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<p>Suppose I roll 10 dice (six-sided). What is the probability of seeing k faces that appear more than (or equal to) 3 times?</p>
<p>For example, if I see 1, 1, 1, 2, 2, 2, 3, 4, 5, 6 then I have observed 2 faces that appeared more than (or equal to) 3 times (1 and 2).</p>
<p>I guess I can try running a simulation but was wondering if there is a neat way to calculate this analytically.</p> | 73,498 | [
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<p>Consider four random complex vectors $\mu_i$ of length $K$ whose entries are drawn from the complex normal distribution $\mathcal{CN}(\mathbf{0},\mathbb{1})$ centered in zero and of unit variance.
Then form the $2 \times 2$ complex "normalized" matrix
$$
\chi=\frac{1}{\sum_i|\mu_i|^2}\begin{pmatrix}\mu_1^*\\\mu_2^*\end{pmatrix}\cdot\begin{pmatrix}\mu_3&\mu_4\end{pmatrix}=\frac{1}{\sum_i|\mu_i|^2}
\begin{pmatrix}\mu_1^*\cdot\mu_3&\mu_1^*\cdot\mu_4\\\mu_2^*\cdot\mu_3&\mu_2^*\cdot\mu_4\end{pmatrix}
$$
where the asterisk stands for conjugate.
I need to calculate the mean value of the largest eigenvalue of $\chi^\dagger\chi$ as a function of $K$ (where the dagger stands for conjugate transpose).</p>
<p>...I don't even know where to start from. I found <a href="https://space.zeo.net/g/1ud3t" rel="nofollow">this paper</a> which I hoped could be of help, but the four entries of the matrix $\chi$, above, are not independent of each other, so I'm not sure how to use that characteristic function (eq. 2 in the paper).</p> | 73,499 | [
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<p>We get vendor quotes (their prices for different financial securities) every month. I want to compare which of these vendors 'agree' (i.e come close to) with our internal prices [the gold standard].</p>
<p>I also want to test these for sub categories - say Vendor A agrees with us on bonds, but vendor B agrees better with equities.</p>
<p>What statistical tests can I perform to rank the vendors (in terms of being close to our prices) and quantify the gap.</p>
<p>thanks.</p> | 31,100 | [
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<p>Does the Gibbs sampler converge to a global maximum in the presence of multiple modes? For example in case of a Gaussian mixture distribution?</p> | 73,500 | [
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<p>To fit a simple AR(5) model, I use SAS <code>PROC AUTOREG</code>. I called the option <code>ARCHTEST=(QLM)</code> which provides Engle’s Lagrange Multiplier Test for ARCH Disturbances and the Portmanteau Q Test. The statistics SAS provided for Engle’s Lagrange Multiplier Test were:</p>
<pre><code>Order LM p-value
1 0.018 0.893
2 0.079 0.961
3 0.881 0.830
4 1.088 0.896
5 1.105 0.954
6 4.114 0.661
7 4.179 0.759
8 4.182 0.840
9 4.528 0.873
10 4.528 0.920
11 5.565 0.901
12 6.318 0.899
</code></pre>
<p>I took the residuals of this model and used the <code>ArchTest</code> function of the <code>FinTS</code> package of <code>R</code> ... <code>library(FinTS)</code>.</p>
<pre><code># A function to do ArchTest at multiple lags where 'data' are the residuals of
# the model.
library(FinTS)
arch.m1 = matrix(NaN,12,3)
for(i in 1:12) {s = ArchTest(data,lags=i)
arch.m1[i,1] = s$statistic
arch.m1[i,2] = s$parameter
arch.m1[i,3] = s$p.value}
</code></pre>
<p>The test results from the <code>ArchTest</code> function of the <code>FinTS</code> package are very different:</p>
<pre><code>Order LM p-value
1 0.014 0.907
2 0.092 0.955
3 0.835 0.841
4 2.194 0.700
5 3.189 0.671
6 13.013 0.043
7 19.157 0.008
8 23.161 0.003
9 18.780 0.027
10 18.467 0.048
11 22.565 0.020
12 23.828 0.021
</code></pre>
<p>I thought the SAS <code>ARCHTEST=(QLM)</code> option and the <code>ArchTest</code> function of the <code>FinTS</code> package were performing the same test but apparently not. Does anyone know what the difference is between the tests in SAS and <code>FinTS</code>?</p>
<p><strong>Note</strong>, the data used by Tsay (2005) in the example on page 102 is available <a href="http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/m-intc7303.txt" rel="nofollow">here</a>.</p>
<p>Using the following <code>R</code> code:</p>
<pre><code>library(FinTS)
intc=read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/m-intc7303.txt", quote="\"")
ArchTest(intc$V2,lags=12)
</code></pre>
<p>I find:</p>
<p>$\chi^2$= 48.0425, df = 12, <em>p</em>-value = 3.073e-06</p>
<p>However, Tsay (2005:102) reports:</p>
<p>$\chi^2$= 43.5041, df = 12, <em>p</em>-value = 0.0000</p>
<p>If I use:</p>
<pre><code>ArchTest(log(1+intc$V2),lags=12)
</code></pre>
<p>Then:</p>
<p>$\chi^2$= 43.5041, df = 12, <em>p</em>-value = 0.0000</p>
<p>However, this doesn't explain the differences observed between SAS and <code>FinTS</code>.</p>
<p><strong>Thanks for your reply @mpiktas.</strong></p>
<p>The statistic calculated by the <code>FinTS library</code> using the <code>ArchTest function</code> is based on the second statistic provided by @mpiktas:</p>
<p>$$TR^2=T\left(1-\frac{SSR_1}{SSR_0}\right)$$</p>
<pre><code>intc=read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/m-intc7303.txt",quote="\"")
intc.ln=log(1+intc$V2)
lag=12
mat=embed(intc.ln^2,lag+1)
m=summary(lm(mat[,1]~mat[,-1]))
SSR1=sum(residuals(lm(mat[,1]~mat[,-1]))^2)
SSR0=sum(residuals(lm(mat[,1]~1))^2)
stat=length(mat[,1])*(1-(SSR1/SSR0))
</code></pre>
<p>which gives $\chi^2$ = 43.5041.</p>
<p>If I apply the first statistic provided by @mpiktas and shown on page 102 of Tsay (2005):</p>
<p>$$F=\frac{(SSR_0-SSR_1)/m}{SSR_1/(T-2m-1)}$$</p>
<pre><code>intc=read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/m-intc7303.txt",quote="\"")
intc.ln=log(1+intc$V2)
