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<p>Let $\mathbf{X}=(X_1,X_2,\ldots,X_{2k})$ be a sample from normal distribution $\mathcal{N}(0,1)$.</p>
<p>Prove that distribution of sample median (for even sample) is symmetric around 0.</p>
<p>$me(\mathbf{X}) = \frac{X_{k:2k}+X_{(k+1):2k}}{2}$, where $X_{i:2k}$ is $i$-th order statistic.</p>
<p>I was trying to use formula $f_{X_{k:n}}(x) =\frac{n!}{(k-1)!(n-k)!}[F_X(x)]^{k-1}[1-F_X(x)]^{n-k} f_X(x)$, but without any success (maybe there is a way to do it differently than finding $me(\mathbf{X})$ distribution?</p> | 74,468 | [
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<p>In a multiple regression analysis (with 4 continuous predictors and 2 categorical factors), we mean centered the data (for each continuous variable) due to issues of multicollinearity when the interaction terms are included. </p>
<p>My question is whether I can center the response variable too. </p>
<p>More specifically, the response variable and the 4 continuous predictors are all averaged survey responses (using scales of 1 to 5). I originally thought that if I were centering the explanatory variables, I might as well center the response.. but I realize that most references to centering seem to only apply to the predictor variables. </p>
<p>Any help is much appreciated.</p>
<p><strong>Update</strong>: as I mentioned in an earlier comment, I questioned the validity of centering my response variable due to having different ANOVA results using centered versus non-centered response. My interpretation is that the linear model for a non-centered response (using the lm() in R) uses the mean of the 'reference level' of my two factors for computing the intercept. When I centered the response variable, it's being subtracted from the grand mean of this variable, rather than the mean of the reference level. Now I've verified that by 'centering' using the reference level mean and it does indeed yield identical results for p-values as the non-centered response model. I hope I am interpreting this issue correctly. IF someone could further confirm/clarify this I would really appreciate that. </p> | 74,469 | [
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<p>I am studying disturbance caused by ship traffic to a small seabird. I observed focal animals for a set amount of time and record whether or not they fly from the water during the observation. This particular bird does not fly at high probabilities when not disturbed (about 10% of the time). Post hoc, I have appended the distance to the nearest ship to every observation (ships of interest had GPS locators recording a point every 5 sec).</p>
<p>I have plotted the cumulative distribution function for ALL observations and for observations where the bird flew from the water as a function of distance to nearest ship. As expected, the majority of the observations in which the bird flew were observed when the ship was close.</p>
<p><img src="http://i.stack.imgur.com/0X3MY.jpg" alt="ecdf plot, fly, did not fly, all obs"></p>
<p>Can I use the Kolmogorov-Smirnov test to test if there is a statistical difference in the distribution of the flight observations and the total observations? My thought is if these two distributions are different it would suggest that ship distance has an influence on flight. I worry as these distribution functions are not independent as the flight observations are a subset of the total observations.</p>
<p>Thoughts? </p>
<p>Having read a bit further on this site I think I can test the distribution of the observations in which flight occurred (F) against the distribution of the observations in which it did not (NF) as these are independent. If these distributions are the same F=NF, then we can assume that the distribution of (F) and (TOT = all observations) are the same as we know that the distribution of (F) is equal to itself and (F)+(T) = (TOT). Right?</p>
<p><strong>UPDATE: 2/12/14</strong></p>
<p>Following the suggestions from @Scortchi I investigated the relationship of the incidence of flight vs distance to the nearest ship in a logistic regression framework. There was slight relationship present (negative slope) but the p-value was non-significant suggesting that the true slope could be zero. Based on the desciptive statistics (including the ecdf plots) I suspected that the effect of close ships was being drowned out by the many observations when the ship was not affecting behavior. I then used the R package segmented (<a href="http://cran.r-project.org/web/packages/segmented/segmented.pdf" rel="nofollow">http://cran.r-project.org/web/packages/segmented/segmented.pdf</a>) to try and find a break-point in the model. The program found that breaking the data at 2.6 km from the ship and fitting two separate coefficients was better than the single coefficient model. The coefficient for the slope of the close ship approaches was negative and suggests that ships affect the flight response until about 2.6 km (p-value < 0.001). The coefficient for the second slope was slightly positive but the p-value was non-significant at the 0.05 alpha level (p-value = 0.11). So in summary, the segmented regression line was able to detect a threshold difference at which flight probability increases. The estimate for the probability of flight when the ship is further than 2.6 km is 0.11. Fittingly, I observed 79 birds in when no ships were even in the study bay (> 100 km away) and 9 of these individuals flew from the water during the observation (0.113 probability).</p>
<p>Thanks for all the suggestions. I hope this question along with the suggestions and answers helps others. </p> | 37,358 | [
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<p>How do you check ergodicity of a wide-sense stationary stochastic processes from its
sample path(s)? </p>
<p>Can we check ergodicity from a single sample path? Or do we need multiple sample paths? </p>
<p>One motivation of checking ergodicity is in time series to ensure that you could safely use the average of a sample path over time as estimate for the population mean?</p>
<p>Thanks!</p> | 74,470 | [
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<p>A new treatment for strokes is put on trial. There are two equal size groups,one group is given a drug, and one a placebo. The 95% confidence interval for the difference between the two proportion of subjects having strokes was (0.07,0.12). What can we infer:</p>
<p>1) not enough info to infer</p>
<p>2)at most 12% of the people had strokes</p>
<p>3) at least 4% had strokes</p>
<p>I'm not sure on this. I think it might be 1). Can anyone help out?</p>
<p>Thanks</p> | 74,471 | [
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<p>Is it possible to apply hurdle models (like the Craggit, probit and truncated models) to panel data, preferably with fixed effects to control for unobserved heterogeneity? </p>
<p>In Stata, the user-written command <code>craggit</code> only allows to use pooled panel data, but not to control for unobserved heterogeneity...
In general, the hurdle model is quite simple to estimate: probit model on the binary outcome variable, truncated (or lognormal) on the (positive values of the) outcome variable.
Is it possible to just estimate a RE probit (or even FE logit) in a first step, and a fixed effects model in the second – and to actually call that a hurdle model?
Thanks.</p> | 37,360 | [
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<p>Asked a version of this question before but realized it needed some clarification.</p>
<p>I have a dataset with identical twin pairs and fraternal twin pairs. I want to examine the relationship between an outcome variable ("DEPENDENT") and a number of other variables ("INDEPENDENT"). However, I can't run a normal OLS regression because each twin's dependent variable is correlated with their co-twin's (for my purposes, this is a nuisance). </p>
<p>The way I have been dealing with this is to use SAS PROC MIXED and include a random intercept defined by twin cluster (FAMILYID). Here is my syntax: </p>
<pre><code>proc mixed method=ml covtest noclprint;
class FAMILYID;
model DEPENDENT = INDEPENDENT/solution;
random intercept/sub=FAMILYID type=un gcorr;
run;
</code></pre>
<p>When I do this, I get a covariance parameter estimate for the "FAMILYID" as well as a residual term. My understanding is that this decomposes the variability into "within-cluster" variability (residual estimate) and "between-cluster" variability (FAMILYID estimate). I get the intraclass correlation coefficient for the family clusters by computing: (FAMILYID estimate) divided by (FAMILYID estimate + residual estimate). </p>
<p>However! I realized that I probably have a heteroskedasticity issue, because fraternal twin pair members are likely to be markedly less similar to each other than identical twin pair members are. </p>
<p>I revised my syntax to this: </p>
<pre><code>proc mixed method=ml covtest noclprint;
class FAMILYID;
model DEPENDENT = INDEPENDENT/solution;
random intercept/sub=FAMILYID group=TWINTYPE type=un gcorr;
run;
</code></pre>
<p>Where TWINTYPE signifies whether the twin pair is identical or fraternal.
When I do this, I get one FAMILYID estimate for fraternal twins, one FAMILYID estimate for identical twins, and one residual term. So -- this means that I have told my model to make different "between-cluster" estimates for identical and fraternal twins -- but isn't it still disregarding twin type when it calculates the residual term? And if the residual term reflects the "within-cluster" variability, isn't it a problem that it doesn't take into consideration that there is likely to be LESS variability within an identical twin pair on the outcome variable and MORE variability within a fraternal twin pair? </p>
<p>So I guess my question is one of two things: </p>
<ol>
<li>is there syntax that would leave me with a between-cluster
estimate for identical twins and fraternal twins, and then a
residual term for identical twins and fraternal twins? or </li>
<li>Am I
fundamentally confused about how all of this works, and if so, what
am I missing?</li>
</ol>
<p>Help/clarification much much appreciated. </p> | 74,472 | [
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<p>Suppose that we have two data sets, <code>R</code> and <code>P</code>. <code>R</code> is larger than or equal to <code>P</code>. <code>R</code> can be negative or positive. <code>P</code> can be negative or positive. So in all cases <code>(R-P)</code> is positive or zero. the <code>R</code>and <code>P</code> are dollars and the scale of them is high likes <code>1,000,000,0</code> or <code>500,000,000</code>. Now I want create a criterion looks like :</p>
<pre><code>E = 1 - ( 1 / 1+(R-P) )
</code></pre>
<p>So I have <code>E</code> that is between <code>0</code> and <code>1</code>. I want <code>E</code> scale between <code>0</code> and <code>1</code>. But the problem is this. <code>(R-P)</code> is has high values,so all outputs of this criterion is near <code>1</code>. I revise this function, Now i have this :</p>
<pre><code>E = 1 - ( 1 / 1+ ( (R-P)/mean(R-P) ) )
</code></pre>
<p>Now because of <code>mean(R-P)</code> we have better scaling between <code>0</code> and <code>1</code>. This is distribution of <code>R-P</code> for more help : </p>
<p><img src="http://i.stack.imgur.com/2XOxM.jpg" alt="enter image description here"></p>
<p>After doing that I want classify data with this <code>E</code> . For example samples that have a <code>E</code> greater than <code>0.7</code> are in <code>class 1</code> and samples with <code>E</code> lower than <code>0.7</code> are in <code>class 2</code> (In a binary classification problem). For a better distribution in final version of this criterion i will remove outliers of <code>(R-P)</code> with this function to have more robust criterion :</p>
<pre><code>abs(X-mean(X)) >=2*std(X)
</code></pre>
<p><code>(X here is (R-P)</code></p>
<p>After removing outliers in <code>R-P</code> i will calculate <code>mean(R-P)</code>what is your recommendations for better transformation of this problem? I have a theoretical problem with using <code>mean</code> in this criterion so i need comments about this.</p>
<p><strong>PS.</strong> We should have this behavior in this criterion:</p>
<ul>
<li>if <code>P=constant</code> and <code>R=increase</code> >> <code>E=increase</code></li>
<li>if <code>R=constant</code> and <code>P=increase (towards negative)</code> >> <code>E=increase</code></li>
<li>if <code>P=constant</code> and <code>R=decrease</code> >> <code>E=decrease</code></li>
<li><p>if <code>R=constant</code> and <code>P=decrease(towards positive)</code> >> <code>E=decrease</code></p></li>
<li><p>a sample that have highest <code>(R-P)</code> should have highest <code>E</code> (<code>1</code> or near <code>1</code>)</p></li>
<li>a sample that have lowest <code>(R-P)</code> should have lowest <code>E</code> (<code>1</code> or near <code>1</code>)</li>
