathscr{l}( heta)end{split}L ( θ ) = arg max θ ln ( P A l l ) = arg max θ ln ∏ i = 1 n [ ( h θ ( x ( i ) ) ) y ( i ) ⋅ ( 1 − h θ ( x ( i ) ) ) 1 − y ( i ) ] = arg max θ ∑ i = 1 n [ y ( i ) ln ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) ] = arg min θ [ − ∑ i = 1 n [ y ( i ) ln ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) ] ] = arg min θ l ( θ ) egin{split} L( heta) &= argmax_{ heta}ln(P_{All})\ &= argmax_{ heta}lnprod_{i=1}^{n} left[ (h_{ heta}(mathbf{x}^{(i)}))^{mathbf{y}^{(i)}}cdot (1-h_{ heta}(mathbf{x}^{(i)}))^{1-mathbf{y}^{(i)}} ight]\ &= argmax_{ heta}sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight]\ &= argmin_{ heta} left[ -sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight] ight]\ &= argmin_{ heta}mathscr{l}( heta) end{split} L(θ)=argθmaxln(PAll)=argθmaxlni=1∏n[(hθ(x(i)))y(i)⋅(1−hθ(x(i)))1−y(i)]=argθmaxi=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]=argθmin[−i=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]]=argθminl(θ)
上例中,在=前添加了&,因此实现等号对齐;
egin{split} end{split}语法默认为右对齐,也就是说如果不在任何地方添加&符号,则公式默认右侧对齐,例如:
egin{split}L( heta)=argmax_{ heta}ln(P_{All})\=argmax_{ heta}lnprod_{i=1}^{n} left[ (h_{ heta}(mathbf{x}^{(i)}))^{mathbf{y}^{(i)}}cdot (1-h_{ heta}(mathbf{x}^{(i)}))^{1-mathbf{y}^{(i)}} ight]\=argmax_{ heta}sum_{i=1}^{n}left[mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) +(1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight]\=argmin_{ heta}left[ -sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight] ight]\=argmin_{ heta}mathscr{l}( heta)end{split}上述LATEX代码没有添加&符号,则公式右对齐: L ( θ ) = arg max θ ln ( P A l l ) = arg max θ ln ∏ i = 1 n [ ( h θ ( x ( i ) ) ) y ( i ) ⋅ ( 1 − h θ ( x ( i ) ) ) 1 − y ( i ) ] = arg max θ ∑ i = 1 n [ y ( i ) ln ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) ] = arg min θ [ − ∑ i = 1 n [ y ( i ) ln ( h θ ( x ( i ) ) ) + ( 1 − y ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) ] ] = arg min θ l ( θ ) egin{split} L( heta) = argmax_{ heta}ln(P_{All})\ = argmax_{ heta}lnprod_{i=1}^{n} left[ (h_{ heta}(mathbf{x}^{(i)}))^{mathbf{y}^{(i)}}cdot (1-h_{ heta}(mathbf{x}^{(i)}))^{1-mathbf{y}^{(i)}} ight]\ = argmax_{ heta}sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight]\ = argmin_{ heta} left[ -sum_{i=1}^{n} left[ mathbf{y}^{(i)}ln(h_{ heta}(mathbf{x}^{(i)})) + (1-mathbf{y}^{(i)})ln(1-h_{ heta}(mathbf{x}^{(i)})) ight] ight]\ = argmin_{ heta}mathscr{l}( heta) end{split} L(θ)=argθmaxln(PAll)=argθmaxlni=1∏n[(hθ(x(i)))y(i)⋅(1−hθ(x(i)))1−y(i)]=argθmaxi=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]=argθmin[−i=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]]=argθminl(θ)
如果希望左对齐,例如
egin{split}&ln h_{ heta}(mathbf{x}^{(i)})=lnfrac{1}{1+e^{- heta^T mathbf{x}^{(i)}}}= -ln(1+e^{ heta^T mathbf{x}^{(i)}})\&ln(1-h_{ heta}(mathbf{x}^{(i)}))=ln(1-frac{1}{1+e^{- heta^T mathbf{x}^{(i)}}})= - heta^T mathbf{x}^{(i)}-ln(1+e^{ heta^T mathbf{x}^{(i)}})end{split}ln h θ ( x ( i ) ) = ln 1 1 + e − θ T x ( i ) = − ln ( 1 + e θ T x ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) = ln ( 1 − 1 1 + e − θ T x ( i ) ) = − θ T x ( i ) − ln ( 1 + e θ T x ( i ) ) egin{split} &ln h_{ heta}(mathbf{x}^{(i)}) = lnfrac{1}{1+e^{- heta^T mathbf{x}^{(i)}}} = -ln(1+e^{ heta^T mathbf{x}^{(i)}})\ &ln(1-h_{ heta}(mathbf{x}^{(i)})) = ln(1-frac{1}{1+e^{- heta^T mathbf{x}^{(i)}}}) = - heta^T mathbf{x}^{(i)}-ln(1+e^{ heta^T mathbf{x}^{(i)}}) end{split} lnhθ(x(i))=ln1+e−θTx(i)1=−ln(1+eθTx(i))ln(1−hθ(x(i)))=ln(1−1+e−θTx(i)1)=−θTx(i)−ln(1+eθTx(i))