lag=12
mat=embed(intc.ln^2,lag+1)
m=summary(lm(mat[,1]~mat[,-1]))
SSR1=sum(residuals(lm(mat[,1]~mat[,-1]))^2)
SSR0=sum(residuals(lm(mat[,1]~1))^2)
stat=((SSR0-SSR1)/lag)/(SSR1/(length(mat[,1])-(2*lag)-1))
</code></pre>
<p>This version of the statistic gives $\chi^2$ = 3.837.</p>
<p>These are very different answers? I have provided the residuals for the AR(5) model from SAS below:</p>
<pre><code>c(0.14167,0.14019,-0.42499,0.0222,0.04388,-0.03157,0.00897,
-0.04567,0.29171,-0.06735,0.05235,-0.04062,-0.18694,
-0.2407,0.03766,-0.15011,0.11023,-0.30812,-0.02399,
0.13434,0.05836,0.09368,-0.01832,-0.15669,-0.00127,
0.19431,0.10366,-0.09682,-0.04597,0.07162,0.16411,-0.02181,
-0.00459,-0.16325,-0.09615,-0.04581,0.10441,0.12845,-0.0398,
-0.03349,-0.01575,0.06923,-0.05006,-0.05827,
-0.0317,0.08914,0.07323,0.06519,0.16141,0.0537,0.08213,
0.01073,-0.06628,0.01297,0.00622,-0.10171,0.05598,-0.05615,
0.10322,-0.04991,-0.01223,-0.04975,-0.06043,0.01794,
-0.00071,0.05425,-0.05462,0.01659,0.08656,0.02482,-0.04214,
0.01221)
</code></pre>
<p>For lag 5, SAS gives $\chi^2$ = 1.105, <code>FinTS</code> gives $\chi^2$ = 3.189, and the F-statistic version gives $\chi^2$ = 0.560.</p>
<p><strong>References:</strong></p>
<p>Engle, R. F. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987-1007.</p>
<p>Tsay, R. S. 2005. Analysis of financial time series. Second edition. John Wiley & Sons, Inc., Hoboken, New Jersey, USA.</p> | 38,088 | [
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] |
<p>So its like saying 95% of the items sold in this website costs between $$25 - $150. While some items might cost less than $25 and other items might cost more than $150.</p>
<p>Is there a way to find this? Is this something related to CI - confidence Interval?</p> | 73,501 | [
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<p>I'm looking for correlations between the answers to different questions in a survey ("umm, let's see if answers to question 11 correlate with those of question 78"). All answers are categorical (most of them range from "very unhappy" to "very happy"), but a few have a different set of answers. Most of them can be considered ordinal so let's consider this case here.</p>
<p>Since I don't have access to a commercial statistics program, I must use R.</p>
<p>I tried <a href="http://rattle.togaware.com/">Rattle</a> (a freeware data mining package for R, very nifty) but unfortunately it doesn't support categorical data. One hack I could use is to import in R the coded version of the survey which has numbers (1..5) instead of "very unhappy" ... "happy" and let Rattle believe they are numerical data.</p>
<p>I was thinking to do a scatter plot and have the dot size proportional to the number of numbers for each pair. After some googling I found <a href="http://www.r-statistics.com/2010/04/correlation-scatter-plot-matrix-for-ordered-categorical-data/">http://www.r-statistics.com/2010/04/correlation-scatter-plot-matrix-for-ordered-categorical-data/</a> but it seems very complicated (to me).</p>
<p>I'm not a statistician (but a programmer) but have had some reading in the matter and, if I understand correctly, <em>Spearman's rho</em> would be appropriate here. </p>
<p>So the short version of the question for those in a hurry: <strong>is there a way to quickly plot
Spearman's rho in R</strong>? A plot is preferable to a matrix of numbers because it's easier to eye ball and also can be included in materials.</p>
<p>Thank you in advance. </p>
<p>PS I pondered for a while whether to post this on the main SO site or here. After searching both sites for R correlation, I felt this site is better suited for the question.</p> | 73,502 | [
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<p>I would like to have a reality check of my understanding of the MH statistic. I have been trying to reproduce an example of the Mantel-Haenszel test provided in Conover (1999, p. 192-194). The data is structured as other examples provided for the procedure in R.</p>
<p>The statistic produces is the $T_{4}$ value, which is described as approximately normal when $H_{0}$ is true. Conover describes $T_{4}$ as:</p>
<hr>
<p>$$ T_{4} = \frac{\sum x_{i} - \sum{\frac{r_{i}\cdot c_{i}}{N_{i}}}}{\sqrt{\sum{\frac{r_{i}c_{i}(N_{i} - r_{i})(N_{i}- c_{i})} {N^2_{i} (N_{i}-1 ) }} }} $$</p>
<hr>
<p>The specification of the example on p. 194 is below:</p>
<pre><code>cancer.in <- array(c(10,12,1,1,9,11,0,1,8,7,0,3), dim = c(2,2,3),
dimnames = list(status = c("treat", "control"), outcome = c("success", "failure"),
group = c("group-1", "group-2", "group-3")))
cancer <- as.table(cancer.in)
cancer
##### continuity correction applied
mantelhaen.test(cancer)
##### no correction applied
mantelhaen.test(cancer, correct = FALSE)
</code></pre>
<p>When examining the structure of the table, it seems to follow the example in the text, but the statistic provided does not match. Conover gets at $T_{4}$ value of $\frac{1.6786 }{1.1719} = 1.4323 $. The $\chi^{2}$ value that is returned in R is 2.0515 (without continuity correction), which equals the value $T_{4}^2$ . </p>
<p>From what I understand, $z^2$ is distributed $\chi^{2}_{1}$, so the methods are producing the same results, but represented in a different way.</p>
<p>Am I on the right track with this?</p>
<p>Any guidance is greatly appreciated.</p> | 35,898 | [
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<p>Let's say I have some kind of survival data - i.e. I'm giving a drug that may cause mortality. So I have three patients: A, B and C. All are given the drug at Time t1.</p>
<p>Let's say patient A dies at time t2. Patient B dies at time t3. And Patient C dies at time t100.</p>
