Update vignette equations.

vignettes/smartid_Demo.Rmd

@@ -115,19 +115,20 @@ The basic version of TF, IDF and IAE can be termed as:
 
 $\mathbf{TF_{i,j}}=\frac{N_{i,j}}{\sum_j{N_{i,j}}},$
 $\mathbf{IDF_i} = log(1+\frac{n}{n_i+1}),$
-$\mathbf{IAE_i} = log(1+\frac{n}{\hat N_{i,j}+1})$
+$\mathbf{IAE_i} = log(1+\frac{n}{\sum_j^n\hat N_{i,j}+1})$
 
 Where $N_{i,j}$ is the counts of feature $i$ in cell $j$; $\hat N_{i,j}$ is $max(0,N_{i,j}-threshold)$;
 $n$ is the total number of documents(cells); $n_i$ is $\sum_{j = 1}^{n} sign(N_{i,j} > threshold)$.
 
 Here for labeled data, we can choose logTF * IDF_prob * IAE_prob for marker identification:
-$\mathbf{score}=logTF*IDF_{prob}*IAE_{prob}$
+$$\mathbf{score}=logTF*IDF_{prob}*IAE_{prob}$$
 
 The probability version of IDF can be termed as:
 $\mathbf{IDF_{i,j}} = log(1+\frac{\frac{n_{i,j\in D}}{n_{j\in D}}}{max(\frac{n_{i,j\in \hat D}}{n_{j\in \hat D}})+ e^{-8}}\frac{n_{i,j\in D}}{n_{j\in D}})$
 
 And the probability version of IAE can be termed as:
 $\mathbf{IAE_{i,j}} = log(1+\frac{mean(N_{i,j\in D})}{max(mean(N_{i,j\in \hat D}))+ e^{-8}}*mean(N_{i,j\in D}))$
+
 Where $D$ is the category of cell $j$; $\hat D$ is the category other than $D$.
 
 TF here stands for gene frequency, which is similar to CPM, while IDF represents the inverse cell/sample frequency for scRNA-seq data, and IAE is the inverse average expression of each gene across all cells or cells in each labeled group.
@@ -255,7 +256,7 @@ upset(fromList(c(marker_ls, marker_ls_new)), nsets = 6)
 While for the unlabeled data, `smartid` also provides the score methods with no need for label information.
 
 Here we choose logTF * IDF_sd * IAE_sd for for gene-set scoring as a use case:
-$\mathbf{score}=logTF*IDF_{sd}*IAE_{sd}$
+$$\mathbf{score}=logTF*IDF_{sd}*IAE_{sd}$$
 
 Where $\mathbf{IDF_i} = log(1+SD(TF_{i})*\frac{n}{n_i+1})$,
 $\mathbf{IAE_i} = log(1+SD(TF_{i})*\frac{n}{\sum_{j=1}^{n}N_{i,j}+1})$