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<p>I am new to this area of machine learning. I am just walking myself through <code>UCB1</code> algorithm which seems to assume that the payoff can be learnt only for action that is taken. What I am curious about is if there is already analysis of the multi-armed bandit problem to the cases where the payoff can also be learnt for actions not taken (may be the term for this is "full reward vector"?), and that information used in subsequent steps. </p>
<p>I could modify <code>UCB1</code> algorithm to update payoff for actions not taken as well. But, I would like to know if there are any drawbacks/pitfalls for doing this. I haven't done closed-form analysis of the regret bounds when reward vector is fully available.</p>
<p>I don't know what the terminology is for such cases. So, I have a hard time finding work on such cases. I understand that when full reward vector is available, it is a supervised reinforcement learning problem. What I am curious about is if multi-armed bandit work can be used well in supervised reinforcement learning problem or not. And whether it makes sense to do this or not. I will very much appreciate pointers to work/paper in this area.</p> | 74,473 | [
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<p>Given $n$ samples from a (continuous) distribution X, the obvious thing to do is sort them, and distribute them equally across $[0,1]$ by taking $(x_{(k)}, (k-1/2)/n)$ as estimates of particular points on the CDF, and doing some sort of interpolation between points, as necessary.</p>
<p>Is this the "right" way to make this estimate? How do I get error bars for the estimated points? It doesn't seem like they'd necessarily be symmetric.</p> | 74,474 | [
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<p>I'm doing a meta-analysis of some studies that report results variously as odds ratios, hazards ratios or rate ratios (all with confidence intervals). Is there any way to combine these together/convert between them so that I can do a meta-analysis of all the studies?</p> | 49,202 | [
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<h3>Overview:</h3>
<p>I want to test if "emotional numbing" is a significantly better predictor of "lower intimate relationship functioning" than "reexperiencing" or "hyperarousal"</p>
<p>It was suggested to me to use stepwise regression to analyze the data. However, one of my dissertation committee members suggested Hotelling's T. </p>
<h3>My hypothesis:</h3>
<ul>
<li>Hypothesis 1: Emotional numbing will be a better predictor of lower relationship functioning than re-experiencing or hyperarousal symptoms. </li>
<li>Hypothesis 1a: Emotional numbing will be a better predictor of lower dyadic adjustment than re-experiencing or hyperarousal symptoms. </li>
<li>Hypothesis 1b: Emotional numbing will be a better predictor of intimacy than re-experiencing or hyperarousal symptoms. </li>
<li>Hypothesis 1c: Emotional numbing will be a better predictor of communication than re-experiencing or hyperarousal symptoms.</li>
</ul>
<h3>Questions:</h3>
<ul>
<li>Is Hotelling's T the most appropriate way to determine whether emotional numbing is a significantly better predictor than the other 2 variables?</li>
<li>Or is there is a better statistical test for this task?</li>
</ul> | 293 | [
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<p>I am looking for a thesis topic for my Applied Economics degree where I can use Data Mining/Machine Learning for some kind of prediction.</p>
<p>Do you guys have any good ideas about what I could do?</p>
<p>My first thought was credit scoring, but that seems a little boring considering the number of papers out there and the lack of available data.</p>
<p>I don't want to write about stock market forecasting because that's what I did for my mathematician thesis.</p>
<p>A good topic would be predicting some sort of an economical indicator but I don't really have any good ideas so far.</p>
<p>Thank you!</p> | 74,475 | [
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<p>Suppose that there is a Cricket match scheduled on Sunday, this weekend. We know that people do not arrive at the stadium at constant rate. Few hours before scheduled start of the match, people start arriving at very slow rate. This rate then, in general, keeps on increasing. A large number of people arrive just before start of the match.</p>
<p>I would like to know which distribution should I use to model this, so that I could tell <strong>What is the probability of arrival of x number of people in a particular time interval?</strong></p> | 37,366 | [
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<p>I fitted the Generalized Pareto distribution (GPD) using the <code>POT</code> package in <code>R</code>.</p>
<p>The fitted object provides shape and scale parameters, but not the location parameter. I need it to evaluate the quantile function with <code>qgpd(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, lambda = 0)</code>.</p>
<p>I am sure this is very basic question (maybe it is mean value, but from which values - all, or only that which are bigger than threshold?) but I am begginer in this area and help is highly appreciated.</p> | 74,476 | [
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<p>I have 4 latent variables each with 3-7 indicators. I created a composite for each variable by summing the data and then mean-centering (variables have some collinearity issues, highly related). Thus I have Var1, Var2, Var3, Var4.</p>
<p>I then wanted to create a master composite of all four variables. I thus summed all the individual indicators and mean-centered the total, for Var1234.</p>
<p>I'm trying to test the theory of complementarity in which these four variables are greater together than individually (i.e. var1 + var2 + var3 + var4 = 4 ... but Var1234 >4)</p>
<p>I ran two linear regressions.<br>
MODEL 1: All the individual indicators + Individual interaction terms + 4 way interaction term = DV
MODEL 2: Var1234 = DV</p>
<p>In the 2nd model Var1234 is significant, however in the 1st model the 4 way interaction term is insignificant. I believe this is a problem, but I am not sure. Is it related to the way I created my composite variables, or the fact that there is a 2nd order variable that is a composite of other composites? </p>
<p>Is there a way to explain why a 4 way interaction term is insignificant when the composite of those 4 terms IS significant?</p> | 74,477 | [
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<p>I have an experiment with two treatments. It is a split plot experiment, with the structure Block/Treatment1/Treatment2. Each treatment has 2 levels. The dependent variable is presence/absence data for functional speceis groups. I am analysing the data using mixed models via lme4 and afex.</p>
<p>The model structure is as follows:</p>
<pre><code>m <- lmer (DV ~ Treatment 1 * Treatment 2 + (1|Block/Treatment1),
family = binomial)
</code></pre>
<p>For graphical purposes however, I want to plot the treatment means for each of the 4 treatment combinations (Treatment 1 level A and Treatment 2 level A; Treatment 1 level B Treatment 2 level A; Treatment 1 level B and Treatment 2 level A; Treatment 1 level B and Treatment 2 level B). I couldn't find an intuitive way to extract these mean and standard error values in R for data with a binomial error distribution, thus I am calculating these values by hand. I appreciate that the standard errors I calculate will not take into account the random effect structure in my dataset, but I cannot find a way to calculate them to include this random effect structure (using a mcmc simulation in language R to create HPD confidence intervals only works for Gaussian data).</p>
<p>As such, I want to calculate treatment means for the proportion of a particular functional group in a particular treatment combination, and then standard errors of these 4 treatment means.</p>
<p>Consider that the proportions I have for a particular treatment are:</p>
<pre><code>Plot 1: 24/65
Plot 2 26/64
Plot 3: 25/65
Plot 4: 22/62
Plot 5: 30/66
Plot 6: 29/65
</code></pre>
<p>I understand that to calculate the average proportion I need to sum the total number of hits (sum of all the numerators above) and divide by the total sum of hits and misses (sum of all the denominators above). This gives me 0.40.</p>
<p>My question then is, how does one calculate the standard error of the resulting proportion?</p>
<p>Calling the proportion of hits p and misses q, I have the following formula for the standard error of a proportion:</p>
<pre><code>sqrt((p*q)/n)
</code></pre>
<p>Is this correct? In addition, what is n? Is n the number of proportions that make up the mean proportion (in this case 6) or the total number of hits and misses (i.e. sum of all the denominators above, which here is 387)?</p>
<p>I'm sorry that this is a simple question, I just cannot find an answer anywhere! Also i'm sorry if this is not the correct forum to post this in...</p>
<p>Many thanks.</p> | 74,478 | [
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<p>I'm trying to optimise model parameters for times series data using k-fold cross-validation. I'm using the forward chaining strategy described here:</p>
<p><a href="http://stats.stackexchange.com/questions/14099/using-k-fold-cross-validation-for-time-series-model-selection">Using k-fold cross-validation for time-series model selection</a></p>
<p>i.e.</p>
<p>fold 1 : optimise [1], test [2]</p>
<p>fold 2 : optimise [1 2], test [3]</p>
<p>fold 3 : optimise [1 2 3], test [4]</p>
<p>fold 4 : optimise [1 2 3 4], test [5]</p>
<p>fold 5 : optimise [1 2 3 4 5], test [6]</p>
<p>Then averaging the resulting optimised parameters from all five folds to give the final model parameters. My question is: does this not put an overly strong weighting on fold 1 which would be included in every optimisation run? Is there a way to avoid this, or a better cross-validation strategy? </p> | 74,479 | [
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<p>Let's say I'm computing some model parameters my minimizing the sum squared residuals, and I'm assuming my errors are Gaussian. My model produces analytical derivatives, so the optimizer does not need to use finite differences. Once the fit is complete, I want to calculate standard errors of the fitted parameters.</p>
<p>Generally, in this situation, the Hessian of the error function is taken to be related to the covariance matrix by:
$$ \sigma^2 H^{-1} = C $$
where $\sigma^2$ is the variance of the residuals.</p>
<p>When no analytical derivatives of the error are available, it is typically impractical to compute the Hessian, so $J^TJ$ is taken as a good approximation.</p>
<p>However, in my case, I've got an analytical J, so it's relatively cheap for me to compute H by finite differencing J.</p>
<p>So, my question is this: Would it be more accurate to approximate H using my exact J and applying the above approximation, or to approximate H by finite differencing J?</p> | 74,480 | [
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<p>My study design involves a control and 2 test groups plus some covariates. Each group consists of around 20 observations. In total I look at around 1,000 variables.</p>
<p>I created a linear model using the <code>lm()</code> function in R including 2 covariates. After that I thought I would include another covariate because doing a PCA plot earlier showed a slight effect on that covariate. However, after adding this covariate to the model 50% of the significant hits are now different. I was actually assuming that it would pretty much identical as the effect was hardly seen in the PCA.</p>
<p>Could it be that I have overfitted the model? Or is the effect simple just not shown in the PCA plot but is there?</p>
<p>I just compared the two models using <code>anova(lm1, lm2)</code> and the p-value is significant which I think means that the third covariate adds significant information to the model?</p>
<pre><code>lm1 <- lm(var ~ factor_of_interest + cov1 + cov2)
lm2 <- lm(var ~ factor_of_interest + cov1 + cov2 + cov3)
anova(lm1, lm2)
</code></pre> | 49,662 | [
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<p>I have a list of records with fields that can be seen as two sets of variables: dependent and independent. There are say 100 fields in each record and we consider 10 of them as dependent variables and 90 as independent. </p>
<p>What are the methods to find all the correlations that exist between the two sets?</p>
<p>(At introductory statistics course that I've taken we have been dealing with simple correlations between two variables but the task like this was not covered.)</p> | 49,217 | [
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<p>Please first have a look at the following little problem:</p>
<blockquote>
<p>There are two indistinguishable light bulbs A and B. A flashes red
light with prob .8 and blue with prob .2; B red with .2 and blue .8.