<p>Clearly, the likelihood that the drug caused the death of patient A and B is higher than the likelihood that the drug cause patient C's death (i.e. patient C likely died from natural causes whereas patients A and B probably died because of the drug).</p>
<p>Are there any techniques in survival analysis/related fields that allow one to quantify the probability/likelihood that the treatment caused the effect. For example, if I could calculate the probability that the treatment caused the death given that patient was treated at time X and died at time Y.</p>
<p>Thanks!</p> | 73,503 | [
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<p>I have a Q-learning algorithm with a finite number of states. Each state, however, is represented by the usual 32 bit Q-value. These Q-values allow to carry information about the previous experience far beyond the state of the Q-learner (in my case the state of the Q-learner is the sequence of his own choices and rewards). </p>
<p>What if the Q-value could be only binary? Then Q-learner wouldn't be able to carry that information and wouldn't learn much. I am very interested in how much memory, in bits, does the Q-learner need to learn different rules. (For example. Q-learner always has two choices, 1 and 0. If he chooses 1 1 0 in a row he gets the reward.)</p>
<p>However, I have problems with implementation. Q values are bounded only by experience, so how exactly do I implement the update rule for Q-value if I want to constrain that Q-value can be only one of n different values? </p>
<p>My update rule
Then $Q_{new} = Q_{old} + \alpha (R + \gamma * \max(Q_{future\ choice\ 1} - Q_{future\ choice\ 0}))$</p>
<p>How do I now map this new value on the limited set that Q-value can be?
If $\alpha = 1 , \gamma = 1; R,Q $ are elements of $\{0,1\}$ then $Q_{new}$ is element of $\{-1, 0, 1, 2\}$ And mapping is easy. But what if it's different?</p>
<p>Has anybody studied that? Papers and any suggestions are appreciated.</p>
<p>P.S. I am not sure that this is the best place for machine learning questions. If you know another Stack Exchange site that deals with that or any other site, please let me know.</p> | 73,504 | [
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<p>I have a set of data from two nights worth of monitoring.</p>
<p>The monitoring picks up how busy a server is in terms of the number of threads processing.</p>
<p>A change has been made and we want to analyse if this has had an impact of reducing the threads.</p>
<p>How can I compare the two data sets given that the load on the server is going to be a variable (but comparable).</p>
<p>I have been thinking of ANOVA, is this a good method? Any others people can suggest?</p>
<p>The data seems to display a positively skewed normal distribution. Is ANOVA still valid?</p>
<p>As an example (dummy data)</p>
<pre><code> Thread Count
Night Before Change Night After Change
00:00 341 223
00:05 365 321
00:10 465 445
.
.
.
08:50 512 243
08:55 314 556
09:00 256 354
</code></pre> | 73,505 | [
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<p>I have a database with different variables which contains information such as age, date of vaccination, number of doses as well as the number of antibodies (which is my target variable). If the number of antibodies is under 100 and 10, the patient's antibodies level is regarded as low and very low accordingly. My goal is to create a model by which I would be able to determine WHEN a patient should receive his next vaccination based on these facts. Thus far I have only generated decision trees using R studio. My mathematical knowledge is limited to come up with a solution. How could this be achieved?</p> | 73,506 | [
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<p>Suppose we have a training data set with variables $a,b$ and $c$ and binary outcome variable $y$. We fit a logistic regression model to this data set:</p>
<p>$$\text{logit}(p) = \hat{\beta_0}+ \hat{\beta_{1}}a + \hat{\beta_{2}}b + \hat{\beta_{3}}c$$</p>
<p>When we get the predicted probabilities from the training data set using the logistic regression model, are we supposed to get perfect classification? Or this depend on the threshold we use? </p> | 73,507 | [
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<p>What is the <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" rel="nofollow">skew normal</a> approximation to Poisson($\lambda$)?</p>
<p><a href="http://www.wolframalpha.com/input/?i=Skewness%5BSkewNormalDistribution%5B%CE%BC%2C%CE%BC%2C%CE%B1%5D%5D%3DSkewness%5BPoissonDistribution%5B%CE%BC%5D%5D%20solve%20for%20%CE%B1" rel="nofollow">Am I doing this wrong?</a></p> | 73,508 | [
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<p>Assume following data set representing each month of the year 2013 with the corresponding consumption of natrual gas to heat my flat and the respective mean temperature.</p>
<p>How can I seasonal adjust/normalize this time series?</p>
<hr>
<pre><code>1 156 m³ 1,4 °C
2 199 m³ 0,3 °C
3 173 m³ 2,3 °C
4 69 m³ 9,6 °C
5 63 m³ 12,2 °C
6 9 m³ 16,9 °C
7 9 m³ 20,8 °C
8 9 m³ 18,5 °C
9 15 m³ 14,3 °C
10 19 m³ 11,4 °C
11 83 m³ 5,2 °C
12 62 m³ 4,2 °C
</code></pre>
<hr>
<p>I would like to take into account that it is colder in winter and warmer in summer and know if I needed more or less energy in comparison. (especially in terms of comparing with the year 2014 for example)</p>
<p>I know how to use R, if that makes it easier for you to explain.</p> | 73,509 | [
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] |
<p>My major is CS and I have a question about Markov decision process.</p>
<p>I have been reading a book, planning with markov decision process an AI perspective.
