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<p>I am reading the Hidden Markov Model note <a href="http://cs229.stanford.edu/section/cs229-hmm.pdf" rel="nofollow">http://cs229.stanford.edu/section/cs229-hmm.pdf</a> from Stanford and have problem finding the meaning of the notation described below.</p>
<p>What does the semicolon (;) means in below (Page 2 in above link)?</p>
<p>$$P( z_t, z_{t-1},\ldots, z_1; A )$$</p>
<p>$A$ is the transition matrix. How should it be read? For example, I know $P( x | y )$ is "probability of $x$ GIVEN $y$"; $P( x=s_1 | y=o_1 )$ is "probability of $x$ being $s_1$ GIVEN that $y$ is $o_1$".</p>
<p>Thanks in advance for any help for this possibly trivial, but surprisingly hard to find answer.</p>
<p><strong>Further question on same notation issue.</strong></p>
<p>In Wikipedia article on <a href="http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm" rel="nofollow">EM algorithm</a>, They have</p>
<p>$$L( \theta;X ) = p( X,Z | \theta )$$</p>
<p>In the above, X is the set of observed data, Z is the set of observed data and $\theta$ is a vector of parameters. Based on my current understanding of the notation, I will read the above as "the likelihood of theta given X is equal to the joint probability of X and Z given theta". In this context, X contains observed data, which to me appears to be random variables rather than fixed parameters. Why is semicolon ';' used here?</p>
<p><strong>Update</strong></p>
<p>I have just noticed a few other threads asking the same question, when these threads shown up as related threads on right pane. My apology that I couldn't find them earlier. The following two threads have elaborated answers, in agreement with the accepted answer of this thread.</p>
<p><a href="http://stats.stackexchange.com/questions/10234/meaning-of-probability-notations-pzd-w-and-pzd-w">Meaning of probability notations $P(z;d,w)$ and $P(z|d,w)$</a></p>
<p><a href="http://stats.stackexchange.com/questions/30825/what-is-the-meaning-of-the-semicolon-in-fx-theta">What is the meaning of the semicolon in $f(x;\Theta)$?</a></p> | 49,858 | [
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<p>It seems that WinBUGS has problems <strong>if it has only zero draws from one binomial distribution</strong>:</p>
<p><strong>1. case - simple model</strong></p>
<pre><code>for (i in 1:sites) {
N[i] ~ dpois(lambda)
for (j in 1:sample) {
y[i, j] ~ dbin(p, N[i])
}
}
</code></pre>
<p>If all values <code>y[i,]</code> are zero, then the following error message appears:</p>
<blockquote>
<p><strong>value of order of binomial y[53,1] must be greater than zero</strong></p>
</blockquote>
<p><strong>EDIT:</strong> note that <code>N[i]</code> was not zero in generated data upon failure.</p>
<p><strong>2. case - advanced removal model</strong></p>
<pre><code>for (i in 1:sites) {
M[i] ~ dpois(lambda)
for (j in 1:sample) {
y[i, j, 1] ~ dbin(p, M[i])
after_removal[i, j] <- M[i] - y[i, j, 1]
y[i, j, 2] ~ dbin(p, after_removal[i, j])
}
}
</code></pre>
<p>Here it seems that if some <code>y[i, j, 2]</code> is zero then the error message (same as in case 1) appears!</p>
<p>Note that in this case, I found a workaround: I reparametrized the model to different one which <strong>I hope</strong> is equivalent:</p>
<pre><code>for (i in 1:sites) {
M[i] ~ dpois(lambda)
for (j in 1:sample) {
y[i, j, 1] ~ dbin(p, M[i])
obs[i, j] <- y[i, j, 1] + y[i, j, 2]
obs[i, j] ~ dbin(p_komb, M[i])
}
}
p_komb <- p + (1 - p) * p
</code></pre>
<p>... anyway an explanation, clean solution and solution for case 1 wanted:</p>
<h2>Why this happens? Is it a WinBUGS bug? How can it be overriden?</h2>
<p>I have WinBUGS 1.4.3 (August 2007) with immortality patch installed.
Below is complete <strong>reproducible code for R and package R2WinBUGS (with data generation)</strong>:</p>
<p><strong>1. case</strong></p>
<pre><code>require(vcd)
sites <- 120 # 60
mean_N <- 16
N <- rpois(sites, mean_N)
p <- 0.4
sample <- 3 # 3
y = matrix(nrow = sites, ncol = sample)
for (i in 1:sites) {
y[i,] = rbinom(sample, N[i], p)
}
y[20,] = 0
sink("tmp_bugs/model.txt")
cat("
model {
# likelihood
for (i in 1:sites) {
N[i] ~ dpois(lambda)
for (j in 1:sample) {
y[i, j] ~ dbin(p, N[i])
}
}
# derived parameters
Ntot <- sum(N[])
# priors
p <- 1/(1+exp(-logit_p))
tau <- 1/(4 * 4)
logit_p ~ dnorm(0, tau)
lambda ~ dunif(0, 100)
}
")
sink()
win.data = list(y = y, sample = sample, sites = sites)
#win.data = list(y = y)
#inits = function () { list(N = apply(y, 1, max), p = mean(apply(y, 1, mean)/apply(y, 1, max))) }
inits = function () { list(N = apply(y, 1, max), logit_p = rnorm(1, 0, 4)) }
params = c("N", "p", "Ntot", "lambda")
ni <- 500
nt <- 12
nb <- 200
nc <- 3
out <- bugs(win.data, inits, params, "model.txt",
nc, ni, nb, nt, bugs.directory = bugs.dir,
working.directory = paste(getwd(), "/tmp_bugs/", sep = ""),
debug = TRUE
)
print(out, dig = 3)
par(mfrow = c(2, 2))
hist(out$sims.list$p, breaks = 100)
abline(v = out$mean$p, col = "red", lwd = 2)
abline(v = p, col = "green", lwd = 2)
#lines(quantile(out$sims.list$p, c(0.025, 0.975)), rep(sum(par("usr")[3:4]*c(0.9,0.1)), 2), lwd = 2)
lines(quantile(out$sims.list$p, c(0.025, 0.975)), rep(par("usr")[3], 2), lwd = 4)
legend("topleft", c("estimated", "real", "95% cred. int."), col = c("red", "green", "black"), lty = 1, box.lty = 0, cex = 0.7)
#hist(mean(y) / out$sims.list$p, breaks = 1000, xlim = c(0, 50))
hist(out$sims.list$Ntot, breaks = 100)
abline(v = out$mean$Ntot, col = "red", lwd = 2)
abline(v = sum(N), col = "green", lwd = 2)
#lines(quantile(out$sims.list$Ntot, c(0.025, 0.975)), rep(sum(par("usr")[3:4]*c(0.9,0.1)), 2), lwd = 2)
lines(quantile(out$sims.list$Ntot, c(0.025, 0.975)), rep(par("usr")[3], 2), lwd = 4)
legend("topright", c("estimated total N", "real total N", "95% credible int."), col = c("red", "green", "black"), lty = 1, box.lty = 0, cex = 0.7)
</code></pre>
<p><strong>2. case</strong></p>
<p>(one of the <code>y[i, j, 2]</code> will likely be zero; if not, it can be assigned directly)</p>
<pre><code>require(vcd)
sites <- 120 # 60
mean_M <- 16
M <- rpois(sites, mean_M)
p <- 0.4 #0.64
sample <- 2 # 3
y = rep(NA, sites * sample * 2)
dim(y) = c(sites, sample, 2)
for (i in 1:sites) {
# obs[i,] = rbinom(sample, M[i], p)
for (j in 1:sample) {
y[i,j,1] = rbinom(1, M[i], p)
y[i,j,2] = rbinom(1, M[i] - y[i,j,1], p)
}
}
y_sample_total = apply(y, c(1, 2), sum)
obs = y_sample_total
############################################
sink("tmp_bugs/model.txt")
cat("
model {
# likelihood
for (i in 1:sites) {
M[i] ~ dpois(lambda)
for (j in 1:sample) {
y[i, j, 1] ~ dbin(p, M[i])
after_removal[i, j] <- M[i] - y[i, j, 1]
y[i, j, 2] ~ dbin(p, after_removal[i, j])
}
}
# derived parameters
Mtot <- sum(M[])
# priors
tau <- 1/(4 * 4)
p <- 1/(1+exp(-logit_p))
logit_p ~ dnorm(0, tau)
lambda ~ dunif(0, 100)
}
")
sink()
win.data = list(y = y,
#obs = y_sample_total,
sample = sample, sites = sites)
inits = function () { list(
M = apply(y_sample_total, 1, max),
logit_p = rnorm(1, 0, 4)
) }
#params = c("M", "M2", "p", "p2", "Mtot", "M2tot", "lambda", "lambda2")
params = c("M", "p", "Mtot", "lambda")
ni <- 2500
nt <- 16
nb <- 1000
nc <- 3
date()
out <- bugs(win.data, inits, params, "model.txt",
nc, ni, nb, nt, bugs.directory = "C:/Program Files/WinBUGS14/",
working.directory = paste(getwd(), "/tmp_bugs/", sep = ""),
debug = TRUE
)
date()
#############################
print(out, dig = 3)
par(mfrow = c(2, 2))
hist(out$sims.list$p, breaks = 100)
abline(v = out$mean$p, col = "red", lwd = 2)
abline(v = p, col = "green", lwd = 2)
#lines(quantile(out$sims.list$p, c(0.025, 0.975)), rep(sum(par("usr")[3:4]*c(0.9,0.1)), 2), lwd = 2)
lines(quantile(out$sims.list$p, c(0.025, 0.975)), rep(par("usr")[3], 2), lwd = 4)
legend("topleft", c("estimated", "real", "95% cred. int."), col = c("red", "green", "black"), lty = 1, box.lty = 0, cex = 0.7)
#hist(mean(y) / out$sims.list$p, breaks = 1000, xlim = c(0, 50))
hist(out$sims.list$Mtot, breaks = 100)
abline(v = out$mean$Mtot, col = "red", lwd = 2)
abline(v = sum(M), col = "green", lwd = 2)
#lines(quantile(out$sims.list$Mtot, c(0.025, 0.975)), rep(sum(par("usr")[3:4]*c(0.9,0.1)), 2), lwd = 2)
lines(quantile(out$sims.list$Mtot, c(0.025, 0.975)), rep(par("usr")[3], 2), lwd = 4)
legend("topright", c("estimated total M", "real total M", "95% credible int."), col = c("red", "green", "black"), lty = 1, box.lty = 0, cex = 0.7)
hist(out$sims.list$lambda)
</code></pre> | 74,349 | [
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<p>When given samples of a discrete random variable, the entropy of the distribution may be estimated by $- \sum \hat{P_i} \log{\hat{P_i}}$, where $\hat{P_i}$ is the sample estimate of the frequency of the $i$th value. (this is up to a constant determined by the base of the log.) This estimate should not be applied to observations from a continuous distribution, at least naively, because it would yield a value which depends only on the sample size.</p>
<p><a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.87.5281" rel="nofollow">Beirlant <em>et al</em></a> describe a number of approaches for the continuous problem, including estimates based on empirical CDF, nearest neighbor distances and the $m$-spacing estimate, which is given by $$\frac{1}{n}\sum_{i=1}^{n-m}\log{(\frac{n}{m}(X_{(i+m)} - X_{(i)}))}$$, where $X_{(i)}$ is the $i$th order statistic of the sample, and $m$ varies in a certain way with $n$. It is not clear how this estimate is to be computed in the presence of ties, <em>i.e.</em> it does not appear to be applicable to discrete distributions. (a naive correction for ties (drop terms which have $\log{0}$) appears to give an estimator which does not depend on the relative frequency of the classes, only their values, which seems wrong.) </p>
<p>The question: is there an 'all-purpose' estimator which can deal with both discrete and continuous (or even mixed) distributions?</p> | 37,205 | [
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<p>I want to run ordinal logistic regression (OLR) in SPSS. My data include 6 predictor variable (two continuous and 4 categorical ) but my outcome variables are also 6 (categorical-likert scale).