Now with .5 prob you are presented with either A or B. You're supposed
to observe its flash color to make a best guess (maximizing
probability of correct guessing) which bulb it is. Before you start to
make observations, however, you must decide how many times you want to
observe it (say n times, then you observe it flashing n times and make
your guess). Suppose flashes are independent.</p>
</blockquote>
<p>Intuitively, one would think the more observations one makes, the better one's chances will be. Curiously though, it is easy calculation to show that n=2 doesn't improve upon n=1, and n=4 doesn't improve upon n=3. I didn't go further but I speculate n=2k doesn't improve upon n=2k-1. I'm not able to prove it for the general case. But is it true? If so, how can one intuitively understand the result? </p> | 37,370 | [
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<p>I'm reading the results of an individual patient data (IPD) meta-analysis. The outcomes of the study is expressed in standard deviations (SD's). Comparisons between groups are stated as 0,23 (95% CI 0.15-0.33), SD 0.34, etc. I can't grasp how to interpretate this. Can somebody help?</p> | 74,481 | [
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<p>Documentation states that R gbm with distribution = "adaboost" can be used for 0-1 classification problem. Consider the following code fragment:</p>
<pre><code>gbm_algorithm <- gbm(y ~ ., data = train_dataset, distribution = "adaboost", n.trees = 5000)
gbm_predicted <- predict(gbm_algorithm, test_dataset, n.trees = 5000)
</code></pre>
<p>It can be found in the documentation that predict.gbm </p>
<blockquote>
<p>Returns a vector of predictions. By default the predictions are on the scale of f(x).</p>
</blockquote>
<p>However the particular scale is not clear for the case of distribution = "adaboost". </p>
<p><strong>Could anyone help with the interpretation of predict.gbm return values and provide an idea of conversion to the 0-1 output?</strong></p> | 74,482 | [
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<p><em>I am not a statistician, so please excuse my lack of statistics knowledge/terminology.</em> </p>
<p>I have bunch of network nodes that I want to run cluster analysis on and identify clusters. So as far as I understand, I can follow the following steps to run a hierarchical agglomerative analysis (HAC):</p>
<ol>
<li>Identify variables</li>
<li>Define a distance function</li>
<li>Run the algorithm to join closer clusters and create one big cluster</li>
<li>Cut the dendrogram tree at the height that makes meaningful clusters based on context</li>
</ol>
<p>My question is related to the second step even though it is not yet clear for me how I am going to do the last step.</p>
<p>The data I want to analyse is bunch of computer network nodes which are down (not responding). I have the following information for each network node:</p>
<ul>
<li>location (longitude, latitude)</li>
<li>time node went down</li>
<li>network provider</li>
</ul>
<p>These are the most relevant information I believe I have to take into consideration for my clustering. Basically I want to cluster the nodes that went down probably because of the same reason in a region. </p>
<p>For example if bunch of nodes went down at about the same time and physically they are close to each other and they have the same provider, so probably they fall into the same cluster.</p>
<p>Now the question is how do I derive my distance function and include all these variables in it such that it would make sense? In other words <strong>what is the mechanism to derive a distance function based on multiple variables?</strong></p>
<p>Also as you notice the variables are of different types. So, in the distance function should I take care of this by considering <strong>Gower's coefficient of similarity</strong>? How?</p>
<p>Any examples or suggestion regarding whether I am in the right direction or not can be very helpful too.</p> | 37,373 | [
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<p>I have a techincal issue with Stata: I need to calculate the SSE of a regression model, but the automatic output just gives me RMSE; I need SSE of this constrained model because I have to test if this model is better than the completely-free model.</p>
<p>SPSS gives me the SSE value in the regression output, but I don't know how to calculate a constrained linear regression with this model.</p> | 40,405 | [
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<p>I have a large dataset from a survey that describes what web pages people use. So for each person I have a list of pages that they visit and how frequently they visit them. What methods can be used to cluster both the webpages (based on similar users who use them) and the users (who use similar webpages).</p>
<pre><code>Example>
User A visited web pages a, b, a, c
User B visited a, b, d, b
User C visited page e, e
</code></pre>
<p>Conclusion from that would be that pages <code>a</code>,<code>b</code>,<code>c</code>,<code>d</code> are somewhat similar and belong to the same cluster. The other cluster is page <code>e</code>. Also, users a,b are similar, the other user cluster is the one containing <code>User C</code>, who visits different pages.</p> | 74,483 | [
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<p>I would be interested in finding ways in R for efficiently updating a linear model when an observation or a predictor is added. biglm has an updating capability when adding observations, but my data are small enough to reside in memory (although I do have a large number of instances to update). There are ways to do this with bare hands, e.g., to update the QR factorization (see "Updating the QR Factorization and the Least Squares Problem", by Hammarling and Lucas), but I am hoping for an existing implementation.</p> | 74,484 | [
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<p>Suppose I observe <em>i.i.d.</em> $x_i \sim \mathcal{N}\left(\mu,\Sigma\right)$, and wish to test $H_0: A\ $<a href="http://en.wikipedia.org/wiki/Vectorization_%28mathematics%29#Half-vectorization">vech</a>$\left(\Sigma^{-1}\right) = a$ for a conformable matrix $A$ and vector $a$. Is there known work on this problem? </p>
<p>The obvious (to me) attempt would be via a likelihood ratio test, but it seems like maximizing the likelihood subject to the constraints of $H_0$ would require a <a href="http://en.wikipedia.org/wiki/Semidefinite_programming">SDP solver</a> and could be pretty hairy. </p> | 74,485 | [
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<p>I am confused how gaussian processes and kriging are related? Can anyone please give me some simple explanation. I tried to go through the wiki but I didn't get it</p> | 74,486 | [
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<p>I wanted to know what the <em>n.minobsinnode</em> parameter means in the GBM package. I read the manual, but it is not clear what it does.
Should that number be small or large to improve the results? </p> | 49,224 | [
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<p>I heard that the F-test is to advice you whether to use fixed effects or pooled OLS. However, I didn't find any details about it in books. Only in a very few studies. What is the hypothesis of the test? After I run my fixed model I got</p>
<pre><code>F test that all u_i=0: F(100, 389) = 2.98
Prob > F = 0.0000
</code></pre>
<p>As I understand <code>u_i=0</code> means I shall use fixed effect (am I right?), in general how accurate is this F-test compared to the Hausman test and LM test given the fact that we will end to knowing whether to use fixed effects or pooled OLS?</p> | 7,034 | [
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<p>I am reviewing my time series knowledge and looking for a document that has the commonly-used time series tests, what they are used for, how to use them, etc. e.g. Augmented Dickey–Fuller test, PACF tests, etc. </p>
<p>I found a wikipedia page of common statistical tests, but I am looking for a list of such that is specific for time series analysis. </p>
<p><a href="http://en.wikipedia.org/wiki/Statistical_hypothesis_testing#Common_test_statistics" rel="nofollow">http://en.wikipedia.org/wiki/Statistical_hypothesis_testing#Common_test_statistics</a></p>
<p>Thanks! </p> | 74,487 | [
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<p>I was reading <a href="http://en.wikipedia.org/wiki/Gaussian_process_regression" rel="nofollow">this wikipedia article</a> related to kriging . I didn't understand the part when it says that</p>
<p>Kriging computes the best linear unbiased estimator, $\hat Z (x_0)$, of $Z(x_0)$ such that kriging variance of is minimized with the unbiasedness condition. I didn't get the derivation and also how the variance is minimized. Any suggestions?</p>
<p>Specially, I didn't get the part where applies minimized subject to unbiasedness condition.</p>
<p>I think it should have been</p>
<p>E[Z'(x0)-Z(x0)] instead of E[Z'(x)-Z(x)] isn't it. ' is equivalent to hat in the wiki article. Also I didn't get how the kriging error is derived</p> | 74,488 | [
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<p>I have been looking for an alternative to Google Docs as a web based spreadsheet application as my employer's corporate firewall now blocks this application. I kicked around a few alternatives such as Zoho.com. I found the latter practically unusable. All its applications took a long time to load and really did not work well if at all most of the time. </p>
<p>When I came across EditGrid, I was delighted. It feels pretty much like Excel on a lot of counts. However, there is no apparent support besides a Help Center that is inactive and invariably does not address the problems I have. For instance, I have no idea how to include titles and legends within charts. I also don't see how one can create folders. </p>
<p>I thought maybe someone would be familiar with EditGrid and either being able to help me out directly or point me in a direction of a support group somewhere. Thanks in advance for any info. </p> | 74,489 | [
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<p>I am looking for methods which can be used to estimate the "OLS" measurement error model.</p>
<p>$$y_{i}=Y_{i}+e_{y,i}$$
$$x_{i}=X_{i}+e_{x,i}$$
$$Y_{i}=\alpha + \beta X_{i}$$</p>
<p>Where the errors are independent normal with unknown variances $\sigma_{y}^{2}$ and $\sigma_{x}^{2}$. "Standard" OLS won't work in this case.</p>
<p><a href="http://en.wikipedia.org/wiki/Measurement_error_model" rel="nofollow">Wikipedia</a> has some unappealing solutions - the two given force you to assume that either the "variance ratio" $\delta=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$ or the "reliability ratio" $\lambda=\frac{\sigma_{X}^{2}}{\sigma_{x}^{2}+\sigma_{X}^{2}}$ is known, where $\sigma_{X}^2$ is the variance of the true regressor $X_i$. I am not satisfied by this, because how can someone who doesn't know the variances know their ratio?</p>
<p>Anyways, are there any other solutions besides these two which don't require me to "know" anything about the parameters? </p>
<p>Solutions for just the intercept and slope are fine.</p> | 37,379 | [
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<p>I'm looking at doing text classification/spam filtering using naive Bayesian classifiers with the e1071 or klaR package on R. Is there a good tutorial out there to describe this? I'm kind of stuck because I'm not sure what to use as the data to input into the NaiveBayes function.</p>
<p>Some help very much appreciated, thanks!</p> | 74,490 | [
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<p>This is closely related to a question I asked yesterday but I've now got a much more complete answer on which I was hoping to get feedback. The previous question was just looking for conceptual advice and was very helpful. You can find the relevant data and introduction <a href="http://stats.stackexchange.com/questions/62646/pooled-time-series-regression-in-r">here</a>.</p>
<p>I wanted to find period effects for each age group. I've run two regressions using dummies as part of an interaction term. I'm hoping to see if my method is flawed and if my interpretation of the results is correct or not. They first regression is as follows:</p>
<pre><code>> ## Generate YearDummy and AgeGroupDummy using factor()
>
> YearDummy <- factor(YearVar)
> AgeGroupDummy <- factor(AgeGroup)
>
> ## Check to see that YearDummy and CohortDummy are indeed factor variables
>
> is.factor(YearDummy)
[1] TRUE
> is.factor(AgeGroupDummy)
[1] TRUE
> ## Regress on AgeGroup and include AgeGroup*YearDummy interaction terms
>
> PooledOLS1 <- lm(PPHPY ~ AgeGroup + AgeGroup*YearDummy + 0,
data=maildatapooled)
> summary(PooledOLS1)
Call:
lm(formula = PPHPY ~ AgeGroup + AgeGroup * YearDummy + 0, data =
maildatapooled)
Residuals:
Min 1Q Median 3Q Max
-38.852 -10.632 3.298 11.275 26.481
Coefficients:
Estimate Std. Error t value Pr(>|t|)
AgeGroup 26.2212 3.5070 7.477 3.84e-10 ***
YearDummy1 119.8836 15.6840 7.644 1.99e-10 ***
YearDummy2 123.7458 15.6840 7.890 7.55e-11 ***
YearDummy3 103.2660 15.6840 6.584 1.28e-08 ***
YearDummy4 97.7102 15.6840 6.230 5.06e-08 ***
YearDummy5 103.3295 15.6840 6.588 1.26e-08 ***
YearDummy6 103.2330 15.6840 6.582 1.29e-08 ***
YearDummy7 84.8291 15.6840 5.409 1.16e-06 ***
YearDummy8 70.7114 15.6840 4.509 3.09e-05 ***
YearDummy9 90.9566 15.6840 5.799 2.65e-07 ***
YearDummy10 50.0885 15.6840 3.194 0.00224 **
YearDummy11 37.7004 15.6840 2.404 0.01933 *
YearDummy12 33.1947 15.6840 2.116 0.03846 *
AgeGroup:YearDummy2 1.8066 4.9597 0.364 0.71695
AgeGroup:YearDummy3 -3.8022 4.9597 -0.767 0.44632
AgeGroup:YearDummy4 -1.7436 4.9597 -0.352 0.72640
AgeGroup:YearDummy5 -6.0494 4.9597 -1.220 0.22735
AgeGroup:YearDummy6 -6.7992 4.9597 -1.371 0.17552
AgeGroup:YearDummy7 -3.6752 4.9597 -0.741 0.46158
AgeGroup:YearDummy8 -0.4799 4.9597 -0.097 0.92323
AgeGroup:YearDummy9 -9.8190 4.9597 -1.980 0.05232 .
