# Overview for mand

2023-09-12 @Atsushi Kawaguchi

The `mand` package provides functions for multivariate analysis for neuroimaging data.

    library(mand)

    ## Loading required package: msma

# Introduction

## Template

One subject image as the template is available in the mand package. The coad to load it is as follows.

    data(template)

The image is compressed because of storage and computation time. The dimension is confirmed as follows.

    dim(template)

    ## [1] 30 36 30

The image is plotted by the `coat` function.

    coat(template)

Other options with the plane argument (such as “axial,” “coronal,” “sagittal,” and “all”) are available. The default setting is “axial”. If the argument is specified as `plane="all"`, three directional slices at a certain coordinate are plotted.

    coat(x=template, plane="all")

## Image Data Matrix

The `imgdatamat` function reads image files saved in the nifti format and creates data matrix with subjects in row and voxel in column (this example does not work).

    fnames1 = c("data1.nii", "data2.nii")
    imgmat = imgdatamat(fnames1, simscale=1/4)

The first argument is file names with the length equaling the number of subjects (the number of rows in the resulting data matrix). The second argument `simscale` is the image resize scale. In this example, the all sizes (number of voxel) for three direction was reduced into its quarter size. The output is the list form where the “S” is data matrix, the “brainpos” is a binary image indicating brain region, and the “imagedim” is image dimension. The ROI (Region Of Interest) volume is computed in the “roi” if the ROI argument is TRUE.

    imgmat = imgdatamat(fnames1, schange=TRUE, simscale=1/4, ROI=TRUE)

## Overlay

The resulting map from the statistical analysis such as the t statistics map from the SPM is represented with overlapping on the template. For example, the `mand` package has the average difference assuming Alzheimer’s disease and healthy subjects with the array format.

The overlay is implemented by the `coat` function.

    data(diffimg)
    coat(template, diffimg)

## Atlas

Anatomical brain segmentation region is useful for the plot and the interpretation. For example, the Automated Anatomical Labeling (AAL) atlas is used. The data.frame has two columns (“ROIid” and “ROIname”) format.

    data(atlasdatasets)
    atlasname = "aal3"
    atlasdataset = atlasdatasets[[atlasname]]
    head(atlasdataset)

    ##   ROIid                                   ROIname
    ## 1     1                     Left Precentral gyrus
    ## 2     2                    Right Precentral gyrus
    ## 3     3  Left Superior frontal gyrus-dorsolateral
    ## 4     4 Right Superior frontal gyrus-dorsolateral
    ## 5     5                 Left Middle frontal gyrus
    ## 6     6                Right Middle frontal gyrus

It is also neccesary to prepare the array data.

    data(atlas)
    tmpatlas = atlas[[atlasname]]
    dim(tmpatlas)

    ## [1] 30 36 30

It has the ROIid as the element.

    tmpatlas[,15:20,12]

    ##       [,1] [,2] [,3] [,4] [,5] [,6]
    ##  [1,]    0    0    0    0    0    0
    ##  [2,]    0    0    0    0    0    0
    ##  [3,]   90   90   90   90   90   90
    ##  [4,]   94   94   90   90   90   90
    ##  [5,]   94   94   90   90   90   90
    ##  [6,]   94   90   90   90   90   90
    ##  [7,]   94    0    0    0   90   90
    ##  [8,]    0    0    0    0    0    0
    ##  [9,]   60   44   42   42   42    0
    ## [10,]   60   44   42   42   42   42
    ## [11,]   44   44   42   42   42   42
    ## [12,]   44   44   44    0    0   42
    ## [13,]   52    0    0    0    0    0
    ## [14,]  100    0    0  162  162  164
    ## [15,]  114    0    0  166  166    0
    ## [16,]    0    0    0    0    0    0
    ## [17,]   99    0    0  165    0    0
    ## [18,]  101  101    0  163  163  163
    ## [19,]   43    0    0    0    0   41
    ## [20,]    0   43   41   41   41   41
    ## [21,]   43   43   41   41   41   41
    ## [22,]   43   43   41   41   41   41
    ## [23,]    0    0    0    0    0    0
    ## [24,]   93    0    0    0    0    0
    ## [25,]   93    0    0   89   89   89
    ## [26,]   93   89   89   89   89   89
    ## [27,]   89   89   89   89   89   89
    ## [28,]   89   89   89   89   89   89
    ## [29,]    0   89   89   89   89    0
    ## [30,]    0    0    0    0    0    0

