Last updated: 2021-03-19
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Knit directory: soybean_exploration/
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# # Cleanup and Global Settings
# rm(list = ls())
# if (!is.null(sessionInfo()$otherPkgs)) {
# invisible(lapply(paste0('package:', names(sessionInfo()$otherPkgs)),
# detach, character.only=TRUE, unload=TRUE))
# }
# graphics.off()
# options(stringsAsFactors = FALSE)
library(tidyverse)── Attaching packages ────────────────────────────────── tidyverse 1.2.1 ──✔ ggplot2 3.2.1 ✔ purrr 0.3.2
✔ tibble 2.1.3 ✔ dplyr 0.8.3
✔ tidyr 1.0.0 ✔ stringr 1.4.0
✔ readr 1.3.1 ✔ forcats 0.4.0── Conflicts ───────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()First, some notation. Let:
Note that \(\bar{y} = \frac{n_j}{n} \bar{y}_j + \frac{n_{j^*}}{n} \bar{y}_{j^*}\) and that \(n = n_j + n_{j^*}\) for all \(j\).
We can created coded yield data which here we will explore in more detail:
\[ x_{ij} = p_{ij}\bar{y}_j + (1 - p_{ij})\bar{y}_{j^*} \]
Note that \[\begin{align*} \frac{1}{n}\sum_ix_{ij} &= \frac{1}{n}\sum_i(p_{ij}\bar{y}_j + (1 - p_{ij})\bar{y}_{j^*}) \\ &= \frac{\bar{y}_j}{n}\sum_ip_{ij} + \frac{\bar{y}_{j^*}}{n}\sum_i(1 - p_{ij}) \\ &= \frac{n_j}{n} \bar{y}_j + \frac{n_{j^*}}{n} \bar{y}_{j^*} \\ &= \bar{y} \end{align*}\] for all \(j\).
Let \(X_j\) denote the column vector with entries \([x_{ij}]_i\). If we consider the yield data to be pre-centered to have \(\bar{y} = 0\) then the \(X_j\) will also be centered by construction for all \(j\). One important consequence of this centering is that \(\frac{n_j}{n} \bar{y}_j + \frac{n_{j^*}}{n} \bar{y}_{j^*} = 0\) and so \(\bar{y}_{j^*} = -\frac{n_j}{n_{j^*}} \bar{y}_j\). We will consider the \(y_i\) to be centered from here on. The variance of the \(X_j\) would then be proportional to \[\begin{align*} X_j^TX_j &= \sum_ix_{ij}^2 \\ &= \sum_i(p_{ij}\bar{y}_j + (1 - p_{ij})\bar{y}_{j^*})^2 \\ &= \sum_i\left(p_{ij}^2\bar{y}_j^2 + 2p_{ij}(1 - p_{ij})\bar{y}_j\bar{y}_{j^*} + (1 - p_{ij})^2\bar{y}_{j^*}^2\right) \\ &= \sum_ip_{ij}\bar{y}_j^2 + \sum_i(1 - p_{ij})\bar{y}_{j^*}^2 \\ &= n_j\bar{y}_j^2 + n_{j^*}\bar{y}_{j^*}^2 \\ &= n_j\bar{y}_j^2 + n_{j^*}(-\frac{n_j}{n_{j^*}} \bar{y}_j)^2 \\ &= n_j\bar{y}_j^2 (1 + \frac{n_j}{n_{j^*}}) \\ &= \frac{n_j}{n_{j^*}}\bar{y}_j^2 (n_{j^*} + n_j) \\ &= \frac{n_j}{n_{j^*}}n\bar{y}_j^2 \\ \end{align*}\] and \[\begin{align*} X_j^TX_k &= \sum_ix_{ij}x_{ik} \\ &= \sum_i(p_{ij}\bar{y}_j + (1 - p_{ij})\bar{y}_{j^*})(p_{ik}\bar{y}_k + (1 - p_{ik})\bar{y}_{k^*}) \\ &= \sum_i\left(p_{ij}p_{ik}\bar{y}_j\bar{y}_k + (1 - p_{ij})p_{ik}\bar{y}_{j^*}\bar{y}_k + p_{ij}(1 - p_{ik})\bar{y}_j\bar{y}_{k^*} + (1 - p_{ij})(1 - p_{ik})\bar{y}_{j^*}\bar{y}_{k^*}\right) \\ \end{align*}\]
These four terms represent the four outcomes in a two-by-two table, perhaps worth investigating further. Let:
These relate to the earlier quantities in the following ways
This two-way table notation allows us to write
\[\begin{align*} X_j^TX_k &= n_{jk}\bar{y}_j\bar{y}_k + n_{{j^*}k}\bar{y}_{j^*}\bar{y}_k + n_{j{k^*}}\bar{y}_j\bar{y}_{k^*} + n_{{j^*}{k^*}}\bar{y}_{j^*}\bar{y}_{k^*} \\ &= n_{jk}\bar{y}_j\bar{y}_k + n_{{j^*}k}(-\frac{n_j}{n_{j^*}} \bar{y}_j)\bar{y}_k + n_{j{k^*}}\bar{y}_j(-\frac{n_k}{n_{k^*}} \bar{y}_k) + n_{{j^*}{k^*}}(-\frac{n_j}{n_{j^*}} \bar{y}_j)(-\frac{n_k}{n_{k^*}} \bar{y}_k) \\ &= \bar{y}_j\bar{y}_k \left( n_{jk} - n_{{j^*}k}(\frac{n_j}{n_{j^*}}) - n_{j{k^*}}(\frac{n_k}{n_{k^*}}) + n_{{j^*}{k^*}}(\frac{n_j}{n_{j^*}})(\frac{n_k}{n_{k^*}}) \right) \\ \end{align*}\]
Notice that if \(\frac{n_j}{n_{j^*}} = \frac{n_{jk}}{n_{{j^*}k}}\) then \(X_j^TX = 0\). This corresponds to the two genes being independant in presence/absence, which results in a covariance of zero.