While reading it, I have a question regarding the definition of MDP and its generality.</p>
<p>The 2nd order Markov chain can be transformed into 1st order Markov chain.
So any stochastic process that is depend on limited length history can be eventually
1st order Markov chain. </p>
<p>I think Markov Decision Process is the most general one when we consider discrete and finite state space. I don't know if there is non-Markovian decision process.</p>
<p>If we want to find optimal policy of an MDP with respect to maximum total expected utility (M.E.U) using value iteration, dynamic programming; if connectivity graph of MDP is acyclic, value iteration is same as Bellman-Ford shortest path finding algorithm.</p>
<p>I am curious about if there is something different decision process that can't be solved by dynamic programming when we are still finding an optimal policy w.r.t. M.E.U?</p>
<p>The question itself is confusing, but simply my question is MDP is the most general one if the state space is finite?</p>
<p>The additional question is when we are optimizing w.r.t. other metrics than M.E.U, the MDP can be solved by dynamic programming? </p>
<p>I hope someone suggest directions to explore this field.</p>
<p>Thanks in advance.</p> | 73,510 | [
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<p>I have an unknown discrete probability distribution $D$ ($D$ is a probability mass function), defined on an interval $[a,b]$ ($a>0$) and an estimation $\hat{D}$ such that, for all $t\in[a,b]$,
$$(1-\varepsilon)D(t)\le\hat{D}(t)\le(1+\varepsilon)D(t)$$
for some fixed $\varepsilon\in(0,1)$. Moreover $\hat{D}$ is actually a distribution on $[a,b]$ too.</p>
<p>Can anything be said about how well does the index of dispersion $\hat{\sigma}^2/\hat{\mu}$ of $\hat{D}$ approximate the index of dispersion $\sigma^2 / \mu$ of $D$?</p>
<p>Note that it is easy to show that $(1-\varepsilon)\mu\le \hat{\mu}\le (1+\varepsilon)\mu$, but I'm stuck at showing a bound for the deviation between the variances $\hat{\sigma}^2$ and $\sigma^2$</p> | 35,904 | [
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<p>I have a dataset: </p>
<pre><code>5, 10, 11, 13, 15, 35, 50 ,55, 72, 92, 204, 215
</code></pre>
<p>The formula for binning into equal-widths is this (as far as I know)
$$width = (max - min) / N$$</p>
<p>I think N is a number that divides the length of the list nicely. So in this case it is 3.
Therefore:</p>
<p>width = 70</p>
<p>How do I use that 70 to make the bins? </p>
<p><strong>EDIT: Found the answer. Should I answer my own question or just delete it?</strong></p> | 73,511 | [
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<p>For an independent t-test, if my hypothesis states there will be no significant difference between the two groups, is that a one- or two-tailed test?</p> | 49,920 | [
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<p>Could you please tell me how to conduct an event study? </p>
<p>I have 60 companies and 16 events. I am computing the abnormal return within 20 days around the event study (-10,10). </p>
<ul>
<li>Should I do it for each event for all the companies and then take the average? </li>
<li>Or each event for each company and then take the average? </li>
<li>Should I use equal weights or value weights? </li>
</ul>
<p>Also, once I obtain the abnormal returns, I would like to plot it using the 95% confidence interval. </p>
<p>Could you please tell me what is the Stata command to do so?</p> | 73,512 | [
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<p>I have a time series of daily data and am assuming each point in the time series is normally distributed. If I have a distribution of the daily data and want to scale this to cover a month (30 days) I believe I just scale the standard deviation by square root of 30. I think this assumes the data points are independently distributed. If I use an exponentially weighted moving average to calculate the standard deviation can I still scale the daily data to monthly by multiplying by square root 30?</p> | 35,905 | [
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<p>Is there a method to find the right distance function in non-parametric regression?
I use some time series to learn forecasting. Series are nonlinear and non-gaussian.
I can get the right dimension and delay. I can find the right bandwidth with the hdrcde library.
I have no problem with kernel functions.<br>
My problem is with distances. I use Euclidian, Cosine and Correlation weighting functions.
These are the kernel which give good results, but one time, Euclidian is good, then after adding some data, one to 5-7 generally, Euclidian give nothing, even with very little change in statistics.
