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<h1 class="title toc-ignore">Fitting FLASH with an arbitrary error covariance matrix</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>8/22/2018</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-08-23</p>
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<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Here I examine whether it is possible to fit a FLASH model with an arbitrary error covariance matrix using an idea suggested <a href="https://github.com/stephenslab/flashr/issues/17">here</a> by Matthew Stephens.</p>
<p>That is, I want to fit the model <span class="math display">\[ Y = LF' + E,\]</span> where the columns of <span class="math inline">\(E\)</span> are distributed i.i.d. <span class="math display">\[ E_{\bullet j} \sim N(0, V). \]</span> Equivalently, letting <span class="math inline">\(\lambda_{min}\)</span> be the smallest eigenvalue of <span class="math inline">\(V\)</span> and letting <span class="math inline">\(W = V - \lambda_{min} I_n\)</span> (so that, in particular, <span class="math inline">\(W\)</span> is positive semi-definite), write <span class="math display">\[ Y = LF' + E^{(1)} + E^{(2)}, \]</span> with the columns of <span class="math inline">\(E^{(1)}\)</span> distributed i.i.d. <span class="math display">\[ E^{(1)}_{\bullet j} \sim N(0, W) \]</span> and the elements of <span class="math inline">\(E^{(2)}\)</span> distributed i.i.d. <span class="math display">\[ E^{(2)}_{i j} \sim N(0, \lambda_{min}) \]</span> Notice that by taking the eigendecomposition of <span class="math inline">\(W\)</span> <span class="math display">\[ W = \sum_{k = 1}^n \lambda_k w_k w_k' \]</span> and letting <span class="math display">\[ f_i \sim N(0, \lambda_i), \]</span> one can write <span class="math display">\[ E^{(1)}_{\bullet j} = w_1 f_1' + \ldots + w_n f_n'. \]</span></p>
<p>Thus, one should be able to fit the desired model by adding fixed loadings <span class="math inline">\(w_1, \ldots, w_n\)</span>, by fixing the priors on the corresponding factors at <span class="math inline">\(N(0, \lambda_1), \ldots, N(0, \lambda_n)\)</span>, and by taking <span class="math inline">\(\tau = 1 / \lambda_{min}\)</span> (with <code>var_type = "zero"</code>).</p>
</div>
<div id="rank-zero-flash-model" class="section level2">
<h2>Rank-zero FLASH model</h2>
<div id="code" class="section level3">
<h3>Code</h3>
<p>First I need a function that will generate random covariance matrices. I normalize the matrices so that the largest eigenvalue is equal to one. Further, I ensure that the smallest eigenvalue is bounded below by some constant. (If the covariance matrix is poorly conditioned, then the final backfit can be very slow, and in practice, we would not expect these eigenvalues to be terribly small.)</p>
<pre class="r"><code>rand.V <- function(n, lambda.min=0.25) {
A <- matrix(rnorm(n^2), nrow=n, ncol=n)
V <- A %*% t(A)
max.eigen <- max(eigen(V, symmetric=TRUE, only.values=TRUE)$values)
d <- max.eigen * lambda.min / (1 - lambda.min)
# Add diagonal matrix to improve conditioning and then normalize:
V <- (V + diag(rep(d, n))) / (max.eigen + d)
return(V)
}</code></pre>
<p>The next function simulates data from the rank-zero FLASH model <span class="math inline">\(Y = E\)</span>, with <span class="math inline">\(E_{\bullet j} \sim^{i.i.d.} N(0, V)\)</span>.</p>