e.g my dependent variable is Business development and 6 likert scale questions were asked for this, increase in profit, sales, size, asset, marketing and labour.</p>
<p>1-If I composite these 6 variable into one by first sum all the 6 variables and then recode them into again five categories like before (In SPSS in Transform, Compute variable and then recode into different variables). I think in this way I lost my original data and may categorize them wrong.</p>
<p>2-If I think of using Categorical Principle component analysis for reducing the dependent variables, it will give results (object scores) in continuous form on which I have to run linear regression. I do not want to run linear regression as my original data is categorical. </p>
<p>3-So option left is to run OLR without combining the dependent variable on each DV. It means that there is no model for Business development and 6 models for profit, sales.... </p>
<p>My question is that</p>
<ul>
<li>Is it preferable to run OLR with each DV and then summarize the results</li>
<li><p>OR Is there any other method for reducing DV and running OLR.</p></li>
<li><p>OR Is there any better method than ordinal logistic regression for this data.</p></li>
</ul> | 74,350 | [
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<p>I am doing some basic fitting of data and exploring different fits. I understand that the residual is the difference between the sample and the estimated function value. The norm of the residuals is a measure of the deviation between the correlation and the data. A lower norm signifies a better fit.</p>
<p>Suppose a cubic fit has a norm of residuals of 0.85655 and a linear fit has a norm of residuals of 0.89182. What if the norms are 0.17113 and 0.24916? Is the difference between these two significant? Is a norm of residual less than 1 considered good? If not what is generally regarded as "acceptable" norm of residual. </p> | 74,351 | [
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<p>I am working on an assignment for a course in statistics, and a part of the assignment is to give the null hypothesis of an experiment. The experiment is as followed (short summary):</p>
<p>A company is testing 3 interfaces for a website. They test the interfaces with 15 students, 5 for each interface, and all 5 have a different computer knowledge level. They are all given the same task and the time it takes for each of them to complete the task is the data we're using. </p>
<p>I know need to find the null hypothesis of interest. As far as I understand null hypothesis and the alternative hypothesis should be 2 competing propositions. However, as far as I can tell I have 3 proposition: interface 1 is better, interface 2 is better and interface 3 is better. </p>
<p>What null hypothesis should I have here and why?</p>
<p>Thanks in advance!</p> | 37,210 | [
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<p>When I want to apply an ARIMA model like this:</p>
<pre><code>arimax(dataMat, order=c(7,0,0), xreg=xreg)
</code></pre>
<p>it raises an error </p>
<blockquote>
<p>"singular and could not compute QR decomposition of Hessian" for xreg.</p>
</blockquote>
<p>How can I fit proper model by using <code>xreg</code> values?</p> | 74,352 | [
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<p>I am doing a two-way independent ANOVA using type of organisational change (5 levels) and job level (7 levels) within the organisation as the factors (in SPSS). I followed Andy Field's procedure (<em>Discovering Statistics Using SPSS</em>, third edition) and I cannot manage to get the option of Post-Hoc tests for cases where equal variance is not assumed, i.e. Games-Howell. </p>
<p>I searched the syntax help files and somewhere there's a note saying that when using more than one factor Games-Howell cannot be performed. However, Andy Field did it! (please see page 434 of his book). Any ideas how can I do the same?</p> | 74,353 | [
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<p>So I have a multi-label classification problem where the exact number of labels in each test set example is unknown. The loss function is mean F1-score which is </p>
<p><img src="http://i.stack.imgur.com/5bMSS.png" alt="enter image description here"></p>
<p>where p is the precision and r is the recall. I also have posterior (beta) distributions for each of the possible labels. I am wondering how I would calculate the Bayesian expected loss for each possible label. And since it is a multi-label classification problem, I would also need to calculate the expected loss for all possible combinations of labels. Is this even a tractable problem?</p> | 74,354 | [
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<p>I have a time-series forecasting task and would like some input on variable selection and regularisation.</p>
<p>My problem has the following characteristics:</p>
<ul>
<li>2,000,000 sample size.</li>
<li>Most of the time, no change in the response. Change in the response happens 0.3% of the time and is what I'm interested in (don't talk about asymmetric loss functions and the like - I'm purely interested in variable selection/regularlisation).</li>
<li>I will be predicting direction (up/down/no change). </li>
<li>100 predictor variables. Mostly hand engineered from complex non-stationary data. All carry some predictive power - the useless ones with no univariate forecasting power were eliminated early in the process.</li>
<li>There will be clusters of highly correlated predictors - in general if you select two predictors at random the correlation will not be high.</li>
<li>Predictors are heteroscedastic in nature.</li>
<li>Some predictors are heavily serially correlated, many are not. </li>
<li>Predictors all evolve continuously and are decimal processes. They are all bell shaped. </li>
<li>Some could be non-stationary, but are thought to carry critical information about the evolution of the response process that would be lost upon differencing or similar $I(0)$ transforms.</li>
<li>My forecasting model will be linear, but least squares loss function is not certain (perpendicular regression and quantile regression and discrete response regression will be tried).</li>
<li>Standard error unbiasedness and variance is not a direct consideration. All I care about is out of sample forecast accuracy and model generalisation. </li>
</ul>
<p>I know I can use anything from non-negative garotte, ridge, lasso, elastic nets, random subspace learning, PCA/manifold learning, least angle regression and various yucky hacks to select based on out of sample forecast performance. But is there anything specific to forecasting, or to the characteristics of my data, that would push me one way over another (time efficiency is important - trying everything and selecting the best is not practical). </p>
<p>Note that decision boundaries on the forecasts are not intended to be part of the regularisation algorithm, but integration could be possible. </p>
<p>I will also be using a supercomputer. </p> | 74,355 | [
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<p>I work in a problem where:</p>
<ul>
<li>The input data is a collection of time series from different sensor
modalities, where each sensor comes with different acquisition rates
and dynamic ranges</li>
<li>The number of potential features vastly exceeds the number of
instances (i.e. $p \gg N$)</li>
</ul>
<p>My question is: <strong>Has deep learning been applied on this type of data before?</strong> </p> | 74,356 | [
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<p>Suppose a data set contains the numbers $1-100$ (i.e. $\{1, \dots, 100 \}$). In $10$-fold cross validation, the data set is divided into $10$ subsets with one used as the validation data set. For example, one such division is $S_1 = \{1, \dots, 10 \}$, $S_2 = \{11, \dots, 20 \}, \dots, S_{10} = \{91, \dots, 100 \}$. </p>
<p>In "regular" 10-fold cross validation, each subset is used as the test data set. In 10-repeated 10-fold cross validation (for example), the part that is repeated is how $S_1, \dots, S_{10}$ are constructed?</p> | 74,357 | [
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<p>I computed the T-score and P-value using <code>t.test()</code> for my data, and finally I've plotted the density of my p-value and I've got strange plot. I don't know, why I see peak around 0, which it's density is bigger than 1 ?
can anybody help me with the interpretation of this plot ? I used the <code>plot(density(mymatrix))</code>
<img src="http://i.stack.imgur.com/GFWrW.png" alt="enter image description here"></p> | 49,345 | [
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<p>I would like to specify a multivariate model with <code>lme</code> with a random effect for group which is independent across variables. I found <a href="http://scs.math.yorku.ca/index.php/Mixed_Models_with_R/Multivariate_Mixed_Models.Rmd" rel="nofollow">this post</a>,
which explains that the model specified as:</p>
<pre><code>fit.multilevel <- lme( y ~ var -1, dd,
random = ~ var -1| school,
correlation = corSymm( form = ~ v |school/id),
weights = varIdent(form = ~ 1 | v))
</code></pre>
<p>is equivalent to (using <code>MCMCglmm</code>):</p>
<pre><code>fit.w.null <- MCMCglmm( cbind(mathach, ses) ~ trait -1 ,
random = ~us(trait):school,
rcov = ~us(trait):units,
data = hs,
family = c("gaussian","gaussian"))
</code></pre>
<p>However the model I would really like has independent random effects for school across the variables:</p>
<pre><code>fit.w.null <- MCMCglmm( cbind(mathach, ses) ~ trait -1 ,
random = ~idh(trait):school,
rcov = ~us(trait):units,
data = hs,
family = c("gaussian","gaussian"))
</code></pre>
<p>(the random effect is specified with <code>idh()</code>). I cannot find a way to specify this in <code>lme</code>.<br>
Here is some code to generate example data and run the analysis:</p>
<pre><code>gr=rep(c("A","B","C"),each=100)
grp.mean=rep(rnorm(3,5,1),each=100)
var1=rnorm(300,rep(rnorm(3,5,1),each=100),1)
var2=0.8*var1+3*rnorm(300,rep(rnorm(3,5,1),each=100),2)
dat.mcmc=data.frame(var1=var1,var2=var2,group=gr)
dat.lme=data.frame(y=c(var1,var2),var=rep(c("1","2"),each=length(gr)),
group=rep(gr,2),id=rep(1:length(gr),2))
dat.lme$v=as.integer(dat.lme$var)
</code></pre>
<p>Currently this is the best I can do:</p>
<pre><code>fit.multilevel <- lme( y ~ var -1, dat.lme,
random = ~ var -1| group,
correlation = corSymm( form = ~ v |group/id),
weights = varIdent(form = ~ 1 | v))
</code></pre>
<p>This is (I believe) equivalent to:</p>
<pre><code>fit.w.null <- MCMCglmm( cbind(var1, var2) ~ trait -1 ,
random = ~us(trait):group,
rcov = ~us(trait):units,
data = dat.mcmc,
family = c("gaussian","gaussian"))
</code></pre>
<p>However, I would like an <code>lme</code> command equivalent to:</p>
<pre><code>fit.w.null <- MCMCglmm( cbind(var1, var2) ~ trait -1 ,
random = ~idh(trait):group,
rcov = ~us(trait):units,
data = hs,
family = c("gaussian","gaussian"))
</code></pre>
<p>Any help you can give me will be much appreciated.</p> | 74,358 | [
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<p>Through the last year I have been working with support vector machines for a binary text classification task. Having used software such as R and Rapidminer I have not spent much time on understanding what actually goes on inside support vector machines. This I have now started looking into in the hope of getting a better understanding of this classification/regression method. </p>
<p>I have spent a lot of time looking for calculation examples as it tends to enhance my understanding of a concept quite well if I can actually setup a problem in Excel. Therefore I hope to get guidance by asking this question here, as I have not been able to find any step-by-step calculation examples. One can easily find descriptions of the math and optimization problems one need to understand and solve computationally, but a step-by-step calculation example I have not been able to find.</p>
<p>If the forum approves my idea of producing such an example I will do the editing and in the end produce a nice and clear Excel sheet and a guide for future use.</p>
<p>I suggest that we use the <em>Iris</em> dataset (even though it is a multiclass dataset) and simply try to separate <em>Iris setosa</em> from <em>Iris versicolor</em>.</p>
<p>I provide three links. Link one is theory of application of SVMs which I thought one could use as a scaffold. Link two provides a regression example of how I was thinking our product would look in the end. Link three will take you to the <em>Iris</em> dataset.</p>
<p><a href="http://sutikno.blog.undip.ac.id/files/2011/11/SVM-Explained.pdf" rel="nofollow">Theory and application of SVMs</a></p>
<p><a href="http://www.cba.edu.kw/zainal/docs/220Docs/Regression%20Analysis.pdf" rel="nofollow">A guide for regressions</a></p>
<p><a href="http://archive.ics.uci.edu/ml/machine-learning-databases/iris/" rel="nofollow">Link for the Iris dataset</a></p>
<p>Below I will try to formulate the problem more neatly.</p>
<p>Problem description:</p>
<blockquote>
<p>How can one apply Excel and the technique of a linear support vector machine with soft
margins in order to solve a binomial classification task given by
separating <em>Iris setosa</em> and <em>Iris versicolor</em> from the <em>Iris</em> dataset
using all available features?</p>
</blockquote> | 74,359 | [
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<p>I have a set of results of independent measurements of some physical quantity. As an example I give here real expermental data on methanol refractive index at 25 degrees Celsius published in scientific literature from 1960 to 2011:</p>
<pre><code>data={{1960, 1.32652}, {1961, 1.32662}, {1963, 1.32650}, {1963,
1.32750}, {1968, 1.32698}, {1968, 1.32890}, {1970, 1.32657}, {1970,
1.32660}, {1971, 1.3260}, {1971, 1.32610}, {1971, 1.32630}, {1971,
1.3350}, {1972, 1.32640}, {1972, 1.32661}, {1973, 1.32860}, {1975,
1.32515}, {1975, 1.32641}, {1976, 1.32663}, {1977, 1.32670}, {1980,
1.3250}, {1983, 1.32850}, {1985, 1.32653}, {1991, 1.32710}, {1995,
1.32621}, {1995, 1.32676}, {1996, 1.32601}, {1996, 1.32645}, {1996,
1.32715}, {1998, 1.32820}, {1999, 1.32730}, {1999, 1.32780}, {2001,
1.32634}, {2006, 1.32620}, {2011, 1.32667}};
</code></pre>
<p>The first number is the year of first publication, the second is the published value. The authors do not always provide their own estimate for error of the published value and even when they do this, the published estimate is often not accurate.</p>
<p>It is easy to see that the distribution is asymmetrical (other datasets I work with are even more asymmetrical). Here is a density histogram with bins selected by hands:</p>
<pre><code>bins = {1.325, 1.3259, 1.3261, 1.3263, 1.3265, 1.3266, 1.3267, 1.3269,
1.3275, 1.328};
Histogram[dataFirst[[All, 2]], {bins}, "PDF"]
</code></pre>
<p><img src="http://i.stack.imgur.com/x4lne.png" alt="histogram"></p>
<p>Because of asymmetric feature of the distribution I cannot use the mean as an estimate for the true value of methanol refractive index. </p>
<p>The fact is that the probability to get smaller value than the true value in general is not equal to the probability to get higher value. This means that the median in general is also unsuitable.</p>
<p>It may be assumed that each invidual measurement has normal distribution for the instrumental error of the measurement. Additional sources of error are impurities in methanol. The most common impurity is water which increases the refractive index, but authors usually dry methanol before measurement. Probably this is the reason why the density is higher for low values: other impurities such as some common organic solvents (benzene) and (possibly) diluted gases (methane, carbon oxide, carbon dioxide) lower the refractive index. From the other hand alcohols, aldehydes and ketones which may present in industrial methanol and some organic solvents (CCl4) increase the refractive index. So there at least two sources of error: one (instrumental) may be assumed normally distributed and even equal for the all measurements and the second (impurities) is probably asymmetrical (and may be even bimodal) and depends on how methanol was obtained and purified.</p>
<p>What is the best way to estimate the true value and its 95% confidence bands for the measured quantity in such cases? </p>
<p>P.S. References for relevant scientific papers will be appreciated.</p>
<p>P.S.2. The above code is for <em>Mathematica</em> system.</p> | 74,360 | [
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<p>I have problem with real basics. I have variable that has p=.000 in Shapiro-Wilk. As I looked on the histogram it appears that the cause is natural limit for this measurement (value 0, skewness > 1,5).I'm wondering what whould be the best way to present this data as descriptives.