So, my question is if there is a method to choose the right distance. I would like to have advices on that point and on articles that will help solve this problem. What package in R could eventually help?</p>
<p>Thank you. </p> | 28,190 | [
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<p>I am working on SVMs and try to get all the concepts involved. For instance, the kernel mapping. I would like to construct some parts of the algorithm by myself, to understand what is happening.</p>
<p>My goal is to create a mapping as in this picture (taken from <a href="http://www.tristanfletcher.co.uk/SVM%20Explained.pdf" rel="nofollow" title="here">here</a>)
<img src="http://i.stack.imgur.com/7yM2K.png" alt="Example mapping 2D --> 3D"></p>
<p>I do not fully understand what the input and output values of the kernel are; to map the data points to the 3rd dimension, the output should be the Z-values, right? And the input are (vectors of) the X and Y-values?</p>
<p>My (matlab) code to get the z-values is:</p>
<pre><code>z = exp(-( (abs(x-y).^2)./ (2*gamma^2) ));
</code></pre>
<p>But the z-values are just a bell-curve, us such:
<img src="http://i.stack.imgur.com/69m6f.png" alt="enter image description here"></p>
<p>I don't really know what I am doing wrong, but I think I confuse the concepts of kernel and (implicit/explicit) mapping.</p>
<p>How can I construct a (matlab) function that maps the 2D data to 3D space, using the Gaussian Radial Basis Function?</p>
<p>-- Edit --
Thanks to user27840 I made it work, with the following matlab code:</p>
<pre><code>gamma = 2;
D = squareform( pdist(data, 'euclidean') );
D = exp(-(D .^ 2) ./ ( 2*gamma^2));
z = sum(D);
</code></pre>
<p>This results in the following 3D plot, from original 2D data:
<img src="http://i.stack.imgur.com/qQssQ.png" alt="enter image description here"></p>
<p>-- Edit2: --
For those who are interested in one-class support vector machines; I wrote a blog post about it, using the answer from this thread: <a href="http://rvlasveld.github.io/blog/2013/07/12/introduction-to-one-class-support-vector-machines/" rel="nofollow">Introduction to one-class Support Vector Machine</a></p> | 73,513 | [
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<p>Suppose that one runs 8 correlation analyses but in two separate batches: Would the alpha-correction apply to all 8 tests as a family or each of the 4 tests as a family?</p>
<p>For example, if there are two research questions and both require correlation analysis, the researcher conducts the first 4 tests (for research question 1) and then conducts the second 4 tests (for research question 2).
In this case, should the alpha be corrected for a family of 4 tests or for a family of 8 tests?
(Note that the same statistical test--Pearson's $r$--is used in both cases.)</p>
<p>I understand there isn't a consensus on this matter (as with many other matters in statistics as I am learning fast!) but some broad guidelines would be most helpful.</p> | 35,908 | [
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<p>Consider the two-way ANOVA model with mixed effects :
$$
Y_{i,j,k} = \underset{M_{i,j}}{\underbrace{\mu + \alpha_i + B_j + C_{i,j}}} + \epsilon_{i,j,k},
$$
with $\textbf{(1)}$ : $\sum \alpha_i = 0$, the random terms
$B_j$, $C_{i,j}$ and $\epsilon_{i,j,k}$ are independent,
$B_j \sim_{\text{iid}} {\cal N}(0, \sigma_\beta^2)$,
$\epsilon_{i,j,k} \sim_{\text{iid}} {\cal N}(0, \sigma^2)$ ;
and there are two possibilities for the random interactions :
$\textbf{(2a)}$ :
$C_{i,j} \sim_{\text{iid}} {\cal N}(0, \sigma_\gamma^2)$
or $\textbf{(2b)}$ :
$C_{i,j} \sim {\cal N}(0, \sigma_\gamma^2)$ for all $i,j$, the random vectors $C_{(1:I), 1}$, $C_{(1:I), 2}$, $\ldots$, $C_{(1:I), J}$ are independent, and $C_{\bullet j}=0$ for all $j$ (which means that mean of each random vector $C_{(1:I), j}$ is zero). </p>
<p>Model $\textbf{(1)}$ + $\textbf{(2a)}$ is the one which is treated by the nlme/lme4 package in R or the PROC MIXED statement in SAS.
Model $\textbf{(1)}$ + $\textbf{(2b)}$ is called the "restricted model", it satisfies in particular $M_{\bullet j} = \mu + B_j$.
Do you think one of these two models is "better" (in which sense) or more appropriate than the other one ? Do you know whether it is possible to perform the fitting of the restricted model in R or SAS ? Thanks.</p> | 73,514 | [
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<p>A die is rolled twice,</p>
<ul>
<li><p>$X_1$ : the minimum value to appear in the two rolls</p></li>
<li><p>$X_2$ : the maximum</p></li>
</ul>
<p>I would like to derive $\ F_{X_1,X_2}(x_1,x_2)$. </p>
<p>I know that that the CDF of $\ X_1 $ = $\ 1- [1-{F(x)]} ^ n $, CDF of $\ X_2 $ $\ = $ $ \ [{F(x)]} ^ n $
and $\ F_{X_1,X_2}(x_1,x_2) = P(X_1<x_1|X_2<x_2)F_{X_2}(x_2) $</p>
<p>The solution seems to be : </p>
<p>$\ F_{X_1,X_2}(x_1,x_2) = 2F(\min[{x_1,x_2}])F(x_2) - F(\min[{x_1,x_2}])^2 $</p>
<p>I want to understand how such solution can be obtained.</p> | 73,515 | [
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<p>I'm trying to do some exploratory analysis on some weather. I'd like to do a multiple linear regression on my data and then plot the predicted value against the actual value. Here's where I've got so far: </p>
<pre><code>data<-read.csv("Amsterdam.csv", header=TRUE)
data2<-data[SolarAltitude>0,]
data2.lm<-lm(DirectRadiation ~ DryBulbTemperature + RelHum
+AtmosphericPressure, data=data2)
data.data.frame(data2,fitted.value=fitted(data2.lm),residual=resid(data2.lm))
</code></pre>
<p>If you could help, I would be very grateful,</p> | 73,516 | [
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<p>I am running some experiments for a few days now and I am asking myself why this is taking so long. So here is the setup:</p>
