---
title: "A Vignette for DeMixT"
author: "Zeya Wang, Fan Gao, Shaolong Cao and Peng Yang"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
    %\VignetteIndexEntry{DeMixT.Rmd}
    %\VignetteEngine{knitr::rmarkdown}
    \usepackage[utf8]{inputenc}
---


```{r setup, include=FALSE, message=FALSE}
library(ggplot2)
library(DeMixT)
# knitr::opts_chunk$set(echo = TRUE)
```

# 1. Introduction

Transcriptomic deconvolution in cancer and other heterogeneous tissues remains 
challenging. Available methods lack the ability to estimate both 
component-specific proportions and expression profiles for individual samples. 
We develop a three-component deconvolution model, DeMixT, for expression data 
from a mixture of cancerous tissues, infiltrating immune cells and tumor 
microenvironment. DeMixT is a software package that performs deconvolution on 
transcriptome data from a mixture of two or three components.

DeMixT is a frequentist-based method and fast in yielding accurate estimates of 
cell proportions and compart-ment-specific expression profiles for 
two-component \and three-component deconvolution problem. Our method promises 
to provide deeper insight into cancer biomarkers and assist in the development 
of novel prognostic markers and therapeutic strategies.

The function DeMixT is designed to finish the whole pipeline of deconvolution 
for two or three components. The newly added DeMixT_GS function is designed to 
estimates the proportions of mixed samples for each mixing component based on a 
new approach to select genes more effectively that utilizes profile likelihood. 
DeMixT_S1 function is designed to estimate the proportions of all mixed samples 
for each mixing component based on the gene differential expressions to select 
genes. DeMixT_S2 function is designed to estimate the component-specific
deconvolved expressions of individual mixed samples for a given set of genes.

# 2 Feature Description

The DeMixT R-package builds the transcriptomic deconvolution with a couple of 
novel features into R-based standard analysis pipeline through Bioconductor. 
DeMixT showed high accuracy and efficiency from our designed experiment. 
Hence, DeMixT can be considered as an important step towards linking tumor 
transcriptomic data with clinical outcomes.

Different from most previous computational deconvolution methods, DeMixT has 
integrated new features for the deconvolution with more than 2 components. 

**Joint estimation**: jointly estimate component proportions and expression 
profiles for individual samples by requiring reference samples instead of 
reference genes; For the three-component deconvolution considering immune 
infiltration, it provides a comprehensive view of tumor-stroma-immune 
transcriptional dynamics, as compared to methods that address only immune 
subtypes within the immune component, in each tumor sample. 

**Efficient estimation**: DeMixT adopts an approach of iterated conditional 
modes (ICM) to guarantee a rapid convergence to a local maximum. We also design 
a novel gene-set-based component merging approach to reduce the bias of 
proportion estimation for three-component deconvolutionthe. 


**Parallel computing**: OpenMP enables parallel computing on single computer by 
taking advantage of the multiple cores shipped on modern CPUs. The ICM 
framework further enables parallel computing, which helps compensate for the 
expensive computing time used in the repeated numerical double integrations. 

# 3. Installation

## 3.1 Source file

DeMixT source files are compatible with Windows, Linux and macOS. 

DeMixT_1.2.6 is the latest version, which is for a computer that has OpenMP. 
It can be downloaded from Bioconductor website.
https://bioconductor.org/packages/release/bioc/html/DeMixT.html

To install DeMixT_1.2.6 from GitHub, start R and enter:

```{r install_DeMixT}
# devtools::install_github("wwylab/DeMixT")
```

For more information, please visit:
<http://bioinformatics.mdanderson.org/main/DeMixT>


## 3.2 Functions

The following table shows the functions included in DeMixT.