What parameters of descriptive & non-paramteric statistics would be the best to present data like this? Is there equivalent of 95%CI for non-normal distribution ? (for example interquartile range)</p>
<p>Thank you in advance!</p> | 74,361 | [
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<p>I use <a href="http://stat.ethz.ch/R-manual/R-devel/library/stats/html/optim.html" rel="nofollow">optim()</a> in R to do a lot of MLE. I've used the approach for a lot of problems, but the one I'm working on right now consists of fitting the parameters of the generalized extreme value distribution (GEV). There are three parameters (location scale shape), and I want to get the standard errors from this fit. An example of a paper doing this is <a href="https://www.isse.ucar.edu/staff/katz/docs/pdf/ecology.pdf" rel="nofollow">Katz et al. 2005</a>, and here is the <a href="http://www.esapubs.org/archive/ecol/E086/060/rextreme.txt" rel="nofollow">online code</a> for the article. I am unaffiliated with this publication. I should also point out that the GEV question is just an example of the type of approach I take -- most of my MLE fits are time series models (state-space). I could give more examples if needed.</p>
<p>1) Which of the arguments to optim() are of particular relevance to the statistical validity of using the optim() Hessian to computer standard errors? </p>
<ul>
<li>The first thing that comes to mind is the type of algorithm -- In
R, optim() is a wrapper for several types of optimization algorithms
(see above link for reference on optim()). I don't even think all
of them return a Hessian.</li>
<li>I also think that the parameters relating to the <code>reltol</code> and <code>abstol</code> parameters could be important, but I don't fully understand what they do.</li>
</ul>
<p>2) Is there some pre-treatment of the data (e.g., standardizing data before doing the MLE) that is advisable if I want to use the hessian to compute S.E.'s of the fitted parameters?</p>
<p>3) In general, of what do I need to be careful. I just don't want to incorrectly use this approach.</p>
<p>4) Is there anything you can think of that I definitely <strong>should not</strong> do?</p>
<p>Thanks.</p> | 47,570 | [
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<p>Is there anything wrong with regressing a ratio on itself? For example can we regress the log(savings ratio) on log(income) or the log(debt to income) ratio on log(income)? If not, should we use the log transformation to take income to the RHS and regress log(savings) on log(income)? If using a log transformation is the way to go, how do I interpret my results, given that the savings ratio is what I am interest in, not savings?</p> | 74,362 | [
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] |
<p>I am trying to calculate the average of hourly data of three sensors but the hourly timestamps of all three sensors are different. How is it possible to measure the average of hourly data of all three sensors at a regular interval of hours say- 01:00,02:00,03:00 and so on.</p>
<p>And also is it possible to start an extensible time series with a specific time which is before the first timestamp of the given dataset so that the extra timestamps have NA values and i can use na.rm=TRUE while calculating mean.:</p>
<pre><code> library(xts)
library(forecast)
data1$V1<-as.POSIXct(paste(data1$V1, data1$V2), format="%Y-%m-%d %H:%M:%S")
sensid<-as.factor(data1$V4)
#For Temp
for(i in 1:length(levels(sensid)))
{
l[[i]]=data.frame(data1$V6[data1$V4==i])
l[[i]]<-xts(l[[i]],order.by=data1$V1[data1$V4==i])
if(length(which (is.na(l[[i]])))<length(l[[i]]))
{
wAvgt[[i]]<-apply.weekly(l[[i]], function(x) sapply(x,mean,na.rm=TRUE)) #Weekly Average of Temp
dAvgt[[i]]<- apply.daily(l[[i]], function(x) sapply(x,mean,na.rm=TRUE)) #Daily Average of Temp
hrAvgt[[i]] <- period.apply(l[[i]], endpoints(l[[i]], on = "hours", 1), function(x) sapply(x,mean,na.rm=TRUE)) #Hourly Average of Temp
#For forecasting future values
aritw[[i]]<-auto.arima(wAvgt[[i]]) #Weekly Forecast ARIMA model using ARIMA
foretw[[i]]<-forecast.Arima(aritw[[i]],h=5) #Forecast of Weekly Average of Temp
arith[[i]]<-auto.arima(hrAvgt[[i]]) #Hourly Forecast ARIMA model using ARIMA
foreth[[i]]<-forecast.Arima(arith[[i]],h=50)#Forecast of Hourly Average of Temp
aritd[[i]]<-auto.arima(dAvgt[[i]]) #Daily Forecast ARIMA model using ARIMA
foretd[[i]]<-forecast.Arima(aritd[[i]],h=10)#Forecast of Daily Average of Temp
}
else
{
next
}
}
</code></pre>
<p>I have 27 sensors data. so i m trying to caclulate hourly,weekly and daily average of each sensors but each sensor provides 4 critical parameters like - blood pressure, temperature, Pulse and WBC. The above loop provided is only for temperature of 27 sensors. I have a bed connected to some 3 sensors and each sensor has 4 critical parameters to measure. I m concerned only with temperature for now. I need to calculate hourly average of temp for that bed only. So, initially i measure hourly average of any 3 given sensors each and then i intend to take the hourly average between these 3 sensors.</p>
<p>The problem i m facing here is each hourly average has different number of rows and different timestamps. How to normalise all the timestamps?</p>
<p>If the starting timestamp of sensor 1 is 2004-02-28 00:59:16, sensor 2 is 2004-02-28 01:08:23, sensor 3 is 2004-02-28 01:19:34, it is not possible to normalise them by any methods i know of yet.</p>
<p>Please help me find the hourly average of three sensors.</p> | 74,363 | [
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<p>I've the following situation.</p>
<p>I've a binary classifier which classifies input feature vectors into either of two classes 'y' or 'n', along with the probability of it being in either of the classes P_y and P_n. I know that the classifier is right 60% of the time. How do I estimate the true probability of it being in each of the classes? </p>
<p>What I'm doing now is to compute 0.6*P_y / (0.6*P_y + 0.4*(1- P_y)). This is a Bayesian adjustment, but I'm not sure if I'm doing the right thing. Any pointers much appreciated.</p> | 37,217 | [
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<p>I have a GLMM of the form: </p>
<pre><code>lmer(present? ~ factor1 + factor2 + continuous + factor1*continuous +
(1 | factor3), family=binomial)
</code></pre>
<p>When I use <code>drop1(model, test="Chi")</code>, I get different results than if I use <code>Anova(model, type="III")</code> from the car package or <code>summary(model)</code>. These latter two give the same answers. </p>
<p>Using a bunch of fabricated data, I have found that these two methods normally do not differ. They give the same answer for balanced linear models, unbalanced linear models (where unequal n in different groups), and for balanced generalised linear models, but not for balanced generalised linear mixed models. So it appears that only in cases where random factors are included does this discord manifest.</p>
<ul>
<li>Why is there a discrepancy between these two methods? </li>
<li>When using GLMM should <code>Anova()</code> or <code>drop1()</code> be used? </li>
<li>The difference between these two is rather slight, at least for my data. Does it even matter which is used?</li>
</ul> | 47,657 | [
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<p>I need to get as accurate as possible a value for the brightness of a mainly stable light source given twelve sample luminosity values. The sensor is imperfect, and the light can occasionally "flicker" brighter or darker, which can be ignored, hence my need for outlier detection (I think?).</p>
<p>I've done some reading up on various approaches here and can't decide on which approach to go for. The number of outliers is never known in advance and will often be zero. Flicker is generally a very large deviation from the stable brightness (enough to really mess with any average taken with a big one present), but not necessarily so.</p>
<p>Here's a sample set of 12 measurements for completeness of the question:</p>
<p>295.5214, 277.7749, 274.6538, 272.5897, 271.0733, 292.5856, 282.0986, 275.0419, 273.084, 273.1783, 274.0317, 290.1837</p>
<p>My gut feeling is there are probably no outliers in that particular set, although 292 and 295 look a little high.</p>
<p>So, my question is, what would be the best approach here? I should mention that the values come from taking the euclidean distance of the R G and B components of the light from a zero (black) point. It would be programmatically painful, but possible, to get back to these values if required. The euclidean distance was used as a measure of "overall strength" as I'm not interested in the color, just the strength of output. However, there's a reasonable chance that the flickers I mentioned have a different RGB composition to the usual output.</p>
<p>At the moment I am toying with some sort of function that will repeat until a stable membership of allowed measures is reached by:</p>
<ol>
<li>Finding the standard deviation</li>
<li>Putting everything outside say 2 SDs into an ignore list</li>
<li>Recalculating the average and SD with the ignore list excluded</li>
<li>Re-deciding who to ignore based on the new average and SD (assess all 12)</li>
<li>Repeat until stable.</li>
</ol>
<p>Is there any value in that approach?</p>
<p>All comments gratefully accepted!</p> | 74,364 | [
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<p>I am implementing the ARIMAX model for stock price predictions. To select the order, I looped through $0\le p,d,q\le10$ and found the best order to be (1,0,10). I'm a bit confused how to interpret these results. How do I tell if its reasonable, and what does this imply about the data?</p> | 37,220 | [
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<p>I know 3 methods to do parameter estimation, ML, MAP and Bayes approach. And for MAP and Bayes approach, we need to pick priors for parameters, right? </p>
<p>Say I have this model $p(x|\alpha,\beta)$, in which $\alpha,\beta$ are parameters, in order to do the estimation using MAP or Bayes, I read in the book that we'd better <em>pick a conjugate prior $p(\alpha,\beta)$</em>, which is a <strong>joint</strong> probability of $\alpha,\beta$, right?</p>
<p>I have 2 questions:</p>
<ol>
<li><p>Do we have other choices picking the prior other than this conjugate one? </p></li>
<li><p>Can we pick priors for $\alpha$ and $\beta$ respectively like $p(\alpha)$ and $p(\beta)$, other than put them together in a joint one?</p></li>
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<p>I have a dependent variable (in counts, i.e., a number of technicians). My independent variable is a grouping factor (control and treatment). From the literature, the average number of technician required for the procedure is 3 with SD of 1. I would like to calculate a sample size to determine significant changes of 1 SD decrease in technicians.</p>
<p>I think that these count data should follow a Poisson distribution and I could do a sample size calculation for Poisson regression. However, my SD is lower than one assumed for Poison regression (sqrt(mean)=1.73). What alternative tool should I use to determine a 1 SD difference and a sample size?</p>
<p>Thanks!
