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The formatted source code for this file is here.
And a raw version here.
Previous work by Youngser Park can be found here.

1 Introduction

On Fri, Dec 11, 2015 at 11:53 AM, joshua vogelstein jovo@jhu.edu wrote:
we will get n=10^6 points, each in d=25 dimensions.
i want to hierarchically cluster them, in a ways:

  1. recursive k-means on the data, maybe 5 levels
  2. compute approximate k-neighbors, svd in d=dimensions, and then #1
  3. maybe some other ways.

2 Data

This corresponds to 24 channels x 6 features per synapse, ordered like c0f0,c0f1,c0f2,c0f3,c0f4,c0f5,c1f0,c1f1… etc

f0 = integrated brightness
f1 = local brightness
f2 = distance to Center of Mass
f3 = moment of inertia around synapsin maxima
f4,f5 are features that I forget what they are.. would need to ask brad.
i would throw them out, I did so in my kohonen code (which you have, its in matlab).

and

On Feb 8, 2016, at 2:00 PM, Kristina Micheva kmicheva@stanford.edu wrote:

and

On March 10, 2016, 00:29:04 (UTC), Kristina Micheva kmicheva@stanford.edu wrote:

There are 2 different Synap channels (2 different antibodies were used), so that part is fine. And 2 different VGluT1 channels (same antibody but done at different times) The NOS channel is the same, so count it as one even though it appears twice. It is listed two times because it can be found at both excitatory and inhibitory synapses. This is where your count of 25 comes, even though there are 24 channels. I would also add the 2 Synap channels to the Inhibitory presynaptic category - there is supposed to be synapsin there, but at lower levels compared to excitatory presynaptic category.

2.1 Potential Filtering

Following is a discussion on which subset of markers could be used as a subset to explore.

On Thu, Apr 14, 2016 at 3:05 AM, Kristina Micheva kmicheva@stanford.edu wrote:

I suggest: Synap, VGluT1, VGluT2, psd, gad, vgat, gephyr,

Or a bit bigger: Synap, VGluT1, VGluT2, psd, gad, vgat, gephyr, VGlut3, CB1

On Apr 12, 2016, at 9:54 AM, Jesse L. Patsolic studiojlp@gmail.com wrote:

Kristina,

Out of the markers available, which do you think are the best to use as a subset?

This subset has not yet been explored.

featFull <- fread("../Data/synapsinR_7thA.tif.Pivots.txt.2011Features.txt",showProgress=FALSE)

### Setting a seed and creating an index vector
### to select half of the data
set.seed(2^10)
half1 <- sample(dim(featFull)[1],dim(featFull)[1]/2)
feat <- featFull[half1,]
dim(feat)
# [1] 559649    144
channel <- c('Synap_1','Synap_2','VGlut1_t1','VGlut1_t2','VGlut2','Vglut3',
              'psd','glur2','nmdar1','nr2b','gad','VGAT',
              'PV','Gephyr','GABAR1','GABABR','CR1','5HT1A',
              'NOS','TH','VACht','Synapo','tubuli','DAPI')
channel.type <- c('ex.pre','ex.pre','ex.pre','ex.pre','ex.pre','in.pre.small',
                  'ex.post','ex.post','ex.post','ex.post','in.pre','in.pre',
                  'in.pre','in.post','in.post','in.post','in.pre.small','other',
                  'ex.post','other','other','ex.post','none','none')
nchannel <- length(channel)
nfeat <- ncol(feat) / nchannel
ffchannel <- (factor(channel.type,
    levels= c("ex.pre","ex.post","in.pre","in.post","in.pre.small","other","none")
    ))
fchannel <- as.numeric(factor(channel.type,
    levels= c("ex.pre","ex.post","in.pre","in.post","in.pre.small","other","none")
    ))
ford <- order(fchannel)
Syncol <- c("#197300","#5ed155","#660000","#cc0000","#ff9933","mediumblue","gold")
ccol <- Syncol[fchannel]

exType <- factor(c(rep("ex",11),rep("in",6),rep("other",7)),ordered=TRUE)
exCol<-exType;levels(exCol) <- c("#197300","#990000","mediumblue");
exCol <- as.character(exCol)

fname <- as.vector(sapply(channel,function(x) paste0(x,paste0("F",0:5))))
names(feat) <- fname
fcol <- rep(ccol, each=6)
mycol <- colorpanel(100, "purple", "black", "green")
mycol2 <- matlab.like(nchannel)

2.2 Transformations: Considering only f0 Integrated Brightness

We will consider only the f0 (integrated brightness) features, and will transform the feature vector data by scaling and filtering.

f0 <- seq(1,ncol(feat),by=nfeat)
featF0 <- subset(feat, select=f0)
f01e3 <- 1e3*data.table(apply(X=featF0, 2, function(x){((x-min(x))/(max(x)-min(x)))}))

fs <- f01e3

### Taking log_10 on data + 1.
log1f <- log10(featF0 + 1)
slog1f <- data.table(scale(log1f, center=TRUE,scale=TRUE))

We now have the following data sets:

  • featF0: The feature vector looking only at the integrated brightness features.
  • fs: The feature vector scaled between \([0,1000]\).
  • logf1: The feature vector, plus one, then \(log_{10}\) is applied.
  • slog1f: The feature vector, plus one, \(log_{10}\), then scaled by subtracting the mean and dividing by the sample standard deviation.