<pre class="r"><code>sim.E <- function(V, p) {
n <- nrow(V)
return(t(MASS::mvrnorm(p, rep(0, n), V)))
}</code></pre>
<p>The following function fits a FLASH model using the approach outlined above.</p>
<pre class="r"><code>fit.fixed.V <- function(Y, V, verbose=TRUE, backfit=FALSE, tol=1e-2) {
n <- nrow(V)
lambda.min <- min(eigen(V, symmetric=TRUE, only.values=TRUE)$values)
data <- flash_set_data(Y, S = sqrt(lambda.min))
W.eigen <- eigen(V - diag(rep(lambda.min, n)), symmetric=TRUE)
# The rank of W is at most n - 1, so we can drop the last eigenval/vec:
W.eigen$values <- W.eigen$values[-n]
W.eigen$vectors <- W.eigen$vectors[, -n, drop=FALSE]
fl <- flash_add_fixed_loadings(data, LL=W.eigen$vectors, init_fn="udv_svd")
ebnm_param_f <- lapply(as.list(W.eigen$values),
function(eigenval) {
list(g = list(a=1/eigenval, pi0=0), fixg = TRUE)
})
ebnm_param_l <- lapply(vector("list", n - 1),
function(k) {list()})
fl <- flash_backfit(data, fl, var_type="zero", ebnm_fn="ebnm_pn",
ebnm_param=(list(f = ebnm_param_f, l = ebnm_param_l)),
nullcheck=FALSE, verbose=verbose, tol=tol)
fl <- flash_add_greedy(data, Kmax=50, f_init=fl, var_type="zero",
init_fn="udv_svd", ebnm_fn="ebnm_pn",
verbose=verbose, tol=tol)
if (backfit) {
n.added <- flash_get_k(fl) - (n - 1)
ebnm_param_f <- c(ebnm_param_f,
lapply(vector("list", n.added),
function(k) {list(warmstart=TRUE)}))
ebnm_param_l <- c(ebnm_param_l,
lapply(vector("list", n.added),
function(k) {list(warmstart=TRUE)}))
fl <- flash_backfit(data, fl, var_type="zero", ebnm_fn="ebnm_pn",
ebnm_param=(list(f = ebnm_param_f, l = ebnm_param_l)),
nullcheck=FALSE, verbose=verbose, tol=tol)
}
return(fl)
}</code></pre>
</div>
<div id="example" class="section level3">
<h3>Example</h3>
<pre class="r"><code>devtools::load_all("/Users/willwerscheid/GitHub/flashr/")</code></pre>
<pre><code>Loading flashr</code></pre>
<pre class="r"><code>devtools::load_all("/Users/willwerscheid/GitHub/ebnm/")</code></pre>
<pre><code>Loading ebnm</code></pre>
<pre class="r"><code>n <- 20
p <- 500
set.seed(666)
V = rand.V(n=n)
Y <- sim.E(V, p=p)
fl <- fit.fixed.V(Y, V)</code></pre>
<pre><code>Backfitting 19 factor/loading(s) (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -9917.69 Inf</code></pre>
<pre><code> 2 -9917.69 0.00e+00</code></pre>
<pre><code>Fitting factor/loading 20 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -9948.16 Inf</code></pre>
<pre><code> 2 -9932.50 1.57e+01</code></pre>
<pre><code> 3 -9917.69 1.48e+01</code></pre>
<pre><code> 4 -9917.69 0.00e+00</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 20 increases objective by 4.66e-03. Factor zeroed out.</code></pre>
<pre><code> Nullcheck complete. Objective: -9917.69</code></pre>
<p>Here, after backfitting the fixed loadings corresponding to the eigenvectors of <span class="math inline">\(W\)</span>, FLASH (correctly) fails to find any additional structure in the data. In contrast, fitting FLASH without paying attention to the fact that <span class="math inline">\(V \ne I\)</span> gives misleading results:</p>
<pre class="r"><code>bad.fl <- flash_add_greedy(Y, Kmax=50)</code></pre>
<pre><code>Fitting factor/loading 1 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10046.63 Inf</code></pre>
<pre><code> 2 -10034.53 1.21e+01</code></pre>
<pre><code> 3 -10033.07 1.46e+00</code></pre>
<pre><code> 4 -10032.20 8.68e-01</code></pre>
<pre><code> 5 -10031.47 7.23e-01</code></pre>