AgeGroup:YearDummy10 -2.2452 4.9597 -0.453 0.65241
</code></pre>
<p>My interpretation of the interaction term coefficients is that they represent the difference in slope of AgeGroup between the period of the corresponding YearDummy and the AgeGroup slope at the top of the results. This is kind of like the AgeGroup effect across different periods.</p>
<p>My second regression is as follows:</p>
<pre><code>> ## Regress YearVar and Include YearVar*AgeGroupDUmmy
>
> PooledOLS2 <- lm(PPHPY ~ YearVar + YearVar*AgeGroupDummy + 0,
data=maildatapooled)
> summary(PooledOLS2)
Call:
lm(formula = PPHPY ~ YearVar + YearVar * AgeGroupDummy + 0, data =
maildatapooled)
Residuals:
Min 1Q Median 3Q Max
-29.345 -9.325 -0.915 8.540 40.150
Coefficients:
Estimate Std. Error t value Pr(>|t|)
YearVar -7.089 1.252 -5.664 3.04e-07 ***
AgeGroupDummy1 142.292 9.211 15.447 < 2e-16 ***
AgeGroupDummy2 185.508 9.211 20.139 < 2e-16 ***
AgeGroupDummy3 218.170 9.211 23.685 < 2e-16 ***
AgeGroupDummy4 255.733 9.211 27.763 < 2e-16 ***
AgeGroupDummy5 278.180 9.211 30.200 < 2e-16 ***
AgeGroupDummy6 300.910 9.211 32.667 < 2e-16 ***
AgeGroupDummy7 282.325 9.211 30.650 < 2e-16 ***
YearVar:AgeGroupDummy2 -1.737 1.770 -0.981 0.3298
YearVar:AgeGroupDummy3 -2.401 1.770 -1.357 0.1792
YearVar:AgeGroupDummy4 -3.772 1.770 -2.131 0.0366 *
YearVar:AgeGroupDummy5 -2.915 1.770 -1.647 0.1040
YearVar:AgeGroupDummy6 -3.587 1.770 -2.026 0.0465 *
YearVar:AgeGroupDummy7 -2.372 1.770 -1.340 0.1846
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 14.97 on 70 degrees of freedom
Multiple R-squared: 0.9946, Adjusted R-squared: 0.9935
F-statistic: 917.9 on 14 and 70 DF, p-value: < 2.2e-16
</code></pre>
<p>My interpretation of the interaction term coefficients here is that they represent the difference in slope of YearVar between the corresponding AgeGroup in the interaction term and the YearVar result at the very top. That is, they are something like a period effect across the different age groups.</p>
<p>Can anyone see a problem with what I've done here or with my interpretation? This second regression is the closest thing to period effects across distinct age groups that I've been able to muster. Any critiques/new ideas are welcome.</p> | 31,358 | [
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] |
<p>I came in a new post because I’m quite confused with the use of an offset in gam() function (mgcv package). I’m actually working on predicting marine top predator’s distribution using aerial surveys (predators’ observations) and secondary production model outputs. My data are shaped in a 0.25° resolution grid, over large areas such as south-west Indian Ocean. Surveys have been done in several sectors over that grid, that is, all the grid is not sampled. My secondary production covariates are from SEAPODYM model, developed by Lehodey et al. I have a set of about 12 covariates, and wish to find the best combination of them to <em>predict</em> predators’ distribution. </p>
<p>That said, because my goal is to predict predators’ distribution, I performed a model selection based on GCV score, after having excluded all combinations of correlated covariates. I am modelling observations, therefore I used a quasipoisson distribution (overdispersed data) with sampled area per pixel as an offset. I made my model selection with the model written as follows:</p>
<pre><code>mod1<-gam(Y ~ offset(log(sampled area))+ covariate1 + covariate2 + covariate3 + covariate4, family=quasipoisson)
</code></pre>
<p>Until then, everything worked well, but then I started predicting predators’ observations over my full grid. Doing this with the previous model resulted in predicting only over sampled pixel. Following colleagues’ advices, I predicted using a different writing, thinking it would not change anything in the results: </p>
<pre><code>mod2<-gam(Y ~ covariate1 + covariate2 + covariate3 + covariate4, offset(log(sampled area)), family=quasipoisson)
</code></pre>
<p>Indeed, it allows me to predict observation over the entire grid, but I am completely confused since I looked at the summaries of the models. The results are in fact quite different, with both GCV scores and Explained Deviances having higher values: </p>
<pre><code>summary(mod1)
Family: quasipoisson
Link function: log
Formula:
species ~ offset(log(sampled area) + s(covariate1,k=4) + s(covariate2,k=4)
+ s(covariate3,k=4) + s(covariate4,k=4)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.12268 0.08025 -1.529 0.127
Approximate significance of smooth terms:
edf Ref.df F p-value
s(bathy_mnk_B) 2.722 2.951 15.89 4.83e-10 ***
s(meso_mnk_B) 2.995 3.000 17.52 3.52e-11 ***
s(bathy_mnk_P) 2.989 3.000 20.37 6.21e-13 ***
s(mmeso_mnk_P) 2.536 2.855 12.71 6.49e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) = 0.0838 Deviance explained = 16.4%
GCV score = 22.371 Scale est. = 22.188 n = 1491
summary(mod2)
Family: quasipoisson
Link function: log
Formula:
species ~ s(covariate1,k=4) + s(covariate2,k=4) + s(covariate3,k=4) +
s(covariate4,k=4)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.30093 0.07884 16.5 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Approximate significance of smooth terms:
edf Ref.df F p-value
s(bathy_mnk_B) 2.802 2.974 17.37 5.18e-11 ***
s(meso_mnk_B) 3.000 3.000 16.35 1.86e-10 ***
s(bathy_mnk_P) 2.988 3.000 27.75 < 2e-16 ***
s(mmeso_mnk_P) 2.460 2.806 12.47 1.12e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) = 0.0938 Deviance explained = 17.5%
GCV score = 31.809 Scale est. = 31.547 n = 1491
</code></pre>
<p>Given that I based my model selection on GCV scores, the fact that it changed like that puzzled me, and I am wondering whether or not my model selection was correct. I re-performed a model selection with the second kind of model (Y~X1+X2,offset), and resulting selected models are not always the same than with my first procedure. </p>
<p>Seeing this, I am wondering on which writing of the model should I base my modelling and predictions? </p>
<p>Does anyone could explain me why those two models are giving such different results? </p>
<p>Thanks in advance for your help and advices, </p>
<p>Charlotte</p> | 41,855 | [
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<p>It is well known that most model selection algorithms can easily fall into a multiple comparison trap. To quote Friedman:</p>
<blockquote>
<p>Consider developing a regression model in a context where substantive
theory is weak. To focus on an extreme case, suppose that in fact
there is no relationship between the dependent variable and the
explanatory variables. Even so, if there are many explanatory
variables, the R2 will be high. If explanatory variables with small t
statistics are dropped and the equation refitted, the R2 will stay
high and the overall F will become highly significant. This is
demonstrated by simulation and by asymptotic calculation.</p>
</blockquote>
<p>Now, say you've got a multiple regression model based on multiple theories; you expect correlation between $y$ and many of your $X$'s, but you're not entirely sure how those correlations are affected by partial co-linearity between your $X$'s. Further, lets say that you suspect that there might be interactions between some of your $X$'s, but you're not sure. Because of collinearity, you want to estimate a model with all the $X$'s, rather than several separate models.</p>
<p>Now, say you were fitting the model to the population, rather that a sample from the population. Suppose that some subset of your $X$'s and your interactions were in fact poor predictors. Since you're in the population, all coefficient estimates are unbiased, and those poor predictions represent real, but weak, correlations.</p>
<p>But what if you're not sure about the model? Coefficient estimates suffer from omitted variables bias, or misspecification bias, or probably both. </p>
<p>Now go from the population to the sample. You specify the same rich model based on your best guess of theory, expecting that some aspects of your theory are correct, and some are not. If you <strong>do not</strong> select out (say via AIC) some parameters, your model is overfit, and will generalize poorly. If you <strong>do</strong> select out certain terms (say via backwards selection), your model will suffer from a multiple comparison problem -- certain features will be deemed "significant" even if they do not represent real population correlations.</p>
<p>So what is an applied guy to do? I've got a complicated dataset with a bunch of correlated variables, and a bunch of reasonably well-justified theories. I want to test whether there is any support for the theories, and I want to make predictions (with associated measures of uncertainty of predictions) based on the most robust model I can construct. </p>
<p>I've been doing iterative backwards selection -- dropping terms that lower AIC the most when dropped. <strong>Is there a way to correct standard errors and/or prediction intervals for multiple comparison after doing backwards selection</strong>? </p>
<p>(Note that I do NOT want to do ridge regression or lasso, because I am not willing to trade bias for variance. I know that mis-specification can lead to bias, but I am using semi-parametrics to avoid functional form mis-specification on features.)</p> | 74,491 | [
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<p>So I have data like:</p>
<pre><code>Cost 20 30 10 5
Rating 5 3 2 5
</code></pre>
<p>I want to make a chart of rating vs. cost, so the points would be </p>
<pre><code>[(5,20), (3,30), (2,10), (5,5)]
</code></pre>
<p>I can't seem to get excel to do anything other than put the two rows as independent series. Am I missing something, or I do have to pivot the data somehow to make it do that?</p>
<p>(Actually, I'm using an old-ish version of <a href="http://www.apple.com/iwork/numbers/" rel="nofollow">Numbers.app</a> on OS X, but I'm hoping the concept will be the same. I have access to excel if need be.)</p> | 74,492 | [
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<p>How to interpret the Null and Residual Deviance in GLM in R? Like we say the for AIC that smaller AIC is better, is there any similar and quick interpretation for the deviances also?</p>
<p>Null deviance: 1146.1 on 1077 degrees of freedom
Residual deviance: 4589.4 on 1099 degrees of freedom
AIC: 11089</p> | 37,385 | [
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<p>I am using the R 'multcomp' library (<a href="http://cran.r-project.org/web/packages/multcomp/" rel="nofollow">http://cran.r-project.org/web/packages/multcomp/</a>) to calculate Dunnett's test. I am using the script below:</p>
<pre><code>Group <- factor(c("A","A","B","B","B","C","C","C","D","D","D","E","E","F","F","F"))
Value <- c(5,5.09901951359278,4.69041575982343,4.58257569495584,4.79583152331272,5,5.09901951359278,4.24264068711928,5.09901951359278,5.19615242270663,4.58257569495584,6.16441400296898,6.85565460040104,7.68114574786861,7.07106781186548,6.48074069840786)
data <- data.frame(Group, Value)
aov <- aov(Value ~ Group, data)
summary(glht(aov, linfct=mcp(Group="Dunnett")))
</code></pre>
<p>Now if I run this script through the R Console multiple times I get very slightly different results each time. Here's one example:</p>
<pre><code> Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Dunnett Contrasts
Fit: aov(formula = Value ~ Group, data = data)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
B - A == 0 -0.35990 0.37009 -0.972 0.76545
C - A == 0 -0.26896 0.37009 -0.727 0.90019
D - A == 0 -0.09026 0.37009 -0.244 0.99894
E - A == 0 1.46052 0.40541 3.603 0.01710 *
F - A == 0 2.02814 0.37009 5.480 0.00104 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Adjusted p values reported -- single-step method)
</code></pre>
<p>And here's another:</p>
<pre><code> Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Dunnett Contrasts
Fit: aov(formula = Value ~ Group, data = data)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
B - A == 0 -0.35990 0.37009 -0.972 0.7654
C - A == 0 -0.26896 0.37009 -0.727 0.9001
D - A == 0 -0.09026 0.37009 -0.244 0.9989
E - A == 0 1.46052 0.40541 3.603 0.0173 *
F - A == 0 2.02814 0.37009 5.480 <0.001 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Adjusted p values reported -- single-step method)
</code></pre>
<p>As you can see, the above two results differ very slightly, but it's enough to move the final group (F) from two stars to three stars, which I find worrying. </p>
<p>I have several questions regarding this:</p>
<ol>
<li>Why is this happening?! Surely if you put the same data in each time
you should get the same data out. </li>
<li>Is there some kind of random number being used somewhere in the Dunnett's calculation?</li>
<li>Is this slight variation each time actually a problem?</li>
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<p>i have a new data set which is basically as bad as the last (same sort of data) and have been asked to try non linear regression on it, with the focus on partition (I will be using boosting and bagging). The dependent variable is continuous however.</p>
<p>I require the input variables intact (as in not factorized) as i again require identification of what input variables affect the dependent variable of interest and how.</p>
<p>Again the many X input variables are very different in terms of distribution, i have categorical inputs too.</p>
<p>I do not see much information in general with regards to non linear regression and linear correlation.