The anatomical region can be plotted by the `coat` function with regionplot=TRUE.

    coat(template, tmpatlas, regionplot=TRUE,
    atlasdataset=atlasdataset, ROIids = c(1:2, 41:44), regionlegend=TRUE)

The resulting map can be converted into the summary of the anatomical regions.

    atlastable(tmpatlas, diffimg, atlasdataset, ROIids = c(1:2, 41:44))

    ##    ROIid                     ROIname sizepct sumvalue   Min. Mean  Max.
    ## 41    41            Left Hippocampus   0.591   -7.190 -1.000    0 0.448
    ## 42    42           Right Hippocampus   0.733  -12.110 -0.968    0 0.505
    ## 43    43  Left Parahippocampal gyrus   0.732   -5.495 -0.875    0 0.530
    ## 1      1       Left Precentral gyrus   0.617    8.300 -0.568    0 0.802
    ## 44    44 Right Parahippocampal gyrus   0.692   -3.147 -0.750    0 0.690
    ## 2      2      Right Precentral gyrus   0.632   -0.305 -0.735    0 0.750

The outputs are the number of voxel in the `sizenum` column, the percentage of the voxel in the `sizepct` column, and the minimum, mean, and maximum valued of the region of the overlaying map. The order of the table row is in the larger absolute value of the minimum or maximum values.

The brain image corresponding to the region of interest can be extracted as follows. First, we create a mask image in which the hippocampal region is represented by 1 and others by 0. Then the product of the template and the mask image is taken for each voxel.

    hipmask = (tmpatlas == 41) + (tmpatlas == 42)
    template2 = template * hipmask

The images generated by these processes are plotted from left to right in one slice.

    par(mfrow=c(1,3), mar=rep(1,4))
    coat(template, pseq=12, paron=FALSE)
    coat(hipmask, pseq=12, paron=FALSE)
    coat(template2, pseq=12, paron=FALSE)

The template image (left) and the mask image (middle) are multiplied voxel by voxel to obtain the only hippocampus region image (right).

The sum of the voxel values in the region is calculated as follows.

    sum(template[which(hipmask==1, arr.ind = TRUE)])/1000

    ## [1] 0.07090703

Such a value is calculated for each region and a dataset with ROI values is created.

# Principal Component Analysis

## Generate Simulation Data

The `mand` package has a function to generate simulation brain data from the base image, the difference image and the standard deviation image. These basic images are loaded as follows.

    data(baseimg)
    data(diffimg)
    data(mask)
    data(sdevimg)

The number of voxels in the original 3D image is as follows.

    dim(baseimg)

    ## [1] 30 36 30

To understand the result easily, the difference region was restricted to Parahippocampus and Hippocampus.

    diffimg2 = diffimg * (tmpatlas %in% 41:44)

An image data matrix with subjects in the rows and voxels in the columns was generated by using the `simbrain` function.

    img1 = simbrain(baseimg = baseimg, diffimg = diffimg2,
    sdevimg=sdevimg, mask=mask, n0=20, c1=0.01, sd1=0.05)

The base image, the difference image and the standard deviation image were specified in the first three arguments. The out-of-brain region was specified by the mask argument, which was the binary image. The remaining arguments are the number of subjects per group, the coefficient multiplied by the difference image and the standard deviation for the noise.

The data matrix dimension was 40(subject) x 12620(voxel).

    dim(img1$S)

    ## [1]    40 12620

The `rec` function creates the 3D image from the vectorized data (the first subject).

    coat(rec(img1$S[1,], img1$imagedim, mask=img1$brainpos))

The standard deviation image is created from the resulting data matrix.

    sdimg = apply(img1$S, 2, sd)
    coat(template, rec(sdimg, img1$imagedim, mask=img1$brainpos))

## Principal Component Analysis

If the input is a matrix, a principal component analysis is implemented by the `msma` function of the `msma` package. Principal component analysis with the number of components of 2 for the image data matrix is performed as follows.

