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<p>Good day! </p>
<p>I'm a second year psych grad student. From my fascination with formal logic, I wondered (just personal; not homework) about how to statistically establish necessary/sufficient relationships between variables. </p>
<p>Let's suppose two variables, Intelligence and Creativity, whose relationship is summarized by the conditional, "if creative, then intelligent".</p>
<p>So on the high end of Intelligence, we find the full spectrum of Creativity; but on the low end of Intelligence, we find only low levels of Creativity. We can say that Intelligence is a necessary (but not sufficient) condition for Creativity, while Creativity is a sufficient (but not necessary) condition for Intelligence.</p>
<p>Two questions:</p>
<p>(1) I imagine the scatter plot for Intelligence and Creativity would look like a right triangle whose non-hypotenuse legs are perpendicular to the axes. If Intelligence is on the x-axis, then the right angle would be at the +x,-y corner. Is this correct?</p>
<p><a href="http://i.stack.imgur.com/i9Lqw.jpg" rel="nofollow">intelligence on the x-axis; creativity on y-axis</a></p>
<p>(2) If I imagined it correctly, what formula or procedure do I use to objectively measure how well data follows such a relationship? Is there an already-established way for dealing with this? My psych professor once told me that there is actually such a relationship between Intelligence and Creativity, but when I asked how to measure it, he said something only to the effect of "look at the scatter plot". All I know is that Pearson correlation is inappropriate, as the relationship is not linear. But I don't want to just look at the scatter plot. </p>
<p>It's my first time to post here, so please be patient with me. Thank you!</p> | 73,434 | [
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<p>I have a matrix with about 550k elements (2500 x 220) with 100k values known and the rest are unknown.</p>
<p>Would it make sense to use matrix completion in this case, or are there too many values which are missing? What should I be careful of when I use matrix completion with it, if there are enough values there?</p> | 73,435 | [
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<p>I would like to test two relatively small samples against the null hypothesis that both their means and variances are the same. The alternative would be that they in fact differ. I saw a post on this site advocating a ML test but I recall there is also a named test for this case, which I would ideally like to use in R, but whose name I forget. Can anyone help? Normality assumptions might be reasonable but would be difficult to test.</p> | 38,105 | [
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<p>Mathematically an AR model is expressed as:
$$ X_t = -\sum_{i=1}^p a_i X_{t-i} + \eta_t$$</p>
<p>Resources say that $\eta_t$ is a white gaussian noise. Is this $\eta_t$ the residual error? When invoking Matlab commands like <code>ar()</code> or <code>arburg()</code> or <code>aryule()</code> do we take into account this term? What is the difference between the $\eta_t$ and the residuals?</p> | 73,436 | [
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<p>Suppose that I have an assortment of summary statistics on a sample, and some beliefs about the underlying distribution, but no access to the sample itself. Each of the sets of summary statistics is sufficient to allow me to estimate the parameters of the distribution that I believe applies. For, instance, I may have (1) the quintiles (quintile break points) (with published sample variance estimates for each), (2) the inter-quintile sample means (with variance estimates), and, (3) for some finer disaggregation into value categories, the number of observations that fall into each category (with variance estimates). </p>
<p>I know more or less how to do parameter estimates given any one of these summary statistics – though I am not sure how to make best use of the accompanying variance estimates for each. I could produce 3 distinct parameter estimates with a little work. But suppose I want to produce a single set of parameter estimates that makes the best use of all of the summary statistics that are available? </p>
<ol>
<li>How would I combine the three (or however many I have – this is a simplified example) different types of summary statistics to get one set of parameters?</li>
<li>Can I make any further improvement to my parameter estimates by using the variance estimates that accompany each?</li>
</ol>
<p>When I say “best” I am thinking maximum likelihood, but if there is an interestingly different and perhaps better answer for minimum mean square error estimates or MAP estimates or something, I’d love to know that too.</p> | 73,437 | [
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<p>The cost function of my TreeBagger class and fitensemble (Bag method) are both [0 8;1 0] for binary classification. The confusion matrix on fitensemble shows that the classfication tends to turn in the favor of the costy class (like [100 0; 20 80] favoring false negatives) but the same on TreeBagger does not hold. I have tried it on 3 datasets and TreeBagger seems to ignore the cost as I have omitted the cost argument and the confusion matrix is like when you dont give cost at all. Is there a Problem with my code or is it the TreeBagger? (BTW the problem is not misinterpretation of confusion matrix, it just doesn't work)</p>
<p>My Code </p>
<p>(TREEBAGGER WHICH DOES NOT WORK)</p>
<pre><code>RF=TreeBagger(150,Xtrain,Ytrain,'oobpred','on','cost',[0 8;1 0])
</code></pre>
<p>(FITENSEMBLE WHICH WORKS)</p>
<pre><code>Bag=(Xtrain,Ytrain,'Bag',150,'type','classification','cost',[0 8;1 0]);
</code></pre>
<p>Thank you for your help.</p> | 37,790 | [
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<p>I was wondering what's the difference between variance and standard deviation? </p>
<p>If you calculate it its is clear that you get the standard deviation out of the variance...</p>
<p>But what does that mean in terms of the distribution you are looking at?</p>
<p>Why do you really need a standard deviation?</p>
<p>I appreciate your answer!!!</p> | 49,832 | [
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<p>Patients arrive randomly and independently at a doctor's surgery from
8·0 A.M. at an average rate of one in Five minutes.The waiting room holds 2
persons. What is the probability that the room will be full when the doctor
arrives at 9·0 A.M.?</p>
<p>Answer is 53.84%</p>
<p>I have tried Poisson distribution with parameter as 12/Hour? and i am getting like 99% as answers.</p> | 25,010 | [
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<p><b>Summary:</b> The last formula boxed in red (which is a modified log likelihood from logistic regression) is a special non-differentiable loss function that is adapted to contain a Bonferroni correction in the loss function. The function is non-differentiable with respect to theta which is a problem for gradient descend and thus for using it to train regressors and neural networks. If it could become differentiable the function could optimise a classifier specifically designed to <b>increase the power of multiple hypothesis testing procedures in the case there is useful additional information that goes with the tests</b> and would then lead to more significant findings in cancer research. my question thus is how to make the formula boxed in red differentiable with respect to theta (preferably with some theoretical justification! ;P ).
<br><br></p>
<h2> Intro </h2>
<p>For some research into the <b>Biology of Cancer</b> (details excluded for clarity sake)
I devised a method to mitigate the issue finding important drivers of the process, which is in essence an issue of Multiple Hypothesis testing. <br> To do this I combine Machine Learning approaches with Multiple Hypothesis testing correction and aim to end up with valid/significant drivers of the cancer process.
<br><br></p>
<h2> The Challenge </h2>
<p>Consider the following situation: One is supplied with a bunch of independent statistical tests that each have some additional information that comes with them as displayed in figure 1 below. For these statistical test the goals is to perform Multiple Hypothesis Testing (MHT) and call significance for all the 7 tests, using <i>Bonferroni correction</i>:
<br></p>
<p><img src="http://i.stack.imgur.com/rbNSb.png" alt="Figure 1: A bunch of independent statistical tests">
<i><b> Figure 1: Seven independent statistical tests, with additional information </b></i></p>
<p>Confronted with the situation in figure 1 one would perhaps like to make use of the fact that all the tests with additional information 'A' attached to them have very low p-values and in a way <i>pre-select</i> them for Multiple Hypothesis Testing (MHT). In this example case the number of statistical test is small (7), but there are of course situations were one has to correct for many more (>10^6). The use of the additional information could in those cases really increase the overall power of the procedure. <br></p>
<p>Note that in this case the additional information is comprised of only one bit of information (A/B), but could just as well be a complex feature vector (this is where the <b>Machine Learning</b> comes in !)
<br><br></p>
<h2>Possible Solution 1</h2>
<p>Given that the statistical tests are <em>independent</em> (note: in my case they actually aren't fully independent but that is perhaps something for another post) one could use Machine Learning algorithms to try to predict the -log10(p) value for for instance test 1 by training a linear regression model on tests 2 till 7.
One could do this for all the tests, effectively a leave-one-out cross validation and arrive at an unbiased prediction of the -log10(p) and use this information to select the really juicy 'A' type tests over the 'B' type, thereby decreasing the number of test in the MHT, increasing the power.<br></p>
<p>--> In machine learning terms: The additional information becomes the feature vectors (x), The individual tests are the samples and the -log10(p-values) of the tests become the labels (y). After this the tests/samples can freely be distributed in training and test sets for cross validation because all the samples are stated to be <i>independent</i>.</p>
<p>There is only one issue with the approach above: It does not optimize for the desired end result being to find as much as possible significant results. In small sets like here this does not constitute a problem, but in large MHT problems with only a few small p-values this leads in effect to regression methods fitting the large 'uninteresting' part of the distribution, leading to poor performance on the small p-values it should actually focus on. A possible solution could be weighting of the samples, but this is problematic since it is not clear what an optimal weight should be. <br></p>
<p>Therefore another solution is needed that results in a model that actually optimizes
the desired end result, being the largest possible number of significant results after MHT correction. This is the target of the next possible solution (2).
<br></p>
<h2>Possible Solution 2</h2>
<p>To have a Machine Learning algorithm try to maximize the number of significant results on a test set, the method should be supplied with an <i>appropriate cost function</i>. If we assume for a moment that the dataset essentially consists of 2 classes being the significant tests and non significant test and given that these classes were known one could maximize the following log likelihood (cross entropy) as cost function to arrive at a model:</p>
<p><img src="http://i.stack.imgur.com/gYMhc.png" alt="Cross Entropy"></p>
<p>Which is a nice differentiable cost function that can be used for gradient descend. In this equation n is the total number of samples. y<sup>i</sup> represents the i'th example in the training set which can have the value 1 for the 'significant class' or 0 for the 'non-significant class' and h<sub>theta</sub>(x<sup>i</sup>) represents the output by the model, which has the following structure:</p>
<p><img src="http://i.stack.imgur.com/QJhdY.png" alt="Hypothesis h_theta(x)/Model"></p>
<p>With Theta being the model parameters and x being the feature vector. Now,.. If one reckons that the second term of the loss function (1st eq.) is concerned with assigning a cost to wrongly classified samples of the non-significant class removing it makes sense because the classification of non-significant samples is not of interest. One could also take the Bonferroni MHT correction into account in the cost function and one could modify the cost function into: </p>
<p><img src="http://i.stack.imgur.com/TfLq4.png" alt="Modified loss function"></p>
<p>Indeed the strikethrough is not part of the formula anymore. I{A} here is an indicator function that is 1 if A is true and 0 otherwise. In the brackets of the first indicator function above one can see the Bonferroni correction for the i'th p-value, in fact determining its class (sig. or non-sig). This class label also depends on m (the number of statistical test performed) which is determined by the sum of the next indicator function over the sample hypotheses. If h<sub>theta</sub>(x) reaches over 0.5 it is more likely according to the model that the sample is significant so it should thus be included in the final MHT correction.
(note: The value of m/the cost function should eventually be corrected somewhat for the fact that the procedure excluded the one sample in the leave-one-out cross validation).