2.2.1 Kernel Density Estimates of the marginals

df <- melt(as.matrix(log1f))
names(df) <- c("ind","channel","value")
df$type <- factor(rep(ffchannel,each=dim(fs)[1]),levels=levels(ffchannel))

lvo <- c(1:5,7:10,19,22,11:16,6,17,18,20,21,23,24)
levels(df$channel)<-levels(df$channel)[lvo]

ts <- 22

gg1 <- ggplot(df, aes(x=value)) + 
    scale_color_manual(values=ccol[lvo]) +
    scale_fill_manual(values=ccol[lvo]) +
    geom_histogram(aes(y=..density..,group=channel,colour=channel),bins=100) +
    geom_density(aes(group=channel, color=channel),size=1.5) +
    facet_wrap( ~ channel, scale='free', ncol=6) +
    theme(plot.title=element_text(size=ts),
          axis.title.x=element_text(size=ts),
          axis.title.y=element_text(size=ts),
          legend.title=element_text(size=ts),
          legend.text=element_text(size=ts-2),
          axis.text=element_text(size=ts-2),
          strip.text=element_text(size=ts), 
          legend.position='none')+
    ggtitle("Kernel Density Estimates of `log1f` data.")

print(gg1)
Figure 1: Kernel density estimates for each channel, on log data.

2.3 Correlations

tmp <- as.numeric(table(fchannel))
cmatslog1f <- cor(slog1f)
corrplot(cmatslog1f[ford,ford],method="color",tl.col=ccol[ford], tl.cex=0.8)
corrRect(tmp,col=Syncol,lwd=4)
Figure 2: Correlation on log_10 data, reordered by synapse type.

2.3.1 PCA on the Correlation Matrix

pcaLog <- prcomp(cmatslog1f,scale=TRUE, center=TRUE)
elLog <- getElbows(pcaLog$sdev, plot=FALSE)

We run K-means for \(K=3\) on the PCA embedded in \(\mathbb{R}^3\) of the correlation matrix.

K1 <- c(3)  ## The set of K's.

## Run kmeans on the pca of the correlation matrix of the slog1f data
kvecCor <- foreach(i = K1) %dopar% {
    set.seed(2^4 - 1)
    kmeans(pcaLog$x[,1:3],centers=i)
}

2.3.2 Plots of embeddings

par(mfrow=c(1,2))

plot(pcaLog$x[,1:2],
     col=ccol[ford],
     pch=as.numeric(exType)+15,
     cex=1.5,
     xlim=c(min(pcaLog$x[,1])-0.2,max(pcaLog$x[,1])+0.7),
     main="Embedding of PCA log_10 correlation data")
text(pcaLog$x[,1:2],label=abbreviate(rownames(pcaLog$x)),offset=1, pos=4)


plot(pcaLog$x[,1:2],
     col=as.numeric(kvecCor[[1]]$cluster)+1,
     pch=as.numeric(kvecCor[[1]]$cluster)-1,
     cex=1.5,
     xlim=c(min(pcaLog$x[,1])-0.2,max(pcaLog$x[,1])+0.7),
     main="Embedding of PCA log_10 correlation data, \ncolor based on 3-means clustering.")
text(pcaLog$x[,1:2],col='black',label=abbreviate(rownames(pcaLog$x)),offset=1, pos=4)
Figure 3: 2d Embeddings.

pca <- pcaLog$x[,1:3]
rgl::plot3d(pca[,1],pca[,2],pca[,3],type='s',col=ccol[ford], size=1, main="Log")
rgl::rgl.texts(pca[,1],pca[,2],pca[,3], abbreviate(rownames(pca)), col=ccol[ford], adj=c(0,1.5))
subid <- currentSubscene3d()
rglwidget(elementId="plot3dLog", width=720, height=720)

2.4 K-Means Level 1

Next we run K-means with \(K=3\).

** Note that a seed is being set for the random initialization of K-means. **

K2 <- c(2)  ## The set of K's.

## Run kmeans on the slog1f data
kvecslog1f <- foreach(i = K2) %dopar% {
    set.seed(2^4 - 1)
    kmeans(slog1f,centers=i)
}

2.4.1 Heat maps: scaled data.

For the following we manualy choose 2 clusters.