除了egin{split} end{split},也可以用egin{align} end{align},用法与split相同,对齐方式也相同;
只有一点不同:采用align环境会默认为每一条公式编号(如下例),split则不会编号。
egin{align}&ln h_{ heta}(mathbf{x}^{(i)})=lnfrac{1}{1+e^{- heta^T mathbf{x}^{(i)}}}= -ln(1+e^{ heta^T mathbf{x}^{(i)}})\&ln(1-h_{ heta}(mathbf{x}^{(i)}))=ln(1-frac{1}{1+e^{- heta^T mathbf{x}^{(i)}}})= - heta^T mathbf{x}^{(i)}-ln(1+e^{ heta^T mathbf{x}^{(i)}})end{align}ln h θ ( x ( i ) ) = ln 1 1 + e − θ T x ( i ) = − ln ( 1 + e θ T x ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) = ln ( 1 − 1 1 + e − θ T x ( i ) ) = − θ T x ( i ) − ln ( 1 + e θ T x ( i ) ) egin{align} &ln h_{ heta}(mathbf{x}^{(i)}) = lnfrac{1}{1+e^{- heta^T mathbf{x}^{(i)}}} = -ln(1+e^{ heta^T mathbf{x}^{(i)}})\ &ln(1-h_{ heta}(mathbf{x}^{(i)})) = ln(1-frac{1}{1+e^{- heta^T mathbf{x}^{(i)}}}) = - heta^T mathbf{x}^{(i)}-ln(1+e^{ heta^T mathbf{x}^{(i)}}) end{align} lnhθ(x(i))=ln1+e−θTx(i)1=−ln(1+eθTx(i))ln(1−hθ(x(i)))=ln(1−1+e−θTx(i)1)=−θTx(i)−ln(1+eθTx(i))
但可以在align后加一个*号,则align环境也可以取消公式自动编号,如下: (也就是说align*和split的用法完全相同)
egin{align*}&ln h_{ heta}(mathbf{x}^{(i)})=lnfrac{1}{1+e^{- heta^T mathbf{x}^{(i)}}}= -ln(1+e^{ heta^T mathbf{x}^{(i)}})\&ln(1-h_{ heta}(mathbf{x}^{(i)}))=ln(1-frac{1}{1+e^{- heta^T mathbf{x}^{(i)}}})= - heta^T mathbf{x}^{(i)}-ln(1+e^{ heta^T mathbf{x}^{(i)}})end{align*}ln h θ ( x ( i ) ) = ln 1 1 + e − θ T x ( i ) = − ln ( 1 + e θ T x ( i ) ) ln ( 1 − h θ ( x ( i ) ) ) = ln ( 1 − 1 1 + e − θ T x ( i ) ) = − θ T x ( i ) − ln ( 1 + e θ T x ( i ) ) egin{align*} &ln h_{ heta}(mathbf{x}^{(i)}) = lnfrac{1}{1+e^{- heta^T mathbf{x}^{(i)}}} = -ln(1+e^{ heta^T mathbf{x}^{(i)}})\ &ln(1-h_{ heta}(mathbf{x}^{(i)})) = ln(1-frac{1}{1+e^{- heta^T mathbf{x}^{(i)}}}) = - heta^T mathbf{x}^{(i)}-ln(1+e^{ heta^T mathbf{x}^{(i)}}) end{align*} lnhθ(x(i))=ln1+e−θTx(i)1=−ln(1+eθTx(i))ln(1−hθ(x(i)))=ln(1−1+e−θTx(i)1)=−θTx(i)−ln(1+eθTx(i))
方程组使用egin{cases} end{cases}
例如:
egin{cases} egin{split} p &= P(y=1|mathbf{x})= frac{1}{1+e^{- heta^Tmathbf{X}}}\ 1-p &= P(y=0|mathbf{x})=1-P(y=1|mathbf{x})= frac{1}{1+e^{ heta^Tmathbf{X}}} end{split}end{cases}{ p = P ( y = 1 ∣ x ) = 1 1 + e − θ T X 1 − p = P ( y = 0 ∣ x ) = 1 − P ( y = 1 ∣ x ) = 1 1 + e θ T X egin{cases} egin{split} p &= P(y=1|mathbf{x})= frac{1}{1+e^{- heta^Tmathbf{X}}}\ 1-p &= P(y=0|mathbf{x})=1-P(y=1|mathbf{x})= frac{1}{1+e^{ heta^Tmathbf{X}}} end{split} end{cases} ⎩ ⎨ ⎧p1−p=P(y=1∣x)=1+e−θTX1=P(y=0∣x)=1−P(y=1∣x)=1+eθTX1
注意LATEX语法可以嵌套使用,上例即为egin{cases} end{cases}下嵌套了begin{split} end{split}。
也可以将公式和文字结合起来,例如:
ext{Decision Boundary}=egin{cases} 1quad ext{if } hat{y}>0.5\ 0quad ext{otherwise}end{cases}Decision Boundary = { 1 if y ^ > 0.5 0 otherwise ext{Decision Boundary}= egin{cases} 1quad ext{if}quad hat{y}>0.5\ 0quad ext{otherwise} end{cases} Decision Boundary={1ify^>0.50otherwise 注:quad表示空格。