Leaving the data set as is causes multiple inclusion of what are strongly correlated variables particularly in bootstrapping (the reason seems obvious due to the nature of bootstrap). </p>
<p>to what degree is this a problem? For example; if X1 and X2 are collinear and X1 is Out of Bag with X2 in bag you have effectively not taken that variable out. Is this even an issue?</p> | 37,388 | [
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<p>Say we have $N$ time series $X_t^i$ for $i=1...N$and we want to predict a separate time series $Y_t$. Let's consider the following model: $Y_t = \sum_{i} \beta_i X_{t-1}^i $</p>
<p>I am just trying to figure when such a model makes sense</p>
<ul>
<li>Does this model have a name? Is there an interpretation in terms of random processes? Are there any known applications?</li>
<li>Does it make sense to solve the $\beta_i$ by using regression approaches (ridge, lasso, etc) using the training points $(<x_{t-1}^1... x_{t-1}^N>, y_t)$?</li>
</ul> | 74,494 | [
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<p>Let's say I am using a logistic model to predict whether it rains (yes or no) based on the high temperature and have collected data for the past 100 days. Let's say that it rains 30/100 days. Furthermore, let's say that 70/100 days the temperature is actually 50 degrees. So there were only 30/100 days where the temperature was not 50 degrees. </p>
<p>So in terms of the response variable (rain - yes or no), there is enough variation to develop the model. However, in regard to the predictor (temperature), with so many days where the temperature is 50 degrees, how does a recurring variable "element" (50 degrees) impact the calculations of a logistic model?</p> | 37,390 | [
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<p>I am looking for a certain multivariate probability distribution function to fit to my data, but the usual multivariate normal distribution is unfit, since my data has a dent (antimode) instead of a peak in certain cases. (In other cases the data does have a peak.)</p>
<p>My data consists of data points with multiple variables, which are all constrained to be within the interval [0,1].</p>
<p>In the univariate case I would simply use the beta distribution, but the multivariate analogue, the Dirichlet distribution, seems unfit as well; it constrains the input variables to sum up to 1, instead of demanding all the input variables to lie within the interval [0,1]. Furthermore, it seems that the distribution doesn't depend on the covariance matrix - it seems to be unable to distinguish high covariance from low covariance given the same variances.</p>
<p>What kind of distribution should I use?</p> | 74,495 | [
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<p>In my research I am working on some data (let’s call it sales data), however we almost have no information on the original sample. My goal is to come up with possible explanations on how it could be sampled by consistency arguments. </p>
<p>In more detail, the data provided is composed of sales transactions and it has 2 columns (buyer and receiver). One of the most reasonable ways to sample would be (we believe) to assign a probability (p) to include each transaction or not, and then include a subset of transactions according to random realizations ( sampling without replacement). </p>
<p>This corresponds to binomial sampling. Our information on the original sample is limited (almost none), except that I have an estimate on total number of buyers there may be on overall (circa ~2 million). In addition, we assume that buyers are the customers that belong to a network of original data provider, however sellers may belong to other networks, thus our attention is limited to buyers. </p>
<p>The consistency argument is based on the following: on the original network if the buyer is highly active (there are more transactions) then the probability that that specific buyer gets included in a sample is higher than a less active customer. </p>
<p>Then by counting how many transactions each buyer makes, and grouping them by common number of transactions – we get the number of buyers who make $i$ number of transactions. Then, based on our estimate on the total number of buyers in the network (2 M), my goal to estimate the sampling rate (p). </p>
<p>R code:</p>
<pre><code> set.seed(0);
##generate random sales transactions
ss1= sample.int( 100, size = 100,replace = TRUE, prob = NULL)
ss2= sample.int(10000, size = 100, replace = TRUE, prob = NULL)
buyer= sort(ss1,decreasing=TRUE);
receiver = sort(ss2,decreasing=TRUE);
data= data.frame("buyer"= buyer, "receiver"= receiver);
##compute the number of sales - each buyer makes
count= data.frame(table(buyer));
buyer.sales = count[order(count$Freq,decreasing=TRUE),];
df = data.frame(table(buyer.sales$Freq)); ##number of buyers who make 1 transaction, 2 transactions etc...
## We have a guess on total number of buyers - 2M. What would be the sampling rate(p)?
</code></pre>
<p>Here are the approaches I’ve tried so far: </p>
<p><strong>Approach 1:</strong></p>
<p>Pose Y_i the number of customers who make “i” calls in the sample (data), and similarly X_i the number of customers who make “i” calls in the original network. We know the Y’s and try to get X’s. </p>
<p>Then if the transactions are sampled with a constant p rate, then </p>
<p>$Y_i = sum_{k=i}^{k= max(i)} N_k * (choose(k,i))* p^k * (1-p)^{(n-k)}$;</p>
<p>If we define the matrix A as follows:</p>
<pre><code> A <- function(i,j,prob) ifelse(i > j, 0,(choose(j,i)*((1-prob)^(j-i))*(prob^(i))) );
</code></pre>
<p>The problem would be solve A*x= b, which has an easy solution since A is upper triangular. But for my current problem, the x values that I get do not make any sense – they highly range between plus-minus infinity and ideally I would think that x vector should have some kind of correlation with y’s. </p>
<p>I solved the linear system of equalities by minimizing L1, L2 norms and also with the positivity constraints as well. Still, not correlated results- and the sum of x_i’s are highly different than 2M. </p>
<p><strong>Approach 2:</strong></p>
<p>This approach is on subsampling from the sample (data). I thought I could subsample from the sample with a given p, then count the number of buyers I get in that subsample. The probability value –p, which has generates the following ratio ( the number of buyers original / the number of buyers in sample), equal to (the number of buyers in sample / the number of buyers in subsample) would be the most reasonable choice. </p>
<p>This approach gave me p~0.02% but I don’t think this is correct. I think approach 1 makes more sense but the results do not suggest that.</p>
<p>I should note as well, that the data on hand is composed of monthly transaction data for a period of 12 months. Thus the amount of data I take as the sample is important as well. I tried poisson rate estimation (<a href="http://stats.stackexchange.com/questions/2466/estimate-the-size-of-a-population-being-sampled-by-the-number-of-repeat-observat">Estimate the size of a population being sampled by the number of repeat observations</a>) but the amount of data I include, highly effects the estimate of total number of buyers – and I am really stuck. </p>
<p>I would think that the original data would have some kind of a power law behavior, but that’s purely a guess without any justification. I am really stuck on which approach makes more sense. Any comments, ideas appreciated. </p> | 37,392 | [
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<p>I want to understand how the sampling distribution of the whole covariance matrix behaves for large $n$. I am trying to use the delta method and multivariate CLT. I am trying to show that when the $X$'s are IID with nonzero finite fourth order moments then the operation (sample variance) takes place, as in this equation:</p>
<p>$$\sqrt{n}(s_X^2 - \sigma^2) \underset{n\rightarrow \infty}{\overset{d}{\longrightarrow}} N(0, \mathbb{E}X_1^4 - (\mathbb{E}X_1^2)^2).$$</p>
<p>(For simplicity I have assumed $\mathbb{E}X_i = 0$). Help is solicited in understanding the steps I need to take. I started with studying the asymptotic behavior of the sample variance and then went on to use the delta method for the following function $g(a,b) = b - a^2$.</p> | 49,248 | [
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<p>After fitting a coxmodel it is possible to make predictions and retrieve the relative risk of new data. What I don't understand is how the relative risk is computed for an individual and what is it relative to (i.e. the average of the population)? Any recommendations for resources to help understand (I not very advanced in survival analysis so the simpler the better)?</p> | 74,496 | [
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<p>Should false discovery be controlled at the data acquisition level, or should this be at the data interpretation level?</p>
<p>I have an experiment in which microarrays were used to quantify the expression of about 30,000 genes (variables) in two groups of biological tissues (group-sizes of 75 and 76). Raw array data was pre-processed to remove background signals and genes whose expression levels are undetectable, and to normalize values across arrays. The final data was then examined using the Mann-Whitney U test to compare gene expression between the two groups to identify differentially expressed genes, with false discovery rate (FDR) controlled with the <a href="http://en.wikipedia.org/wiki/False_discovery_rate#Benjamini.E2.80.93Hochberg_procedure" rel="nofollow">Benjamini-Hochberg</a> procedure. At FDR <5%, no gene is identified as differentially expressed, and I formally conclude that 'there is no differential expression of genes between the two tissues.'</p>
<p>Now, suppose someone is interested in the expression of only one specific gene. Using my pre-processed gene expression data-set and the U test, they compare the two groups for the expression of only this gene and notice a P value <0.05, the commonly used significance threshold in my field of study. As this does not involve multiple testing, there is no false discovery control. Can this observer formally conclude that 'the gene is differentially expressed between the two tissues,' contradicting my conclusion?'</p>
<p>Or should the observer have applied false discovery control because such a control has to be applied at the data acquisition level (as per which, data on multiple variables were collected) and not at the data interpretation level (as per which, data for only one variable was analyzed)?</p> | 49,249 | [
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<p>I am reading the article of wikipedia about <a href="http://en.wikipedia.org/wiki/Bayesian_average" rel="nofollow">bayesian average</a>. This seems very interesting about on developing rating systems, but I have a doubt. How to fix the weight C of the average?</p> | 74,497 | [
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<p>I am doing a regression analysis which troubled me. </p>
<p>My independent variable are 4 interplanetary condition components, and the dependent variable is the latitude of auroral oval boundary.