    (fit111 = msma(img1$S, comp=2))

    ## Call:
    ## msma.default(X = img1$S, comp = 2)
    ##
    ## Numbers of non-zeros for X block:
    ##        comp1 comp2
    ## block1  9865  9865
    ##
    ## Numbers of non-zeros for X super:
    ##         comp1 comp2
    ## comp1-1     1     1

The scatter plots for the score vectors specified by the argument `v`. The argument `axes` is specified by the two length vectors on which components are displayed.

    plot(fit111, v="score", axes = 1:2, plottype="scatter")

The weight (loading) vectors can be obtained and reconstructed as follows.

    midx = 1 ## the index for the modality
    vidx = 1 ## the index for the component
    Q = fit111$wbX[[midx]][,vidx]
    outstat1 = rec(Q, img1$imagedim, mask=img1$brainpos)

The reconstructed loadings as image is overlayed on the template.

    coat(template, outstat1)

The output is unclear; however, this will be improved later.

# Two-steps dimension reduction

## Basis Expansion

This is an example of the two-step dimension reduction.

Generate radial basis function.

    B1 = rbfunc(imagedim=img1$imagedim, seppix=2, hispec=FALSE, mask=img1$brainpos)

Multiplying the basis function to image data matrix.

    SB1 = basisprod(img1$S, B1)

The original dimension was 12620.

    dim(img1$S)

    ## [1]    40 12620

The dimension was reduced to 1604.

    dim(SB1)

    ## [1]   40 1604

## Principal Component Analysis

The Principal Component Analysis (PCA) is applied to the dimension-reduced image.

    (fit211 = msma(SB1, comp=2))

    ## Call:
    ## msma.default(X = SB1, comp = 2)
    ##
    ## Numbers of non-zeros for X block:
    ##        comp1 comp2
    ## block1  1604  1604
    ##
    ## Numbers of non-zeros for X super:
    ##         comp1 comp2
    ## comp1-1     1     1

The loading is reconstructed to the original space by using the `rec` function.

    Q = fit211$wbX[[1]][,1]
    outstat1 = rec(Q, img1$imagedim, B=B1, mask=img1$brainpos)

The plotted loading is smoother than the one without the dimension reduction by the basis function.

    coat(template, outstat1)

## Sparse PCA

If `lambdaX` (>0) is specified, a sparse principal component analysis is implemented.

    (fit112 = msma(SB1, comp=2, lambdaX=0.06))

    ## Call:
    ## msma.default(X = SB1, comp = 2, lambdaX = 0.06)
    ##
    ## Numbers of non-zeros for X block:
    ##        comp1 comp2
    ## block1    53    40
    ##
    ## Numbers of non-zeros for X super:
    ##         comp1 comp2
    ## comp1-1     1     1

The plotted loading is narrower than that from the PCA.

    Q = fit112$wbX[[midx]][,vidx]
    outstat1 = rec(Q, img1$imagedim, B=B1, mask=img1$brainpos)
    outstat2 = outstat1
    coat(template, outstat2)

The region is reported as follows to be compared with the next method.

    atlastable(tmpatlas, outstat2, atlasdataset)

    ##     ROIid                                       ROIname sizepct sumvalue Min.  Mean  Max.
    ## 154   154     Right Anterior cingulate cortex-pregenual   1.000  208.904    0 0.006 7.480
    ## 153   153      Left Anterior cingulate cortex-pregenual   1.000  133.103    0 0.004 6.944
    ## 156   156 Right Anterior cingulate cortex-supracallosal   1.000  136.198    0 0.004 6.880
    ## 155   155  Left Anterior cingulate cortex-supracallosal   1.000  106.948    0 0.003 6.449
    ## 19     19            Left Superior frontal gyrus-medial   1.000  237.574    0 0.007 6.166
    ## 20     20           Right Superior frontal gyrus-medial   1.000  226.415    0 0.007 6.114
    ## 38     38   Right Middle cingulate & paracingulate gyri   0.459   40.319    0 0.001 5.686
    ## 4       4     Right Superior frontal gyrus-dorsolateral   0.845  257.710    0 0.008 5.165
    ## 6       6                    Right Middle frontal gyrus   0.976  127.759    0 0.004 3.423
    ## 152   152     Right Anterior cingulate cortex-subgenual   1.000   13.537    0 0.000 3.414

## Supervised Sparse PCA

The `simbrain` generates the synthetic brain image data and the binary outcome. The outcome Z is obtained.