<pre><code> 6 -10030.91 5.61e-01</code></pre>
<pre><code> 7 -10030.58 3.38e-01</code></pre>
<pre><code> 8 -10030.30 2.79e-01</code></pre>
<pre><code> 9 -10029.90 4.00e-01</code></pre>
<pre><code> 10 -10029.34 5.58e-01</code></pre>
<pre><code> 11 -10028.99 3.52e-01</code></pre>
<pre><code> 12 -10028.89 9.62e-02</code></pre>
<pre><code> 13 -10028.86 2.89e-02</code></pre>
<pre><code> 14 -10028.85 1.47e-02</code></pre>
<pre><code> 15 -10028.84 9.14e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 1 decreases objective by 3.89e+01. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -10028.84</code></pre>
<pre><code>Fitting factor/loading 2 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10000.11 Inf</code></pre>
<pre><code> 2 -9988.22 1.19e+01</code></pre>
<pre><code> 3 -9986.88 1.33e+00</code></pre>
<pre><code> 4 -9986.23 6.52e-01</code></pre>
<pre><code> 5 -9985.81 4.21e-01</code></pre>
<pre><code> 6 -9985.49 3.16e-01</code></pre>
<pre><code> 7 -9985.23 2.58e-01</code></pre>
<pre><code> 8 -9985.01 2.21e-01</code></pre>
<pre><code> 9 -9984.82 1.93e-01</code></pre>
<pre><code> 10 -9984.65 1.68e-01</code></pre>
<pre><code> 11 -9984.51 1.47e-01</code></pre>
<pre><code> 12 -9984.37 1.32e-01</code></pre>
<pre><code> 13 -9984.25 1.23e-01</code></pre>
<pre><code> 14 -9984.13 1.21e-01</code></pre>
<pre><code> 15 -9984.01 1.20e-01</code></pre>
<pre><code> 16 -9983.89 1.16e-01</code></pre>
<pre><code> 17 -9983.79 1.00e-01</code></pre>
<pre><code> 18 -9983.72 7.57e-02</code></pre>
<pre><code> 19 -9983.67 5.04e-02</code></pre>
<pre><code> 20 -9983.64 3.13e-02</code></pre>
<pre><code> 21 -9983.62 1.89e-02</code></pre>
<pre><code> 22 -9983.61 1.14e-02</code></pre>
<pre><code> 23 -9983.60 6.88e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 2 decreases objective by 4.52e+01. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -9983.6</code></pre>
<pre><code>Fitting factor/loading 3 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -9976.87 Inf</code></pre>
<pre><code> 2 -9963.19 1.37e+01</code></pre>
<pre><code> 3 -9961.15 2.03e+00</code></pre>
<pre><code> 4 -9960.17 9.81e-01</code></pre>
<pre><code> 5 -9959.64 5.28e-01</code></pre>
<pre><code> 6 -9959.14 5.03e-01</code></pre>
<pre><code> 7 -9958.78 3.65e-01</code></pre>
<pre><code> 8 -9958.62 1.57e-01</code></pre>
<pre><code> 9 -9958.55 6.56e-02</code></pre>
<pre><code> 10 -9958.52 3.34e-02</code></pre>
<pre><code> 11 -9958.50 1.97e-02</code></pre>
<pre><code> 12 -9958.49 1.26e-02</code></pre>
<pre><code> 13 -9958.48 8.11e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 3 decreases objective by 2.51e+01. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -9958.48</code></pre>
<pre><code>Fitting factor/loading 4 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -9981.09 Inf</code></pre>
<pre><code> 2 -9965.90 1.52e+01</code></pre>
<pre><code> 3 -9964.03 1.87e+00</code></pre>
<pre><code> 4 -9963.31 7.24e-01</code></pre>
<pre><code> 5 -9962.84 4.67e-01</code></pre>
<pre><code> 6 -9962.46 3.87e-01</code></pre>
<pre><code> 7 -9962.12 3.38e-01</code></pre>
<pre><code> 8 -9961.82 2.97e-01</code></pre>
<pre><code> 9 -9961.56 2.59e-01</code></pre>
<pre><code> 10 -9961.34 2.24e-01</code></pre>
<pre><code> 11 -9961.15 1.91e-01</code></pre>
<pre><code> 12 -9960.99 1.60e-01</code></pre>
<pre><code> 13 -9960.86 1.32e-01</code></pre>
<pre><code> 14 -9960.75 1.08e-01</code></pre>
<pre><code> 15 -9960.66 8.79e-02</code></pre>