Table Header  | Second Header
------- | ----------------------------------
DeMixT | Deconvolution of tumor samples with two or three components 
DeMixT_GS | Estimates the proportions of mixed samples for each mixing component based on a new approach to select genes that utilizes profile likelihood 
DeMixT_S1 | Estimates the proportions of mixed samples for each mixing component
DeMixT_S2 |Deconvolves expressions of each  sample for unknown component
Optimum_KernelC | Call the C function used for parameter estimation in DeMixT

# 4. Methods

## 4.1 Model

Let \(Y_{ig}\) be the observed expression levels of the raw measured data from 
clinically derived malignant tumor samples for gene \(g, g = 1, \cdots, G\) and 
sample \(i, i = 1, \cdots, My\). \(G\) denotes the total number of probes/genes 
and \(My\) denotes the number of samples. The observed expression levels for 
solid tumors can be modeled as a linear combination of raw expression levels 
from three components:
\[ {Y_{ig}} = \pi _{1,i}N_{1,ig} + \pi _{2,i}N_{2,ig} + 
(1 - \pi_{1,i} - \pi _{2,i}){T_{ig}} \label{eq:1} \]

Here \(N_{1,ig}\), \(N_{2,ig}\) and \({T_{ig}}\) are the unobserved raw 
expression levels from each of the three components. We call the two components 
for which we require reference samples the \(N_1\)-component and the 
\(N_2\)-component. We call the unknown component the T-component. We let 
\(\pi_{1,i}\) denote the proportion of the \(N_1\)-component, \(\pi_{2,i}\) 
denote the proportion of the \(N_2\)-component, and \(1 - \pi_{1,i}-\pi_{2,i}\) 
denote the proportion of the T-component. We assume that the mixing proportions 
of one specific sample remain the same across all genes.

Our model allows for one component to be unknown, and therefore does not 
require reference profiles from all components. A set of samples for 
\(N_{1,ig}\) and \(N_{2,ig}\), respectively, needs to be provided as input 
data. This three-component deconvolution model is applicable to the linear 
combination of any three components in any type of material. It can also be 
simplified to a two-component model, assuming there is just one 
\(N\)-component. For application in this paper, we consider tumor (\(T\)), 
stromal (\(N_1\)) and immune components (\(N_2\)) in an admixed sample (\(Y\)).

Following the convention that \(\log_2\)-transformed microarray gene expression 
data follow a normal distribution, we assume that the raw measures \(N_{1,ig} 
\sim LN({\mu _{{N_1}g}},\sigma _{{N_1}g}^2)\), \(N_{2,ig} 
\sim LN({\mu _{{N_2}g}},\sigma _{{N_2}g}^2)\) and \({T_{ig}} 
\sim LN({\mu _{Tg}}, \sigma _{Tg}^2)\), where LN denotes a \(\log_2\)-normal 
distribution and \(\sigma _{{N_1}g}^2\),\(\sigma _{{N_2}g}^2\),
\(\sigma _{Tg}^2\) reflect the variations under \(\log_2\)-transformed data. 
Consequently, our model can be expressed as the convolution of the density 
function for three \(\log_2\)-normal distributions. Because there is no closed 
form of this convolution, we use numerical integration to evaluate the complete 
likelihood function (see the full likelihood in the Supplementary Materials).

## 4.2 The DeMixT algorithm for deconvolution

DeMixT estimates all distribution parameters and cellular proportions and 
reconstitutes the expression profiles for all three components for each gene 
and each sample. The estimation procedure (summarized in Figure 1b) has two
main steps as follows.

1. Obtain a set of parameters \(\{\pi_{1,i}, \pi_{2,i}\}_{i=1}^{My}\), \(\{\mu_T,
\sigma_T\}_{g=1}^G\) to maximize the complete likelihood function, for which
\(\{\mu_{N_{1,g}}, \sigma_{N_{1,g}}, \mu_{N_{2,g}}, 
\sigma_{N_{2,g}}\}_{g=1}^G\) were already estimated from the available 
unmatched samples of the \(N_1\) and \(N_2\) component tissues. 
(See further details in our paper.)

2. Reconstitute the expression profiles by searching each set of \(\{n_{1,ig},
n_{2,ig}\}\) that maximizes the joint density of \(N_{1,ig}\), \(N_{2,ig}\) and
\(T_{ig}\). The value of \(t_{ig}\) is solved as \({y_{ig}} - 
{{\hat \pi }_{1,i}}{n_{1,ig}} - {{\hat \pi }_{2,i}}{n_{2,ig}}\).