<br></p>
<h2>Concluding</h2>
<p>This last MHT cost function is a function that in fact optimizes the desired goal of finding the largest possible number of significant tests, but because of the indicator functions it is now non-differentiable, which is a big issue of gradient descent optimisation.
The <b>Final Challenge</b> is to modify the MHT cost function to make it differentiable in such that it can directly be used for (stochastic) gradient descend (SGD). (One of the reason this is desired is because of the fact that that number of test is VERY large, but that SGD implementations on GPUs can crack the problem. Another reason is also the fact that it could be used to train a deep neural network, which has the benefit that it can model the highly non-lin representation that are likely to be found in the data.<br>
I think that this cost function would really help to improve the number of discovered cancer drivers.
<br><br></p>
<p>Hope you guys can help me out. Really curious though :) Perhaps there is even a better approach to this problem than my suggested path.
<br><br></p> | 73,438 | [
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<p>I have collected temperature readings for 4 locations on a beach in Cornwall UK using data loggers. The loggers recorded temperature every 15 mins accurate to .1 of a degree (Celsius). The loggers collected data for 10 days with a one day break to calibrate the sensors. In total I used 30 sensors with 6-7 sensors in each position.</p>
<p>I have now downloaded the data and have produced time-series for temperature over the 10 day period. Visual inspection of the time-series shows how the temperature changes with tides and night and day. I want to investigate whether the time-series differ significantly from one another. Please could someone point me in the direction of a technique to do this in SPSS?</p> | 49,621 | [
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<p>We are extracting features from EEG, which is a time dependent signal.</p>
<p>We have signals of 10,000 datapoints over 64 channels, and we extract 10 features per timestamp per channel, so at the end we have a feature data set of 64x10 features for each time step.</p>
<p>We are applying PLS to regress against hand movements (X,Y,Z). </p>
<p>We are using cross validation to verify the results, but some of our colleagues are pointing out that using interleaved sampling from the datasets feels like cheating.</p>
<p>Their suggestion is to use 2/3rds of contiguous data for training and 1/3rd for testing.</p>
<p>I would like to hear the overall opinion on this setting, my intuition tells me that since it is time dependent data, just dividing the datasets like that would yield bad results (which it does)</p> | 73,439 | [
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<p>I have 5 different computational tasks. I executed both tasks on two laptops named LA and LB. Out of 5 tasks, 4 tasks were computed in less time on LB, while 1 task took less time on LA. I want to measure, that the achieved performance gain is significant or not. Below is my data example:</p>
<pre><code> LA LB
Task1 2.3 4.1 (mean values of 10 executions)
Task2 5.6 2.3
Task3 10.5 3.4
Task4 15.2 4.6
Task5 11.3 3.1
</code></pre>
<p>I have the following questions:
1. Which test should I apply to find the significance?
2. Can I apply that test on mean values rather than real 10 execution values?
3. Which parameters (Alpha, P, t) of the test must be presented in table and discussed, as I cannot present/discuss all parameters due to space limitation.</p>
<p>Thank you</p> | 27,661 | [
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<p>I'm trying to find out the name of a distribution that is like negative binomial, only for finite population and without replacement. Or like Hypergeometric distribution where the last event has to be a success.</p>
<p>That is:</p>
<p>Let's say we have N balls in an urn, where W of them are white balls and B are black balls. I want to know what are the chances of needing to draw exactly n balls so it is the first time I got k of the n drawn balls to be black (e.g: that the last ball drawn was black, and a total of k out of n are black).</p>
<p>I can't seem to find the name of such a distribution, so if you could also advise me on where to find names of distributions, that would also be lovely.</p>
<p>Thanks.</p> | 48,371 | [
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<p>Can some of you tell me why the following holds true?:</p>
<p>$ P(A \& B^C )= P(A)-P(A\&B)$ in which: $ B^C$ is $B$ complement.
In which: $A$ and $B$ are independent.</p>
<p>thanks..</p> | 35,823 | [
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<p>I'm working on a data set for order prediction/classification and a close deadline upcoming. Fortunately, my university has a super-computer with restricted access. I was thinking of using a few nodes (each node is a 16-core processor, 2.3 GHz each and 32 GB of memory running on linux) to overcome the time limitation I'm facing. </p>
<p>Basic software that I have in mind is WEKA, RapidMiner, MATLAB and maybe R. But as it's my first confrontation to parallel data mining, I need some help on this. I think the software needs to have a parallel nature in order to get most out of multi-core processors and using a lot of cores doesn't make any significant speed-up on serial programs. </p>
<p>Best result of my searches was WEKA-parallel that is a modification of WEKA on <em>distributed</em> systems. </p>
<p>At last I'm open to any suggestion on getting most out of this computational power specially using WEKA. New software, new method, anything... </p> | 35,824 | [
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<p>To my understanding, "Control" can have two meanings in statistics.</p>
<ol>
<li><p>Control group: In an experiment, no treatment is given to the member of the control group.
Ex: Placebo vs Drug: You give drugs to one group and not to the other (control), which is also referred as "controlled experiment". </p></li>
<li><p>Control for a variable: Technique of separating out the effect of a particular independent variable. Some of the other names given to this techniques are, " accounting for", "holding constant", "controlling for", some variable.
For example: In a Football viewing study (like or not like), you may want to take out the effect of gender as we think gender causes bias, that is, male may like it more than female. </p></li>
</ol>
<p>So, my question is for point (2). Two questions:</p>
<p>How do you "control"/ "account for" variables, in general. What techniques are used? (In terms of regression, ANOVA framework). </p>
<p>In above example, does choosing male and female <em>randomly</em> constitutes control? That is, is "randomness" one of the techniques for <em>controlling</em> other effects?</p> | 73,440 | [
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] |
<p><img src="http://i.stack.imgur.com/9JFmv.gif" alt="enter image description here"></p>
<p>In the chart above, how can I derive the area between the two graphs, taking into consideration positive and negative areas?</p>
<p><img src="http://i.stack.imgur.com/lP3EJ.gif" alt="enter image description here"></p> | 73,441 | [
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<p>Assuming that the distribution has finite variance (a condition not required for the LLN), then doesn't the LLN follow from the CLT?</p> | 35,827 | [
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<p>Let's say I am building a logistic regression model where the dependent variable is binary and can take the values $0$ or $1$. Let the independent variables be $x_1, x_2, ..., x_m$ - there are $m$ independent variables. Let's say for the $k$th independent variable, the bivariate analysis shows a U-shaped trend - i.e., if I group $x_k$ into $20$ bins each containing roughly equal number of observations and calculate the 'bad rate' for each bin - # observations where y = 0 / total observations in each bin - then I get a U shaped curve.</p>
<p>My questions are:</p>
<ol>
<li>Can I directly use $x_k$ as input while estimating the beta parameters? Are any statistical assumptions violated which might cause significant error in estimating the parameters?</li>
<li>Is it necessary to 'linearize' this variable through a transformation (log, square, product with itself, etc.)?</li>
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<p>Beta stability in linear regression with high multi-collinearity?</p>
<p>Let's say in a linear regression, the variables $x_1$ and $x_2$ has high multi-collinearity (correlation is around 0.9).</p>
<p>We are concerned about the $\beta$ coefficient stability so we have to treat the multi-collinearity.</p>
<p>The textbook solution would be to just throw away one of variables. </p>
<p>But we don't want to lose useful information by simply throwing away variables.</p>
<p>Any suggestions? </p> | 73,443 | [
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<p>I came across a topic I am somewhat confused about: the merging of two variables.</p>
<p>Assume we have two measurements from the same subjects. The two variables ($x_1$ and $x_2$) are measuring something similar, but not exactly the same thing. The variables (or combined variable, called $x_{12}$) will later be used as an explanatory variable ($X$) of some other variable ($Y$).</p>
<p>For example, let's say we want to estimate the IQ of someone, and we only have the IQ of his father and mother (and we don't know the child's gender).</p>
<p>What statistical (and non-statistical) issues are relevant for deciding on whether or not to combine the two measurements into one?</p>
<p>Some issues to consider:</p>
<ol>
<li>Let us say that we would later fit a linear regression of the type $Y$~$X$ (where $X$ is either $x_1$ and $x_2$ or the combination of the two), is there a time we would rather merge the two variables ($x_1$, $x_2$) into one?</li>
<li>How does the association of the two variables ($x_1$ and $x_2$) relevant to the decision on whether or not to merge them?</li>
<li>Is there any relation between $x_1$/$x_2$ and $Y$ that might influence the merging decision?</li>
<li>What if $x_1$ and $x_2$ are ordinal variables, or forced integer variables, does that make a difference on the value of collapsing them in order to be merged?</li>
<li>Is there any other issue to consider on this topic that I didn't mention?</li>
</ol> | 47,228 | [
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<p>We're creating a chart showing traffic by time of day over a given period. So the y-axis is traffic, the x-axis is midnight, 1am, 2am, etc. It could also be days of the week. What's the generic name for this type of chart? I've come up with "cycle chart". Is that the standard? Is there one?
<img src="http://i.stack.imgur.com/IDu3k.png" alt="example"></p>
<p>Update:</p>
<p>Just to add a bit more clarity, what's being shown in the top chart is <em>not</em> one day, it's an aggregation of many days. E.g. over the last month, 6am has on average been lower than noon. Similarly, in the bottom chart, over the last year, traffic dips on Saturdays.</p> | 73,444 | [
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<p>I am required to analyse samples a compositional data set. However my samples come from seven different locations. Is there any way for me to check if there is a location/mean effect in the data which I can remove from the Isometric Log Ratio (ILR) transformation of the data before analysing? Can I carry out the test on the ILR data, which is obviously straight-forward, or must the test be on the compositional data?</p> | 73,445 | [
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<p>I would like to test how well my model fits the data. The response is binary and the Chi-Squared Test cannot be applied for the residual deviance because the $n_i$ are $1$. To use the Chi-Squared GOF Test, the $n_i$ need to be $\geq 5$. What's an alternative method can I use to test for goodness of fit?</p> | 73,446 | [
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<p>We know that density for a student-t distribution is given as</p>
<p>$$\frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} \left[1+\frac{\lambda(x-\mu)^2}{\nu}\right]^{-\frac{\nu+1}{2}}$$</p>
<p>with
$\text{E}(X) = \mu$,
$\text{var}(X) = \frac{1}{\lambda}\frac{\nu}{\nu-2}$</p>
<p>where the three parameters are shape factor $\nu$, location $\mu$ and dispersion $\lambda$.</p>
<p>Now if I standardize X to Z as $\frac{X-\text{E}(X)}{\text{Std}(X)}$, what is the distribution of $Z$. If I standardize my input data to $Z$ as above, what is the density function for $f_Z(z)$?</p>
<p>I came across this question in Stephen Taylor's Asset Dynamics book. I am a bit confused after reading this question. As I understand, $\frac{X-\mu}{\sigma}$ where $\sigma=1/\sqrt(\lambda)$ will give the standardized t distribution with center 0 and disperson 1. But not sure what $Z$ above will look like.</p> | 35,831 | [
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<p>I like to find the feature weights in a <a href="http://en.wikipedia.org/wiki/Structured_support_vector_machine" rel="nofollow">structured SVM</a> for ranking the features w.r.t. importance. I know that in a binary SVM the weight vector can be written as a <a href="http://pyml.sourceforge.net/doc/howto.pdf" rel="nofollow">linear combination of examples</a>. But how do you compute the same for an SSVM?</p> | 73,447 | [
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<p>Let's say I'm performing regularized regression and I want to validate the results using holdout. (I'm choosing holdout instead of cross-validation because my dataset is fairly large, so computational power is an issue and the difference between holdout and cross-validation estimates will likely be negligible.) I want to put a confidence interval on the $R^2$ score for the holdout set. Is the following procedure valid?</p>
<ol>
<li><p>Compute the regression coefficients on the training data.</p></li>
<li><p>Compute the predictions on the held out data. Treat each prediction as if it were an observed random variable.</p></li>
<li><p>Compute the Pearson correlation of the predictions with the actual $Y$ values for the held-out data. Compute a confidence interval for this correlation using Fisher's Z-Transformation.</p></li>
<li><p>If the confidence interval of this correlation is some set of real numbers, then the confidence interval of $R^2$ is just the set of the squares of these numbers.</p></li>
</ol>
<p>I can't think of any reason why this procedure would be wrong, but I've never heard of it being done before, so I wanted to run it by some more experienced statisticians.</p> | 73,448 | [
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<p>Why the expected value $\int h(x)f(x)$ is desired for inference in Dirichlet Process Mixture?