<pre><code> 16 -9960.59 7.14e-02</code></pre>
<pre><code> 17 -9960.53 5.80e-02</code></pre>
<pre><code> 18 -9960.48 4.72e-02</code></pre>
<pre><code> 19 -9960.44 3.86e-02</code></pre>
<pre><code> 20 -9960.41 3.16e-02</code></pre>
<pre><code> 21 -9960.39 2.59e-02</code></pre>
<pre><code> 22 -9960.37 2.13e-02</code></pre>
<pre><code> 23 -9960.35 1.76e-02</code></pre>
<pre><code> 24 -9960.33 1.46e-02</code></pre>
<pre><code> 25 -9960.32 1.21e-02</code></pre>
<pre><code> 26 -9960.31 1.00e-02</code></pre>
<pre><code> 27 -9960.30 8.33e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 4 increases objective by 1.82e+00. Factor zeroed out.</code></pre>
<pre><code> Nullcheck complete. Objective: -9958.48</code></pre>
</div>
</div>
<div id="rank-one-flash-model" class="section level2">
<h2>Rank-one FLASH model</h2>
<div id="code-1" class="section level3">
<h3>Code</h3>
<p>The following function simulates data from the rank-one FLASH model <span class="math inline">\(Y = \ell d f' + E\)</span>. <code>pi0.l</code> and <code>pi0.f</code> give the expected proportion of null entries in <span class="math inline">\(\ell\)</span> and <span class="math inline">\(f\)</span>. Since <span class="math inline">\(\ell\)</span> and <span class="math inline">\(f\)</span> are normalized to have length one, <span class="math inline">\(d\)</span> measures how large the factor/loading pair is, and thus, how easy it is to find (recall that <span class="math inline">\(V\)</span> is normalized so that its largest eigenvalue is equal to one).</p>
<pre class="r"><code>sim.rank1 <- function(V, p, pi0.l=0.5, pi0.f=0.8, d=5^2) {
E <- sim.E(V, p)
n <- nrow(V)
# Nonnull entries of l and f are normally distributed:
l <- rnorm(n) * rbinom(n, 1, 1 - pi0.l)
# Nonnull entries of f are all equal:
f <- rnorm(p) * rbinom(p, 1, 1 - pi0.f)
# Normalize l and f:
l <- l / sqrt(sum(l^2))
f <- f / sqrt(sum(f^2))
LF <- outer(l, f) * d
return(list(Y = LF + E, l = l, f = f))
}</code></pre>
</div>
<div id="example-1" class="section level3">
<h3>Example</h3>
<p>Here, the procedure outlined above correctly finds the additional rank-one structure. Running FLASH as is, however, yields structure of higher rank:</p>
<pre class="r"><code>set.seed(999)
V = rand.V(n=n)
data <- sim.rank1(V, p=p)
fl <- fit.fixed.V(data$Y, V)</code></pre>
<pre><code>Backfitting 19 factor/loading(s) (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10626.81 Inf</code></pre>
<pre><code> 2 -10626.81 0.00e+00</code></pre>
<pre><code>Fitting factor/loading 20 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10296.26 Inf</code></pre>
<pre><code> 2 -10294.19 2.07e+00</code></pre>
<pre><code> 3 -10294.15 4.51e-02</code></pre>
<pre><code> 4 -10294.14 1.80e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 20 decreases objective by 3.33e+02. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -10294.14</code></pre>
<pre><code>Fitting factor/loading 21 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10326.73 Inf</code></pre>
<pre><code> 2 -10299.97 2.68e+01</code></pre>
<pre><code> 3 -10294.15 5.82e+00</code></pre>
<pre><code> 4 -10294.15 0.00e+00</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 21 increases objective by 2.53e-03. Factor zeroed out.</code></pre>
<pre><code> Nullcheck complete. Objective: -10294.14</code></pre>
<pre class="r"><code>bad.fl <- flash_add_greedy(data$Y, Kmax=50)</code></pre>