Julia</p> | 74,365 | [
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<p>An experiment has two possible outcomes: A or B. In 20 experiments, A happened 2 times, and B happens 18 times. What would I expect from a larger number of experiments? With what confidence can I say that the mean number of outcome A is 10%?</p> | 74,366 | [
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<p>I have to incorporate a number of line plots in a Latex document and I want to get a very high quality.</p>
<p>Are there any website that show some templates or default plots styles that I can use. Myself I am not good in designing this kind of plot style, so I am searching some good basic settings for the tool I want to use.</p>
<p>About the tools, I think I would like to use Tikz or gnuplot as I want to build tex document and plots in a single run. My data is anyway stored in text files.</p> | 74,367 | [
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<p>Let's say we created a plot, with labels:</p>
<pre><code>plot(1, xlab = "text", ylab = "text times three") # text too small
</code></pre>
<p>In this plot there is still a lot of free region to enlarge the text. We could do so by using cex.lab, like this:</p>
<pre><code>par(cex.lab = 2)
plot(1, xlab = "text", ylab = "text times three")
</code></pre>
<p>Questions:</p>
<ol>
<li>Is there a way to know what is the largest cex.lab possible, without the text getting over the borders?</li>
<li>Is it possible to do that separately for each component of the axis? (so to have a different fitting of cex.axis to the xlab vs the ylab) ?</li>
</ol> | 49,011 | [
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<p>Despite this being of a program-related question I was asked to submit this here instead stackoverflow.</p>
<p>Is it possible to have confint use the robust variance-covariance matrix obtained by vcovHC (from the sandwich package) after fitting a model? I\ve been looking around without finding it..</p> | 49,012 | [
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<p>consider the following sorted sample: </p>
<p>$x_1<x_2<\ldots<x_n$</p>
<p>and the kernel function:</p>
<p>$h(j)=\log(x_j-x_1), j>1$. </p>
<p>Now, it turns out that when I draw data
from many continuous distributions, sort
them and compute $h(j)$, the plot of the
$h(j)$ as a function of $j$ is concave in
this sense: $h((j+k)/2)>(h(j)+h(k))/2$
for $1<j<j+1<k<n$ </p>
<p>My question is the following: will this
observation (concavity of $h(j)$ as a
function of $j$) be true for all continuous
distributions? If not is there a class
of distributions for which it will always be
true?</p>
<p>Thanks in advance, </p>
<p>At this point, given the useful comments from @Glen_b and @Momo,
I think I should give an example.</p>
<p>this is a vector of values of $x_j-x_1,j>1$ (where the $x$'s are drawn from
a continuous distribution, but have been rounded here to the third digit
so that they don't take too much place.)</p>
<p>My problem, is that when I plot $h(j)$ for this data-set, the result is clearly
nor concave (in the sense I outlined above). </p>
<pre><code>x=c(1.403, 1.406, 1.408, 1.416, 1.417, 1.42, 1.427, 1.441, 1.448,
1.456, 1.458, 1.465, 1.466, 1.472, 1.472, 1.477, 1.477, 1.479,
1.482, 1.491, 1.5, 1.504, 1.518, 1.52, 1.52, 1.53, 1.544, 1.545,
1.561, 1.573, 1.595, 1.595, 1.599, 1.603, 1.605, 1.612, 1.617,
1.618, 1.618, 1.628, 1.64, 1.644, 1.646, 1.653, 1.679, 1.682,
1.682, 1.693, 1.694, 1.71, 1.71, 1.711, 1.741, 1.75, 1.756, 1.773,
1.794, 1.799, 1.804, 1.808, 1.86, 1.882, 1.895, 1.955, 1.992,
1.995, 2.009, 2.009, 2.063, 2.123, 2.329, 2.356, 2.405, 2.535,
2.632, 2.635, 2.725, 2.763, 2.783)
</code></pre> | 37,231 | [
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<p>My background is in labor economics/applied micro but I am currently working with data collected from a field experiment. My experimental/micro skills are rusty to say the least.</p>
<p>In my data, I observe individuals choosing one of the choices $A, B, C, D$. The value of the choice is realized after the choice. I know that </p>
<p>$$E(A) <= E(B) <= E(C) <= E(D)$$
$$VAR(A) <= VAR(B) <= VAR(C) <= VAR(D)$$</p>
<p>If I assume that $A, B, C, D$ are continuous, independent and normally distributed and individuals are aware of expected payoffs $E(i)$ and variances $VAR(i)$. </p>
<ul>
<li>Is there a way to estimate a expected risk aversion parameter (e.g. CRRA) for a group of people? </li>
<li>Would it make sense to compare these risk estimates from two groups of people, say men and women or high educated and low educated?</li>
</ul>
<p>I would appreciate if you guys could come up with some relevant literature.</p> | 74,368 | [
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<p>I have a set of Likert type items asking respondents their attitude on different topics. For example, attitudes toward drugs/alcohol where 1=strongl disagree and 5=strongly agree to the statement "I think it is okay for someone my age to smoke marijuana." The 6 items in this particular scale are then scored using the mean of non-missing items. Is there some advantage to first rescaling the items so that the range of responses goes from 0-4 rather than 1-5? The only difference is in interpretation of the scale where the average "attitude" is 3.5 on a scale of 0-4 and 4.5 on a scale of 1-5. From an interpretation standpoint, what is easier for a lay audience? Or is there some standard of practice in terms of reporting? </p> | 36,204 | [
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<p>Near the middle of page 16 of Andrew Ng's <a href="http://cs229.stanford.edu/notes/cs229-notes3.pdf" rel="nofollow">notes on SVM</a>, he explained an intuitive view of kernel as measuring similarity between observations, but then added the caveat that</p>
<blockquote>
<p>there are things wrong with this intuition, but nevermind</p>
</blockquote>
<p>The notes don't really explain what's wrong with this intuition. Does anyone know what he might be referring to?</p> | 74,369 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://stats.stackexchange.com/questions/26650/how-do-i-reference-a-regression-models-coefficients-standard-errors">How do I reference a regression model's coefficient's standard errors?</a> </p>
</blockquote>
<p>If I have a dataset:</p>
<pre><code>data = data.frame(xdata = 1:10,ydata = 6:15)
</code></pre>
<p>and I run a linear regression:</p>
<pre><code>fit = lm(ydata~.,data = data)
out = summary(fit)
Call:
lm(formula = ydata ~ ., data = data)
Residuals:
Min 1Q Median 3Q Max
-5.661e-16 -1.157e-16 4.273e-17 2.153e-16 4.167e-16
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.000e+00 2.458e-16 2.035e+16 <2e-16 ***
xdata 1.000e+00 3.961e-17 2.525e+16 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.598e-16 on 8 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 6.374e+32 on 1 and 8 DF, p-value: < 2.2e-16
</code></pre>
<p>How do I extract the standard errors of the regression coefficients from either fit or out? I can't seem to figure it out. Thanks!</p> | 49,930 | [
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<p>A massive problem in communicating the results of statistical calculations to the media and to the public is how we communicate uncertainty. Certainly most mass media seems to like a hard and fast number, even though except in a relatively small number of cases, numbers always have some uncertainty.</p>
<p>So, how can we, as statisticians (or scientists describing statistical work), best communicate our results, while keeping the uncertainty in tact, and making it meaningful to our audience?</p>
<p>I realise that this isn't actually a statistics question, rather a psychology question about statistics, but it's certainly something that most statisticians and scientists will be concerned about. I'm imagining that good answers might reference psychological research more than stats textbooks...</p>
<hr>
<p><strong>Edit:</strong> As per user568458's suggestion, a case-study may be useful here. If possible, please keep answers generalisable to other areas.</p>
<p>The particular case that I'm interested in serves as a nice example: <em>the communication of climate science to politicians and the general public, through mass media</em>. In other words, as a scientist, it is your job to convey information to a journalist in such a way that they have little difficulty in accurately conveying that information to the public - that is, the truth, although not necessarily the <em>whole</em> truth, which won't usually fit in a news-bite. </p>
<p>Some particularly common examples might be the communication of the uncertainty in the estimate of degree of warming over the remainder of the century, or in the increased likelihood of a specific extreme weather event (i.e. in response to a "was this storm caused by climate change" type question).</p> | 74,370 | [
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<p>I want to estimate</p>
<p>$\log Y_t=\beta_0+\beta_1\log X_{1t}+\beta_2\log X_{2t}+e_t$</p>
<p>Using ADF-tests, I have found that both $\{\log Y_t\}$ and $\{\log X_{1t}\}$ are I(1) (i.e. they contain a unit root), while $\{\log X_{2t}\}$ is I(2). I have checked that $\{\Delta \log Y_t\}$, $\{\Delta \log X_{1t}\}$ and $\{\Delta_2\log X_{2t}\}$ are all stationary. Now my question is what regression do I run using these differences, so that I can still say something about the original relationship above? Can I simply take second differences of $\{\log Y_t\}$ and $\{\log X_{1t}\}$ and estimate</p>
<p>$\Delta_2\log Y_t=\beta_1\Delta_2\log X_{1t}+\Delta_2\beta_2\log X_{2t}+v_t$</p>
<p>or does taking the second difference of an I(1) series introduce some kind of problem?</p> | 37,238 | [
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<p>Consider a kind of auction where you are presented with, say, 1000 prospective clients. Based on information about these prospectives--age, sex, race, income, educational achievement and the like--you may 'bid' to pitch your product to some fraction of them, say 250. (Ignore the cost of bidding.) To maximize your chances of selecting the right subset, I would probably use a model of our product's 'likability' built using logistic regression. I can fumble my way through this part well enough.</p>
<p>However, to <em>build</em> the model of product likability, I have to do some market research, testing the product pitch on subjects that we may recruit from the population at large. This is actually fairly costly. Moreover, it should probably be tuned to the demographics of the population from which the prospectives are drawn. For example, a factorial design, say, might ask us to recruit test subjects in equal proportion across the levels of the race factor, when in reality we are likely to encounter very few Native Americans, say, in the 1000 prospectives, and can simply choose not to pitch to them at all as a rule. (Sad, but true.)</p>
<p>How should such an experiment be designed? To be concrete, the design variables are all categorical and ordinal factors, the bidding fraction is an input parameter (1/4 in the example quoted above), as is the maximum number of subjects that can be recruited. It seems like maybe some mix of experimental design and random sampling might be appropriate, but I am open to all reasonable suggestions and pointers. </p>
<p>I should also note that given the likely small effect sizes and the small sample recruitment pools that we can afford, it is unlikely that the market research will yield statistically significant regression coefficients. And so overoptimizing the experimental design is probably silly, and any reasonably non-insane procedure will suffice. </p> | 31,335 | [
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<p>I'm trying to fashion survey experiments for which the demographic composition of the samples needs to be as close as possible to a 'golden' composition. For concreteness, let's say the 'golden' composition is <code>14% middle class, 35% upper-middle class, and 51% upper class</code>.</p>
<p>I have a whole bunch of samples, each with slightly different compositions. For example:</p>
<pre><code>S1 = 10% middle class, 41% upper-middle class, 49% upper class
S2 = 33% middle class, 33% upper-middle class, 34% upper class
</code></pre>
<p>I'm trying to find a good statistical measure to describe how close the distribution in each sample is to the ideal, 'golden' composition. Is there a simple metric here that I could use?</p> | 74,371 | [
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<p>I would like to compare the effect of one IV ("sensation seeking") on a DV ("intended infidelity") based on another IV ("gender"). Actually I really just want to compare the means. So I would like to compare the mean effect of sensation seeking (a continuous variable with 5 levels) for males and females on intended infidelity (a continuous variable with 4 levels). Why can't I seem to do this in SPSS? Do I have to do a moderation analysis? Can I re-compute the variables and into what? Thanks very much for any suggestions! </p> | 74,372 | [
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<p>I am drafting a literature review on a current public health issue where data are confounded: </p>
<p><strong>What are common historical case-studies that are used in public health/epidemiology education where invalid or confounded relationships or inferences were intentionally or erroneously employed in public health policy and legislation?</strong> </p>
<p>The automobile fatality surge of the 1960s and subsequent evidence-based, government-led study which determined seatbelts and eventually airbags should be required by law is a great example of <strong>HOW</strong> public health policy should be driven by statistically powerful inferences and models. </p>
<p>I am looking more for examples of cases of the opposite type (bad science to make policy hastily). However, if nothing else I would like to learn of more cases similar to the previous example of powerful studies for successful public health benefit.</p>
<p>I want to use these as examples to demonstrate how evidence-based statistical public health research is important to the making of policy.</p> | 74,373 | [
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<p>I am estimating a linear probability model with a hierarchical structure. I am bootstrapping standard errors to test the significance of the coefficients in order to double check the parametric estimates of the standard errors. </p>
<p>In the bootstrapping process I am finding that the coefficients are all positive (which I expected) but there are a few extreme positive outliers. The outliers make the standard errors huge and subsequent t-statistics small. This is misleading because an empirical p-value has a value under 0.01. </p>
<p>My questions are</p>
<ol>
<li><p>What is the best way to convey that this coefficient is statistically different from zero? </p></li>
<li><p>What would be the best way to report these results? Normally in a LaTex table I report coefficients plus standard errors. </p></li>
</ol> | 37,240 | [
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] |
<p>I have a set of data for companies in a certain industry. The data set spans two consecutive years, 2000 and 2001. </p>
<p>Let's say I have 100 unique companies in the dataset and I calculate average service time for each company and for each year. </p>