So far, the specific relationship is still unknown in physical principle, what we want to do is to get a model (function expression) from the massive data which shows how these independent variables affect the dependent variable.</p>
<p>I used the Matlab statistical toolbox to do the regression analysis, but the results were very bad. The p values of the F statistic and t statistic are very small, but the R² is also very low, about 20%. </p>
<p>So how should I improve the R²? Are there good methods? I see that SVM (or LS-SVM) can do regression anaysis, is it a good way to manage the massive data, multiple independent variables regression anaysis?</p>
<p>The following are the results:</p>
<pre><code>mdl =
Linear regression model:
y ~ 1 + x1*x2 + x1*x3 + x1*x4 + x2*x3 + x2*x4 + x1^2 + x2^2 + x3^2 + x4^2
Number of observations: 18471, Error degrees of freedom: 18457
Root Mean Squared Error: 2.44
R-squared: 0.225, Adjusted R-Squared 0.225
F-statistic vs. constant model: 413, p-value = 0
</code></pre>
<p>when we add another predictor, i.e., the independent variables become 5, the resuts of the regression analysis are:</p>
<p>Linear regression model:</p>
<pre><code>y ~ 1 + x1*x2 + x1*x3 + x1*x4 + x1*x5 + x2*x3 + x2*x4 + x2*x5 + x3*x5 + x2^2 + x3^2 + x4^2 + x5^2
</code></pre>
<p>Number of observations: 18457, Error degrees of freedom: 18439
Root Mean Squared Error: 2.21
R-squared: 0.366, Adjusted R-Squared 0.366
F-statistic vs. constant model: 627, p-value = 0</p> | 74,498 | [
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<p>This is a problem from the "7th Kolmogorov Student Olympiad in Probability Theory":</p>
<blockquote>
<p>Given one observation $X$ from a $\operatorname{Normal}(\mu,\sigma^2)$ distribution with both parameters unknown, give a confidence interval for $\sigma^2$ with a confidence level of at least 99%.</p>
</blockquote>
<p>It seems to me that this should be impossible. I have the solution, but haven't read it yet. Any thoughts?</p>
<p>I'll post the solution in a couple days.</p>
<p>[Follow-up edit: Official solution posted below. Cardinal's solution is longer, but gives a better confidence interval. Thanks also to Max and Glen_b for their input.]</p> | 37,394 | [
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<p>Stats noob here. I've been trying to find out if there is a technique to perform a certain type of analysis but so far a lot of searching hasn't yielded anything.</p>
<p>My question is, if a large peak or trough is identified in an organisation's overall rate, is there a way of identifying which of the organisation's production sites has contributed the most to this rate change?</p>
<p>The data describes an organisation and it's performance against a rate based performance measure (let's say % wastage on a washer production line). The dataset contains 4 fields. These fields are:</p>
<ol>
<li>Site (There are 100+ of these)</li>
<li>Month</li>
<li>Numerator (Number of washers lost to wastage)</li>
<li>Denominator (Total number of washers produced)</li>
</ol>
<p>The organisation monitors it's performance based on its overall rate (sum of all numerators / sum of all denominators) by month. This gives a trend line. If there is a sudden large increase/decrease in rate then the organisation wants to target the sites which have most contributed to this rate change.</p>
<p>I'm not certain if there is an answer to this question, as one site could see a decrease in denominator and another could see an increase in its numerator value, meaning these sites both show a small change, whilst the organisation as a whole may see a larger scale change when the figures are combined and the overall rate is calculated.</p>
<p>I was originally thinking about something like a logistic regression with weighting but wasn't sure if this was quite the way to go...</p> | 74,499 | [
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<p>I want to know I often I need to crawl certain webpages, to make sure I have the newest content as fast as possible.
The naïve approach to this, is to simply calculate the average over a certain period of time. However, different web pages are updated at different rates, so this wouldn't be a good fit.
I did some googling, and apparently, the Poisson distribution is a good fit for this kind of estimation. My question is, how should I calculate lambda? And it should be constantly updated, correct?</p>
<p><strong>Edit:</strong>
John's answer helped me out; for further reference, check this article: <a href="http://oak.cs.ucla.edu/~cho/papers/cho-freq.pdf" rel="nofollow">Estimating Frequency of Change</a></p>
<p>Thank you</p> | 74,500 | [
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<p>We have a small dataset (about 250 samples * 100 features) on which we want to build a binary classifier after selecting the best feature subset. Lets say that we partition the data into:</p>
<p>Training, Validation and Testing</p>
<p>For feature selection, we apply a wrapper model based on selecting features optimizing performance of classifiers X, Y and Z, separately. In this pre-processing step, we use training data for training the classifiers and validation data for evaluating every candidate feature subset.</p>
<p>At the end, we want to compare the different classifiers (X, Y and Z). Of course, we can use the testing part of the data to have a fair comparison and evaluation. However in my case, the testing data would be really small (around 10 to 20 samples) and thus, I want to apply cross-validation for evaluating the models.</p>
<p>The distribution of the positive and negative examples is highly ill-balanced (about 8:2). So, a cross-validation could miss-lead us in evaluating the performance. To overcome this, we plan to have the testing portion (10-20 samples) as a second comparison method and to validate the cross-validation.</p>
<p>In summary, we are partitioning data into training, validation and testing. Training and validation parts are to be used for feature selection. Then, cross-validation over the same data is to be applied to estimate the models. Finally, testing is used to validate the cross-validation given the imbalance of the data.</p>
<p><strong>The question is:</strong> If we use the same data (training+validation) used in selecting the features optimizing the performance of classifiers X, Y and Z, can we apply cross-validation over the same data (training+validation) used for feature selection to measure the final performance and compare the classifiers?</p>
<p>I do not know if this setting could lead to a biased cross-validation measure and result in un-justified comparison or not.</p> | 49,252 | [
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<p>My understanding is that a hazard ratio from a Cox <a href="http://en.wikipedia.org/wiki/Proportional_hazards_model" rel="nofollow">proportional hazards model</a> compares the effect on the hazard rate of a given factor to a reference group. Does that actual group have to exist in the data?</p>
<p>Say we enroll people in a study of how long before they buy a couch. We right-censor at 3 years. For this example we have two factors: age < 30 or >= 30, whether they own a cat. It turns out the hazard ratio of "owns cat" to the reference group (age < 30, "doesn't own cat") is 1.2, and significant (say p<0.05). Do I know that the data includes people with age < 30 who don't own a cat?</p>
<p>For this example it seems a little crazy, but if there are many factors in the model, it becomes easier to imagine that there might be no data for some combination of factors, and that might end up as the reference group if chosen arbitrarily. In fact I think I've had that happen (in R, if it matters).</p> | 74,501 | [
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<p><em>Mathematics and Statistics are quite far behind in my educational curriculum, and yet now I seem to need them again, so this set of questions is very basic I suppose:</em></p>
<h2>What is a trend line? What various types of trend lines are there? And what are their respective use or relevance?</h2>
<h3>Background:</h3>
<p>I'm working on a thesis and I am producing some visual rendering of a survey I ran, in which I tried to measure public knowledge of (colonial) history and heritage in a town of Kerala, India.</p>
<p>I have data for instance, representing the result of a test, the average result of which is compared to various criteria of the population (age, education, etc.) With regard to criteria such as age-group, income or education, it is also interesting to find out whether their increase improves the results.</p>
<p>I am working on this data with LibreOffice, which offers me the possibility to show a "trend line", and I suppose the trend-line is adequate to show the relation between <em>x</em> and <em>y</em>, i.e. criteria and result. However, there are 4 types of trend lines available: <em>linear, logarithmic, exponential</em> and <em>power</em>.</p>
<p>Which one should I chose? Which difference does it make? How would I figure out a trend line if LibreOffice would not do it for me?</p>
<p><a href="http://en.wikipedia.org/wiki/Trend_line" rel="nofollow">Wikipedia</a> brings some light on those questions. Although I am not sure whether I should look at linear regression or technical trend line analysis.</p>
<p>Could you please provide a short "Trend Line for Dummies" introduction? Thanks :)</p> | 37,397 | [
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] |
<p>As science must be reproducible, by definition, there is increasing recognition that data and code are an essential component of the reproduciblity, as discussed by the <a href="http://reproducible-research.googlegroups.com/web/CISE-12-5-News.pdf">Yale Roundtable for data and code sharing</a>.</p>
<p>In reviewing a manuscript for a journal that does not require data and code sharing, can I request that the data and code be made available </p>
<ol>
<li>to me at the time of review</li>
<li>publicly at time of publication (the journal supports supplements)</li>
</ol>
<p>also, how might I phrase such a request?</p>
<hr>
<p><em>update</em>: although I am interested in the general case, this particular case consists of a meta-analysis with all previously published data, and the code is simple linear models in SAS</p>
<p><em>side note</em> the ability to make cross-study inference (as is the goal of meta-analysis) would be greatly enhanced if more studies provided raw data </p>
<p><em>update 2</em>: </p>
<p>I requested the data and code from the editor for purposes of review, the editor considered the request reasonable, and I have received the requested material (sufficient but with cryptic variable names, no metadata, and few inline comments) within a day.</p> | 74,502 | [
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<p>I am using SVM to predict diabetes. I am using the <a href="http://www.cdc.gov/BRFSS/">BRFSS</a> data set for this purpose. The data set has the dimensions of $432607 \times 136$ and is skewed. The percentage of <code>Y</code>s in the target variable is $11\%$ while the <code>N</code>s constitute the remaining $89\%$.</p>
<p>I am using only <code>15</code> out of <code>136</code> independent variables from the data set. One of the reasons for reducing the data set was to have more training samples when rows containing <code>NA</code>s are omitted.</p>
<p>These <code>15</code> variables were selected after running statistical methods such as random trees, logistic regression and finding out which variables are significant from the resulting models. For example, after running logistic regression we used <code>p-value</code> to order the most significant variables.</p>
<p>Is my method of doing variable selection correct? Any suggestions to is greatly welcome. </p>
<p>The following is my <code>R</code> implementation. </p>
<pre><code>library(e1071) # Support Vector Machines
#--------------------------------------------------------------------
# read brfss file (huge 135 MB file)
#--------------------------------------------------------------------
y <- read.csv("http://www.hofroe.net/stat579/brfss%2009/brfss-2009-clean.csv")
indicator <- c("DIABETE2", "GENHLTH", "PERSDOC2", "SEX", "FLUSHOT3", "PNEUVAC3",
"X_RFHYPE5", "X_RFCHOL", "RACE2", "X_SMOKER3", "X_AGE_G", "X_BMI4CAT",
"X_INCOMG", "X_RFDRHV3", "X_RFDRHV3", "X_STATE");
target <- "DIABETE2";
diabetes <- y[, indicator];
#--------------------------------------------------------------------
# recode DIABETE2
#--------------------------------------------------------------------
x <- diabetes$DIABETE2;
x[x > 1] <- 'N';
x[x != 'N'] <- 'Y';
diabetes$DIABETE2 <- x;
rm(x);
#--------------------------------------------------------------------
# remove NA
#--------------------------------------------------------------------
x <- na.omit(diabetes);
diabetes <- x;
rm(x);
#--------------------------------------------------------------------
# reproducible research
#--------------------------------------------------------------------
set.seed(1612);
nsamples <- 1000;
sample.diabetes <- diabetes[sample(nrow(diabetes), nsamples), ];
#--------------------------------------------------------------------
# split the dataset into training and test
#--------------------------------------------------------------------
ratio <- 0.7;
train.samples <- ratio*nsamples;
train.rows <- c(sample(nrow(sample.diabetes), trunc(train.samples)));
train.set <- sample.diabetes[train.rows, ];
test.set <- sample.diabetes[-train.rows, ];
train.result <- train.set[ , which(names(train.set) == target)];
test.result <- test.set[ , which(names(test.set) == target)];
#--------------------------------------------------------------------
# SVM
#--------------------------------------------------------------------
formula <- as.formula(factor(DIABETE2) ~ . );
svm.tune <- tune.svm(formula, data = train.set,
gamma = 10^(-3:0), cost = 10^(-1:1));
svm.model <- svm(formula, data = train.set,
kernel = "linear",
gamma = svm.tune$best.parameters$gamma,
cost = svm.tune$best.parameters$cost);
#--------------------------------------------------------------------
# Confusion matrix
#--------------------------------------------------------------------
train.pred <- predict(svm.model, train.set);
test.pred <- predict(svm.model, test.set);
svm.table <- table(pred = test.pred, true = test.result);
print(svm.table);
</code></pre>
<p>I ran with $1000$ (training = $700$ and test = $300$) samples since it is faster in my laptop. The confusion matrix for the test data ($300$ samples) I get is quite bad.</p>
<pre><code> true
pred N Y
N 262 38
Y 0 0
</code></pre>
<p>I need to improve my prediction for the <code>Y</code> class. In fact, I need to be as accurate as possible with <code>Y</code> even if I perform poorly with <code>N</code>. Any suggestions to improve the accuracy of classification would be greatly appreciated.</p> | 13,014 | [
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<p>I am working on a stock prediction model which predicts next day Open High Low and Close stock prices, my question is how to use RMSE and MAD to measure the error or accuracy for the entire model say 500 stocks? Or to clarify is the concept of RMSE and MAD useful only when looking at a single stock prediction error ? Does it make sense to average all RMSE or MAD for all 500 stocks ?</p>
<p>Thanks,</p>
<p>Deepesh</p> | 74,503 | [
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<p>I have about $50$ Bernoulli random variables $X_i$ whose joint distribution is unknown, but I can generate a sample of size on the order of $10^4$.