<pre><code>Fitting factor/loading 1 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10290.91 Inf</code></pre>
<pre><code> 2 -10276.06 1.48e+01</code></pre>
<pre><code> 3 -10274.56 1.49e+00</code></pre>
<pre><code> 4 -10274.09 4.75e-01</code></pre>
<pre><code> 5 -10273.93 1.55e-01</code></pre>
<pre><code> 6 -10273.88 4.93e-02</code></pre>
<pre><code> 7 -10273.87 1.58e-02</code></pre>
<pre><code> 8 -10273.86 5.21e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 1 decreases objective by 1.71e+02. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -10273.86</code></pre>
<pre><code>Fitting factor/loading 2 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10229.69 Inf</code></pre>
<pre><code> 2 -10218.30 1.14e+01</code></pre>
<pre><code> 3 -10217.19 1.11e+00</code></pre>
<pre><code> 4 -10216.49 6.96e-01</code></pre>
<pre><code> 5 -10215.86 6.26e-01</code></pre>
<pre><code> 6 -10215.26 6.00e-01</code></pre>
<pre><code> 7 -10214.68 5.81e-01</code></pre>
<pre><code> 8 -10214.12 5.61e-01</code></pre>
<pre><code> 9 -10213.59 5.35e-01</code></pre>
<pre><code> 10 -10213.08 5.03e-01</code></pre>
<pre><code> 11 -10212.62 4.66e-01</code></pre>
<pre><code> 12 -10212.19 4.26e-01</code></pre>
<pre><code> 13 -10211.81 3.84e-01</code></pre>
<pre><code> 14 -10211.47 3.41e-01</code></pre>
<pre><code> 15 -10211.17 3.01e-01</code></pre>
<pre><code> 16 -10210.90 2.63e-01</code></pre>
<pre><code> 17 -10210.68 2.28e-01</code></pre>
<pre><code> 18 -10210.48 1.97e-01</code></pre>
<pre><code> 19 -10210.31 1.69e-01</code></pre>
<pre><code> 20 -10210.16 1.46e-01</code></pre>
<pre><code> 21 -10210.04 1.25e-01</code></pre>
<pre><code> 22 -10209.93 1.08e-01</code></pre>
<pre><code> 23 -10209.84 9.37e-02</code></pre>
<pre><code> 24 -10209.75 8.13e-02</code></pre>
<pre><code> 25 -10209.68 7.09e-02</code></pre>
<pre><code> 26 -10209.62 6.21e-02</code></pre>
<pre><code> 27 -10209.57 5.47e-02</code></pre>
<pre><code> 28 -10209.52 4.84e-02</code></pre>
<pre><code> 29 -10209.48 4.30e-02</code></pre>
<pre><code> 30 -10209.44 3.83e-02</code></pre>
<pre><code> 31 -10209.40 3.43e-02</code></pre>
<pre><code> 32 -10209.37 3.09e-02</code></pre>
<pre><code> 33 -10209.34 2.78e-02</code></pre>
<pre><code> 34 -10209.32 2.51e-02</code></pre>
<pre><code> 35 -10209.30 2.28e-02</code></pre>
<pre><code> 36 -10209.28 2.06e-02</code></pre>
<pre><code> 37 -10209.26 1.87e-02</code></pre>
<pre><code> 38 -10209.24 1.70e-02</code></pre>
<pre><code> 39 -10209.22 1.55e-02</code></pre>
<pre><code> 40 -10209.21 1.41e-02</code></pre>
<pre><code> 41 -10209.20 1.28e-02</code></pre>
<pre><code> 42 -10209.19 1.16e-02</code></pre>
<pre><code> 43 -10209.18 1.06e-02</code></pre>
<pre><code> 44 -10209.17 9.59e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 2 decreases objective by 6.47e+01. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -10209.17</code></pre>
<pre><code>Fitting factor/loading 3 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10173.09 Inf</code></pre>
<pre><code> 2 -10161.29 1.18e+01</code></pre>
<pre><code> 3 -10160.15 1.15e+00</code></pre>
<pre><code> 4 -10159.47 6.78e-01</code></pre>
<pre><code> 5 -10158.92 5.47e-01</code></pre>
<pre><code> 6 -10158.46 4.61e-01</code></pre>
<pre><code> 7 -10158.07 3.89e-01</code></pre>
<pre><code> 8 -10157.75 3.25e-01</code></pre>
<pre><code> 9 -10157.48 2.68e-01</code></pre>
<pre><code> 10 -10157.26 2.17e-01</code></pre>