<p>How do I test whether the average waiting time for company x in 2001 is less than the average waiting time for company x in 2000?</p>
<p>The data set contains all service transactions performed by company x in 2000 and 2001, and this is what confuses me. The way I understand hypothesis test is that you test the null hypothesis using sample mean. However, with this data I have the population mean so what does it mean when your professor ask you to test whether the average service time in 2001 is less than the average service time in 2000 using "simple mean comparison test such as t-statistic"?</p> | 74,374 | [
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<p>I currently have a gradient boosting model that uses the <code>gbm</code> package in R that classifies observations at the end of a year. Daily behaviors are logged for each observation (with most of them zero). The model is built on observations which have a year's worth of complete data and performs well out of sample. </p>
<p>For illustrative purposes, assume the complete data look like this:</p>
<pre><code>id day1 day2 day3 day4 day5, ..., day365
1 1 1 2 0 0, ..., 0
2 3 0 1 0 0, ..., 0
3 1 1 2 0 0, ..., 0
4 1 0 0 0 0, ..., 1
</code></pre>
<p>However, I want to predict the probability of each class for new individuals who have not yet been in the sampling frame for a year. Thus, all of the days past the present day will be NA. This precludes the use of many algorithms, but treating them as 0s also implies having more information than I actually have.</p>
<p>What is the best way to treat the unobserved, but observable missing data when making predictions? I will not be re-estimating the model on a daily basis but rather using the model fit from the existing training set.</p> | 74,375 | [
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<ol>
<li><p>What is the difference between quasi-likelihood estimating equations, and GEE?</p>
<p>From <a href="http://www.biostat.umn.edu/~xianghua/8452/note/05GEE.pdf" rel="nofollow">a note</a></p>
<blockquote>
<p>Quasi-score function is for independent, over-dispersed data (Poisson or binomial), while GEE1 is for correlated data.</p>
</blockquote>
<p>But it seems to me that in the note, their equations both don't take
into account the correlation between different responses. So how
shall I understand them correctly?</p></li>
<li>How is independence estimating equations (IEE) different from the
above two?</li>
</ol>
<p>Thanks!</p> | 1,035 | [
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<p>The method that I'm familiar with for constructing a random forest is as follows:
(from <a href="http://www.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm" rel="nofollow">http://www.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm</a>)</p>
<p>To build a tree in the forest we:</p>
<ol>
<li>
Bootstrap a sample of size N where N is the size of our training set. Use this bootstrapped sample as the training set for this tree. </li>
<li>At each node of the tree randomly select m of our M features. Select the best of these m features to split on. (where m is a parameter of our Random Forest)</li>
<li>Grow each tree to largest extent possible -- i.e. no pruning. </li>
</ol>
<p>While this algorithm makes sense at a procedural level and certainly produces good results, I'm not clear what the theoretical motivation is behind the steps 1, 2, and 3. Could someone explain what motivated someone to come up with this procedure and why it works so well?</p>
<p>For example: why do we need to perform step 1? It doesn't seem like we're bootstrapping for its usual purpose of variance-reduction.</p> | 74,376 | [
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<p>Suppose I have repeated observations on individuals, which I've analyzed using a generalized estimating equation (GEE). What is the procedure for computing the prediction interval for future observations given values of the predictors? I would appreciate either instructions or references. Thank you.</p> | 74,377 | [
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<p>When should I use weights when performing a logistic regression? The weights I'm referring to are sampling weights from a survey? Or should I just use the unweighted data?</p> | 74,378 | [
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<p>It is clear to me, and well explained on multiple sites, what information the values on the diagonal of the hat matrix give for linear regression.</p>
<p>The hat matrix of a logistic regression model is less clear to me. Is it identical to the information you get out of the hat matrix applying linear regression? This is the definition of the hat matrix I found on another topic of CV (source 1):</p>
<p>$H=VX ( X'V X)^-1 X' V$</p>
<p>with X the vector of predictor variables and V is a diagonal matrix with $\sqrt{(π(1−π))}$.</p>
<p>Is it, in other words, also true that the particular value of the hat matrix of an observation also just presents the position of the covariates in the covariate space, and has nothing to do with the outcome value of that observation?</p>
<p>This is written in the book "Categorical data analysis" of Agresti:</p>
<blockquote>
<p>The greater an observation’s lever- age, the greater its potential
influence on the fit. As in ordinary regression, the leverages fall
between 0 and 1 and sum to the number of model parameters. Unlike
ordinary regression, the hat values depend on the fit as well as the
model matrix, and points that have extreme predictor values need not
have high leverage.</p>
</blockquote>
<p>So out of this definition, it seems we can not use it as we use it in ordinary linear regression?</p>
<p>Source 1: <a href="http://stats.stackexchange.com/questions/8126/how-to-calculate-the-hat-matrix-for-logistic-regression-in-r">How to calculate the hat matrix for logistic regression in R?</a></p> | 74,379 | [
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<p>I'm trying to model a real-world random variable that behaves approximately as a Gaussian, so a Normal distribution fit is reasonable but far from perfect.</p>
<p>However, I only care about its tail, that is, I only care about finding $x$ such as $P(X > x) < p$, with $p$ anywhere between 1% and 5%. I'm modelling the random variable as $X\sim\mathcal{N}(\mu, \sigma^2)$, with parameters estimated from all the samples.</p>
<p>I tried to search for specific ways to handle this and I got a lot of results regarding Extreme Value Theory, but it seems to apply mostly to even smaller probabilities than I care or to a heavy-tailed distributions, which is not my case.</p>
<p>Any pointers?</p> | 74,380 | [
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<p>Is the power of a logistic regression and a t-test equivalent? If so, they should be "data density equivalent" by which I mean that the same number of underlying observations yields the same power given a fixed alpha of .05. Consider two cases:</p>
<ol>
<li>[The parametric t-test]: 30 draws from a binomial observation are made and the resulting values are averaged. This is done 30 times for group A (which has a binomial Pr of .70 of occurring) and 30 times for group B (which has a binomial Pr of .75 of occurring). This yields 30 means per group that represent a summary of 1,800 draws from a binomial distribution. A 58df t-test is performed to compare the means.</li>
<li>[The logistic regression]: A logistic regression is performed with a dummy coded slope representing group membership and each of the 1,800 draws.</li>
</ol>
<p>My question has two parts:</p>
<ol>
<li>Given a set alpha of .05, will the power of these methodologies be the same or different? Why? How can I prove it?</li>
<li>Is the answer to question 1 is sensitive to the sample sizes going into the t-test, sample size of each group in the t-test, underlying binomial probabilities, or some other factor? If so, how can I know (without simulation) that the power is indeed different and what sort of changes will produce what sort of changes in power? Alternatively, provide worked out R code that solves the issue using simulation.</li>
</ol> | 74,381 | [
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<p>Basically, I want to manually compute what <code>predict(princomp(somedata))</code> produces.</p>
<p>In order to do that I need to have coefficients matrix so that</p>
<pre><code>newdata %*% coefficients
</code></pre>
<p>would produce the predicted values of each component.</p>
<p>Can anybody give me a clue to get the coefficients?</p> | 74,382 | [
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<p>As you know, there are many password generators out there to increase computer security.</p>
<p>Suppose I am given such a password (say, a string of letters, numbers, symbols, etc.), is there a way for me to test how random it is? Is there some sort of index for this measurement?</p>
<p>Thanks!</p> | 37,247 | [
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<p>I know PCA and PLS are considered as alternative method to each other. But I am thinking about a kind of combination of the two in case of lots of predictors with little variability.</p>
<p>In that case, when I run 1-component PLS with original predictors, it does not produce a meaningful model in terms of prediction. But if I first compute 10-20 PCA components and run 1-component PLS with those PC scores as predictors, practically, the model is quite good in terms of prediction power. But I would like to know why.</p>
<p>Can anybody explain why this is better than 1-component PLS with original predictors?</p> | 74,383 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://stats.stackexchange.com/questions/25211/simple-combination-probability-question-based-on-string-length-and-possible-char">Simple combination/probability question based on string-length and possible-characters</a> </p>
</blockquote>
<p>Suppose I have a collection of 1,000,000 unique strings where each string is exactly 10 characters long, the characters being A-Z, a-z, 0-9. What is the probability of having a string colliding with existing data -- and -- how do I calculate this probability.</p> | 49,931 | [
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<p>The alpha-expansion algorithm described <a href="http://books.google.co.in/books?id=7dzpHCHzNQ4C&pg=PA592&lpg=PA592&dq=alpha+expansion+algorithm&source=bl&ots=pu1CCm--AR&sig=PIN3b33Wm02_FWi3f1dJO_np0W8&hl=en&sa=X&ei=Z2d8T4veCsrnrAetw6THAg&redir_esc=y#v=onepage&q=alpha%20expansion%20algorithm&f=false" rel="nofollow">here</a> (Probabilistic Graphical Models: Principles and Techniques By Daphne Koller, Nir Friedman, Page 593: algorithm 13.5), is designed for graphs where each vertex has the same set of labels. I am interested in a more general problem where each vertex can have different set of labels having different cardinalities.</p>
<p>Can the alpha-expansion algorithm be used in such cases?</p>
<p>For example, let the graph have 3 vertices. The first vertex can assume labels (a1, a2, a3). The second vertex can assume labels (b1, b2) and the third one can have (c1, c2, c3, c4, c5). The node and edge potentials are defined accordingly. Can alpha-expansion still help me in finding an optimal (approximate) label combination?</p> | 74,384 | [
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<p>It's well known that members of the public have difficulty comprehending numbers over a certain size. A common tricky issue for people communicating demographic trends is helping people make sense of rates out of 100,000. This is an important issue in helping people make sense of changing health and crime statistics, for example.</p>
<p>For those of you who do statistical communication for the public, what frames of reference or other analogies and aids have you found to be successful in helping people to put rates out of 100,000 in perspective? There's <a href="http://stats.stackexchange.com/questions/12810/why-do-demographers-give-rates-per-100-000-people">another question with a discussion alluding to 100,000 people being a small to medium sized city</a>, and it seems there's disagreement over whether this analogy is helpful or not. </p>
<p>Answers citing evidence would be great, personal experience and ideas are helpful too.</p>
<p><em>(p.s. I'm also <a href="http://meta.stats.stackexchange.com/questions/1099/statistical-communications-questions">starting a discussion on meta</a> on whether to tag/allow/encourage stats communications questions like this on this site)</em></p> | 74,385 | [
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<p>Let's say I have a $m \times n$ matrix where $m$ is the number of points and $n$ is the number of dimensions. I would like to give a target dimension parameter which is let's say d. d can be a set of values like $\{2,4,6,\ldots,n\}$. I would then approach the problem using Fisher's linear discriminant analysis to give me output matrix in $m \times d$. Is this possible to do? I still don't understand how LDA reduces dimensions from let's say 10,000 to 1,000.</p> | 74,386 | [
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<p>From <em>Robert (The Bayesian Choice, 2001)</em>, it is proposed that the Bayes Estimator associated with the prior distribution $\pi$ and the multilinear loss is a $(k_2/(k_1+k_2))$ fractile of $\pi(\theta|x)$.</p>
<p>The proof follows that</p>
<p>$$
E^\pi[L_{k_1,k_2}(\theta,d|x)] = k_1\int\limits_{-\infty}^{d} (d-\theta)\pi(\theta|x)d\theta + k_2\int\limits_{d}^{+\infty}(\theta-d)\pi(\theta|x)d\theta
$$</p>
<p>Then, using the identity</p>
<p>$$
\int\limits_{c<y} (y-c)f(y)dy = P(y>c)
$$</p>
<p>I would get to</p>
<p>$$
E^\pi[L_{k_1,k_2}(\theta,d|x)] = k_2P^\pi(\theta>d|x) - k_1P^\pi(\theta<d|x)
$$</p>
<p>But he gets to</p>
<p>$$
E^\pi[L_{k_1,k_2}(\theta,d|x)] = k_1\int\limits_{-\infty}^{d} P^\pi(\theta<y|x)dy + k_2\int\limits_{d}^{+\infty} P^\pi(\theta>y|x)dy
$$</p>
<p>And then takes the derivative in $d$. What am I'm missing, and why he does this last step? Thanks!</p> | 49,050 | [
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<p>Since OLS tries to measure E[Y|X], and regression trees try to partition the data into different branches, then take means of the response under different branches, is it reasonable to say that regression trees strictly dominate OLS in prediction? Also, since OLS is not robust to outliers, would it be reasonable to say that regression trees are a better first, off-the-shelf algorithm to try in a prediction problem?</p>
<p>For interpretation, it definitely makes sense to me that knowing $R^2$, regression coefficients, p-values, etc. are more useful than seeing the tree structure.</p> | 1,036 | [
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<p>I use Andrew F. Hayes' macro for SPSS (HCREG at <a href="http://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html" rel="nofollow">http://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html</a>) to perform multiple regression analyses with robust standard errors.</p>
<p>The information I get in the output is limited compared to the one SPSS provides when applying multiple regression. For example, there are no standardized coefficients (Beta) for predictors and there is no adjusted R-squared.</p>
<p>Can I assume that the standardized coefficients will be the same as in the model without robust standard errors? If not, does anyone know how can I go about computing them by hand?</p>
<p>I have the same question for the adjusted R-squared.</p>
<p>Any help would be greatly appreciated!