What is the intuition for MCMC in Dirichlet Process Mixture?</p>
<p>$f(x)$ is the probability density function, and $h(x)$ is the outcome of the Dirichlet Process Mixture (I am not sure).</p>
<p>This formula is originally from page 40 of <a href="http://people.cs.pitt.edu/~milos/courses/cs3750/lectures/class23.pdf" rel="nofollow">slides</a>.</p> | 73,449 | [
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<p>I have a 2D plane (a large rectangle) with a finite size in the x and y direction, which is the field of my problem.</p>
<p>The field is covered by $n$ smaller rectangles that are located randomly within its bounds (the smaller rectangles are free to overlap each other).</p>
<p>If I "sample" the field at some random location I will intersect some number $i$ of smaller rectangles, which happen to overlap this location.</p>
<p>My goal is to estimate the expected number of intersections, with a confidence interval (say 95%).</p>
<p>Is there a way to do this analytically? Any help is appreciated. Lots of detail and laymans terms would be appreciated. As you can probably tell, statistics is not my strong suit.</p>
<p>Edit: Please see comments below for important additional information.</p> | 73,450 | [
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<p>I'm trying to generate random integers which have Poisson distribution. <a href="https://www.gnu.org/software/gsl/manual/html_node/The-Poisson-Distribution.html#The-Poisson-Distribution" rel="nofollow">The open source library GSL has one such distribution</a>.</p>
<blockquote>
<p>Function: <code>unsigned int gsl_ran_poisson (const gsl_rng * r, double mu)</code>
This function returns a random integer from the Poisson distribution with mean <code>mu</code>. The probability distribution for Poisson variates is <code>p(k) = {mu^k over k!} exp(-mu)</code></p>
</blockquote>
<p>I want to know what it actually returns. Does it return <code>k</code> for a particular value of <code>p(k)</code>?</p>
<p>For a particular value of <code>mu</code>, <code>k</code> can take values from <code>0,1,2,3...</code>
So if I specify a value for <code>mu</code>, do I have an upper bound on the values returned by this function?</p> | 73,451 | [
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<p>Do you know <a href="http://en.wikipedia.org/wiki/Bradley_Efron" rel="nofollow">Bradley Efron</a>? He's a great man.<br>
How did Efron imagine or think about "bootstrap" for the first time?</p> | 73,452 | [
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<p>I'm a fairly inexperienced statistician fighting a huge deadline and just need some peace of mind that I'm not making a massive error here. I'd be most grateful for any pointers.</p>
<p>I've been working on fitting an <code>lmer()</code> model that explores the spatial relationships between crime and socio-economic factors. The ICC demonstrates a 40% attribution of variances to group factors, which is encouraging. The model's evolution has seen a steady improvement of AIC scores. I am now a little confused about how to demonstrate its effectiveness against other non-HLM techniques. Is this purely by standard error residuals, or am I missing something important?!</p>
<p>Many thanks</p> | 73,453 | [
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<p>I have an interesting modelling problem in which I am trying to forecast the occurrence of a type of weather event using an empirical model driven by measurements of a number of different physical quantities in different locations.</p>
<p>There are a number of complications. First, the predictor measurements are not independent in the sense that the driving force that generates these weather events (which we don't measure directly) has an impact on all the quantities that we measure, driving them from their normal values and giving us warning of the weather event to follow.</p>
<p>The second complication is that the training data is relatively sparse, so it cannot support highly complex models. Using simple linear models is problematic though, because of the stated non-independence of the predictors. We could use an approach like principal component regression to solve this, except for the third complication, which is that for both the training data we have and in operations, data is frequently missing due to sensors being broken or offline at the time.</p>
<p>So, what strategies might be useful here? There are a few basics approaches I can see. Say for example we want to forecast the probability of event F occurring, based on the unreliable predictors A and B. We could train 3 completely separate models:</p>
<p>$$P(F|AB), P(F|A), P(F|B)$$</p>
<p>using whatever approach works best for each model and then use whichever model is appropriate given the available data. This however does not generalise for many predictors (we have at least 50 different quantities we could use) due to the curse of dimensionality.</p>
<p>The next simplest approach I've considered would be to construct separate models for each predictor alone</p>
<p>$$ P(F|A), P(F|B) $$</p>
<p>and then learn weights for each model so that we have
$$ P(F|AB) = \frac{w_A P(F|A) + w_B P(F|B)}{w_A + w_B} $$</p>
<p>the weights would be learned over the whole training set, setting the relevant weight to zero for each instance where a predictor is missing. This is a much more tractable solution, but I fear it would miss important interactions between the predictors and would sacrifice too much performance for the gain in simplicity.</p>
<p>A third approach I can vaguely conceive of but I'm not sure if it is valid, is to try and express</p>
<p>$$P(F|AB)$$</p>
<p>in terms of the individual models</p>
<p>$$P(F|A), P(F|B) $$</p>
<p>and the conditional probabilites e.g.</p>
<p>$$P(A|B)$$</p>
<p>the form of which would need to be learned from the training data. I haven't managed to come up with a useful form for this yet, although maybe it isn't possible (or advisable). The curse of dimensionality is likely to be a factor here as well I guess.</p>
<p>I realise this could be phrased as a data imputation problem but I'm looking for a solution that gives a good trade-off between practicality and performance.</p> | 73,454 | [
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<p>I have monthly usage data (spanning 3 years) for a customer base of around 200K, and I need to generate 1-month ahead forecasts for each of them. There are a couple of exogenous variables that would need to be included too.</p>
<p>One way to go about is to build an "Arima with exogenous variables" model for each customer. In which case, I would be looking to manage 200K models. I'm wondering if there is an alternate approach to solving this -- Assuming we are able to segment the customer base into groups that are "homogenous" in terms of usage, then would it make sense to create one model for each group? Would it be a good approximation to have the usage data time series for each group correspond to the mean values across the customers in that group? </p>
<p>Any pointers to literature related to this problem is appreciated. Additionally, I'm looking to use R, so pointers to specific packages will be of help as well.</p>
<p>Thanks!</p> | 73,455 | [
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<p>I am trying to test whether means of two populations with lots of zeros are different. Here is the following python code example:</p>
<pre><code>from scipy.stats import mannwhitneyu
import numpy as np
a = np.random.random(100)
b = np.random.random(100) * 2
aa = np.hstack((a, np.zeros(1000)))
bb = np.hstack((b, np.zeros(1000)))
np.random.shuffle(aa)
np.random.shuffle(bb)
mw_stat_1, p_value_1 = mannwhitneyu(a, b) # 100 obs in a and b
mw_stat_2, p_value_2 = mannwhitneyu(aa, bb) # 1000 zeors added to each a and b
# I take mean of sample of 20 elements, a and b become an array of 55 elements
samp_sum_aa = aa.reshape(-1,20).mean(axis=1)
samp_sum_bb = bb.reshape(-1,20).mean(axis=1)
mw_stat_3, p_value_3 = mannwhitneyu(samp_sum_aa, samp_sum_bb)
</code></pre>
<p>Result:</p>
<pre><code>>>> p_value_1 # no zeros
2.5956488654494193e-09
>>> p_value_2
0.42124151395226317
>>> p_value_3 # using sampling
0.0020853586023447269
</code></pre>
<p>I find that if I do Mann-Whitney test on raw populations (after zeros are added), my p-value is large; however, if I take random samples, I get a small enough p-value, such that I can reject the null hypothesis for all practical purposes. </p>
<p>Is sampling a proper technique here? If so, how do I know what is the right sample size? Are there other methods to address this problem?</p> | 12,805 | [
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<p>I recently read a paper in which the authors used Cronbach's $\alpha$ and $\omega$ as measures for unidimensionality and homogeneity. In case of $\alpha$ I know that it is wrong, but I am not so familiar with $\omega$. The literatur I read said it is the proportion of test variance due to a general factor. Does that mean a high value in $\omega$ would imply unidimensionality and homogeneity, because it seems some researchers seem to make a point to differentiate between $\omega$ and unidimensionality and homogeneity, but I do not see why. </p> | 35,841 | [
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<p><strong>Question:</strong>
Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as</p>
<p>$E(\mathcal{O}^n_1)= \displaystyle\int_{-\infty}^{+\infty}nx\Phi(x)^{n-1}\phi(x)\:dx$.</p>
<p>I wish to show that for $n_1<n_2<n_3$,</p>
<p>$E(\mathcal{O}^{n1}_1)<E(\mathcal{O}^{n2}_1)<E(\mathcal{O}^{n3}_1)$, and </p>
<p>$E(\mathcal{O}^{n3}_1)-E(\mathcal{O}^{n2}_1)<E(\mathcal{O}^{n2}_1)-E(\mathcal{O}^{n1}_1)$, </p>
<p>where $\mathcal{O}^{n1}_1$ is the first order statistic (maximum) for a sample of $n1$.</p>
<p><strong>Progress so far</strong>
I've managed to prove the part for first-order derivative by invoking the concept of first-order stochastic dominance. But still no progress on the second-order...</p>
<p>Any help will be greatly appreciated. Thanks!</p> | 73,456 | [
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<p>I have relatively large (100k items) dataset which I need to split in two groups. So far I've tried knn and the results are not good mainly because I have disproportion in my training data: 90% of points belong to the first group. The same proportion is expected to be in test data.</p>
<p>Is there a way to improve prediction quality with this kind of data? Performance is not important while quality of prediction is paramount. </p> | 73,457 | [
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<p>Wikipedia <a href="http://en.wikipedia.org/wiki/Sample_%28statistics%29" rel="nofollow">defines a sample</a> as:</p>
<blockquote>
<p>a subset of a population.</p>
</blockquote>
<p>While exploring the reason why we divide by $(n-1)$ instead of $n$ when calculating standard deviation (discussed in <a href="http://stats.stackexchange.com/questions/3931/intuitive-explanation-for-dividing-in-n-1-when-calculating-sd">this question</a>), I came across <a href="http://nebula.deanza.edu/~bloom/math10/m10divideby_nminus1.pdf" rel="nofollow">this PDF</a> demonstrating why $(n-1)$ is better.</p>
<p>When listing all possible samples of $n=2$ from a population of three cards numbered 0, 2, and 4, it includes the samples (0,0), (2,2), and (4,4). I am having trouble reconciling this with the definition of a sample that I thought I knew (and that is given by Wikipedia).</p>