<p>I am using Weka api and have written some code in Java which automates the process of classifcation. I have a dataset consisting of 17,553 samples that contain some labels. For the top-100 most frequent labels I wish to train a binary classifier and test my algorithm with 5-Fold Cross Validation.</p>
<p>This is all running on a dell laptop with centrino dual core with the latest Ubuntu 14.04 LTS and a total of 4GB RAM. It takes roughly about 40-45 minutes to get results of one label. The laptop is now running for more than 2 days and I do not even have all results yet.</p>
<p>And also, I am training the SMO classifier found in WEKA with its default parameters.</p>
<p>I was just wondering, if this normal?</p>
<p>FYI before starting to run 5-fold cross validation, I have made an .ARFF file containing the data and read it in with a BufferedReader. However this had to be done only once in the beginning when the program starts.</p>
<p>And also, I am training an SMO classifier as found in WEKA with its default parameters.</p> | 35,912 | [
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<p>Suppose I have sets A and B of normally distributed data so that:</p>
<pre><code> A: mean=250, SD=200, N=25
B: mean=248, SD=200, N=20
</code></pre>
<p>Clearly, there is no statistically significant difference between the means (p=0.9736). But does this mean that the means are equal or that we have no evidence to suggest otherwise? Considering the huge standard deviation it seems intuitively unlikely that the means are <strong>equal</strong>. But this is what the null hypothesis says and based on the the t-test we didn't reject that.</p>
<p>Having such noisy data, how can one quantify how equal the means are?</p>
<p><strong>Update</strong>
This question arose from a discussion between myself and a colleague who was testing the effect of two drug treatments on biological samples. As the test revealed an insignificant difference between treatments, my colleague assumed this allows him to claim that the treatments have the same effect. With the example above I tried to show that this is not necessarily the case. But I didn't know the proper terminology to make things more quantitative. Hence this question.</p> | 73,517 | [
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<p>I read that 'Euclidean distance is not a good distance in high dimensions'. I guess this statement has something to do with the curse of dimensionality, but what exactly? Besides, what is 'high dimensions'? I have been applying hierarchical clustering using Euclidean distance with 100 features. Up to how many features is it 'safe' to use this metric?</p> | 73,518 | [
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<p>This is a question that arose from studying Hogg and Craig "Introduction to Mathematical Statistics",7th edition, pg 568.</p>
<p><img src="http://i.stack.imgur.com/v5FWG.jpg" alt="enter image description here"></p>
<p>It is assumed that we have taken a random sample of $X_1,\ldots,X_{n1}$ and a random sample of $Y_1,\ldots, Y_{n_2}$, $n$ denotes the combined sample size, i.e. $n=n_1+n_2$ and <strong>sgn(.)</strong> is $1$ if the quantity in the brackers is positive,$0$ if it is zero and $-1$ if it is negative. We also rank the variables from low to high.</p>
<p>Could you please help me understand how the authors reach the conclusion that the difference in medians estimator solves that equation? There is no explanation from their part and the word "easily" seems out of place :)</p>
<p>Thank you.</p> | 73,519 | [
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<p>Scikit Learn's <a href="http://scikit-learn.org/stable/tutorial/statistical_inference/model_selection.html" rel="nofollow">page on Model Selection</a> mentions the use of nested cross-validation: </p>
<blockquote>
<pre><code>>>> clf = GridSearchCV(estimator=svc, param_grid=dict(gamma=gammas),
... n_jobs=-1)
>>> cross_validation.cross_val_score(clf, X_digits, y_digits)
</code></pre>
<p>Two cross-validation loops are performed in parallel: one by the GridSearchCV estimator to set gamma and the other one by cross_val_score to measure the prediction performance of the estimator. The resulting scores are unbiased estimates of the prediction score on new data.</p>
</blockquote>
<p>From what I understand, <code>clf.fit</code> will use cross-validation natively to determine the best gamma. In that case, why would we need to use nested cv as given above? The note mentions that nested cv produces "unbiased estimates" of the prediction score. Isn't that also the case with <code>clf.fit</code>?</p>
<p>Also, I was unable to get the clf best estimates from the <code>cross_validation.cross_val_score(clf, X_digits, y_digits)</code> procedure. Could you please advise how that can be done?</p> | 73,520 | [
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<p>My dataset contains a lot of variables that appear to me as practically categorical on a continuous scale to differing degrees.</p>
<p>Many have a large chunk of zeros or specific value followed by one or more apparent separate chunks. In some cases this is obvious where there are literally 2 specific single effectively on/off. Others are much more potential candidates where there are almost 2 or more separate distributions.</p>
<p>I am trying to model a continuous normally distributed dependent on a number of potential variables (collected on a continuous scale). Most of these are likely not to contribute to the model. I will be using various modelling methods to explore what is best (i.e. I will be trying tree methods where the apparent binomial appearance isn't a problem). I am not assuming a good model can be produced.</p>
<p>In these situations is are there any hard and fast rules/techniques as to whether to categorise or not?
Also having potentially performed the transformations what considerations/measures might I have to be aware of?