These two steps can be separately implemented using the function DeMixT_S1 or 
DeMixT_GS for the first step and DeMixT_S2 for the second, which are combined 
in the function DeMixT(Note: DeMixT_GS is the default function for first step).

In version 1.2.6, DeMixT added simulated normal reference samples, i.e., spike-in, 
based on the observed normal reference samples. It has been shown to improve 
accuracy in proportion estimation for the scenario where a dataset consists of 
samples where true tumor proportions are skewed to the high end.

```{r Algorithm, echo=FALSE, out.width='100%'}
knitr::include_graphics(path = paste0("Algorithm.png"))
```


# 5. Examples

## 5.1 Simulated two-component data 

```{r, sim_2comp_GS, results="hide", message=FALSE}
data("test.data.2comp")
# res.GS = DeMixT_GS(data.Y = test.data.2comp$data.Y, 
#                     data.N1 = test.data.2comp$data.N1,
#                     niter = 30, nbin = 50, nspikein = 50,
#                     if.filter = TRUE, ngene.Profile.selected = 150,
#                     mean.diff.in.CM = 0.25, ngene.selected.for.pi = 150,
#                     tol = 10^(-5))
load('Res_2comp/res.GS.RData')
```

```{r sim_2comp_GS_res}
head(t(res.GS$pi))
head(res.GS$gene.name)
```

```{r, sim_2comp_S2, results="hide", message=FALSE}
data("test.data.2comp")
# res.S2 <- DeMixT_S2(data.Y = test.data.2comp$data.Y, 
#                     data.N1 = test.data.2comp$data.N1,
#                     data.N2 = NULL, 
#                     givenpi = c(t(res.S1$pi[-nrow(res.GS$pi),])), nbin = 50)
load('Res_2comp/res.S2.RData')
```

```{r sim_2comp_S2_res}
head(res.S2$decovExprT[,1:5],3)
head(res.S2$decovExprN1[,1:5],3)
head(res.S2$decovMu,3)
head(res.S2$decovSigma,3)
```

## 5.2 Simulated two-component data with DeMixT Spike-in Normal 

In the simulation, 
````markdown
## Simulate MuN and MuT for each gene
  MuN <- rnorm(G, 7, 1.5)
  MuT <- rnorm(G, 7, 1.5)
  Mu <- cbind(MuN, MuT)
## Simulate SigmaN and SigmaT for each gene
  SigmaN <- runif(n = G, min = 0.1, max = 0.8)
  SigmaT <- runif(n = G, min = 0.1, max = 0.8)
## Simulate Tumor Proportion
  PiT = truncdist::rtrunc(n = My,
                          spec = 'norm', 
                          mean = 0.55,
                          sd = 0.2,
                          a = 0.25,
                          b = 0.95)

## Simulate Data
  for(k in 1:G){
    
    data.N1[k,] <- 2^rnorm(M1, MuN[k], SigmaN[k]); # normal reference
    
    True.data.T[k,] <- 2^rnorm(My, MuT[k], SigmaT[k]);  # True Tumor
    
    True.data.N1[k,] <- 2^rnorm(My, MuN[k], SigmaN[k]);  # True Normal
    
    data.Y[k,] <- pi[1,]*True.data.N1[k,] + pi[2,]*True.data.T[k,] # Mixture Tumor
    
  }
````
where $\pi_i \in (0.25, 0.95)$ is from truncated normal distribution. In general, the true distribution of tumor proportion does not follow a uniform distribution between $[0,1]$, but instead skewed to the upper part of the interval.

```{r read_data_2comp,  warning=FALSE, message=FALSE}
# ## DeMixT_S1 without Spike-in Normal
# res.S1 = DeMixT_S1(data.Y = test.data.2comp$data.Y, 
#                    data.N1 = test.data.2comp$data.N1,
#                    niter = 30, nbin = 50, nspikein = 0,
#                    if.filter = TRUE, 
#                    mean.diff.in.CM = 0.25, ngene.selected.for.pi = 150,
#                    tol = 10^(-5))
# ## DeMixT_S1 with Spike-in Normal
# res.S1.SP = DeMixT_S1(data.Y = test.data.2comp$data.Y, 
#                      data.N1 = test.data.2comp$data.N1,
#                      niter = 30, nbin = 50, nspikein = 50,
#                      if.filter = TRUE, 
#                      mean.diff.in.CM = 0.25, ngene.selected.for.pi = 150,
#                      tol = 10^(-5))
# ## DeMixT_GS with Spike-in Normal
# res.GS.SP = DeMixT_GS(data.Y = test.data.2comp$data.Y, 
#                      data.N1 = test.data.2comp$data.N1,
#                      niter = 30, nbin = 50, nspikein = 50,
#                      if.filter = TRUE, ngene.Profile.selected = 150,
#                      mean.diff.in.CM = 0.25, ngene.selected.for.pi = 150,
#                      tol = 10^(-5))
load('Res_2comp/res.S1.RData'); load('Res_2comp/res.S1.SP.RData'); 
load('Res_2comp/res.GS.RData'); load('Res_2comp/res.GS.SP.RData'); 
```