<p>A sample of 2 playing cards from a population of 52 would not include the Three of Hearts twice, would it? Likewise, I'd guess a survey of a sample of voters would not include the same voter multiple times.</p>
<p>Other sources back the method described in the PDF. What am I misunderstanding here?</p> | 73,458 | [
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<p>I'm reading "The Drunkard's Walk" now and cannot understand one story from it.</p>
<p>Here it goes:</p>
<p><i>Imagine that George Lucas makes a new Star Wars film and in one test market decides to perform a crazy experiment. He releases the identical film under two titles: "Star Wars: Episode A" and "Star Wars: Episode B". Each film has its own marketing campaign and distribution schedule, with the corresponding details identical except that the trailers and ads for one film say "Episode A" and those for the other, "Episode B". </p>
<p>Now we make a contest out of it. Which film will be more popular? Say we look at the first 20,000 moviegoers and record the film they choose to see (ignoring those die-hard fans who will go to both and then insist there were subtle but meaningful differences between the two). Since the films and their marketing campaigns are identical, we can mathematically model the game this way: Imagine lining up all the viewers in a row and flipping a coin for each viewer in turn. If the coin lands heads up, he or she sees Episode A; if the coin lands tails up, it’s Episode B. Because the coin has an equal chance of coming up either way, you might think that in this experimental box office war each film should be in the lead about half the time. </p>
<p>But the mathematics of randomness says otherwise: the most probable number of changes in the lead is 0, and it is 88 times more probable that one of the two films will lead through all 20,000 customers than it is that, say, the lead continuously seesaws"</i></p>
<p>I, probably incorrectly, attribute this to a plain Bernoulli trials problem, and must say I fail to see why the leader won't seesaw on average! Can anyone explain?</p> | 73,459 | [
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<p>I have a data set with ~ <code>7 million</code> rows, of which ~ <code>100k</code> are positives. I'm looking to shrink the data by keeping all the positives and then randomly sampling <code>several hundred thousand</code> negative examples to round out the data set.</p>
<p>I'm uncertain what the guidelines are though. Are there any rules of thumb for what % needs to be positive / negative? </p>
<p>Any suggestions would be appreciated.</p> | 35,844 | [
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<p>I am currently analysing data where the outcome variable is 'U' shaped. The outcome variable asks 'how many of the last seven days have you smoked'. Most responses to this fall in the first (none) and last (all seven) categories. Because of this I do not think a count data model is appropriate.</p>
<p>What would be a good approach to modelling this variable?</p> | 73,460 | [
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<p>We often face singular matrices in practice: OLS with singular (X'X), GMM with singular weighting matrix, singular matrix in Wald statistics. I'm wondering how can we overcome this issue. I've seen two solutions used in different contexts:</p>
<p>1) Ridge regression: <a href="http://en.wikipedia.org/wiki/Tikhonov_regularization" rel="nofollow">http://en.wikipedia.org/wiki/Tikhonov_regularization</a></p>
<p>2) Generalized inverse: <a href="http://en.wikipedia.org/wiki/Generalized_inverse" rel="nofollow">http://en.wikipedia.org/wiki/Generalized_inverse</a></p>
<p>What do you think would be preferable to do when the estimator or test statistics depends on the inverse of a matrix which is singular?</p> | 73,461 | [
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<p>I'm a fairly inexperienced statistician fighting a huge deadline and just need some peace of mind that I'm not making a massive error here. I'd be most grateful for pointers.</p>
<p>I've been playing around with different model combinations using the <code>lmer()</code> function, and have managed to get the AIC scores down from over 10,000 with the most basic combinations to -180 by factoring in various random effects. But I'm concerned I might be creating bias by using variables that already correlate. Is this a valid concern or does the model mitigate for this?</p>
<p>Many thanks in advance.</p> | 73,462 | [
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<p>One of the often cited issues in RNN training is the vanishing gradient problem [1,2,3,4].</p>
<p>However, I came across several papers by Anton Maximilian Schaefer, Steffen Udluft and Hans-Georg Zimmermann (e.g. [5]) in which they claim that the problem doesn't exist even in a simple RNN, if shared weights are used. </p>
<p>So, which one is true - does the vanishing gradient problem exist or not? </p>
<hr>
<ol>
<li><p><a href="http://dx.doi.org/doi:10.1109/72.279181" rel="nofollow">Learning long-term dependencies with gradient descent is difficult </a> by Y.Bengio et al. (1994)</p></li>
<li><p><a href="http://dx.doi.org/10.1142/S0218488598000094" rel="nofollow">The Vanishing Gradient Problem During Learning Recurrent Neural Nets and Problem Solutions</a> by S.Hochreiter (1997)</p></li>
<li><p><a href="ftp://ftp.idsia.ch/pub/juergen/gradientflow.pdf" rel="nofollow">Gradient Flow in Recurrent Nets: the Difficulty of Learning Long-Term Dependencies</a> by S.Hochreiter et al. (2003)</p></li>
<li><p><a href="http://arxiv.org/abs/1211.5063" rel="nofollow">On the difficulty of training Recurrent Neural Networks</a> by R.Pascanu et al. (2012)</p></li>
<li><p><a href="http://dx.doi.org/10.1016/j.neucom.2007.12.036" rel="nofollow">Learning long-term dependencies with recurrent neural networks</a> by A.M. Schaefer et al. (2008)</p></li>
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<p>I have two algorithms A and B, used to automatically classify each of N elements into K categories, N and K both being in the millions. Neither A or B is perfect, but it is relatively easy for a human to do the classification, or put another way, to select which one of the algorithms A and B did a better job for a given element.</p>
<p>I want to show that algorithm A is better than B, but am not sure how to approach this problem. As A and B most often say the same thing, I want to focus on the cases where they don't. My question is now, how many elements do I need to check to show (with a certain degree of confidence) that A is better than B? Or put another way, given a random sample from where A and B differ where n_a elements were correctly classified by A, and n_b elements were correctly classified by B, how likely is it that A is better than B?</p> | 73,463 | [
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] |
<p>I have a space of 35 dimensions (attributes). My analytic problem is a simple classification one.</p>
<p>Out of 35 dimensions, more than 25 are categorical and each attribute takes more than 50+ types of values.</p>
<p>In that scenario, introducing a dummy variable also will not work for me.</p>
<p><strong>How can I run an SVM on a space which has a lot of categorical attributes?</strong> </p> | 73,464 | [
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<p>While I was reading about Bayesian networks, I run into "<a href="http://en.wikipedia.org/wiki/Markov_blanket" rel="nofollow">Markov blanket</a>" term and got severely confused with its independency in a Bayesian network graph. </p>
<p>Markov blanket briefly says that every node is only dependent on its parents, children and children's parents [it is gray area for node A in the picture].</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/e/ea/MarkovBlanket.png" alt="Markov blanket"></p>
<p>What is the joint probability of this BN, $P(M,S,G,I,B,R)$?</p>
<p><img src="http://www.aiqus.com/upfiles/IMG_0095_copy.png" alt="alt text"></p>
<p>If I follow the step parent only independency rule, it is:</p>
<p>$$ P(M | S)P(S | G,I)P(I | B)P(R | B)P(G)P(B)$$</p>
<p>However, if I follow the <a href="http://en.wikipedia.org/wiki/Markov_blanket" rel="nofollow">Markov Blanket independency</a>, I end up with this (notice $P(I|\mathbf{G},B)$ is different):</p>
<p>$$P(M | S)P(S | G,I)P(I | \mathbf{G},B)P(R | B)P(G)P(B)$$</p>
<p>So which is the correct joint probability of this BN?</p>
<p>Update: <a href="http://www.aiqus.com/questions/36374/what-is-the-joint-probability-of-this-bayesian-network" rel="nofollow">Crosslink of this question in AIQUS</a></p>
<p>and</p>
<p>Respective chapter and diagrams are below:</p>
<p><img src="http://img828.imageshack.us/img828/9783/img0103s.png" alt="alt text"></p>
<p><img src="http://img406.imageshack.us/img406/3788/img0104l.png" alt="alt text"></p> | 35,849 | [
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<p>Essentially, I have some covariate data <code>X</code>, and a dependent variable <code>Y</code> consisting of proportions of a sample that shown a certain response (i.e. between 0 and 1). I suspect I want to proceed via a GLM approach, but the thing is, I don't know the sizes of each of those samples!</p>
<p>My thoughts are to proceed by a quasibinomial methodology, estimating the dispersion parameter. Assuming the sizes of each sample are not too different, I can thus keep the logit link between the linear predictor and the proportion, but disregard the contribution of <code>n</code> in the usual binomial variance of <code>np(1-p)</code>. Then I can do hypothesis testing the usual way?</p>
<p>Does this make any sense?</p>
<p>Some R code:</p>
<pre><code>#simulate some data
X = rnorm(500)
Z = rnorm(500)
p = exp(X*0.1 + 2)/(1+exp(X*0.1 + 2))
n = 50
Y = NULL
for (i in 1:length(X)){
Y = c(Y,sum(runif(n) < p[i])/n)
}
Y2 = cbind(Y*n, n-Y*n)
#glm, binomial model is 'true' ?
summary(glm(Y~X+Z, family = "quasibinomial"))
summary(glm(Y2~X+Z, family = "binomial"))
anova(glm(Y~X+Z, family = "quasibinomial"), test = "Chisq")
anova(glm(Y2~X+Z, family = "binomial"), test= "Chisq")
</code></pre>
<p>Seems to work, but am I missing something? Surely someone's done something like this before?</p> | 73,465 | [
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<p>Suppose I have 3 eggs, with independent hatching times (in minutes) following an $\text{Exp}(0.05)$ distribution, and A is defined as the earliest hatching time. <br> If I were to plot A, how would I explain its distribution? <br> From running a simulation on R, I can see that the histogram of A follows an exponential distribution but I'm not sure why? </p> | 35,851 | [
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<p>I have a nominal and an ordinal variable and I would like to look measure the association that is between the two. For now, I have done Chi-square test and non-parametric Kruskall-Wallis Test and I have come to the conclusion that there is an association/relationship between the two variables. How can I further measure the effect of this association? I have read about ETA, which SPSS suggests using when you have a nominal and a quantitative variable, and I have also read that you can use it with ANOVA (which is basically the parametric version of Kruskall-Wallis). Could I use this statistic (ETA) as a measure of size effect and trust the value that Crosstabs option in SPSS provides me?
As a secondary question, how can I interpret the values of ETA? I came upon different guidelines on the net. Would you describe a ETA=0.321 as a medium effect?