<pre><code> 11 -10157.09 1.75e-01</code></pre>
<pre><code> 12 -10156.95 1.39e-01</code></pre>
<pre><code> 13 -10156.84 1.10e-01</code></pre>
<pre><code> 14 -10156.75 8.74e-02</code></pre>
<pre><code> 15 -10156.68 6.92e-02</code></pre>
<pre><code> 16 -10156.63 5.47e-02</code></pre>
<pre><code> 17 -10156.58 4.32e-02</code></pre>
<pre><code> 18 -10156.55 3.50e-02</code></pre>
<pre><code> 19 -10156.52 2.99e-02</code></pre>
<pre><code> 20 -10156.49 2.66e-02</code></pre>
<pre><code> 21 -10156.47 2.43e-02</code></pre>
<pre><code> 22 -10156.44 2.25e-02</code></pre>
<pre><code> 23 -10156.42 2.08e-02</code></pre>
<pre><code> 24 -10156.40 1.89e-02</code></pre>
<pre><code> 25 -10156.39 1.68e-02</code></pre>
<pre><code> 26 -10156.37 1.43e-02</code></pre>
<pre><code> 27 -10156.36 1.17e-02</code></pre>
<pre><code> 28 -10156.35 9.28e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 3 decreases objective by 5.28e+01. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -10156.35</code></pre>
<pre><code>Fitting factor/loading 4 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10146.50 Inf</code></pre>
<pre><code> 2 -10134.17 1.23e+01</code></pre>
<pre><code> 3 -10133.44 7.32e-01</code></pre>
<pre><code> 4 -10133.29 1.47e-01</code></pre>
<pre><code> 5 -10133.24 5.09e-02</code></pre>
<pre><code> 6 -10133.22 2.32e-02</code></pre>
<pre><code> 7 -10133.21 1.20e-02</code></pre>
<pre><code> 8 -10133.20 6.59e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 4 decreases objective by 2.31e+01. Factor retained.</code></pre>
<pre><code> Nullcheck complete. Objective: -10133.2</code></pre>
<pre><code>Fitting factor/loading 5 (stop when difference in obj. is < 1.00e-02):</code></pre>
<pre><code> Iteration Objective Obj Diff</code></pre>
<pre><code> 1 -10156.31 Inf</code></pre>
<pre><code> 2 -10142.29 1.40e+01</code></pre>
<pre><code> 3 -10141.41 8.88e-01</code></pre>
<pre><code> 4 -10141.26 1.48e-01</code></pre>
<pre><code> 5 -10141.22 4.25e-02</code></pre>
<pre><code> 6 -10141.20 1.89e-02</code></pre>
<pre><code> 7 -10141.19 1.09e-02</code></pre>
<pre><code> 8 -10141.18 7.15e-03</code></pre>
<pre><code>Performing nullcheck...</code></pre>
<pre><code> Deleting factor 5 increases objective by 7.98e+00. Factor zeroed out.</code></pre>
<pre><code> Nullcheck complete. Objective: -10133.2</code></pre>
<p>To check that the new approach gives reasonable results, one can calculate the angle between the estimated <span class="math inline">\(l\)</span> and the true <span class="math inline">\(l\)</span> (and likewise for <span class="math inline">\(f\)</span>):</p>
<pre class="r"><code>ldf <- flash_get_ldf(fl, drop_zero_factors=FALSE)
l.angle <- acos(abs(sum(ldf$l[, n] * data$l)))
f.angle <- acos(abs(sum(ldf$f[, n] * data$f)))
round(c(l.angle, f.angle), digits=2)</code></pre>
<pre><code>[1] 0.43 0.37</code></pre>
<p>These results are not terrible, but an additional backfit can improve upon them:</p>
<pre class="r"><code>fl.b <- fit.fixed.V(data$Y, V, verbose=FALSE, backfit=TRUE)
ldf <- flash_get_ldf(fl.b, drop_zero_factors=FALSE)
l.angle <- acos(abs(sum(ldf$l[, n] * data$l)))
f.angle <- acos(abs(sum(ldf$f[, n] * data$f)))
round(c(l.angle, f.angle), digits=2)</code></pre>
<pre><code>[1] 0.16 0.33</code></pre>
</div>
</div>
<div id="code-for-experiments" class="section level2">
<h2>Code for experiments</h2>
<p>I include code below that can be used to verify that the above results are typical. Since they can take a long time, I do not run them here.</p>