This simulation was designed to compare previous DeMixT resutls with DeMixT spike-in results under both gene selection method.
```{r sim_2comp, fig.height = 4, fig.width = 6, fig.align='center', warning=FALSE} 
res.2comp = as.data.frame(cbind(round(rep(t(test.data.2comp$pi[2,]),3),2), 
                                round(c(t(res.S1$pi[2,]),t(res.S1.SP$pi[2,]), t(res.GS.SP$pi[2,])),2),
                                rep(c('S1','S1-SP','GS-SP'), each = 100)), num = 1:2)
res.2comp$V1 <- as.numeric(as.character(res.2comp$V1))
res.2comp$V2 <- as.numeric(as.character(res.2comp$V2))
res.2comp$V3 = as.factor(res.2comp$V3)
names(res.2comp) = c('True.Proportion', 'Estimated.Proportion', 'Method')
## Plot
ggplot(res.2comp, aes(x=True.Proportion, y=Estimated.Proportion, group = Method, color=Method, shape=Method)) +
  geom_point() + 
  geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "black", lwd = 0.5) +
  xlim(0,1) + ylim(0,1)  +
  scale_shape_manual(values=c(seq(1:3))) +
  labs(x = 'True Proportion', y = 'Estimated Proportion')  
```

## 5.3 Simulated three-component data 

In this simulation, 
````markdown
G <- G1 + G2
## Simulate MuN1, MuN2 and MuT for each gene
  MuN1 <- rnorm(G, 7, 1.5)
  MuN2_1st <- MuN1[1:G1] + truncdist::rtrunc(n = 1, 
                                             spec = 'norm',
                                             mean = 0,
                                             sd = 1.5,
                                             a = -0.1, 
                                             b = 0.1)
  MuN2_2nd <- c()
  for(l in (G1+1):G){
    tmp <- MuN1[l] + truncdist::rtrunc(n = 1, 
                                       spec = 'norm',
                                       mean = 0,
                                       sd = 1.5,
                                       a = 0.1, 
                                       b = 3)^rbinom(1, size=1, prob=0.5)
    while(tmp <= 0) tmp <- MuN1[l] + truncdist::rtrunc(n = 1, 
                                                       spec = 'norm',
                                                       mean = 0,
                                                       sd = 1.5,
                                                       a = 0.1, 
                                                       b = 3)^rbinom(1, size=1, prob=0.5)
    MuN2_2nd <- c(MuN2_2nd, tmp)
  }
## Simulate SigmaN1, SigmaN2 and SigmaT for each gene
  SigmaN1 <- runif(n = G, min = 0.1, max = 0.8)
  SigmaN2 <- runif(n = G, min = 0.1, max = 0.8)
  SigmaT <- runif(n = G, min = 0.1, max = 0.8)
## Simulate Tumor Proportion
  pi <- matrix(0, 3, My)
  pi[1,] <- runif(n = My, min = 0.01, max = 0.97)
  for(j in 1:My){
    pi[2, j] <- runif(n = 1, min = 0.01, max = 0.98 - pi[1,j])
    pi[3, j] <- 1 - sum(pi[,j])
  }
## Simulate Data
  for(k in 1:G){
    