Thank you very much in advance.</p> | 1,037 | [
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<p>I have seen many places where they have input/output datasets where they first create a linear regression line, correct the bias, and then only use that data for their model. I didn't get what this bias correction is?</p> | 74,387 | [
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<p>A person is trying to claim that psychics exist by flipping a coin 10 times for each person behind a screen. n=10 p=1/2
Its created into a one tailed test with significance level at .001. The first 999 people do not get enough right to reject the null hypothesis. But the 1000th person gets all 10 right.
if Y is the number of people out of 1000 who get all 10 coin flips right, would the distribution of y be binomial too?</p>
<p>Probability that one or more of the 1000 people would correctly guess 10 coin flips?</p> | 74,388 | [
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<p>These are the values:</p>
<pre><code>18.2 17.6
------------------------------------------------------------------------------------------------------------
18.9 18.6
------------------------------------------------------------------------------------------------------------
20 19.7 19.8 19.6
------------------------------------------------------------------------------------------------------------
21.4 20.6 21
------------------------------------------------------------------------------------------------------------
21.9 22.1 22.2 22.2 21.6
------------------------------------------------------------------------------------------------------------
23.1 22.6 23.2 23 22.9 23.2 23.2 23 22.5 22.5
------------------------------------------------------------------------------------------------------------
23.8 24.1 23.8 23.6 24.3 23.7 24 24.2 24.4 23.8 23.5
------------------------------------------------------------------------------------------------------------
24.7 24.9 25 25.4 25.1 25.3 25.3 25.2 25.2 24.7 24.6 24.5 24.5
------------------------------------------------------------------------------------------------------------
25.6 26 25.6 25.6 25.6 25.7 25.6 26.3 26.3 26.2 26 25.9 26.2 26
------------------------------------------------------------------------------------------------------------
27 27.2 27 27.2 26.7 27 26.6 27.3 26.8 26.6 26.9 27 26.5
------------------------------------------------------------------------------------------------------------
28.2 27.7 28.1 27.9 27.6 27.7 28.1 28.2 27.5
------------------------------------------------------------------------------------------------------------
28.6 29.1 28.7 29.1
------------------------------------------------------------------------------------------------------------
30.3 29.6 29.7 29.5
------------------------------------------------------------------------------------------------------------
30.9 31 31 30.5
------------------------------------------------------------------------------------------------------------
32.3 31.6
------------------------------------------------------------------------------------------------------------
</code></pre>
<p><br></p>
<p>These are the groups:</p>
<pre><code> - 17.5 - 18.5
- 18.5 - 19.5
- 19.5 - 20.5
- 20.5 - 21.5
- 21.5 - 22.5
- 22.5 - 23.5
- 23.5 - 24.5
- 24.5 - 25.5
- 25.5 - 26.5
- 26.5 - 27.5
- 27.5 - 28.5
- 28.5 - 29.5
- 29.5 - 30.5
- 30.5 - 31.5
- 31.5 - 32.5
</code></pre>
<p>I'm supposed to calculate the standard deviation and the expected value ...
How is it different comparing to having only 15 values and not divided them into groups?
If I only have 15 values, I know how to calculate the stand.deviation and the exp.value.
I use the formulas that are well-known: </p>
<pre><code>Exp.value: ............(1/n)*Sum(xi)..................where xi = all the 15 values <br>
Stand.deviation: ...(1/(n-1))*Sum( xi - u)^2...where xi = all the 15 values and u = exp.value
</code></pre> | 49,053 | [
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<p>I have three site temperature types: cold, medium, and hot. This is a fixed effect - I am specifically am testing if temperature affects my response variable. I believe temperature type is my 'block'. </p>
<p>I will have three sites for each temperature type. I think site is a random effect. Within each site, I will have a factorial design, where two other fixed factors, A and B, are crossed. </p>
<p>I am interested in the interaction of all three fixed effect (tempxAxB). </p>
<p>From reading my stats book, I think this is a split plot design, but my stats book does not have information about factorial design (the cross of A and B) within blocks. <strong>Is what I just described a split plot design or a three way ANOVA?</strong> And if it is a funky split plot, could someone point me to an ANOVA table (how to calculate sum of squares) with a similar set up?</p>
<p>Here is a schematic of the setup: </p>
<p><img src="http://i.stack.imgur.com/E60hm.png" alt="Set up"></p> | 74,389 | [
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<p>I don't know what such plots are called and thus I just gave this question a stupid title.</p>
<p>Let's say I have an ordered dataset as follows</p>
<pre><code>4253 4262 4270 4383 4394 4476 4635 ...
</code></pre>
<p>Each number corresponds to the amount of postings a certain user contributed to a website. I am empirically investigating the "participation inequality" phenomenon as defined <a href="http://www.nngroup.com/articles/participation-inequality/">here</a>.</p>
<p>In order to make it easy to grasp I would like to produce a plot which allows the reader to quickly deduce statements such as "10% of the users contribute 50% of the data". It should probably look similar to this admittedly pretty lousy paint sketch:</p>
<p><img src="http://i.stack.imgur.com/ysrzN.png" alt="enter image description here"></p>
<p>I have no clue how this is called thus I don't know where to look for. Also, if somebody had an implementation in <code>R</code>, that would be awesome.</p> | 74,390 | [
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<p>If I'm watching a series of accounts for transactions going in and transactions going out, I want to notice unusually large or transactions for any particular account on any particular day.</p>
<p>So if account A typically moves a few hundred dollars and one day moves five thousand dollars, that's a clear outlier.
If account B typically moves a few million dollars in or out and one day moves 20 million dollars, that's a clear outlier. </p>
<p>What I'd like to do is present a measure that should highlight outliers - I was thinking number of standard deviations versus a population of the rolling last 60 days, but I'm wondering if that's correct. I'm checking to see if it's a gaussian distribution, but are there better ways to hit what I'm looking for?</p>
<p>I think this poses a different set of questions than <a href="http://stats.stackexchange.com/questions/1223/robust-outlier-detection-in-financial-timeseries">Robust outlier detection in financial timeseries</a>. </p> | 37,254 | [
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<p>I am trying to work out Example 2 from Casella and George's paper "Explaining the Gibbs Sampler" in R.</p>
<p>The example is:</p>
<pre><code>f(x | y) = y*exp(-yx)
f(y | x) = x*exp(-xy)
where x, y are defined on [0,B]
</code></pre>
<p>The idea is to generate the distribution of f(x).</p>
<p>My code (below) works, but what I do not understand is the role of the interval on which x and y are defined. In the paper this interval is [0, B]. In their example, the authors say that their simulation uses B = 5, but I am not clear where to incorporate this into my code...</p>
<pre><code>X = rep(0, 500)
Y = rep(0, 500)
k = 15
for (i in 1:500) {
x = rep(1, k)
y = rep(1, k)
for (j in 2:k) {
x[j] = y[j-1]*rexp(1, y[j-1])
y[j] = x[j]*rexp(1, x[j])
}
X[i] = x[k]
Y[i] = y[k]
}
print(max(X))
print(max(Y))
hist(X, breaks=40, freq=F)
</code></pre> | 74,391 | [
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<p>I have time series, for example, gdp and unemployment(<code>unemp</code>), freq= 4. </p>
<p>What if I interpret it as cross-sectional data and do cross-sectional analysis instead of time series? </p>
<p>My task is to test how unemployment affects gdp. </p>
<p>Is it allowed to do that kind of analysis? </p>
<p>Do the coefficients in the model <code>lm(gdp~unemp)</code> have an economic explanation?</p> | 74,392 | [
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<p>I have a spectrum. Specifically, my data is relative intensity $[I_{\tilde{\nu}}]$ versus wavenumbers $[\tilde{\nu}]$. The wavenumbers are equally sampled so that ${d\tilde{\nu}} = c$, where $c$ is some constant.</p>
<p>However, I need the data to be given in terms of wavelength $\lambda = \frac{1}{\tilde{\nu}}$.
When I perform this transformation, the step size of wavelength $d \lambda$ decreases with increasing wavelength. However, I need the step size to remain constant for the entire spectrum.</p>
<p>I am guessing this calls for some kind of interpolation or fitting. How can an even sampling of wavelengths $\lambda$ be accomplished?</p> | 74,393 | [
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<p>I am trying to find out if I can somehow assess if one model fits my data significantly better than another model? They are both ordinal regressions and we have introduced an additional interaction term in the second model. Now we want to know if the model has improved. Do I only look at the model fit indices? How do I know if the model has significantly improved?</p> | 37,261 | [
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<p>So I've been reading various online tutorials and I've not come across and example which is similar to what I'm trying to do. I've thought up this example because I was struggling to explain with words alone. It's really basic but hopefully illustrates the concept:</p>
<pre><code>movie_eg=data.frame(score_children=c(1.1,2.5,3.9,1.5,4.3),
score_adults=c(12.5,13.1,18.1,13.1,18.8),
brad_hrs=c(0.1,0.2,0.3,0.1,0.6),
angelina_hrs=c(0.2,0.6,0.8,0.9,0.1),
cgi_budget=c(0.9,0.1,0.2,0.3,0.8),
seat_cost=c(0.1,0.3,0.6,0.1,0.9))
rownames(movie_eg)=c("shrek","bourne","lockstock","scream","sharknado")
</code></pre>
<p>So in reality I have over 400 "movies" each with 2 continuous outcomes ("scores") and 4 predictive measurements ("brad","ang","budget","seats") which are also continuous. The two "scores" are moderately correlated across "movies".</p>
<p>I'd like to see if there is a strong connection between any/all of the predictive values and the two outcomes - so here, what (if anything) influences both "scores"?</p>
<p>Therefore, I believe that the "dependent variables" are the two outcomes and the "independent variables" are the four predictions.</p>
<p>So I would do:
<code>summary.aov(manova(cbind(brad_hrs,angelina_hrs,cgi_budget,seat_cost)~cbind(score_children,score_adults),data=movie_eg))</code></p>
<p>and then see all P values are greater than 0.05, so nothing had a strong effect on score? But that seat_cost was the best predictor (although still not significant)? </p>
<pre><code>Response seat_cost :
Df Sum Sq Mean Sq F value Pr(>F)
cbind(score_children, score_adults) 2 0.45417 0.227085 17.583 0.05381 .