Let \(\eta\) be a \(d\)-dimensional unit vector i.e. with entries \(\eta_j \in \mathbb{R}\) such that \(\sum_j \eta_j^2 = 1\), and let \(\mathbb{X}\) be the data matrix with entries \(x_{ij}\), then consider \[\begin{align*} (\eta \mathbb{X})^T \eta \mathbb{X} &= \sum_i \left( \sum_j \eta_j x_{ij} \right)^2 \\ &= \sum_i \sum_j \sum_k \eta_j \eta_k x_{ij} x_{ik} \\ &= \sum_j \eta_j^2 \sum_i x_{ij}^2 + 2\sum_j \sum_{k < i} \eta_j \eta_k \sum_i x_{ij} x_{ik} \\ &= \sum_j \eta_j^2 X_j^TX_j + 2\sum_j \sum_{k < i} \eta_j \eta_k X_j^TX_k \\ \end{align*}\]
# Read tidy data saved at the of initial_data_organisation.Rmd
load(file.path('data', 'pav_data.RData'))
# Reduce to NBS genes
nbs_table = pav_table[, c(TRUE, names(pav_table)[-1] %in% nbs$Name)]
# Merge on Yield and reduce to lines for which we have yield data
meta.df = subset(meta.df, !is.na(Yield))
nbs_table = merge(meta.df[, c('Line', 'Yield')], nbs_table)
# Reduce to genes with some decent level of variation
nbs_table = nbs_table[, c(TRUE, TRUE, colMeans(nbs_table[, -(1:2)]) <= 0.98 & colMeans(nbs_table[, -(1:2)]) >= 0.02)]
# Simplify gene names slightly
names(nbs_table) = sub('00.1.p$', '', names(nbs_table))
names(nbs_table) = sub('^GlymaLee.', 'GL', names(nbs_table))
names(nbs_table) = sub('^UWASoyPan', 'UWA', names(nbs_table))
# Move Line information into row.names
row.names(nbs_table) = nbs_table$Line
nbs_table$Line = NULL
sessionInfo()R version 3.6.3 (2020-02-29)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.5 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/libopenblasp-r0.2.20.so
locale:
[1] LC_CTYPE=en_AU.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_AU.UTF-8 LC_COLLATE=en_AU.UTF-8
[5] LC_MONETARY=en_AU.UTF-8 LC_MESSAGES=en_AU.UTF-8
[7] LC_PAPER=en_AU.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_AU.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] forcats_0.4.0 stringr_1.4.0 dplyr_0.8.3 purrr_0.3.2
[5] readr_1.3.1 tidyr_1.0.0 tibble_2.1.3 ggplot2_3.2.1
[9] tidyverse_1.2.1
loaded via a namespace (and not attached):
[1] tidyselect_1.1.0 xfun_0.10 haven_2.3.1 lattice_0.20-41
[5] colorspace_1.4-1 vctrs_0.3.1 generics_0.0.2 htmltools_0.4.0
[9] yaml_2.2.0 rlang_0.4.6 later_1.0.0 pillar_1.4.2
[13] withr_2.1.2 glue_1.3.1 modelr_0.1.5 readxl_1.3.1
[17] lifecycle_0.1.0 munsell_0.5.0 gtable_0.3.0 workflowr_1.6.2
[21] cellranger_1.1.0 rvest_0.3.4 evaluate_0.14 knitr_1.25
[25] httpuv_1.5.2 broom_0.5.2 Rcpp_1.0.3 promises_1.1.0
[29] backports_1.1.5 scales_1.0.0 jsonlite_1.6 fs_1.3.1
[33] hms_0.5.1 digest_0.6.23 stringi_1.4.3 grid_3.6.3
[37] rprojroot_1.3-2 cli_1.1.0 tools_3.6.3 magrittr_1.5
[41] lazyeval_0.2.2 crayon_1.3.4 whisker_0.4 pkgconfig_2.0.3
[45] xml2_1.2.2 lubridate_1.7.4 assertthat_0.2.1 rmarkdown_1.16
[49] httr_1.4.1 rstudioapi_0.10 R6_2.4.0 nlme_3.1-149
[53] git2r_0.26.1 compiler_3.6.3