They are not independent, but I think the dependence is rather weak.
I would like to get some idea of ${\rm Pr}({\rm all}\ X_i = 1)$, which I expect to be
considerably smaller than $10^{-4}$, maybe around $10^{-6}$ (so it's unlikely that the sample will
contain outcomes with all $X_i = 1$). If I assumed they were independent, I could estimate their means and multiply these estimates. What can I do that
takes correlations into account?</p> | 74,504 | [
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<p>I'm looking for insights on how to test the accuracy of a model I built to predict next days stock price - Open, High, Low and Close next day prices.</p>
<p>Is there a unique indicator of accuracy or standard approach that should be used to represent the accuracy of this system ? </p>
<p>This is a predictive model for prices, not a trading system.</p>
<p>Input to the model is the historic time series - end of day stock prices-Open, High, Low, Closing.</p>
<p>I've run some initial stats by calculating the ratio of predicted prices to the actual end of day prices i.e if PC represents predicted close price for next day I am simply taking the ratio of PC / C where C is the actual close price for the predicted day and representing it as a %.</p>
<p>Have researched some forecasting system models online and understand that the common approach is to use Mean Absolute Percent Error, Mean Absolute Deviation, R-Squared, Correlation, Cumulative Forecast Error. As an example my prediction model is giving me values for Correlation and R-Squared 0.995 0.9924 0.9912 0.9885 & 0.9905 0.983 0.981 0.976 respectively for the O H L C predictions..would these be considered good values ?</p>
<p>Any suggestions on what would be the best approach?</p> | 74,505 | [
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] |
<p>We have the following solar meteorology parameters through satellite data with resolution of 100 km$^2$ (10 km x 10 km). </p>
<p>Insolation on horizontal surface </p>
<p>Diffuse radiation on horizontal surface </p>
<p>Direct normal radiation </p>
<p>Like this we can have about 4-5 more parameters which are associated with Solar Power generation. These same parameters are also measured at 23 ground stations.</p>
<p>We also have same parameter data which are taken at ground station at respective long-latitude locations. Our objective will be to estimate all those solar parameters for every 1 km x 1 km area. By interpolating satellite data and ground station data in given vicinity area with best possible accuracy. My question is Which is best interpolation technique we can use: please provide some references and the software which can be used.</p> | 74,506 | [
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<p>Let's say I have a grid of cells from a satellite image that take on the value of 0 or 1, with 1 being forest.</p>
<p>There are many spatial pattern metrics that measure things about this pattern such as some dimension of forest fragmentation.</p>
<p>A common one is the join-count statistic which is simple enough that its expectation and variance are calculable for a random allocation process. This allows one to estimate the probability of a join-count with an equal or greater value than that observed, given a random process. One only needs to know the relationship among grid cells (are they spatially connected/neighbors or not) and the probability of any cells taking on a specific value (e.g. the probability that a cell is forest and probability that it is not).</p>
<p>More complex spatial patterns do not typically have a standardized version. For instance, the perimeter of all forest patches can be divided by the surface area of all patches to calculate a perimeter-area ratio. The farther from one large circle (the minimum) then the more fragmented.</p>
<p>It would be easy to calculate the probability of any given spatial pattern (they would all be the same, I believe, if they contain the same number of forested/non-forested cells), however I am interested in knowing the expected perimeter-area ratio and variance so that measures such as this can be standardized.</p>
<p>Are there any texts someone can suggest? </p>
<p>I have read quite extensively within landscape ecology and related fields, but I do not believe this has been addressed, so I need a math text or even area of study to research of which I am not currently aware.</p>
<p>A simple approach would be Monte Carlo simulation, but this would be very computationally intensive for even moderately large collections of grid cells. Further, the patterns observed are typically quite rare in comparison to a random process and so you could allow the simulation to run quite a while and I believe never fill in the tails of the distribution created by it so as to get a good handle on the probability of observing a pattern with the same value or greater.</p>
<p>Any guidance related to obtaining my objective is appreciated.</p>
<p>The primary motivation of this is not to test "straw man" null hypotheses of random allocation but instead to be able to compare the patterns from two areas while removing some of the confounding due to differences in the number of cells allocated to each category.</p> | 74,507 | [
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<p>I was just recently using the Student t-Test to check whether values from two samples could have an identical mean or not. I was wondering whether there is a complementary technique in bayesian statistics that does a similar thing.</p>
<p>Especially I am wondering about the $P$ values that I obtain as a result (<a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_ind.html" rel="nofollow">http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_ind.html</a>). With a sample size of 2 in my two samples, I figure that the resulting p value would probably fluctuate quite a bit if I added some more measurements or take some away.</p> | 74,508 | [
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<p>I am working on multi class imbalance data.I want to divide my classes based on one verus one method and then perform sampling techinque
How can I do that I just want to divide the classes not classification.
Is it possible in libsvm</p> | 22,166 | [
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<p>Suppose that the sample space S of some experiment is finite.
Show that the collection of all subsets of S satisfies the three conditions required to be called the collection of events:</p>
<ol>
<li>$S \in \beta $</li>
<li>$ if \; A \in \beta, \; A^\complement\in\beta$</li>
<li>$if A_{1},A_{2},\cdots,A_{n}\in\beta,\; A_{1}\cup A_{2}\cdots\cup A_{n}\in\beta$</li>
</ol>
<p>How can I proove axioms above? I.e., how can I show that the collection of all subsets of S satisfies the three conditions required to be called the collection of events.</p>
<p>A detailed explanation with Borel field and its full proof will do help me improve my statistical insight and skills.</p> | 43,009 | [
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<p>In many financial models we are interested in measuring the correlation between variables, returns etc. However, research shows that during crises times we observe "Correlation Breaks" where previously un-correlated variables become correlated.</p>
<p>What is the best way to quantify "stability of correlation" by examining the historic time series of the variables on which I am interested?</p>
<p>Would some kind of bootstrapping be a good way to start examining the sample distribution of the correlation? Is there any other method that I could apply?</p> | 37,407 | [
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] |
<p>My data is as follows. I have two groups of patients. Patients in each group had a different type of eye surgery. 5 variables were measured on patients in each group. I want to compare those variables between the two groups using a permutation test or MANOVA. The eye on which the surgery was done does not really matter in the analysis. However, Patient 2 in group A for example, had the surgery on both eyes and therefore has those 5 variables measured twice, one time on each eye. Can I consider patient 2 Left and patient 2 Right as two different observations? Same for patient 31 in group B.
$$
\begin{array}
\hline
\text{Patient} & \text{Surgery type} & \text{Side} & \text{V1}& \ldots & V5\\
1 & \text{A} & \text{Left} & 91 & \ldots & 22\\
2 & \text{A} & \text{Left} & 87 & \ldots & 19\\
2 & \text{A} & \text{Right} & 90 & \ldots & 23\\
. & . & . &\\
31 & \text{B} & \text{Left} & 90 & \ldots & 17\\
31 & \text{B} & \text{Right} & 88 & \ldots & 19\\
32 & \text{B} & \text{Right} & 91 & \ldots & 24\\
. & . & . &\\
\hline
\end{array}
$$
Thank you</p> | 37,408 | [
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<p>I am creating a small webapp and a part of it is generating a number (I like to call it diversity coefficient) that would represent how "diverse" someones listening habits are. The problem is - I am not sure how to create a valid formula for what I want. I will try my best to explain what information I have, and what I need as a result, and hopefully someone will be able to provide me with a solution. I guess it's nothing hard or complicated, but I am not that deep into math, so I am not sure how to get what I want (tried playing with different formula ideas for a few day, but could not get what I wanted)...</p>
<p>What I do is - I gather information from someones <a href="http://www.last.fm/" rel="nofollow">lastfm</a> profile, which is a website that lets you track the music you listen to, and also lets you use all the data that they have stored on their servers to create something with that data (you can see a sample of a profile and what it is about <a href="http://www.last.fm/user/pootzko" rel="nofollow">here</a> if you like, just to better understand what it's all about). So let's say you listen to a certain band, and you listen to a number of their songs. </p>
<p>The data that I have is the following:</p>
<ul>
<li>total count of how many times you listened to a certain band </li>
<li>number of different songs (from that band) that you listened to</li>
<li>total count of how many times you listened to each of that different songs</li>
</ul>
<p>For example:</p>
<ul>
<li>someone listened to a band called FooBar for 1012 times</li>
<li>he listened to 17 different songs from that band</li>
<li>he listened to the first of that 17 songs for 100 times, the second one of that 17 for 110 times, the third one 120 times, the fourth one 170 times, the fifth one 500 times and the remaining 12 songs only once per song.</li>
</ul>
<p>What I would like to have as a result is a number that ranges from 0-100. That number would represent if a person listens to all the songs evenly or if he just listens to one or two song from a band all the time, and all the other song only once or twice and then forgets about them...</p>
<p>A perfect 100 score would be for a person that listened to FooBar for 500 times, 5 different song, each one 100 times... That is as even as it gets.</p>
<p>A score of 0 (or some very low score) would be for a person that listened to FooBar for 500 times, 5 different songs - one of them for 496 times, and the remaining four songs only once per song. </p>
<p>I think that it is some kind of "density" that I want, but how to get to that number - I am not sure.. And that is where I need your help. I would be happy to provide any additional info to help you help me.</p>
<p>The thing looks like this (if you take a look at these bars each one representing a different song (and its share of total count), that is why I say "density"):</p>
<p><img src="http://i.stack.imgur.com/IoOer.jpg" alt="bars sample"></p>
<p>In this example it is clear that Dredg is more evenly spread than Deftones, and should have a higher score (in my case the score is almost the same which is not good, it's just something that remained from my experimenting with the formula).</p>
<p>Thanks!</p>
<p>PS - feel free to edit tags, I really have no clue what to put there..</p> | 74,509 | [
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<p>I'm trying to make sense of these two statements about UMP (uniformly most powerful) tests: </p>
<ol>
<li><p>If $g(t\mid\theta)~$ is a UMP then $~g(t\mid\theta_1)>k g(t\mid\theta_0)~\forall~t\in C$ and $g(t\mid\theta_1)<k g(t\mid\theta_0)~\forall~t\in C^c$</p></li>
<li><p>For $\mathbf X~i.i.d.~f(x\mid\theta): \theta\in\Omega\subset\mathcal R$ if $f(x\mid\theta)$ has an MLR in $T(x)$ and any $k$, a test that rejects $H_0 \iff T>k~$ is a UMP test of size $\alpha$ with $\alpha=P_{\theta_0}(T>k)$</p></li>
</ol>
<p>Why are these statements not tautological? How are they not the definition of <em>any</em> one-tailed statistical test? What would be an example of a one-tailed statistical test that isn't a UMP? </p>
<p>Moreover, what is the relationship between LRT and UMP tests? I'm reviewing old exams where sometimes a question asks for an LRT and sometimes a UMP... Aren't all simple LRT tests UMP?</p>
<p>Thanks.</p> | 74,510 | [
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<p>In the following example:</p>
<pre><code>x <- rnorm(100)
y <- rpois(100, exp(1+x))+10
g1 <- glm(y ~ x, family = Gamma(link = "log"))
par(mfrow = c(2,2))
plot(y~x)
plot(y,resid(g1))
plot(y,resid(g1$lm))
</code></pre>
<p>I am trying to look at the pattern in the predictor variable, i.e. if the model is 'correct' there should be no patterns left in the residuals when plotted against the predictor. From the example shown, which residual am I suppose to use, <code>resid(g1)</code> or <code>resid(g1$lm)</code>, or something else? </p>
<p>Same goes when using a <code>gam</code>, is that <code>resid(g1$lm)</code>, <code>resid(g1$gam)</code>?