Residuals 2 0.02583 0.012915
</code></pre>
<p>(Probably worth mentioning in reality the "scores" are measures of drug response and the predictions are measures of different cellular functions, so really nothing to do with movies!)</p> | 74,394 | [
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<p>I have read several textbook descriptions on RVM and none of them provide an adequate (plain English) explanation of how RVM achieves sparsity.</p>
<p>I am left feeling like the authors left out a paragraph of text that would have connected the dots and instead decided to replace (rather than supplement) it with mathematical derivations.</p>
<p>Could someone please explain the basic idea as to how RVM works in relation to learning sparse regression models?</p> | 37,263 | [
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<p>I have run an online poll on two candidates. Participants are shown both candidates and are asked to determine which one they like better. Here are the labels on my Likert scale:</p>
<ul>
<li>Strongly for Romney</li>
<li>Leaning towards Romney</li>
<li>Neutral</li>
<li>Leaning towards Obama</li>
<li>Strongly for Obama</li>
</ul>
<p>I want run hypothesis testing on the results and see which candidate is more desirable. What is the proper way of designing the analysis? So far I have been just comparing the ones that are strongly for either candidate and reported the count value as the winner. But I want to be more systematic and do more rigorous hypothesis testing. The distribution of votes is like a binomial distribution. </p> | 74,395 | [
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<p>I have a dependent variable that includes four categories. I also have four continuous independent variables. The 3rd category covers the largest group in the sample. Is it important to choose the category covering largest group as a reference factor or can I choose another one that would make sense for comparisons?</p>
<p>Many thanks for response</p> | 37,264 | [
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<p>I am using proc mixed for multilevel hierarchical data. What should be the acceptable range for Standard Error values for Fixed Effects?</p> | 74,396 | [
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<p>I have a question regarding the best practice for dealing with cross-loadings on factors after conducting an exploratory factor analysis using a promax rotation. Just to give a bit of background information, I am trying to determine the factor structure of a set of 20 questions that I created about spirituality (based on PCA which identified 4 components). After conducting the EFA with 184 participants, I noticed that quite a few of the questions are cross-loading onto two or three factors. I read online that you should eliminate questions whose cross-loadings are less than or equal to .2, but I am confused because doesn't an oblique rotation (promax) imply that there is inter-correlation? For example, if question 2 loads onto factor 1 (.234), factor 2 (.346), and factor 3 (.212), what should I do?</p> | 47,975 | [
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<p>I am looking to do a linear regression on two independent variables that will be present in varying proportions.</p>
<p>For example trying to do a linear regression on $Y$ which is payment behavior (payback rate) of customers based on the the quality (let's say Gini coefficient) of the new and existing customer credit scores ($X_1$ and $X_2$, respectively) adjusted for the proportion of new and existing customers in the sample.</p>
<p>Existing customers will be present in proportion $p$ and new customers in proportion $1-p = q$.</p>
<p>$Y$, payback rate is the percentage of total customers who pay back. It could be expressed as the weighted average $Y = Y_1q + Y_2p$ where $Y_i$ is the payback rate of new/existing customers.</p>
<p>In general more new customers, $q$, has a negative effect. Better scoring ($X_1, X_2$) and more existing customers p have a positive effect.</p>
<p>What is a good way to model this? </p>
<p>Would something like the following be a good solution trying to use $p$ and $q$ as some sort of interaction effect? </p>
<p>$Y = X_1+X_2+\frac{X_1}{q}+X_2 p$</p>
<p>Would it be better to include p and q as variables themselves as well?</p> | 37,266 | [
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<p>I am trying to implement the ID3 algorithm on a data set. However, all attributes are continuous and can have values between 1-10. I found that we have to specify the bin intervals for discretization but couldn't understand how to do this exactly. </p>
<p>Can some one explain on how to do this? The <a href="http://archive.ics.uci.edu/ml/machine-learning-databases/breast-cancer-wisconsin/breast-cancer-wisconsin.data" rel="nofollow">data set</a> I am using is Breast Cancer Data from Wisconsin hospitals.</p> | 74,397 | [
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<p>What are some typical approaches to handling incomplete data in the Kalman Filter? I'm talking about the situation where some elements of the observed vector $y_t$ are missing, distinct from the case where an entire observed vector $y_t$ is missed. Another way of thinking about this would be that the dimension $p$ of the observed vector is different for each time point.</p>
<p>To explain my context a bit further, the observations are estimated parameters from a logistic regression performed at each time point. While each logistic regression includes the same covariates, sometimes the estimates are undefined due to collinearities in the data for that time point.</p> | 4,246 | [
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<p>I want to evaluate a Gaussian PDF that does not exist because the determinant of sigma is either approx 0 or -inf (depends on parameters), and the condition number is of order 10^20.</p>
<p>I need this density because I want to find the posterior mixing coefficients in an inverse problem with a Gaussian mixture prior. The formula for the posterior mixing coefficients looks like this:</p>
<p>$\lambda_{(x|y)}^ j= \frac{\pi_j N(\mathbf{y}; \mu_{yj}, \Sigma_{yj})}{\sum_{k=1}^n\pi_k N(\mathbf{y}; \mu_{yk}, \Sigma_{yk})}$</p>
<p>Does anyone have a tip on how I can approximate these coefficients/densities?</p>
<p>I do not have a sample from (x|y). But I know $\mu_{x|y}^i$ and $\Sigma_{x|y}^i$.</p>
<p>The full form of the posterior is</p>
<p>$f(x|y)=\sum \lambda_{(x|y)}^ j N_m( \mathbf{\boldsymbol \mu_{x|y}^j,\boldsymbol \Sigma_{x| y}^j})$</p>
<p>Where</p>
<p>$\boldsymbol \mu_{x|y}^i =\boldsymbol \mu_{x}^i + \Gamma_{xy}^i (\Sigma_y^i )^{-1}(y-\mu_{y}^i)$</p>
<p>$\boldsymbol \Sigma_{x|y}^i =\boldsymbol \Sigma_{x}^i + \Gamma_{xy}^i (\Sigma_y^i )^{-1}(\Gamma_{xy}^i)^T$</p> | 74,398 | [
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<p>I want to estimate the survival function $S(t_0)=1-F(t_0)$, at a specific point $t_0$ where $F$ is a continuous cumulative distribution function, based on an uncensored sample $x_1,..,x_n$ and a nonparametric estimator. So far, I am considering two estimators $\hat S(t_0)=1-\hat F(t_0)$, where $\hat F$ is</p>
<p>(1) Kernel density estimator</p>
<p>(2) The empirical CDF.</p>
<p>What other methods could I consider using?</p> | 37,268 | [
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<p>I am having the following problem. I have a group of words which are all permutations of each other. On a random dataset of DNA sequences I can calculate the distribution of the number of word occurrences. On my biological dataset I would like to compare the word occurrences with the distribution of those on random data. My final result should a list of permutations that are significantly enriched. (the tail of the real-data distribution versus the random data distribution will be slightly different)</p>
<p>Given the random distribution (type unknown, basically a positive valued function with extremum around 1 and a long tail on the right side) I would be able to calculate p-values for any word in the group, but I think this might imply that I need a multiple hypothesis correction, although you can state that the hypotheses are clearly dependent.</p>
<p>We can cast this problem into a dice problem. Suppose we have a die with N sides and we roll it a large number T of times. Next we try to assess whether some of the sides occur much more then expected. This leads to N hypotheses where we check if the occurrence of side N is much higher than expected. These hypothesis are not independent since the occurrence of one side influences the other.
In this case is it possible to state that the die has a bias towards a subset of sides?
If it's not possible use the p-values, (due to multiple hyp corr), would it make sense to use the p-value simply as a ranking mechanism?</p>
<p>Thanks a lot for any help provided!</p> | 37,269 | [
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<p>I've successfully implemented a RBM and trained it with CD-k, but I now have some issues doing the same with Convolutional RBM. </p>
<p>The visible and hidden biases gradients are easy to compute: v1 - v2 for visible bias and h1_k - h2_k for the hidden bias. But I failed to see how to compute the gradients for the weights. The weights are NH x NH matrices. In RBM it was easy since the weight matrix was NH * NV dimension and with multiplication of h1 * v1 - h2 * v2 we could have the gradients, but the dimension are not equivalent in CRBM. </p>
<p>What is the rule to compute the gradients ? Or is there a different way to update the weights ? It's probably a convolution, but I fail to see how...</p> | 74,399 | [
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<p>I have a series of descriptors, some continuous, some discrete and an output variable which is dichotomous.</p>
<p>I have several parameters, but for the sake of simplicity let's say my data look like:</p>
<pre><code> Sex | Age | Genotype | Dose 1 | Dose 2 | Outcome
------|-------|------------|---------|-----------|------------
M | 32 | AABB | 150 | 30 | YES
F | 65 | aaBb | 110 | 30 | YES
M | 42 | AaBb | 200 | 50 | NO
...
</code></pre>
<p>I would like to make a predictive model to determine the optimal combination of <code>Dose 1</code> and <code>Dose 2</code> to have a desired outcome.</p>
<p>So my question is, if I have a new male subject of given genotype and age, what is the best combination of doses that will give a positive outcome with the highest probability? Or, to see things the other way around, given the other parameters, what are the odds of having a positive outcome with a given set of doses?</p>
<p>I thought I could use R to generate a linear model with <code>glm</code>, and then use <code>predict</code> to predict the outcome. However, I never really dealt with this type of problems before and I am a bit confused on how to use these functions and interpret their results. Also, I am not sure if this is the correct way to deal with the problem.</p>
<p>For instance, let's generate some random data:</p>
<pre><code>set.seed(12345)
num.obs <- 50
sex <- sample(c("M", "F"), num.obs, replace=T)
age <- sample(20:80, num.obs, replace=T)
genotype <- sample(LETTERS[1:8], num.obs, replace=T)
dose.1 <- sample(100:200, num.obs, replace=T)
dose.2 <- sample(30:70, num.obs, replace=T)
outcome <- sample(0:1, num.obs, replace=T)
data <- data.frame(sex=sex, age=age, genotype=genotype,
dose.1=dose.1, dose.2=dose.2, outcome=outcome)
</code></pre>
<p>Which gives 50 observation such as</p>
<pre><code>> head(data)
sex age genotype dose.1 dose.2 outcome
1 F 78 C 183 54 0
2 F 70 E 156 66 1
3 F 39 H 180 35 0
4 F 32 E 135 51 0
5 M 64 E 121 57 1
6 M 50 H 179 61 1
</code></pre>
<p>Now, I generate a model with</p>
<pre><code>model <- glm(outcome ~ sex + age + genotype + dose.1 + dose.2,
data=data, family="binomial")
</code></pre>
<p>First question: without any <em>a priori</em> knowledge of the interactions between the descriptors, <strong>how do I choose the correct formula</strong>? Should I try various interactions and see which models gives the best fit e.g. looking at residual deviance or AIC? Are there functions to do this for me or should I try all of the combinations manually? </p>
<p>OK, let's say I found the model is good, now I use <code>predict</code></p>
<pre><code>new.data <- list(sex=factor("M", levels=c("M", "F")), age=35,
genotype=factor("C", levels=LETTERS[1:8]),
dose.1=150, dose.2=30)
outcome <- predict(model, new.data, se=T)
</code></pre>
<p>Which gives:</p>
<pre><code>$fit
1
-2.774538
$se.fit
[1] 1.492594
$residual.scale
[1] 1
</code></pre>
<p>So... what do I do with this? <code>?predict.glm</code> says <code>$fit</code> is the prediction but obviously that is not a yes/no type of prediction... what I would ideally need is something on the lines of "89% YES / 11% NO".</p>
<p><strong>How do I interpret the result of <code>predict</code> and how would I go about having the type of result I want?</strong></p>
<p>Finally, <strong>are there functions to explore the parameter space so that I get a graph with the outcome in the dose1 vs dose2 space?</strong></p>
<p>EDIT: just to clarify: I do not have a specific reason to use a generalized linear model, it is just something that came to my mind as a possible solution. If anyone can think of other ways to solve this type of problem I would gladly welcome their suggestion!</p> | 49,250 | [
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<p>I have dependent variable matrix $Y_{i,j}$ and feature matrix $X_{i,k}$. My objective is to predict each element of the vector $[y_{i,0},...,y_{i,J}]$ by using new observations of the features, $[x_{i,0},...,x_{i,K}]$, i.e. to estimate $Y_{i,j} = f(X_{i,k})$.</p>
<p>In my problem, all the dependent variables are closely related, and any given dependent variable is explained by a large number of the available features (so for any dependent variable, there aren't many redundant features). In addition, interactions between the features are likely to be important in predicting the response vector.</p>
<p>An example of the type of problem I'm facing would be if each dependent variable was the expected life span of various species of fish, and the features were water temperature, salt/fresh water binary, lake/ocean binary, binary for existence of particular predators, etc etc. We want a fast model that can mix these to predict each element of the vector.</p>
<p>There are many ways to tackle this problem. What I am interested in is what multivariate models exist that are both fast at run time (it's for a real time application) and can handle interactions. I am also looking for a model that is automated; automated variable selection and automated interaction selection. I am also looking an approach where the multivariate version generalizes better than simply fitting separate models for each $y_{.,j}$ (which could be achieved with splines, glms, etc).</p>
<p>The obvious answer is multivariate forests or multi-output neural nets, and so on. I'm looking for a faster approach; i.e. a model that doesn't have any sort of "deep" or iterative architecture that is computationally burdensome for large data (a multivariate model that's additive in transforms of features or feature interactions is ideal). A good answer is multivariate splines, looking for others.</p> | 37,270 | [
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<p>I don't really know what's possible, and would like a pointer in the right direction. Many thanks.</p>
<p>I have measurements of time and position which could be anything from a person walking, a vehicle on a road, or in a car park, or a printer in an office. I need to work out journey times for vehicles between two points. It may be they take a meandering route, or even take days to get from A to B. Or they may be a pedestrian, or an emergency service vehicle.</p>