~Thanks.</p> | 73,466 | [
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<p>I'm currently training a support vector machine with a second order Markov transition matrix. These probabilities are taken from a computer image, and are the probabilities of adjacent pixels having the difference between them being specific values - so each value can be represented as a fraction $a/b$, where a and b are the cardinality of sets.</p>
<p>Now, I've come across a curious phenomenon, which I'm trying to explain. I can remove a significant part of my input features with no loss in the accuracy of the support vector machine.</p>
<p>My current theory as for why this occurs is because there are functional dependencies between certain statistics. For example, the statistics for one sample might be</p>
<p>$(n_1/d_1, n_1/d_2, n_2/d_2, n_2/d_3, n_3/d_3, n_3/d_4, n_4/d_4, \dots, n_n/d_1)$</p>
<p>Each $d_i$ is used seven times in total, whilst each $n_1$ is used either once (all parameters identical) or twice.</p>
<p>Now, I'm not as familiar with support vector machines as many, so I'd like to ask - is there a theorem or something which would confidently allow me to state that the above feature set is no better a feature set than:</p>
<p>$(n_1/d_1, n_2/d_2, n_3/d_3, n_4/d_4, \dots, n_n, d/n)$ in terms of the constructed hyperplane?</p> | 73,467 | [
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<p>Subject says it all - AR/MA/ARMA/ARIMA are often described as workhorses of time series analysis. But what are some real-world examples where these methods gave great results, and another more modern method wouldn't have been superior?</p> | 35,852 | [
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<p>I calculated a sequence of multi-level models with stata (xtmixed). I have a sample of 800 kids in 46 classes and I want to explain their individual level of prejudice by the proportion of fully prejudiced respondents in the class. This is based on a three categorical index, the level of prejudice is measured by a principal component.
In the empty model the ICC is 0.12. In a model where I control for the usual suspects (age, gender, school type, etc), it is reduced to 0.08.
In a third model where I include the "proportion of fully prejudiced students in the class" variable it has a significant** positive effect, just as I excepct it to. The ICC is reduced to 0.0000 (23 0-s in a row).
My colleagues tell me that this is not usual, they've never seen anything like this before.
In order to test for validity I calculated a linear regression with cluster and robust options.
To me it seemed reassuring: Model with control variables had R square of 7%, model with the "proportion of fully prejudiced students in the class" variable had 15% of explained variance. This suggested me that the variable "proportion of fully prejudiced students in the class" has an additional explanatory power of 8%. And as the ICC reduced from 8% to 0 in the ML model, this seemed logical to me.
Have you ever seen an ICC of 0.0000000000000000? Is it OK?
Thanks</p> | 73,468 | [
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<p>I have about 3 hours worth of 3-axis accelerometer, gyroscope and compass data taken at subsecond intervals. I'm fully aware that this data needs to be filtered and that there are a shedload of methods for filtering it (FFT, ARMA, et al). How can I tell from my data or a plot thereof, which is the "best" filter to use? </p>
<p>For example, if I have 3 hours of a jog on the street and I'd like to track the length of the jogger's stride. I'm thinking I would integrate the gyroscope to get the absolute direction and then double integrate the raw accelerometer data to get distance. Finally, divide the result of this by the number of troughs in the spline encompassing the points (as they would represent mainly downward acceleration). Is this the correct approach? Many thanks! </p>
<p><img src="http://analysis.d8u.us/~hdiwan/accelAndGyro.png" alt=""> </p> | 35,853 | [
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<p>I want to compare the <strong>delays</strong> of packets belonging to some kind of traffic to the delays of packets belonging to some other (different and larger) traffic, all generated from the same machine.</p>
<p>I would like to see if the <strong>distributions</strong> of the two groups of delays <strong>are the same</strong>. Samples for the first group are between <strong>size 20</strong> and <strong>2 000</strong> (bust most of the time between 20 and 75), while the comparison group is always of the order of <strong>a few thousands</strong> (up to 15 000). </p>
<p>When the first group has <strong>more than 250</strong> values, I just (randomly) <strong>subsample</strong> it in smaller groups of size 50, since it I noticed that it really takes nothing for large samples to get rejected with Kolmogorov-Smirnov. If the <strong>majority</strong> of subsamples gets accepted by the test, the original sample also does.</p>
<p>Now, due to the precision of timestamping, there are lots of duplicate delays (i.e. <strong>ties</strong>) in my observations. For instance, out of <strong>15000</strong> delays in my second group, I have <strong>only 1000 unique values</strong>, with a mean of 0.7 milliseconds and a standard deviation of around 7 milliseconds. 200 values appear up to 10 times; 90 values between 11 and 100; 30 between 101 and 300; 4 between 301 and 500.</p>
<p>What should I take into account in order to <strong>choose a non-parametric test</strong> that best suits my case?</p>
<p>Also, is my subsampling correct? </p>
<p>I've been using a significance level of 0.01, but I'm getting <strong>20% of false positives</strong>, of which roughly half greater, half smaller than the second group (after the two-sided test, I always used the two one-sided ones just to check).</p>
<p>Any ideas?</p> | 73,469 | [
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<p>I know this is probably a basic question... But I don't seem to find the
answer.</p>
<p>I'm fitting a GLM with a Poisson family, and then tried to get a
look at the predictions, however the offset does seem to be taken into
consideration:</p>
<pre><code>model_glm=glm(cases~rhs(data$year,2003)+lhs(data$year,2003),
offset=(log(population)), data=data, subset=28:36, family=poisson())
predict (model_glm, type="response")
</code></pre>
<p>I get cases not rates...</p>
<p>I've tried also</p>
<pre><code>model_glm=glm(cases~rhs(data$year,2003)+lhs(data$year,2003)+
offset(log(population)), data=data, subset=28:36, family=poisson())
</code></pre>
<p>with the same results. However when I predict from GAM, using mgcv, the
predictions consider the offset (I get rates).</p>
<p>I'm missing something?</p>
<p>I would appreciate any comment,</p>
<p>thanks</p>
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<p>I'm trying to plot the predictions of a linear model with multiple numeric predictors as a collection of plots such that $\hat Y$ is on the y-axis, each $X_j$ in turn is on the x-axis, and the remaining $X_{i\ne j}$ are all set to plausible values given $X_j$. Here is some example R code (syntax might look strange, it's that way for conciseness, but it does parse correctly in R):</p>
<pre><code>mdata<-data.frame(x1=x1<-rnorm(100,1.5),
x2=x2<-x1+2*rnorm(100,0.5),
x3=x3<-x1*2+rnorm(100,sd=.2),
y=x1-x2+x3-0.5*x1*x2+0.5*x2*x3-0.5*x1*x3+rnorm(100,sd=.1));
mfit<-lm(y~x1*x2*x3,mdata);
</code></pre>
<p>Note that in the above dataset, x2 and x3 are both correlated with X1, and all three are predictors for y.</p>
<p>The widespread approach of setting each $X_{i\ne j}=0$ or to the population mean or median is misleading if there is the possibility of multicollinearity among the $X$s because no distinction is made between commonly-occurring combinations of variables and ones that are completely out of the support range of the data. Example using above objects:</p>
<pre><code>predictAtX1<-seq(min(mdata$x1),max(mdata$x1),len=100);
## Do you trust this? I don't.
plot(predict(mfit,newdata=data.frame(x1=predictAtX1,
x2=mean(mdata$x2),
x3=mean(mdata$x3)))~predictAtX1,
xlab='x1',ylab='y-hat');
## Ditto.
plot(predict(mfit,newdata=data.frame(x1=predictAtX1,
x2=median(mdata$x2),
x3=median(mdata$x3)))~predictAtX1,
xlab='x1',ylab='y-hat');
## Ditto, even worse, and yet the most common value at which to fix x_js.
plot(predict(mfit,newdata=data.frame(x1=predictAtX1,
x2=0,
x3=0))~predictAtX1,
xlab='x1',ylab='y-hat');
</code></pre>
<p>I thought of fitting each of the $X_j$ to a given $X_i$ and using the predicted values for a given sequence of $X_i$....</p>
<pre><code>plot(predict(mfit,newdata=data.frame(x1=predictAtX1,
x2=predict(lm(x2~x1,mdata),
newdata=list(x1=predictAtX1)),
x3=predict(lm(x3~x1,mdata),
newdata=list(x1=predictAtX1))))~predictAtX1,
xlab='x1',ylab='y-hat')
</code></pre>
<p>...and that doesn't look as obviously nonsensical as the plots for globally fixed values of $X_2$ and $X_3$ but given that there are additional interactions involving $X_2$ and $X_3$ that are being ignored, these predictor values are not representative of the population either.</p>
<p>But. For a given sequence of values for $X_1$ it should be possible to construct a smoothed path through the variable space corresponding to the most likely values of $X_2$ and $X_3$. I can think of brute-force optimization approaches, but given how obvious and universal this problem is for any real-life regression model with multiple predictors, there must be an already-implemented and/or simpler approach to dealing with it. </p>
<p>So, my revised question is, does anybody know the name of such a method or the name of the subfield of statistics that deals with it, or the name of an R package that implements it?</p>
<p>Thanks.</p> | 73,471 | [
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<p>I was told to run a changepoint analysis in R on my data. But the data shows a decreasing trend, so the changepoint analysis won't be of help (I don't think?). If I differenced the data, to get rid of the trend, and then perform a changepoint analysis on the differenced data, does that make sense? And how would I interpret the results?
Thanks so much</p> | 73,472 | [
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<p>I struggle performing a MANOVA for a mixed model. I have the following columns:</p>
<p>S - subject (1,...,N)
B - in between group (B1, B2)
W - repeated measurement - within (W1, W2)
M - measured dependent variable 1
N - measured dependent variable 2</p>
<p>My data is organized the following way:</p>
<pre><code>S; B; W; M; N
1;B1;W1;12.3;14.4
1;B1;W2;35.4;13.5
2;B1;W1;11.1;22.2
2;B1;W2;33.3;11.1
3;B2;W1;15.3;12.3
3;B2;W2;20.3;18.8
...
</code></pre>
<p>If I had only one dependent variable (M), I could obtain the results of an ANOVA in one of these ways:</p>
<pre><code>dat=read.csv(file="data.csv", sep=";")
library(ez)
ezANOVA(dat, dv=.(M), wid=.(S), between=.(B), within=.(W), type=3, detailed=TRUE)
library(lmerTest)
results = lmerTest::lmer(M ~ B*W + (1 | S), data=dat)
anova(results, ddf="Kenward-Roger", type=3, method.grad="Richardson")
library(car)
options(contrasts=c("contr.sum", "contr.poly"))
mod1 <- lme4::lmer(M ~ B*W + (1 | S), data=dat)
car::Anova(mod1, test.statistic="F", type=3)
</code></pre>
<p>I wonder, how the code for the MANOVA has to look like? I tried</p>
<pre><code>dat=read.csv(file="vor_actions_new.csv", sep=";")
results = manova(cbind(M, N) ~ B*W, data=dat)
summary.aov(results)
summary(results)
</code></pre>
<p>but I doubt, the within-factor is handled correctly. I also tried</p>
<pre><code>mod1 <- lme4::lmer(cbind(M, N) ~ B*W + (1 | S), data=dat)
</code></pre>
<p>but, this gave an error, probably because multiple responses are not supported. ezAnova does also not support multiple responses, I think. Is there a way to execute the planned MANOVA?</p> | 73,473 | [
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<p>In SAS enterprise miner we have the optimal binning feature which allows you to transform continuous variables into an ordered set of bins. The binning, as I read from one of their docs, is done so that the log odds of the predicted categorical variable (good/bad) is monotonically increasing or decreasing. </p>
<p>Can we use classification trees to kind of achieve the same result?. Like build a tree with only one variable and how many which splits we can get with sufficiently different log odds. </p> | 35,858 | [
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] |
<p>Is an implementation of a density function for a logit-normal distribution available in R?</p>
<p>I have not found one in a package or in the CRAN task view of probability distributions.</p>
<p>This is for a MLE fitting of a function to data.</p> | 35,861 | [
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<p>In ecology, we often use the logistic growth equation:</p>
<p>$$ N_t = \frac{ K N_0 e^{rt} }{K + N_0 e^{rt-1}} $$ </p>
<p>or</p>
<p>$$ N_t = \frac{ K N_0}{N_0 + (K -N_0)e^{-rt}} $$ </p>
<p>where $K$ is the carrying capacity (maximum density reached), $N_0$ is the initial density, $r$ is the growth rate, $t$ is time since initial.</p>
<p>The value of $N_t$ has a soft upper bound $(K)$ and a lower bound $(N_0)$, with a strong lower bound at $0$.</p>
<p>Furthermore, in my specific context, measurements of $N_t$ are done using optical density or fluorescence, both of which have a theoretical maxima, and thus a strong upper bound.</p>
<p>The error around $N_t$ is thus probably best described by a bounded distribution.</p>
<p>At small values of $N_t$, the distribution probably has a strong positive skew,
while at values of $N_t$ approaching K, the distribution probably has a strong negative skew.