## Formatting data for heatmap
aggslog1f <- aggregate(slog1f,by=list(lab=kvecslog1f[[1]]$cluster),FUN=mean)
aggslog1f <- as.matrix(aggslog1f[,-1])
rownames(aggslog1f) <- clusterFraction(kvecslog1f[[1]])

ford <- order(fchannel)
heatmap.2(as.matrix(aggslog1f[,ford]),dendrogram='row',Colv=NA,trace="none", col=mycol,colCol=ccol[ford],cexRow=0.8, keysize=1.25,symkey=FALSE,symbreaks=FALSE,scale="none", srtCol=90,main="Heatmap of `slog1f` data.") 
#  [1] "#197300"    "#197300"    "#197300"    "#197300"    "#197300"   
#  [6] "#5ed155"    "#5ed155"    "#5ed155"    "#5ed155"    "#5ed155"   
# [11] "#5ed155"    "#660000"    "#660000"    "#660000"    "#cc0000"   
# [16] "#cc0000"    "#cc0000"    "#ff9933"    "#ff9933"    "mediumblue"
# [21] "mediumblue" "mediumblue" "gold"       "gold"
Figure 4: Heatmap of the cluster means vs channels. Rows and columns are rearranged according to synapse type.

Percentage of data within cluster is presented on the right side of the heatmap.

3 Exploring pair-wise relationships

### Sampling to reduce size
set.seed(2^13 - 2)
s1 <- sample(dim(slog1f)[1],2.5e5)
dlog1f <- data.table(log1f[s1,])

3.1 GABABR

## re-formatting data for use in lattice 
dlog1f2 <- data.table(stack(dlog1f, select=-GABABRF0))[,.(values)]
dlog1f2$GABABR <- dlog1f$GABABRF0

### Adding relationship factor variables
nd <- paste0("GABABR","~",abbreviate(channel[-which(channel=="GABABR")]))

dlog1f2$ind <- factor(rep(nd,each=dim(dlog1f)[1]), ordered=TRUE,levels=nd)
names(dlog1f2) <- c("x","y","g")

rg1 <- xyplot(y ~ x | g, data=dlog1f2,
       as.table=TRUE,
       colramp=BTC,
       pch='.',
       scales = list(y = list(relation = "free"),x = list(relation = "free")),
       panel=function(x,y,...){
           panel.hexbinplot(x,y,..., type='g')
           panel.loess(x,y,col='red', lwd=2,...)
        }
       )
Figure 5: Lattice plot of pairwise regressions involving GABABR

4 ARI

Here we consider the channel types to be the “ground truth” and computer the Adjusted Rand Index of between that and the output from k-means.

4.1 Approximate permutation test.

levels(ffchannel) <- c(rep("ex", 2), rep("in", 2), rep("other", 3))
levels(ffchannel)
# [1] "ex"    "in"    "other"
truth <- as.numeric(ffchannel)

Making a data.table of the permutation data for ggplot.

DT <- data.table(avd(pcaLog, elLog[2], truth),key='Embedding_Dimension') 

DT <- DT[,pval := sum(permARI>=ari)/length(permARI),by=Embedding_Dimension]

ua <- DT[,unique(ari),by=Embedding_Dimension]
arid <- data.frame(Embedding_Dimension=as.numeric(ua$Emb),
                   ARI=ua$V1,
                   pval=DT[,unique(pval),by=Embedding_Dimension]$V1)
gg3 <- ggplot(data=DT,aes(x=permARI, y=..density..,color=Embedding_Dimension,label=pval)) + 
        #geom_histogram(binwidth=3.49*sd(DT$permARI)*length(DT$permARI)^(-1/3)) +
        geom_histogram(bins=25)+
        geom_vline(aes(xintercept=ari),colour='darkred',size=1.2)+
        #geom_text(aes(x=(ari-ari/2),y=7))+
        theme(axis.text=element_text(size=18),
                  title=element_text(size=16),  
                  strip.text.x=element_text(size=16)) + 
        facet_wrap(~Embedding_Dimension+pval,scale='free',labeller=label_both)
print(gg3)
Figure 6: ARI Permutation Tests

gg5 <- ggplot(data=arid,aes(x=Embedding_Dimension,y=ARI,label=pval,colour=pval)) + 
        geom_line(size=1.5,colour='salmon') + geom_point(size=3) +
            #geom_text(hjust='right',vjust='top',nudge_x=0.5,nudge_y=0.01,size=5)+
            theme(axis.text=element_text(size=18),
                  title=element_text(size=16)) + 
            ggtitle("ARI vs. DIM with estimated p-values")

gg6 <- 
    ggplot(data=arid, aes(x=Embedding_Dimension, y=pval, colour=pval))+
        geom_line(size=1.5, colour='salmon') + 
        geom_point(size=3) + 
       # geom_hline(yintercept=0.05, 
       #            colour='forestgreen',
       #            size=1.5) +
        scale_y_log10(breaks=c(1e-4,1.53-4,1e-3), na.value=0) +
        theme(axis.text=element_text(size=18),
                  title=element_text(size=16)) + 
        ggtitle("P-values")

grid.arrange(gg5,gg6,nrow=2)
Figure 7: ARI and P-Values