<p>I want the estimated journey time for a normal vehicle along the main route.</p>
<p>The detections are whenever someone is near enough to the detector, which has a particular radius. Sometimes there are very few detections, which probably means the road is empty and journey time would be good, although it could indicate the road is closed, and the journey time would be terrible. Or there could be lots of detections showing traffic not moving, and it could be queueing to turn off the road, but other vehicles are travelling at normal speeds.</p>
<p>The plots look like random noise.</p>
<p>EDIT:</p>
<p>At the moment I am looking at two methods:</p>
<ol>
<li>Use the interquartile range to discard outliers</li>
<li>Use a Kalman filter.</li>
</ol>
<p>I think the filter is the wrong way to go as I don't have a model for the journey time other than from moment to moment I don't expect it to change much.</p> | 49,251 | [
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<p>I injected three different cRNAs constructs (construct A,B and C) into oocytes and the oocytes expressed the respective cRNAs as respective ion channels (ion channel A,B and C). My aim was to test whether these ion channels function similarly or not. Then, I measured the concentration-response relationships to determine EC50 for each ion channel (x-axis is cAMP concentration (µM) , y-axis is I/Imax;
I is the the actual current amplitude
Imax is the maximum current amplitude at saturating cAMP</p>
<p>I got the results EC50 for ion channel A = 59.01
ion channel B = 56.17
ion channel C = 61.24
which test should I use to find out whether they are statistically different ? In theory, they should not be different. Can I use one way ANOVA.</p>
<p>Thanks in advance,</p> | 74,400 | [
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<p>I'm stuck with this problem on quality control and p-charts. </p>
<p>There is a given process, monitored periodically taking samples of 100 items each and counting the number that are defective. In the last 50 samples we found a total of 622 defective items.</p>
<p>I'm OK for the 3-sigma limits of the p-chart and in particular: $$\bar{p} = \frac{622}{50*100} = 0.1244$$
$$\text{UCL} = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}} = 0.1244+ 3\sqrt{\frac{0.1244 (1-0.1244)}{100}} = 0.2234$$
$$\text{LCL} = \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}} = 0.1244+ 3\sqrt{\frac{0.1244 (1-0.1244)}{100}} = 0.0254$$
where $\bar{p}$ represents the estimated proportion of defects from the process, used to calculate the control limits.</p>
<p>Now, how am I supposed to find the <em>average number of samples</em> before having a point falling outside the control limits (signaling an of out-of-control), given an actual shift of the estimated proportion of defects from $\bar{p}$ to $p' = \bar{p} + 0.06 = 0.1844$ due, for example, to something new that is now influencing the process?</p>
<p>I think this has to do with the average run length $\text{ARL}_1 = \frac{1}{1-\beta}$, but I'm not quite sure on the steps leading to its value.</p> | 74,401 | [
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<p><strong>Brief intro:</strong> I'm not really sure how to appropriately treat the dependent variables in a set of hierarchical linear models that I'm trying to run. In my models, Level 1 units are children and Level 2 units are schools (we don't have classroom info so there is no nesting level for this).</p>
<p><strong>Background:</strong> There are five sets of models, one for each dependent variable. The DVs are all behavioral or cognitive outcome measures that are measured at Level 1. I'm using the HLM 7 software, but I could just as easily use Mplus or R. Each of the five DVs were assessed in the same way. Let's take depression as an example of one: this construct had about 6-8 items (presented to the children as scenarios), and for each item there were 2 response options that could be classified as "the depression options," and 2 response options that could be classified as "not the depression options." The 2 items in each set are treated as equivalent in terms of measurement -- that is, no ordinal ranking from least-to-most depressed. The final form of the depression outcome (as with the other 4 outcomes) is a proportion, which can be interpreted as the proportion of items/scenarios to which the child responded in a way indicative of being depressed. Personally, I hate that the assessment was done in this way, but there's nothing I can do about it now.</p>
<p><strong>Problem:</strong> However, I may be able to get permission to score the assessments in a different way, provided I can get access to the item-level data. Right now I only have access to the final proportions. Obviously the default in HLM and in other software is to treat the outcomes as normally distributed. The only distribution appropriate for the [0,1] interval that I know of is the beta, and I do not know of any R packages to implement HLM for beta-distributed variables, nor do I think Mplus or HLM can handle this. I tried using the logit transformation to put the proportions back onto a scale that I could theoretically treat as normal, but with only 6-8 items per DV I still get something that is rather discrete. Not to mention that the 0 and 1 responses, even when converted first to .0001 and .9999 before logit transforming, give me these extreme values that are quite far away from the other transformed values. I've attached histograms to illustrate what I mean.</p>
<p><strong>Question:</strong> What are some appropriate ways to treat an outcome variable on the [0,1] interval that takes on only a small handful of observed values when I want to run a set of hierarchical linear models? I'd REALLY like to use the HLM 7 software for this (the P.I. prefers it), but if the best way to treat these types of variables can only be implemented in R then I'm comfortable with that option as well.</p>
<p><img src="http://i.stack.imgur.com/K6YT1.png" alt="Distribution of depression on original scale"></p>
<p><img src="http://i.stack.imgur.com/R2hh9.png" alt="Distribution of depression after logit transformation"></p> | 74,402 | [
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<p>I have a posterior distribution over a set of parameters denoted by $\theta = \{w, \phi, \lambda\}$ and the posterior is the joint distribution given the observed data i.e. $P(\theta | D)$. Now, this posterior is a multivariate normal distribution (due to the approximate way it is estimated).</p>
<p>Now I want to find out the marginal distributions from this joint. So, for example, $P(\lambda)$ could be found my marginalizing over $w$ and $\phi$. Is there an easy way to do this for such MVN distributions? Also, can I find the mode for this marginal distribution easily?</p> | 74,403 | [
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<p>I try to fit a model of the following form
$$
Y = (\beta X - Z)^+ + \epsilon,
$$
where $Y,Z \ge 0$ and $X,Y,Z \in \mathbb{R}$ and $(x)^+ = \max(x,0)$.
Note that $Y,X$ and $Z$ come from a sample, especially $Z$ is not a constant.
I did this in R in the following way:
<pre><code>
positivePart<- function(x){
x[x<0]<-0
return(x)
}
my.optim = nls(y~ positivePart(beta*x-z) , start = list(beta=5))
</pre></code>
I am getting good results, this is not the problem.</p>
<p>My question: is there a name, a class for this kind of model? If so, which R function is the natural candidate to solve this problem, ie. to formulate the model in R? Thank you!</p> | 49,070 | [
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<p><em><strong>Task:</strong></em><br>
I have e.g. 1 control group and 7 experimental groups which are treated differently. All groups have a dependent <strong>binomial</strong> variable: 1 = success, 0 = no success. </p>
<p>I want to conduct <strong>sequential tests</strong>, which version is "performing" better, between the control group and the experimental groups and stop the tests, when one of the experimental group is better that the control group (or it is proven, that the Null hypothesis is true). </p>
<p><strong><em>Problem:</em></strong><br>
So in my opinion <strong>two problems</strong> arise: <strong>alpha I error inflation</strong> by <strong>sequential testing</strong> and alpha I error inflation by testing <strong>multiple hypothesis</strong> (per timepoint)</p>
<p><strong><em>My research so far:</em></strong><br>
I found <strong>group sequential methods</strong>, <strong>Alpha-Spending approaches</strong> and <strong>SPRT</strong> to solve the sequential testing problem. And I read the <strong>sequential updating approach of Bayesian Methods</strong> in the book of Kruschke.</p>
<p>For the multiple hypothesis testing (per time), there are some techniques which control the overall alpha error rate (per time), like <strong>Bonferroni</strong>, <strong>Hochberg's step-up procedure</strong> and others.</p>
<p>But I <strong>haven't found how to combine the methods</strong>. Nevertheless, group sequential methods as well as Alpha-Spending approaches seem to be a bit inflexible. SPRT und Bayesian Updating seem to fit best for the need to have a small sample number, being able to test after every participant and not to have a predefined endpoint of the study.</p>
<p><strong>Bayesian updating</strong> (with ROPE), as far a I know, does <strong>not</strong> have something like a <strong>predefinable alpha error rate</strong> (and no correction for multiple variables?). SPRT also does not support multiple variables. Please correct me if I am wrong. Do you have any suggestions where to dig deeper?</p>
<p>Best Regards
Andreas</p> | 49,072 | [
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<p>I have a short question regarding interaction variables:</p>
<ol>
<li><p>In a logit regression with 2 independent dichotomous variables (A and B), both variables are significant. By including the interaction (AxB) in the regression, B becomes insignificant.
Is the high (0.6) and significant correlation between (AxB) and B the reason for this?
And in conclusion, has B a significant influence on the dependent variable or not?</p></li>
<li><p>By including demographics and their interaction variables, all initial significant variables become insignificant (enter method). Is it smarter to simply use the stepwise forward method? By doing so, the initial significant variables remain significant. However, all new added varialbes are not included in the model.</p></li>
</ol>
<p>Thanks a lot in advance for your help.</p> | 74,404 | [
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<p>I am performing some regression task, where I try to discover the underlying multivariate Gaussians from a set of $n$, $p$-dimensional vectors. For example, given a split of the set into $S_i$ and $S_j$ I will calculate the sample means and covariance matrices (${\sum}_{i,j}$)and decide which is the best split based on the information gain(defined by the entropy ($log(det({\sum}_{i,j}))$). And then we recurse on the subsets $S_i$ and $S_j$.</p>
<p>I am trying to define some stopping criteria for this algorithm, which basically should be that when the variance of the distribution is small enough (not sure how to define this threshold) stop to avoid over-fitting to the training data. </p>
<p>So, my questions are:</p>
<p>1) How can I get a measure of overall variance, is it just $trace({\sum})$?</p>
<p>2) How can I choose a suitable threshold?</p>
<p>Thanks</p> | 49,075 | [
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] |
<p>I have run an ACF on a function in R and it returns:</p>
<pre><code>Autocorrelations of series ‘sine.curve’, by lag
</code></pre>
<blockquote>
<pre><code> 0 1 2 3 4 5 6 7 8 9 10 11
1.000 0.257 -0.712 -0.677 0.249 0.825 0.243 -0.632 -0.633 0.242 0.731 0.246
</code></pre>
</blockquote>
<p>My question is how can I find out where this function crosses the x axis or how to predict at what lag the correlation is equal to zero using these values</p> | 37,629 | [
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] |
<pre><code>import sympy
sympy.init_printing()
x, y, mu, sigma, density1, density2 = sympy.symbols('x y mu sigma density1 density2')
eq1 = sympy.Eq(density1, 1/(sympy.sqrt(2*sympy.pi)*sigma)
*sympy.exp(-(x-mu)**2/(2*sigma**2))) # normal
eq2 = sympy.Eq(y, sympy.exp(x)) # substitution
eq3 = sympy.Eq(density2, 1/(y*sympy.sqrt(2*sympy.pi)*sigma)
*sympy.exp(-(sympy.ln(y)-mu)**2/(2*sigma**2))) # lognormal
[eq1, eq2, eq3]
</code></pre>
<p>Output:</p>
<pre><code>⎡ 2 2⎤
⎢ -(-μ + x) -(-μ + log(y)) ⎥
⎢ ────────── ───────────────⎥
⎢ 2 2 ⎥
⎢ ___ 2⋅σ ___ 2⋅σ ⎥
⎢ ╲╱ 2 ⋅ℯ x ╲╱ 2 ⋅ℯ ⎥
⎢density₁ = ─────────────────, y = ℯ , density₂ = ──────────────────────⎥
⎢ ___ ___ ⎥
⎣ 2⋅╲╱ π ⋅σ 2⋅╲╱ π ⋅σ⋅y ⎦
</code></pre>
<p>How can I tell SymPy take the first (normal) density, apply the x to y substitution and output the desired second (lognormal) density?</p> | 74,405 | [
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<p>So I often have the situation where I want to compare the runtime of two computer processes. Even when measuring only user+system time (as opposed to wall clock time) there is a random element in the measured time. I'm going to suppose that this randomness is poisson distributed (task switches and interrupts from random events on the computer system). </p>
<p>My hypothesis is that a single program with identical input will run in some amount of time which is composed of a constant amount of time to execute the instructions and some random amount of time generated by one or more poisson process. </p>
<p>So let's simplify for the moment and assume that there's one poisson process and each time it happens it accounts for the same amount of time. We don't know lambda, nor t(lambda), the time associated with it. I can, however, run the same program on the same input many many times. </p>
<p>How can I estimate what the constant part of the run time is?
How can I estimate lambda and t(lambda)?</p>
<p>The deeper issue is...</p>
<p>If I have two programs that do the same thing and I want to estimate the probability that one runs faster than the other how would I do this? It seems that blindly applying the normal distribution could be problematic. The random part is (I believe) poisson; but before applying the poisson distribution I'd have to factor out the constant portion. </p> | 37,275 | [
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<p>In some real-life problems such as authentication, we only have training data for one label (x is authenticated) while the other label doesn't have any data or only few entries (x is an imposter).</p>
<p>What kind of changes should we do in order to adjust a classifier to deal with a label targeted for other/unknown entries?</p> | 74,406 | [
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<p>I have a two stage model where coefficients of model 1 become the $y$-vector for model 2. I have standard errors for those coefficients, and I want to weigh the observations in model 2 according to the standard error that the corresponding coefficient had in model 1. I believe the correct way to do this is weigh the observations in 2 by 1/SE, where SE is the standard error of the corresponding coefficient in model 1. </p>
<p>Now, the data for model 1 actually gets artificially created by me. I therefore know that coefficients with a SE of $\geq X$ are completely useless (I know this given my setup). Now, in model 2, giving those <em>useless</em> observations a weight of $1/\text{SE}$, with $\text{SE}\geq X$, seems like giving them too much weight, since their actual weight should be 0</p>
<p>What is the best method to weigh observations in such a case?</p> | 74,407 | [
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