The distribution thus probably has a shape parameter that can be linked to $N_t$.</p>
<p>The variance may also increase with $N_t$.</p>
<p>Here is a graphical example</p>
<p><img src="http://i.stack.imgur.com/EpTp9.jpg" alt="enter image description here"></p>
<p>with</p>
<pre><code>K<-0.8
r<-1
N0<-0.01
t<-1:10
max<-1
</code></pre>
<p>which can be produced in r with</p>
<pre><code>library(devtools)
source_url("https://raw.github.com/edielivon/Useful-R-functions/master/Growth%20curves/example%20plot.R")
</code></pre>
<ul>
<li><p>What would be the theoretical error distribution around $N_t$ (in consideration of both the model and the empirical information provided)?</p></li>
<li><p>How doe the parameters of this distribution relate to the value of $N_t$ or time (if using parameters were the mode can not be directly associated with $N_t$ eg. logis normal)?</p></li>
<li><p>Does this distribution have a density function implemented in $R$?</p></li>
</ul>
<p>Directions explored so far:</p>
<ul>
<li>Assuming normality around $N_t$ (leads to over estimates of $K$)</li>
<li>Logit normal distribution around $N_t/max$, but difficulty in fitting shape parameters alpha and beta</li>
<li>Normal distribution around the logic of $N_t/max$</li>
</ul> | 73,474 | [
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<p>I have a couple questions about Cox survival regression:</p>
<p>1) Is it true that the hazard function h(t) is not available (even WITHOUT <a href="http://en.wikipedia.org/wiki/Time-varying_covariate" rel="nofollow">time dependent covariates</a>)- and if not, is it because the baseline hazard is not defined (only the cumulative hazard is and you cant go from the cumulative hazard to the hazard)?</p>
<p>2) When you have time dependent covariates is it not possible to estimate the survival curve for a chosen set of covariate values (I'd like to plot prototypical values)? Can anyone explain why not? </p> | 73,475 | [
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<p>From an answer in a <a href="http://stats.stackexchange.com/questions/24605/increase-the-sample-size-of-a-latin-hypercube-study">previous question</a>, I was directed toward the Halton sequence, for creating a set of vectors that covered a uniform sample space fairly evenly. But the <a href="http://en.wikipedia.org/wiki/Halton_sequence" rel="nofollow">wikipedia page</a> mentions that higher primes especially are often highly correlated early in the series. This seems to be the case for any pair of high primes with a relatively short sample size - and even when the variables aren't correlated, the sample space isn't evenly sampled, rather there are <a href="http://stats.stackexchange.com/a/7927/9007">diagonal bands</a> of high sample density across the space. </p>
<p>Because I'm using vectors of length 6 or more, I will inevitably have to use some primes for which this is a problem (although not as bad as in the example aboved), and some pairs of variables will be non-uniformly sampled in their sample plane. Using the Sobol' sequence to generate a similar set seems to me (just from looking at graphs) to generate samples between pairs of variables that are much more evenly distributed, even for relatively small numbers of samples. This seems much more useful, and so I'm wondering when a Halton sequence would be more beneficial? Or is it just the Halton sequence is easier to compute?</p>
<p>Note: discussion of other multi-dimensional low-discrepancy sequences is also welcome.</p> | 73,476 | [
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<p>I try to predict values for regression in LIBSVM. My data is in time series. I use gridregression.m file in LIBSVM to find optimal parameters c, g and p. Gridregression.m file use cross validation to find optimal parameters, but is it ok to use cross validation in time series? </p>
<p>When I use parameters from gridregression.m, sometimes the MSE is not better then the default values. ( cmd= '-s 3 -t 2' is sometimes better )</p> | 73,477 | [
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<p>I ran an experiment and obtained the points shown in black below. </p>
<p>I would like to smooth the curve or fit (something like the red curve) in order to identify the elbow. </p>
<p>The problem is that the experiment is too expensive and there is a large amount of variance in the obtained points.
I tried fitting a second degree polynomial and an exponential curve without sucess (the curve is too smooth and the elbow is lost). Total precision is not necessary. </p>
<p><strong>What is the best way to fit a curve in order to identify the elbow?</strong></p>
<p><img src="http://i.stack.imgur.com/gOtGQ.png" alt="enter image description here"></p> | 47,272 | [
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<p>I have a large survey in which students were asked, among other things, their mother's level of education. Some skipped it, and some answered wrongly. I know this, because there a sub-sample of the initial respondents mother's were later interviewed, and asked the same question. (I'm sure there is some, smaller amount, of error associated with the mothers' responses as well.)</p>
<p>My challenge, is to decide how to best take advantage of this second, more reliable source of data. At the very least I can use it to impute missing data more intelligently than I would be able to if I could only rely on complete cases. But if 3/4 of the children whose data I can cross-check, who respond "My mother never finished elementary school" are contradicting their mother's answer, then it would seem I should use imputation to create multiple datasets to capture the uncertainty there. [added: I said 3/4 to make a point, but now that I've checked the data I might as well tell you that closer to 40% are discrepant]</p>
<p>I will personally be using mother's education as a predictor in a mixed model, but if anyone has something to say about other situations I'd love to learn about them as well.</p>
<p>I would love to receive advice in broadstrokes or in the specifics. Thank you! </p>
<p><strong>Update</strong>: I'm leaving the question unsolved for now, though I appreciate Will and Conjugate_Prior's responses, I'm holding out hope for more specific and technical feedback. </p>
<p>The scatterplot below will give you an idea of how the two variables are related in the 10,000 cases where both exist. They are nested, in more than 100 schools. They correlate at 0.78, Student's Answer- mean:5.12 s.d.=2.05, Mom's answer, mean=5.02, s.d.=1.92 The student's answer is missing in about 15% of cases. </p>
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<p>Cohen's d is a measure of effect size calculated as:</p>
<p>$d = (x_1-x_2) / \sigma_{\text{pooled}}$</p>
<p>where $x_1$ is the mean of one group, $x_2$ is the mean of a second group, and $\sigma_{\text{pooled}}$ is the pooled standard deviation.</p>
<p>Let us say that one treatment has an effect size of 0.6 and another treatment has an affect size of 1.2.
<strong>How can I tell whether this difference is statistically significant?</strong> I.e. does one treatment provide a significantly different outcome from another?</p>
<p>EDIT: The samples are not independent; each sample consists of the same 7 participants.</p>
<p><em>I've briefly searched around but didn't find an answer for this question.</em></p> | 73,479 | [
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<p>I am working on an assignment involving a logistic regression model, where I need to plot the pearson standardized residuals against one of the predictors. Here's the basic setup:</p>
<pre><code>model <- glm(outcome ~ predictor1 + predictor2, family=binomial(logit))
res <- residuals(model, "pearson")
</code></pre>
<p>When looking at the residuals' distribution, I see something totally different than my colleagues who use Stata (using predict and rstandard). Their residuals are more or less normal, whereas in mine there is a gap in the values (not a singe residual is between -0.05 and 1.15). That does make sense in the context of logistic regression, especially that the maximum predicted probability is not so high (38%). </p>
<p>I'd like to understand what's happening here... What is Stata doing that R isn't, with those residuals? </p> | 35,871 | [
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<p>I am trying to derive the expression for the variance of $\hat{\beta_0}$ in simple linear regression. I substitute $\bar{y} - \hat{\beta_1} \bar{x}$ for $\hat \beta_0$, but in the intermediate steps the covariance term $\text{Cov}(\bar{y}, \hat{\beta_1})$ comes up and I don't know how to deal with it. Any help would be appreciated!</p> | 49,649 | [
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<p>I want to estimate $Y$ using different independent variables.</p>
<p>$$Y = d_1 + aX_1\\
Y= d_2 + bX_2\\
Y= d_3 + cX_3$$</p>
<p>Then I want to combine above 3 equations such that it gives the following relation:</p>
<p>$$Y = d + a_1X_1 + b_1X_2 + c_1X_3$$</p>
<p>How can I use the above 3 regression equations to obtain a single relationship as above?</p> | 73,480 | [
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<p>Let $\mathcal{E(h)}$ a function that given some hypothesis $h$ returns the generalization error for that fixed $h$.</p>
<p>I was reading some notes about model selection and generalization error and it said:</p>
<blockquote>
<p>"If we had access to $\mathcal{E(h)}$, there would be no model
selection problem either. We would simply select the larges
$\mathcal{H}$ so as to find a classifier that minimizes the error."</p>
</blockquote>
<p>I was not sure if I fully appreciated or understood that statement or actually agreed with the statement. The reason is that, even if we had access to $\mathcal{E(h)}$ (which I think they mean an oracle that takes $h$ and just says its true generalization error) I think it would still be problematic to find the model that has the hypothesis that generalizes well. The reason is, say that the model classes $\mathcal{H}$ is infinite (i.e. there is an infinite set of models to choose from). We don't really know when $\mathcal{E(h)}$ has reached its minimum unless we check for every $\mathcal{H}$ that is possible. i.e. even if we had such a thing I don't think the problem is eliminated so easily because how can we ever be sure we have truly found the best $\mathcal{H}$ (in polynomial time)? Basically I feel the question assumes that we have an oracle for determining when generalization is minimum too. Furthermore, as I pointed out, the algorithm/turning machine the suggested is decidable and not in P (i.e. it might run forever...)</p>
<p>The main issue/doubt that I have with this question is that even with such an Oracle, I am not convinced that model selection has been trivialized, an answer that tries to tackles this specific issue, has higher chances to tackle my question better.</p> | 35,874 | [
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<p>Is there a way of using the linear model api to add the lasso penalty for a subset of the parameters I am regressing? For example, consider a linear separable decomposition that I want to fit to some sparse data subject to smooth constraints in one of the (separable) dimension.</p>
<p>This question indicates that this is somewhat possible through R's glm.</p>
<p><a href="http://stats.stackexchange.com/questions/62267/is-there-lasso-type-model-in-which-only-some-of-the-regressors-are-regulated">Is there LASSO type model in which only some of the regressors are regulated?</a></p>
<p>Of course, a DIY method will not be too hard. The question is rather whether it is possible through the scikit learn api. </p> | 35,875 | [
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<p>I have two questions. </p>
<p>1) If $X_1,X_2,X_3,...,X_n$ constitute a random sample of size $n$ from an exponential distribution, show that $\bar X$ is a consistent estimator of the parameter $\lambda$. </p>
<p>This is my attempt:<br>
Mean of exponential distribution is $\lambda^{-1}$. Thus by central Limit theorem $E(\bar X)=\lambda^{-1}$. Using mean square consistency, if $\lim_{n\to \infty} E[(\bar X -\lambda )^2]=0$ then $\bar X $ is consistent for $\lambda$.