I find this very confusing. </p> | 74,511 | [
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] |
<p>I came across a paper that uses the abbreviation "p.e.":</p>
<p><a href="http://www.jstor.org/stable/10.2307/2984884" rel="nofollow">Khatri and Madria, The Von Mises-Fisher Matrix Distribution in Orientation Statistics. 1976.</a></p>
<p>It's in Section 7 on page 105. I'm including a short excerpt:</p>
<blockquote>
<p>Using the density function of $K^{\frac{1}{2}}XX'K^{\frac{1}{2}} = S$ given in Khatri (1975a), we find that the p.e. $X$ given $XX'=I_n$ is</p>
</blockquote>
<p>What does the abbreviation "p.e." mean in this sentence?</p> | 43,010 | [
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<p>Under the GARCH($m$,$s$) model, it can be shown that $ E(\eta_t\eta_{t-j}) = E[(a_t^{2}-\sigma_t^{2})(a_{t-j}^{2}-\sigma_{t-j}^{2})] = 0 $.</p>
<p>In my proof attempt, I came across $ E(\epsilon_t^{2}\epsilon_{t-j}^{2}) = 1 $ where $ \epsilon_t \sim^{iid} (0,1) $.</p>
<p>Is it indeed true that if $ \epsilon_t \sim^{iid} (0,1) $, then $ E(\epsilon_t^{2}\epsilon_{t-j}^{2}) = 1 $?</p> | 37,414 | [
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<blockquote>
<p>Let $X_{1},..,X_{n}$ be a sample from normal distribution with mean ${\theta}$ and variance 1. Let $H_{0}:{\theta}={\theta}_{0}$ and $H_{1}:{\theta}={\theta}_{1}$. Use the Neyman-Pearson lemma to show that the best test for $H_{0}$ vs $H_{1}$ has a critical region of the form mean$(X_{1},..,X_{n}){\geq}k$.</p>
</blockquote>
<p>I have done this part. The next question is: </p>
<blockquote>
<p>Let ${\theta}_{0}=0$ and ${\theta}_{1}=1$. Find the values of $n$ and $k$ so that the power function of the test ${\phi}$ has values ${\phi}{\theta}_{0}=0.01$ and ${\phi}{\theta}_{1}=0.9$. Write down the corresponding decision rule.</p>
</blockquote>
<p>My work thus far: Let $Y= \text{mean}(X_{1},..,X_{n})$. we have $P(Y{\geq}k|{\theta}_{0})=0.01$. Similarly for ${\theta}_{1}$. But how do I go from here? </p> | 74,512 | [
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<p>I am trying to understand the relative behavior of the following rank correlation statistics:</p>
<ol>
<li>Spearman coefficient</li>
<li>Kendall Tau / Concordance percentage</li>
<li>Normalized Gini coefficient (area under curve of percentage captured versus percentage observations)</li>
<li>Normalized Area under ROC curve (for binary classifiers)</li>
</ol>
<p>I don't believe any of these are functionally related to the others. The accepted answer <a href="http://stats.stackexchange.com/questions/3943/kendall-tau-or-spearmans-rho">here</a> references <a href="http://www.amstat.org/publications/jse/v14n3/datasets.winner.html" rel="nofollow">this paper</a> and Spearman and Kendall are highly correlated (as one would expect).</p>
<p>Are there good intuitions behind/discussions of relative (across datasets) or absolute (for a given dataset) differences for (any pair of) these measures?</p> | 74,513 | [
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<p>A rather elementary question: If we have:</p>
<p>$X = X_1 + X_2 + ... + X_n$ and $X_i \sim N(0,\sigma^2)$ and independent, </p>
<p>can we express $X$ as $\sqrt n X_i$ for any $i = 1,...,n$ ? </p>
<p>I think since all the components $X_i$ follow the normal distribution and we are summing $n$ of them up, the resulting rv is $X \sim N(0, n\sigma^2)$ and therefor we can use $X = \sqrt n X_i$. </p>
<p>Is this reasoning correct? If yes, under what condition it can be false? </p> | 74,514 | [
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<p>What is the impact of data dimensionality on computation complexity of SVM? I found on the literature that the complexity of SVM is $O(N^3)$, where $N$ is the number of training examples. If the number of dimension (e.g., $D$) does not impact the training time why it s better to reduce the dimensionality for training high-dimension dataset? Is it just to avoid overfitting? </p>
<p>BTW, I m using SVDD (support vector data description) from RRTools.</p> | 74,515 | [
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<p>I am running a pooled OLS regression and Random effects regression. </p>
<p>I have tested for autocorrelation for both methods. In the pooled OLS model I find serial correlation but for the RE model I find no serial correlation. </p>
<p>How is this possible? Is it possible that the unobserved time invariant variables that are removed in RE model are causing serial correlation in the pooled OLS regression? </p> | 74,516 | [
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<p>In Frank Harrell's <a href="http://biostat.mc.vanderbilt.edu/wiki/Main/RmS" rel="nofollow">RMS Short Course</a> today, I became aware that multiple imputation with <code>Hmisc:aregImpute</code> is not invariant to the ordering of terms in its formula argument, and that one therefore ought to check results of imputation under permutations of these terms. Here's a simple function that generates the necessary permutations:</p>
<pre><code>perms <- function(n, formula) {
rhs <- attr(terms(formula), 'term.labels')
formulae <- list()
for(i in 1:n){
formulae[[i]] <- as.formula(paste("~", paste(sample(rhs), collapse="+")))
}
formulae
}
</code></pre>
<p>For example:</p>
<pre><code>> perms(5, ~ age + sex + bp + death + mi)
[[1]]
~mi + age + bp + sex + death
<environment: 0x10dc9cbb8>
[[2]]
~bp + mi + sex + age + death
<environment: 0x10dc9cbb8>
[[3]]
~mi + bp + age + death + sex
<environment: 0x10dc9cbb8>
[[4]]
~age + death + bp + sex + mi
<environment: 0x10dc9cbb8>
[[5]]
~bp + age + death + sex + mi
<environment: 0x10dc9cbb8>
>
</code></pre>
<p>This could be used with <code>lapply</code> as follows:</p>
<pre><code>as <- lapply(perms(5, ~ age + sex + bp + death + mi)
, aregImpute, data=mydata, n.impute=10)
</code></pre>
<p>But I wonder whether it might make more sense to incorporate this into aregImpute itself.</p> | 37,417 | [
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<p>For an application I'm working on, I need to go from some uniformly distributed variables to an uniform distribution on an n-sphere. The standard way to do this seems to be choose (n+1) normal distributed variables, and then divide by the norm.</p>
<p>However, I need to be able to, given a point on the sphere, extract the uniform values I would have used to get this point. There the standard method fails, at it is not injective. I think I have devised a way to do this, however, but would like someone with more statistical knowledge than me to confirm this. Or if anyone knows of an easier way to do it, that's also appreciated.</p>
<p>Now, here's the idea:</p>
<p>Let $x_1, \ldots, x_n$ be cartesian coordinates for $\mathbb{R}^n$,
and choose spherical coordinates given as in the Wikipedia article for an <a href="http://en.wikipedia.org/wiki/N-sphere" rel="nofollow">n-sphere</a>, i.e.</p>
<p>$\begin{align}
x_1 &= r\cos\phi_1 \\
x_2 &= r\sin\phi_1\cos\phi_2\\
x_3 &= r\sin\phi_1\sin\phi_2\cos\phi_3\\
&\qquad\vdots\\
x_{n-1} &= r\sin\phi_1\ldots\sin\phi_{n-2}\cos\phi_{n-1}\\
x_n &= r\sin\phi_1\ldots\sin\phi_{n-1}
\end{align}$</p>
<p>The volume element of an (n-1)-sphere of radius r is</p>
<p>$\mathrm{d}V = r^{n-1}sin^{n-2}\phi_1\sin^{n-3}\phi_2\ldots\sin\phi_{n-2}
\mathrm{d}\phi_1\mathrm{d}\phi_2\ldots\mathrm{d}\phi_{n-1}.$</p>
<p>If we compute $\mathrm{d}x_i$ and $\mathrm{d}x_1\mathrm{d}x_2\ldots\mathrm{d}x_{n-2}$
I think we get that</p>
<p>$\mathrm{d}V = (-1)^nr\mathrm{d}x_1\ldots\mathrm{d}x_{n-2}\mathrm{d}\phi_{n-1}.$</p>
<p>I am then of the belief (and this is what I want confirmed by someone "in the know") that to choose points uniformly on the sphere, we can choose points
$x_1, \ldots, x_{n-2}$ uniformly on the ball $x_1^2+\ldots x_{n-2}^2 \leq r^2$ and
choose $\phi_{n-1}$ uniformly from $[0,2\pi)$.</p>
<p>If we can, we have reduced the problem to finding points uniformly on a (n-2)-ball. This
can be done by choosing points uniformly on the boundary and multiplying by $u^{1/(n-2)}$, where $u$ is uniformly chosen from $[0,1]$. Choosing points uniformly on the boundary is the same as choosing points uniformly on an $n-3$-sphere, and we're back to the original problem, but now 2 dimensions lower. We can work our way down to either the
1-sphere or the 1-ball, where we can choose a point uniformly with no problem.</p> | 74,517 | [
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<p>$X_{1}$ and $X_{2}$ are independent, chi-square distributed random variables. I've managed to show that $Z=X_{1}+X_{2}$ is independent of $\frac{X_{1}}{Z}$. How do I conclude then that $Z=X_{1}+X_{2}$ is independent of $(\frac{X_{1}}{Z},\frac{X_{2}}{Z})$? </p> | 37,421 | [
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<p>Regression: Wage=b0+b1collegegrad, where collegegrad is a dummy variable. Suppose you want to estimate the wage ratio between college graduates and non-college graduates. Is the estimator theta=b0/b1 consistent?</p>
<p>My thinking is that if we could increase our sample size to infinity, we would cover the entire population and thus get the true ratio, so the estimator is consistent. Am I correct, or am I missing something?</p> | 74,518 | [
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