    data.N1[k,] <- 2^rnorm(M1, MuN1[k], SigmaN1[k]); # normal reference 1
    
    data.N2[k,] <- 2^rnorm(M2, MuN2[k], SigmaN2[k]); # normal reference 1
    
    True.data.T[k,] <- 2^rnorm(My, MuT[k], SigmaT[k]);  # True Tumor
    
    True.data.N1[k,] <- 2^rnorm(My, MuN1[k], SigmaN1[k]);  # True Normal 1
    
    True.data.N2[k,] <- 2^rnorm(My, MuN2[k], SigmaN2[k]);  # True Normal 1
    
    data.Y[k,] <- pi[1,]*True.data.N1[k,] + pi[2,]*True.data.N2[k,] +
      pi[3,]*True.data.T[k,] # Mixture Tumor
    
  }
````
where $G1$ is the number of genes that $\mu_{N1}$ is close to $\mu_{N2}$.

```{r read_data_3comp,  warning=FALSE, message=FALSE}
data("test.data.3comp")
# res.S1 <- DeMixT_S1(data.Y = test.data.3comp$data.Y, data.N1 = test.data.3comp$data.N1,
#                    data.N2 = test.data.3comp$data.N2, if.filter = TRUE)
load('Res_3comp/res.S1.RData'); 
```

```{r sim_3comp, fig.height = 4, fig.width = 6, fig.align='center', warning=FALSE} 
res.3comp= as.data.frame(cbind(round(t(matrix(t(test.data.3comp$pi), nrow = 1)),2), 
                                round(t(matrix(t(res.S1$pi), nrow = 1)),2), 
                                rep(c('N1','N2','T'), each = 20)))
res.3comp$V1 <- as.numeric(as.character(res.3comp$V1))
res.3comp$V2 <- as.numeric(as.character(res.3comp$V2))
res.3comp$V3 = as.factor(res.3comp$V3)
names(res.3comp) = c('True.Proportion', 'Estimated.Proportion', 'Component')
## Plot
ggplot(res.3comp, aes(x=True.Proportion, y=Estimated.Proportion, group = Component, color=Component, shape=Component)) +
  geom_point() + 
  geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "black", lwd = 0.5) +
  xlim(0,1) + ylim(0,1)  +
  scale_shape_manual(values=c(seq(1:3))) +
  labs(x = 'True Proportion', y = 'Estimated Proportion')  
```



## 5.4 PRAD in TCGA Dataset

```{r read_data_PRAD,  warning=FALSE, message=FALSE} 
load('res.PRAD.RData'); 
```

```{r PRAD, fig.height = 4, fig.width = 6, fig.align='center', warning=FALSE} 
res.PRAD.df = as.data.frame(cbind(res.PRAD$res.GS.PRAD, res.PRAD$res.GS.SP.PRAD))
res.PRAD.df$V1 <- as.numeric(as.character(res.PRAD.df$V1))
res.PRAD.df$V2 <- as.numeric(as.character(res.PRAD.df$V2))
names(res.PRAD.df) = c('Estimated.Proportion', 'Estimated.Proportion.SP')
## Plot
ggplot(res.PRAD.df, aes(x=Estimated.Proportion, y=Estimated.Proportion.SP)) +
  geom_point() + 
  geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "black", lwd = 0.5) +
  xlim(0,1) + ylim(0,1)  +
  scale_shape_manual(values=c(seq(1:3))) +
  labs(x = 'Estimated Proportion', y = 'Estimated Proportion After Spike-in')  
```
Tumor proportion estimation remains consistence with and without spike-in normal, when its distribution is roughly symmetric around 0.5.


# 6. Session Info 

```{r}
sessionInfo(package = "DeMixT")
```