I would say most of the dataset is like this.</p> | 73,521 | [
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<p>I have a loss data arising out of Operation risk for some particular bank. The standard procedure for arriving at the capital charge w.r.t. Operational risk needs I fit some continuous distribution to this loss data. </p>
<p>Normally, I am able to fit some standard distributions. Once the distribution is identified, the same distribution is used to simulate future loss amounts.</p>
<p>My question is assuming the data is such that I am just not able to fit any statistical distribution, how do I deal with data to simulate the loss amounts. Is there any <strong>non-parametric method</strong> available to deal with such situations.</p>
<p>Kindly give me some advice.</p> | 31,108 | [
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<p>I am trying to find the optimal estimator for the maximal expected $\Sigma X_i$ where $X_i$ is sampled from an unknown distribution which is chosen to be maximal. To clarify and simplify, there are two sets of data $Y_i$ and $Z_i$ which are drawn from similar but likely different pdf distributions $y$ and $z$. I would like an estimator on $Z_i$ and $Y_i$ which would predict which pdf would produce a larger sum over a future sampling, ie $\Sigma X_i$. </p>
<p>The number of elements in each set $N_y$ and $N_z$ is chosen by our implementation so we have to normalize the number of elements out. Or put another, way we are more interested in the mean $\mu = \Sigma X_i / N_x$ because our future sampling has a fixed $N_x$ independent of the choice of $y$ or $z$. This leads to the standard method of making a choice based on each scenario's mean, $\mu$, and error on mean $\sigma_\mu$. What I am interested in is if the mean is really the best estimator to choose the scenario which leads to the highest expected $\Sigma X_i$. In a simple scenario I am sure the answer is yes but there are some complications.</p>
<p>There is no good model for the underlying distributions which produces $Y_i$ or $Z_i$ but a histogram of either decreases exponentially. Hence the occurrence of a few high large value outliers which make things complicated. The main reason I think the mean may not be the best is because of its sensitivity to these outliers. The solution of removing outliers and doing a bootstrapping method is discussed here: <a href="http://stats.stackexchange.com/questions/72242/bootstrapping-do-i-need-to-remove-outliers-first">Bootstrapping - do I need to remove outliers first?</a> From this I think it is clear that the removal of outliers is ill advised. Additionally, the number of elements, $N_y$ and $N_z$, is in the millions so the CLT asserts we do not need to bootstrap at all to obtain $\sigma_\mu$.</p>
<p>Another complication is that the distribution is quantize/discrete in two ways. To illustrate this lets take the example of optimizing for future revenue in a split test. There are two scenarios that lead to two sets $Z_i$ and $Y_i$ representing the amount spent for each customer. To illustrate that the underlying pdfs are not continuous one must first off note that the smallest unit of money is a penny so the range of each pdf is the integers. Secondly and more importantly the items to buy do not span the space. Let say for example there is one item that costs \$20 but no other items in the price range [\$15,\$25]. Since there are a lot of single purchase customers there will be a peak at \$20.</p>
<p>The simplest choice of an alternative estimator would be the median, $\tilde{x}$ since it is less sensitive to outliers and insensitive to reparameterizations. However, since greater than 95% of customers spend \$0 the median will always be \$0. Also, I am not sure the median is a good estimator of the expected $\Sigma X_i$ in general.</p>
<p>So the question is, is there a better estimator than the mean of $Y_i$ and $Z_i$ for maximum expected $\Sigma X_i$? The word "better" in the context refers to the sense of gambling where the best bet is the best.</p> | 73,522 | [
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<p>My aim is to estimate the 2.5%- and 97.5%-quantile function (to get reference intervals) for a specific score in dependence of age separated by classes of a third variable cag. So first I built 11 categories of cag and for each of these categories I modeled the 2 quantile functions for the association score to age (with software R).
In a second analysis I used cag directly in my model <code>[score ~ factor(cag)+age*factor(cag)]</code> and get also 11 plots, but they are different from the first 11 plots. By the way, I modeled exponential functions via linear quantile regression.</p>
<p>Due to the plots, the first analysis fits the 2.5% quantile function better to the data than the second analysis. And vice versa with the 97.5% quantile function. So I am confused which analysis I should take, and my question is: </p>
<p>May I take the 2.5% quantile function of the first analysis and the 97.5% quantile function of the second analysis? May I mix it up?</p> | 73,523 | [
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<p>i have a data set that is strictly binary. each variable's set of values is in the domain: true, false. </p>
<p>the "special" property of this data set is that an overwhelming majority of the values are "false".</p>
<p>i have already used a bayesian network learning algorithm to learn a network from the data. however, for one of my target nodes (the most important one, being death), the AUC result is not very good; it is slightly better than chance. even the positive-predictive value (PPV), which has been suggested to me on CV, was not competitive with what's reported in the literature with other approaches. note that AUC (ROC analysis) is the typical benchmark reported in this area of clinical research, but i am also opened to suggestions on how to more appropriately benchmark the classification model if there are any other ideas.</p>
<p>so, i was wondering what other classification models i can try for this type of data set with this property (mostly false values).</p>
<ul>
<li>would support vector machine help? as far as i know, SVM only deals with continuous - variables as the predictors (although it has been adapted to multi-class). but my variables are all binary.</li>
<li>would a random forest help? </li>
<li>would logistic regression apply here? as far as i know, the predictors in logistic regression also are continuous. is there a generalized version for binary variables as the predictors? </li>