Since
$$
E[(\bar X -\lambda )^2]= E[(\bar X)^2]+E(\lambda)^2-2\lambda E(\bar X) ,
$$
I get
$$
\lim_{n\to \infty} E[(\bar X -\lambda )^2]
=
\lim_{n\to \infty} {2 \over \lambda ^2 }+\lambda^2-2 .
$$
Clearly this is not equal to $0$.
How can I show this is consistent?
I tried using Chebechev inequality as well. But even using that I couldn't. </p>
<p>2) Is $X_{(n)}$ a consistent estimator for $\lambda$?<br>
If $P[|X_{(n)}-\lambda|<\epsilon]$ approaches $1$ as $n$ goes to infinity for all $\epsilon > 0$ then $X_n$ is a consistent estimator for $\lambda$. </p>
<p>The probability density function of $X_{(n)}$ is $f_n(x) = n \lambda e^{-\lambda x}(1-e^{-\lambda x})^{n-1}$, for $X>0$.</p>
<p>Thus
\begin{align*}
P[|X_{(n)}-\lambda|<\epsilon]
&= P[\lambda -\epsilon<X_{n}<\lambda+\epsilon] \\
&=P[0<X_{n}<\lambda+\epsilon] \\
&=\int_0^{\lambda+\epsilon} n \lambda e^{-\lambda x}(1-e^{-\lambda x})^{n-1}\\
&= (1-e^{- \lambda X_{(n)} }),
\end{align*}
where $X{(n)}$ takes values in $[0 , \lambda+ \epsilon]$.</p>
<p>Thus, can I say $\lim_{n\to \infty}[1-e^{-\lambda(\lambda+\epsilon)}]^n=1$?</p> | 73,481 | [
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<p>MaxDiff scaling is a very popular method for understanding customer preferences. I was wondering if we can create a survey questionnaire for MaxDiff analysis in R. Can anyone shed some light on this. </p> | 35,876 | [
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<p>I wish to run Permanova on categorical data (example CSV structure is below) without any successes. Is Adonis can handle categorical data, an example script will be great.</p>
<p>Thanks a lot</p>
<p>I have a this CSV file:</p>
<pre><code>Specie,Sex,Age,Location,Sensitivity
A,M,1,1,1
A,M,1,1,1
A,M,2,2,2
A,M,2,2,2
A,F,1,1,1
A,F,1,1,1
A,F,1,1,1
A,F,2,2,2
A,F,2,2,2
A,F,2,2,2
B,M,1,2,2
B,M,1,2,2
B,M,1,2,2
B,M,2,2,1
B,F,2,1,1
B,F,2,1,1
B,F,2,1,2
B,F,2,2,2
B,F,1,2,2
B,F,1,2,2
C,M,1,2,2
C,M,2,2,1
C,M,2,2,1
C,M,2,2,1
C,F,2,2,2
C,F,1,1,2
C,F,1,1,2
C,F,2,2,2
C,F,2,2,1
C,F,2,2,1
</code></pre> | 73,482 | [
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<p>If you don't care about using GLM model parameters to predict anything, but simply want to select the best-fitting model for your data, is it necessary to get into the theoretical debate as to which link function to use? Is it OK to simply select the link function that gives you the lowest deviance? </p>
<p>Specifically, I am running an ordinal GLMM, and have found that the log-logistic link function in the software program called SuperMix fits my data the best.</p> | 73,483 | [
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<p>Let $\hat{\Gamma}_k$ be the k dimensional sample autocovariance matrix. I am trying to prove this is nonnegative definite. The first step in the proof is to show that if $\hat{\Gamma}_m$ is nonnegative definite then $\hat{\Gamma}_k$ is nonnegative definite for all $k<m$. How do I prove this?</p>
<p>Many thanks in advance</p>
<p>(So far I can verify it for trivial cases like m=2 but not for general m.)</p> | 73,484 | [
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<p>Suppose:</p>
<ul>
<li>$N \sim {\rm Poisson}(\lambda)$</li>
<li>$\lambda$ is unknown, but we believe that it can be assumed $\sim \exp(1)$</li>
</ul>
<p>If I want to calculate $N | X$, i.e., $P(model | data)$, I need to use the Bayes theorem in the following way:</p>
<p>$P(model|data) \propto P(data|model)*P(model)$</p>
<ul>
<li>$P(data|model)$ is the likelihood function</li>
<li>$P(model)$ is my prior distribution density</li>
</ul>
<p>So:</p>
<p>$P(data|model) = L(\lambda) = \exp\{-n\lambda + \log\lambda \sum k_i - \sum \log (k_i!)\}$</p>
<p>And</p>
<p>$P(model) = g(\lambda = 1) = e^{-\lambda}$</p>
<p>Therefore</p>
<p>$P(model|data) = \exp\{-n\lambda + \log\lambda \sum k_i - \sum \log(k_i!)\} e^{-\lambda}$</p>
<p>And if I had a sample of $k_1 = j$, then</p>
<p>$P(\lambda|k_1 = j) = \exp(-\lambda + \log \lambda j - \log j!) e^{-\lambda}$</p>
<p>Is it correct? How do I calculate an expected value for the parameter?</p> | 73,485 | [
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<p>I have a logistic regression model which predicts win/loss on amount of money paid. I run my model every two hours on new data that I acquire and use it to predict the next two hours. However, I keep finding that my model underpredicts win/loss for each amount of money paid. So I'm in this situation where I have a statistical model but it doesn't seem to predict new data as it comes in. </p>
<p>I'm left wondering, what do I do now? My model doesn't predict for the new data, but I need for it to do so.</p>
<p>What are some general strategies for when a model has no predictive power?</p>
<p>As a side note, I should have mentioned that I actually had two models. One which predicted win/loss on 0 to 5 \$ and another for 5.01 \$ and more. It's possible that this may have been a culprit, and I might just want to utilize a robust regression model instead. Not entirely relevant, but just thought I'd mention.</p> | 73,486 | [
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<p>Lets say I have three search engine e.g. search engine A, search engine B and search engine C.</p>
<p>Each search engine is given a set of queries Q (e.g. apple,banana,carrot....), this set Q remains the same for each engine.</p>
<p>Each engine then provides a set of results for each query in the form a SERP (search engine results page).This means for every query (q) we have three search engine results pages (SERPs).</p>
<p>We have recruited 4 users lets say, and each user is given multiple sets of queries and their paired SERPs, and asked to rank each SERP for a query all of the queries in terms of relevance on a scale of 1-4 (1 being not relevant, 4 being very relevant). </p>
<p>So user one will see and rank accordingly:</p>
<pre><code>query 1(apple), SERP A (relevance 1), SERP B(relevance 1), SERP C(relevance 4)
query 2(banana), SERP A (relevance 2), SERP B(relevance 2), SERP C(relevance 4)
query 3(carrot), SERP A (relevance 3), SERP B(relevance 2), SERP C(relevance 3)
</code></pre>
<p>User two will see and rank as follows:</p>
<pre><code>query 1(apple), SERP A(relevance 1), SERP B(relevance 2), SERP C(relevance 3)
query 2(banana), SERP A(relevance 1), SERP B(relevance 2), SERP C(relevance 4)
query 3(carrot), SERP A(relevance 3), SERP B(relevance 2), SERP C(relevance 3)
</code></pre>
<p>User three will see and rank as follows:</p>
<pre><code>query 1(apple), SERP A(relevance 1), SERP B(relevance 2), SERP C(relevance 3)
query 2(banana), SERP A(relevance 2), SERP B(relevance 2), SERP C(relevance 3)
query 3(carrot), SERP A(relevance 3), SERP B(relevance 3), SERP C(relevance 4)
</code></pre>
<p>I was wondering if there was a measure that I could use to say which search engine produced the most relevant search results over a set of queries, based on the the scores the users gave. Would Chi-squared or an analysis of variance ( ANOVA ) be appropriate?</p>
<p>At a quick glance if this was A/B split testing I guess I would say results generated by search engine C provided better results than A or B, whilst B provided better than A. Am I correct in saying this and what would be the best to test to formally write this down??</p> | 35,879 | [
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<p>Is it safe to do my one-hot encoding of categorical variables before splitting the data into training and test and cross validation sets? I ask because this process alters the dimension of the data, and it would be more complicated to get my test/cv sets into the same schema as the training set if they were one-hot encoded separately. My intuition tells me that it is safe.</p>
<p>For an example, let's say there's a categorical feature for marriage status- it can be Married, Single, Divorced or Widowed. If when I partition into training and test and cv sets, no Widowed samples make it into the training set, then my one-hot encoding will produce only 3 columns (Married, Single, Divorced) in the training data. Then I will need to convert my test and cv data to this format in order to test the model. Whereas if I one-hot encoded before splitting, I would have an extra column in the training set with all zeroes. As far I can tell this shouldn't affect the performance of the NN.</p> | 73,487 | [
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<p>I am wondering which combinations of Naive models can be paired with different vectorizing methods so that it makes sense.</p>
<p>Let's say we have a simple binary spam-classification task.</p>
<h3>Multinomial model:</h3>
<p>If I understand correctly, the Multinomial model is used to calculate the class-conditional probabilities based on the term frequencies. Let's say our input text consists of only 1 word for simplicity. So, the class-conditional probability (or likelihood) would be calculated as follows:</p>
<p>\begin{equation} \hat{P}(x_i \mid \omega_j) = \frac{tf_{xi, y} + \alpha}{tf_y + \alpha \cdot V} \end{equation} </p>
<ul>
<li>$tf_{xi, y}$ = term frequency; the number of times the word $x_i$ occurs in the training dataset for samples of class $\omega_j$</li>
<li>$tf_y$ Sum of all term frequencies in the training dataset for class $\omega_j$</li>
<li>$\alpha$ is the addititive smoothening parameter ($\alpha = 1$ for Laplace smoothening</li>
<li>$V$ is the size of the vocabulary</li>
</ul>
<h3>Multi-variate Bernoulli (Binomial) model:</h3>
<p>Here, we would use document frequencies rather than term frequencies since the distribution used binary values: "does a word occur in a document or not?"</p>
<p>\begin{equation} \hat{P}(x_i \mid \omega_j) = \frac{df_{xi, y} + 1}{df_y + 2} \end{equation} </p>
<ul>
<li>$df_{xi, y}$ is the number of documents in the training dataset that contain $x_i$ and belong to class $\omega_j$</li>
<li>$df_y$ Number of documents in the training dataset that belong to class $\omega_j$</li>
<li>+1 and +2 are the parameters for Laplace smoothening</li>
</ul>
<h2>Question</h2>
<p>So, my question is which vectorizer do I have to use when I want to prepare the dataset for either of the models. </p>
<p>Let's say I am using the <code>scikit-learn</code> package in Python. from </p>
<pre><code>from sklearn.feature_extraction.text import TfidfVectorizer,CountVectorizer
sklearn.naive_bayes import MultinomialNB
from sklearn.naive_bayes import BernoulliNB
from sklearn.pipeline import Pipeline
</code></pre>
<p>When I understand correctly, the count vectorizer produces a "bag of words" and for the term frequencies, so this combination seems to make sense to me:</p>
<pre><code>clf_1 = Pipeline([
('vectorizer', CountVectorizer(ngram_range=(1,2), stop_words="english")),
('classifier', MultinomialNB()),
])
</code></pre>
<p>But in terms of the document frequencies, is it okay to use CountVectorizer in combination with the Bernoulli model? Can this automatically retrieve the correct document frequencies from the count vectorizer results?</p>
<pre><code>clf_2 = Pipeline([
('vectorizer', CountVectorizer(stop_words="english")),
('classifier', BernoulliNB()),
])
</code></pre>
<p>And how would I use the Tf-idf vectorizer? Does this combination make sense?</p>
<pre><code>clf_3 = Pipeline([
('vectorizer', TfidfVectorizer(ngram_range=(1,2), stop_words="english")),
('classifier', MultinomialNB()),
])
</code></pre>
<p>I assume that it doesn't make sense to use <code>TfidfVectorizer</code> with the Bernoulli model, right?</p> | 35,880 | [
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<p>Hi guys I am doing some model selection and need to estmiate both observed and expected Fisher Information Matrix to calculate this <a href="http://myweb.uiowa.edu/cavaaugh/ms_lec_4_ho.pdf" rel="nofollow">link</a>.
