From b8402b14ea58076e0fa48a5052cf71a9077924b5 Mon Sep 17 00:00:00 2001 From: Zhaopudark Date: Tue, 28 May 2024 23:47:07 +0800 Subject: [PATCH] Site updated: 05/28/2024 23:47:07 --- source/_posts/data_dividing.md | 9 +++--- ...74\345\237\237\345\210\206\346\236\220.md" | 31 +++++++++---------- source/sitemap.xml | 4 +-- 3 files changed, 22 insertions(+), 22 deletions(-) diff --git a/source/_posts/data_dividing.md b/source/_posts/data_dividing.md index 861801d..bb18018 100644 --- a/source/_posts/data_dividing.md +++ b/source/_posts/data_dividing.md @@ -11,7 +11,7 @@ tags: - Python title: Discuss the mathematics of apportionment when splitting the machine learning dataset into several parts by proportions -updated: "2024-05-27 14:16:41" +updated: "2024-05-28 23:45:58" --- This article discusses an operation that originated in machine learning, @@ -1319,7 +1319,8 @@ and analysis step by step, until the conclusion is reached. - $h\in \mathbb{Z}^+$ - If $h \ge 2$, - $\exists s,t \in \{1,2,\ldots,h\}, s < t, i_s \in G_s, i_t\in G_t, x_{i_s} < x_{i_t}$, - - $\forall b \in B_3$, \$ b ^n,~\_{i=1}^n b_i = m\$, and + - $\forall b \in B_3$, + $b \in \mathbb{Z}^n,~\sum_{i=1}^n b_i = m$, and $b_{i_s} < b_{i_t}$. - If $h=1$, $B_3 = \emptyset$. - If $h \ge 2$. @@ -1412,7 +1413,7 @@ and analysis step by step, until the conclusion is reached. $D^{*'}_b = \complement_{D^{'}_b}B_1 \cap \complement_{D^{'}_b}B_2 \cap \complement_{D^{'}_b}B_3$ - If $m > 0$, there will be $\exists h^{'} \in \{1,2,\ldots,h\} \sum_{i=1}^{h^{'}-1}g_i \le m < \sum_{i=1}^{h^{'}}g_i$. - - If \$\_{i=1}{h{’}-1}g_i = m \$, + - If $\sum_{i=1}^{h^{'}-1}g_i = m$, - then $\forall i \in G_1 \cup G_2 \cup \ldots \cup G_{h^{'}-1}, b_{i}=1$. - and, @@ -1588,7 +1589,7 @@ and analysis step by step, until the conclusion is reached. - So, $D^{*}_b=\{\theta\}$, which is a single element set. - If $m > 0$, there will be $\exists h^{'} \in \{1,2,\ldots,h\} \sum_{i=1}^{h^{'}-1}g_i \le m < \sum_{i=1}^{h^{'}}g_i$. - - If \$\_{i=1}{h{’}-1}g_i = m \$, + - If $\sum_{i=1}^{h^{'}-1}g_i = m$, - then $\forall i \in G_1 \cup G_2 \cup \ldots \cup G_{h^{'}-1}, b_{i}=1$. - and, diff --git "a/source/_posts/\343\200\220\347\256\227\346\263\225\345\210\206\346\236\220\343\200\221round\345\207\275\346\225\260\347\232\204\345\200\274\345\237\237\345\210\206\346\236\220.md" "b/source/_posts/\343\200\220\347\256\227\346\263\225\345\210\206\346\236\220\343\200\221round\345\207\275\346\225\260\347\232\204\345\200\274\345\237\237\345\210\206\346\236\220.md" index 29fbf03..79ad725 100644 --- "a/source/_posts/\343\200\220\347\256\227\346\263\225\345\210\206\346\236\220\343\200\221round\345\207\275\346\225\260\347\232\204\345\200\274\345\237\237\345\210\206\346\236\220.md" +++ "b/source/_posts/\343\200\220\347\256\227\346\263\225\345\210\206\346\236\220\343\200\221round\345\207\275\346\225\260\347\232\204\345\200\274\345\237\237\345\210\206\346\236\220.md" @@ -8,7 +8,7 @@ tags: - Mathematics - Algorithm title: Analyses of round function -updated: "2024-05-23 01:02:52" +updated: "2024-05-28 23:44:04" --- This article does a mathematical abstraction of the @@ -169,8 +169,8 @@ which is base on $round(\cdot) \mathbb{R}\rightarrow \mathbb{Z}$. - Define $f(r,n,N):\{\mathbb{R}^n,\mathbb{Z}^+,\mathbb{Z}^+\}\rightarrow\{\mathbb{Z}\}$ as: - - $\forall (r,n,N) \in D_{r,n,N}$, there is \$ f(r,n,N)=\[*{i=1}^n - round(r*{i}N)\]-N\$ + - $\forall (r,n,N) \in D_{r,n,N}$, there is + $f(r,n,N)=[\sum_{i=1}^n round(r_{i}N)]-N$ - $\forall n \in \mathbb{Z}^+$, define $D_{r,N}=\{(r,N)|N \in \mathbb{Z}^+, N\ge n,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$. - $\forall n \in \mathbb{Z}^+$, define the function $f(r,n,N)$’s value @@ -230,8 +230,8 @@ the $0$ is also necessary. - $D_{r,n,N}=\{(r,n,N)|n \in\mathbb{Z}^+,N \in\mathbb{Z}^+, n \le N,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$ - $f(r,n,N):\{\mathbb{R}^n,\mathbb{Z}^+,\mathbb{Z}^+\}\rightarrow\{\mathbb{Z}\}$ as: - - $\forall (r,n,N) \in D_{r,n,N}$, there is \$ f(r,n,N)=\[*{i=1}^n - round(r*{i}N)\]-N\$. + - $\forall (r,n,N) \in D_{r,n,N}$, there is + $f(r,n,N)=[\sum_{i=1}^n round(r_{i}N)]-N$. - $\forall n \in \mathbb{Z}^+$, $D_{r,N}=\{(r,N)|N \in \mathbb{Z}^+, N\ge n,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$. - Question: @@ -372,8 +372,8 @@ Take the following steps: - $D_{r,n,N}=\{(r,n,N)|n \in\mathbb{Z}^+,N \in\mathbb{Z}^+, n \le N,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$ - $f(r,n,N):\{\mathbb{R}^n,\mathbb{Z}^+,\mathbb{Z}^+\}\rightarrow\{\mathbb{Z}\}$ as: - - $\forall (r,n,N) \in D_{r,n,N}$, there is \$ f(r,n,N)=\[*{i=1}^n - round(r*{i}N)\]-N\$. + - $\forall (r,n,N) \in D_{r,n,N}$, there is + $f(r,n,N)=[\sum_{i=1}^n round(r_{i}N)]-N$. - $\forall n \in \mathbb{Z}^+$, $D_{r,N}=\{(r,N)|N \in \mathbb{Z}^+, N\ge n,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$. - Question: @@ -575,9 +575,8 @@ Take the following steps: - $D_{r,n,N}=\{(r,n,N)|n \in\mathbb{Z}^+,N \in\mathbb{Z}^+, n \le N,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$ - $f(r,n,N):\{\mathbb{R}^n,\mathbb{Z}^+,\mathbb{Z}^+\}\rightarrow\{\mathbb{Z}\}$ as: - - $\forall (r,n,N) \in D_{r,n,N}$, there is \$ f(r,n,N)=\[*{i=1}^n - round(r*{i}N)\]-N=\[*{i=1}^n truncate(r*{i}N)\]-N=\[*{i=1}^n - floor(r*{i}N)\]-N\$. + - $\forall (r,n,N) \in D_{r,n,N}$, there is + $f(r,n,N)=[\sum_{i=1}^n round(r_{i}N)]-N=[\sum_{i=1}^n truncate(r_{i}N)]-N=[\sum_{i=1}^n floor(r_{i}N)]-N$. - $\forall n \in \mathbb{Z}^+$, $D_{r,N}=\{(r,N)|N \in \mathbb{Z}^+, N\ge n,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$. - Question: @@ -667,17 +666,17 @@ Take the following steps: ## A.4 -- Given $round(x)$, where \$x ^+{0},~round(x)=ceil(x)=x\$. Its domain of - definition is restricted to $\mathbb{R}^+\cup\{0\}$ instead of - $\mathbb{R}$, and in this domain, it is equivalent to +- Given $round(x)$, where + $\forall x \in \mathbb{R}^+\cup\{0\},~round(x)=ceil(x)=\lceil x\rceil$. + Its domain of definition is restricted to $\mathbb{R}^+\cup\{0\}$ + instead of $\mathbb{R}$, and in this domain, it is equivalent to $\eqref{round_4}$. - Define: - $D_{r,n,N}=\{(r,n,N)|n \in\mathbb{Z}^+,N \in\mathbb{Z}^+, n \le N,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$ - $f(r,n,N):\{\mathbb{R}^n,\mathbb{Z}^+,\mathbb{Z}^+\}\rightarrow\{\mathbb{Z}\}$ as: - - $\forall (r,n,N) \in D_{r,n,N}$, there is \$ f(r,n,N)=\[*{i=1}^n - round(r*{i}N)\]-N=\[*{i=1}^n truncate(r*{i}N)\]-N=\[*{i=1}^n - floor(r*{i}N)\]-N\$. + - $\forall (r,n,N) \in D_{r,n,N}$, there is + $f(r,n,N)=[\sum_{i=1}^n round(r_{i}N)]-N=[\sum_{i=1}^n truncate(r_{i}N)]-N=[\sum_{i=1}^n floor(r_{i}N)]-N$. - $\forall n \in \mathbb{Z}^+$, $D_{r,N}=\{(r,N)|N \in \mathbb{Z}^+, N\ge n,r = [r_1,r_2,\ldots,r_n]\in \mathbb{R}_+^n,\|r\|_{1}=1\}$. - Question: diff --git a/source/sitemap.xml b/source/sitemap.xml index 92a138b..a8af704 100644 --- a/source/sitemap.xml +++ b/source/sitemap.xml @@ -5,7 +5,7 @@ https://little-train.com/posts/70807cc5.html - 2024-05-27 + 2024-05-28 monthly @@ -19,7 +19,7 @@ https://little-train.com/posts/34195fcb.html - 2024-05-23 + 2024-05-28 monthly