    Z = img1$Z

If the outcome Z is specified in the `msma` function, a supervised sparse principal component analysis is implemented.

    (fit113 = msma(SB1, Z=Z, comp=2, lambdaX=0.06, muX=0.5))

    ## Call:
    ## msma.default(X = SB1, Z = Z, comp = 2, lambdaX = 0.06, muX = 0.5)
    ##
    ## Numbers of non-zeros for X block:
    ##        comp1 comp2
    ## block1    48    38
    ##
    ## Numbers of non-zeros for X super:
    ##         comp1 comp2
    ## comp1-1     1     1

The plotted loading is located differently from the sparse PCA.

    Q = fit113$wbX[[1]][,1]
    outstat1 = rec(Q, img1$imagedim, B=B1, mask=img1$brainpos)
    outstat2 = -outstat1
    coat(template, outstat2)

The region near the hippocampus, which differs from the sparse PCA (without supervision).

    atlastable(tmpatlas, outstat2, atlasdataset)

    ##     ROIid                      ROIname sizepct sumvalue Min.  Mean  Max.
    ## 83     83           Left Heschls gyrus   1.000   63.252    0 0.002 6.878
    ## 41     41             Left Hippocampus   1.000  129.029    0 0.004 6.666
    ## 85     85 Left Superior temporal gyrus   1.000  297.309    0 0.009 6.196
    ## 145   145         Left Pulvinar medial   1.000   35.446    0 0.001 6.040
    ## 33     33                  Left Insula   0.917  100.941    0 0.003 5.910
    ## 129   129  Left Ventral posterolateral   1.000   39.922    0 0.001 5.809
    ## 149   149       Left Pulvinar inferior   1.000    5.286    0 0.000 5.286
    ## 13     13      Left Rolandic operculum   1.000  125.476    0 0.004 5.222
    ## 43     43   Left Parahippocampal gyrus   1.000   71.732    0 0.002 5.095
    ## 51     51           Left Lingual gyrus   0.853   66.548    0 0.002 4.585

The loading for the second component

    Q = fit113$wbX[[1]][,2]
    outstat1 = rec(Q, img1$imagedim, B=B1, mask=img1$brainpos)
    outstat2 = -outstat1
    coat(template, outstat2)

    atlastable(tmpatlas, outstat2, atlasdataset)

    ##    ROIid                                                               ROIname sizepct sumvalue Min.  Mean  Max.
    ## 55    55                                           Left Middle occipital gyrus   0.988  186.365    0 0.006 7.968
    ## 53    53                                         Left Superior occipital gyrus   1.000  116.745    0 0.004 7.709
    ## 63    63                                          Left Superior parietal gyrus   1.000  322.454    0 0.010 7.646
    ## 71    71                                                        Left Precuneus   1.000  307.242    0 0.009 7.467
    ## 65    65 Left Inferior parietal gyrus-excluding supramarginal and angular gyri   1.000  179.870    0 0.006 6.838
    ## 69    69                                                    Left Angular gyrus   1.000  120.055    0 0.004 6.088
    ## 49    49                                                           Left Cuneus   1.000  118.538    0 0.004 5.685
    ## 37    37                            Left Middle cingulate & paracingulate gyri   0.894   35.244    0 0.001 5.265
    ## 39    39                                        Left Posterior cingulate gyrus   1.000   23.401    0 0.001 3.521
    ## 47    47                         Left Calcarine fissure and surrounding cortex   1.000   45.335    0 0.001 2.301

This is similar to the result from the sparse PCA (without supervision).

The following method can be used to plot the weights of several components simultaneously. It is first reconstructed in three dimensions with the `multirec` function and then plotted with the `multicompplot` function. It is set to display four columns per component.

    ws = multirec(fit113, imagedim=img1$imagedim, B=B1, mask=img1$brainpos)
    multicompplot(ws, template, col4comp=4)