How do you estimate that expected one numerically? </p> | 31,095 | [
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<p>I've read that <strong>Mahalanobis distance</strong> is as effective as the <strong>Euclidean distance</strong> when <strong>comparing</strong> 2 <strong>projected feature vectors</strong> in classification using a <strong>LDA classifier</strong>.</p>
<p>I was wondering if this statement were true?</p>
<p>It would be nice if someone could comment on this.</p>
<p>I'm working on <strong>projecting</strong> a <strong>36 dimensional feature vector</strong> to a <strong>1d feature vector</strong> using a <strong>2 class LDA classifier</strong> and comparing projected feature vectors.</p> | 35,881 | [
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<p>I have a set of measurements for web page response - 2356 6788 3658 5545 1008. How do i determine whether this set even bears evaluation / averaging. By manual inspection, this has no significance for me. </p>
<p>OR if I have a set like (1815 1856 1899 4567 1875), manually inspecting it i would eliminate the outlier 4567, and then use the mean of remaining values in the set.</p>
<p>Unfortunately due to the manual nature of data collection, increasing the size of the data set is not viable beyond 10 or so. Also, the blatant outliers like 4567 in the second set are due to known network issues.</p>
<p>I'm just looking for a way to plug these point into excel and answer the 2 questions above. </p>
<ol>
<li>Is the set statistically significant ?</li>
<li>Are there any outliers that can be eliminated to make it statistically significant.</li>
</ol>
<p>Thanks!</p> | 38,331 | [
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<p>I have 50,000 objects on which I have performed two different types of regression. Using cross validation, I obtained the average $R^2$ score from each model on each of the objects. So now I have a list of 50k pairs of average $R^2$ scores. </p>
<p>I have used a paired Wilcoxon-Mann-Whitney U test and found that their means differ significantly:</p>
<pre><code>> wilcox.test(first$R.2, second$R.2, paired=TRUE)
Wilcoxon signed rank test with continuity correction
data: first$R.2 and second$R.2
V = 34503366, p-value < 2.2e-16
alternative hypothesis: true location shift is not equal to 0
</code></pre>
<p>Is this the right approach? I figure this is the right test because $R^2$ is bounded above by 1 but unbounded below so I can't really make any assumption about the distribution. </p>
<p>How I can report this in my paper? Do I just say that I used the Wilcoxon test and obtained a p-value less than 2.2e-16?</p>
<p>I also wanted to show the mean and median $R^2$s from each of the techniques. But should I just show point estimates of these values, or should I also show SDs and MADs or SEMs or what? Should I bootstrap confidence intervals for them and see whether they overlap? </p>
<p>Any help is appreciated, thanks. </p> | 73,488 | [
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<p>Is it common practice (and adequate) to regroup two binary dependant variables into a single 4-level dependent variable to take advantage of the multinomial regression? For instance, say we have information on two related conditions (outcomes) A and B. A new 4-category variable would be defined such that:</p>
<pre><code>category 1 = Neither conditions A nor B
category 2 = Condition A (only)
category 3 = Condition B (only)
category 4 = Both conditions A and B
</code></pre>
<p>This allows running a single multinomial regression instead of using two binary logistic models that include the same predictors.</p> | 73,489 | [
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<p>I'm implementing sentiment analysis on the set of user comments. All comments are on the same object. At the moment I decided to have three classes - negative, neutral and positive. I got test array of 1500 comments with marked classes. Tried to use SVM for classification on binary feature vectors in which each element refers to the presence of some word in the comment. I got maximum accuracy of 60% correct classes. Known researches had 80% and better accuracy, but they was done on English texts.</p>
<p>One of the problems - numerous errors in the comments, spelling and grammar. Also the Russian language is more complex than English.</p>
<p>I would appreciate advice of any kind. Are there any good tools for the analysis of the Russian language? Maybe SVM isn't the right choice, are there any better algorithms for my case? Or maybe i must choose the more efficient feature space?</p> | 392 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://stats.stackexchange.com/questions/16533/properties-of-logistic-regressions">Properties of logistic regressions</a> </p>
</blockquote>
<p>I have built a logistic regression model with two covariates. Let $y_i$ be the binary responses and $\hat{\pi}_i$ the ML estimated probabilities for $i=1,\ldots,n$. Now it happens that $\sum_i{y_i}\neq\sum_i{\hat{\pi}_i}$. So, computing a chi-square goodness of fit test, sum of expected frequencies is different from sum of observed frequencies. Is this normal, or there is something wrong? </p> | 49,757 | [
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<p>In applying a penalty term with either $l_1$ or $l_2$ norm, why would the former result in variable selection but not the latter? </p> | 49,719 | [
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<p>In Section 2.4 of the book ESL by Hastie etc., it was said that $\hat{f}(x)=\text{Ave}(y_i\mid x_i\in N_k(x))\rightarrow E(Y\mid X=x)$ when $N, k\rightarrow \infty$ and $k/N\rightarrow 0$.</p>
<p>Here $N_k(x)$ is the neighborhood containing the $k$ points and "Ave" denotes average.</p>
<p>Is there a rigorous proof of this statement?</p> | 73,490 | [
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<p>Performing a regression and need to find out if my residuals are normally distributed. </p> | 73,491 | [
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<p>I have three random variables, $X, Y, Z$. I know that</p>
<p>$$\log X - \log Y - \log Z = \log \frac{X}{YZ} \sim \mathcal{N}(0, \sigma^2)$$</p>
<p>How can I recover the distribution of $\frac{X}{YZ}$? Is it as simple as $e^{\mathcal{N}(0, \sigma^2)}$?</p> | 73,492 | [
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<p>I have the following test data:</p>
<p><img src="http://i.stack.imgur.com/NgRMv.png" alt="enter image description here"></p>
<p>How do I correlate the "believes" with "before" to determinate if it affects the "after"?</p>
<p>When believes is set to 1 the "after" will be (randomly) higher than when set to 0.</p> | 73,493 | [
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<p>I'm designing a neural network for binary classification problem in <code>MATLAB</code>. I have different number of inputs and samples in every problem ( I designed a software that end user will use it). I know that finding <code>number of neurons</code> and <code>layers</code> is a critical problem in neural networks.</p>
<p>but is any mathematical or experimental method to find appropriate <strong>minimum</strong> and <strong>maximum</strong> neurons in every layer (one layer and two layer networks) and i will search between these values?</p>
<p>Thanks.</p> | 73,494 | [
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<p>I am interested in exploring how different characteristics of national pension systems are related to each other. I have used MCA for a dataset in which the rows are countries and the columns are different features of pension systems. However, I am not sure how to interpret the distances between points in the SPSS Joint Plot of Category Points. Using a symmetrical normalization, do the distances between points representing categories of different variables say something about how these categories are associated? Does a shorter distance mean a higher level of association?</p> | 73,495 | [
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<p>I have a panel of daily stock prices $\{Y_{it}\}$, $t \in \{1,...,1000\}$ and some event that occurs at $t=700$ which causes the average stock price to decrease by about 10% over the next 15 days.</p>
<p>How do I answer the question "which stocks fell first"? I want to know of the characteristics of stocks that tended to be correlated with very early falls (like, first 1 or 2 days). I have many ideas for sub-sample analysis but none that I want to use involving econometrics. </p>
<p>Broad econometric-based ideas welcome.</p>
<p>Can survival analysis be used here? </p>
<p>Note; I did make an earlier post on this issue that was answered, but I constrained the question way too much and didn't get what I wanted. </p>
<p>EDIT: <em>Please don't absolutely answer the question (up to model specification). I just want ideas/suggestions of areas that might be good and a very general approach/suggestion. Then I want to develop it all myself from there.</em> </p> | 73,496 | [
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<p>I'm using a Cox proportional hazards model, estimate the hazard rate for Levamisole relative to 5-FU, adjusting for Age and Sex.</p>
<pre><code>library(survival)
Colon<-subset(colon,etype==2 & rx!='Obs')
fitcox<-coxph(Surv(time,status)~rx+age+sex,data=Colon)
</code></pre>
<p>Now if I want to make some prediction about the conditional expected lifetime given survival time up to t0, rx, age and sex, are there any convenient functions? If there's no similar function and I have to write my own, then here's a piece of pseudo code expressing what I want, but I don't know how to implement it:</p>
<pre><code>function(rx, sex, age, t0, baseline, fitcox) {
coef <- get_coef_from(fitcox)
//Get survival function from cumulative hazard function
baseline$survival <- exp(-basehaz$hazard*exp(coef[1]*rx+coef[2]*sex+coef[3]*age))
//Calculate expected life time, formula is provided below
survival_t0 <- baseline[t is nearest t0, survival]
1/survival_t0*integrate(survival, t0, inf, data=baseline)
}
</code></pre>
<p>The formula I use is from <a href="http://en.wikipedia.org/wiki/Survival_analysis" rel="nofollow">http://en.wikipedia.org/wiki/Survival_analysis</a>
$$\frac{1}{S(t_0)} \int_0^{\infty} t\,f(t+t_0)\,dt = \frac{1}{S(t_0)} \int_{t_0}^{\infty} S(t)\,dt$$</p>
<p>It's fine you can help me with other hazards models, but it's better to provide example on the same dataset, thanks!</p> | 73,497 | [
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