<pre class="r"><code>rank0.experiment <-function(ntests, n, p, lambda.min=0.25, seeds=1:ntests) {
est.rank <- bad.rank <- rep(NA, ntests)
for (i in 1:length(seeds)) {
set.seed(i)
V <- rand.V(n, lambda.min)
Y <- sim.E(V, p)
fl <- fit.fixed.V(Y, V, verbose=FALSE)
k <- flash_get_k(fl)
est.rank[i] <- k - (n - 1)
bad.fl <- flash_add_greedy(Y, Kmax=50, verbose=FALSE)
bad.rank[i] <- flash_get_nfactors(bad.fl)
}
return(list(est.rank = est.rank, bad.rank = bad.rank))
}
rank1.experiment <-function(ntests, n, p, lambda.min=0.25, d=5^2,
seeds=1:ntests) {
est.rank <- bad.rank <- rep(NA, ntests)
l.angle <- f.angle <- rep(NA, ntests)
for (i in 1:length(seeds)) {
set.seed(i)
V = rand.V(n, lambda.min)
data <- sim.rank1(V, p, d=d)
fl <- fit.fixed.V(data$Y, V, verbose=FALSE, backfit=TRUE)
k <- flash_get_k(fl)
est.rank[i] <- k - (n - 1)
ldf <- flash_get_ldf(fl, drop_zero_factors=FALSE)
if (est.rank[i] >= 1) {
l.angle[i] <- acos(abs(sum(ldf$l[, n] * data$l)))
f.angle[i] <- acos(abs(sum(ldf$f[, n] * data$f)))
}
bad.fl <- flash_add_greedy(data$Y, Kmax=50, verbose=FALSE)
bad.rank[i] <- flash_get_nfactors(bad.fl)
}
return(list(est.rank = est.rank, bad.rank = bad.rank,
l.angle = l.angle, f.angle = f.angle))
}</code></pre>
</div>
<div id="questions-for-further-investigation" class="section level2">
<h2>Questions for further investigation</h2>
<p>I have set parameters <code>lambda.min</code> and <code>d</code> favorably for this investigation. If <code>lambda.min</code> is closer to 1, then errors will be more nearly independent, and the usual FLASH model will not fare so poorly. It would be worthwhile to investigate whether the approach detailed here beats the usual FLASH fit in such cases.</p>
<p>Further, I have set <code>d</code> to be quite large. In the above simulations, the true loading and factor are each five times larger (in terms of Euclidean length) than the largest eigenvalue of the error covariance matrix. It would be interesting to see what the detection threshold is as a function of <code>n</code>, <code>p</code>, and <code>lambda.min</code>.</p>
<p>Finally, notice that when <span class="math inline">\(\lambda_{min} = 1\)</span>, the approach detailed above is just the usual FLASH fit, so both of these proposed investigations would help to establish some continuity between the two.</p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ebnm_0.1-13 flashr_0.5-14
loaded via a namespace (and not attached):
[1] Rcpp_0.12.17 pillar_1.2.1 plyr_1.8.4
[4] compiler_3.4.3 git2r_0.21.0 workflowr_1.0.1
[7] R.methodsS3_1.7.1 R.utils_2.6.0 iterators_1.0.9
[10] tools_3.4.3 testthat_2.0.0 digest_0.6.15
[13] tibble_1.4.2 evaluate_0.10.1 memoise_1.1.0
[16] gtable_0.2.0 lattice_0.20-35 rlang_0.2.0
[19] Matrix_1.2-12 foreach_1.4.4 commonmark_1.4
[22] yaml_2.1.17 parallel_3.4.3 withr_2.1.1.9000
[25] stringr_1.3.0 roxygen2_6.0.1.9000 xml2_1.2.0
[28] knitr_1.20 devtools_1.13.4 rprojroot_1.3-2
[31] grid_3.4.3 R6_2.2.2 rmarkdown_1.8
[34] ggplot2_2.2.1 ashr_2.2-10 magrittr_1.5
[37] whisker_0.3-2 backports_1.1.2 scales_0.5.0
[40] codetools_0.2-15 htmltools_0.3.6 MASS_7.3-48
[43] assertthat_0.2.0 softImpute_1.4 colorspace_1.3-2
[46] stringi_1.1.6 lazyeval_0.2.1 munsell_0.4.3
[49] doParallel_1.0.11 pscl_1.5.2 truncnorm_1.0-8
[52] SQUAREM_2017.10-1 R.oo_1.21.0 </code></pre>
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