</ul>
<p>aside from classification performance, i suspect SVM and random forest might very well outperform the bayesian network, but the problem shifts to how to explain the relationships in these models (especially to clinicians). </p> | 235 | [
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<p>I have data of two random variables X and Y (each of size n x 1) and their joint pdf (probability density function) (of size m x m where m is NOT EQUAL TO n). How can I evaluate their joint moment generating function (MGF)? Importantly, I do NOT know the type of random variables. Thanks</p> | 35,916 | [
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<p>For simple linear regression, the regression coefficient is calculable directly from the variance-covariance matrix $C$, by
$$
C_{d, e}\over C_{e,e}
$$
where $d$ is the dependent variable's index, and $e$ is the explanatory variable's index.</p>
<p>If one only has the covariance matrix, is it possible to calculate the coefficients for a model with multiple explanatory variables? </p>
<p>ETA:
For two explanatory variables, it appears that $$\beta_1 = \frac{Cov(y,x_1)var(x_2) - Cov(y,x_2)Cov(x_1,x_2)}{var(x_1)var(x_2) - Cov(x_1,x_2)^2} $$
and analogously for $\beta_2$. I'm not immediately seeing how to extend this to three or more variables.</p> | 73,524 | [
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<p>I am trying to find the best way to visualize the following data:</p>
<p>I have values for 3 different times/dates, each time/date has the same 20 species. For each species I have the average height and standard deviation (that was obtained from n observations). I don't have the observations from where the average and standard deviation was obtained.</p>
<p>This is how the data looks like. Species are numerated from 1 to 20.</p>
<pre><code>t1 = cbind(seq(1,20,1),runif(20,0.5,5.0),runif(20,0,1.0))
t2 = cbind(seq(1,20,1),runif(20,0.9,6.0),runif(20,0.2,1.5))
t3 = cbind(seq(1,20,1),runif(20,0.1,8.5),runif(20,0.4,1.0))
</code></pre>
<p>Can anyone give any advice on what would be the best plot/graphic to use? I want to show the evolution of the average and standard deviation in time, however I don't want to do a line plot, and for a heat map I have a small dataset.</p> | 73,525 | [
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<p>If the best linear approximation (using least squares) of my data points is the line $y=mx+b$, how can I calculate the approximation error? If I compute standard deviation of differences between observations and predictions $e_i=real(x_i)-(mx_i+b)$, can I later say that a real (but not observed) value $y_r=real(x_0)$ belongs to the interval $[y_p-\sigma, y_p+\sigma]$ ($y_p=mx_0+b$) with probability ~68%, assuming normal distribution?</p>
<p>To clarify:</p>
<p>I made observations regarding a function $f(x)$ by evaluating it a some points $x_i$. I fit these observations to a line $l(x)=mx+b$. For $x_0$ that I did not observe, I'd like to know how big can $f(x_0)-l(x_0)$ be. Using the method above, is it correct to say that $f(x_0) \in [l(x_0)-\sigma, l(x_0)+\sigma]$ with prob. ~68%?</p> | 73,526 | [
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<p>The separate function of SVM is :<br>
$wx+b=0$<br>
The function distance of support vector to the separate plane is :<br>
$|r| = wx_i+b$<br>
And we can normalize the $w$, then the distance can be write as :<br>
$\frac{|r|}{|w|} = \frac{wx_i+b}{|w|}$ </p>
<p>And then approximate $|r|$ by 1. So we should get the maximal $\frac{1}{|w|}$.<br>
Actuarially, I didn't understand <strong>why can the $|r|$ be approximated by 1. Because the $|r|$ is also include the factor $w$.</strong></p> | 73,527 | [
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<p>What is the difference of gaussian HMM and gaussian mixture HMM (the emission is gaussian or gaussian mixture)? I want to know if it is the same thing. What is the point when estimating the parameters using Baum Welch algorithm.</p> | 73,528 | [
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<p>I have often seen the statement that linear separability is more easily achieved in high dimensions, but I don't see why. Is it an empirical fact? An heuristic? Plain nonsense?</p> | 35,921 | [
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<p>1. Which one is NOT a linear regression models? Please give a 1-2 sentences brief
explanation to your choice.<br>
(a) $y_i = β_0 +\exp(β_1x_i)+E_i, i = 1, 2, \ldots, n$<br>
(b) $y_i = β_0 + β_1x_i + β_2 x_{ii} + E_i , i = 1, 2, \ldots, n$<br>
(c) $y_i =β_0\exp(x_i)+β_2x_i^7 +E_i, i=1, 2,\ldots, n$ </p>
<p>2. Suppose $X$ and $Y$ has linear correlation coefficient $r = 0.5$, and there are 77 observations, what is the test statistic for the hypothesis test </p>
<p>$$H0:β_1=0 \quad\text{vs.}\quad Ha:β_1\neq0 $$</p>
<p>where $β_1$ comes from the simple linear regression model below? Please give a 1-2
sentences brief explanation to your choice. $\quad Y = β_0 + β_1X + E$ </p>
<p>(a). Not enough information<br>
(b). 5<br>
(c). 0.25 </p>
<p>3. Which model is more possible to have smaller $R^2$? Please give a 1-2 sentences brief explanation to your choice.<br>
A: $Y=β_0+β_1X_1+E$<br>
B: $Y=β_0^*+β_1^*X_1+β_2^*X_2+E^*$<br>
where $Y$ and $X_1$ in model A and B are the same.</p>
<p>(a). Not enough information<br>
(b). Model A<br>
(c). Model B </p> | 73,529 | [
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<p>What do you think these dummy variables do to the resulting weight values in a BCC/VRS data envelopment analysis? It appears the 1 and -1 relations drive the program to make the weights for the included columns equal, but that's not necessarily true. </p>
<p>The program directions that convex constraints on the weights can be included and that they are conical. This is an example file.</p>
<pre><code>
-------------------------------------------
DMU MktSh-I Salar-I Occ$-I #Hous-I IncHs-I OtherExp-I NetCntr-Out
-------------------------------------------
DMU1 1.332 1.706 1.414 1.326 0.853 3.005 1.003
DMU2 1.264 1.429 0.965 1.218 0.982 1.897 1.570
DMU3 1.544 1.496 1.363 0.803 0.703 2.283 1.415
DMU4 0.926 1.861 2.333 1.965 0.796 1.950 1.134...
Wynnewd 0.733 0.937 0.581 1.583 0.579 0.833 0.566
CasaLind 1.247 1.017 0.563 1.881 0.795 0.962 1.706
JoseyLn 0.823 0.675 0.416 0.769 0.972 0.523 0.500
.
.
DMU32 1.073 0.603 0.499 0.765 1.636 0.656 0.758
Dummy01 -0.950 0.050 0.050 0.050 0.050 0.050 0.000
Dummy02 0.000 1.000 -1.00 0.000 0.000 0.000 0.000
Dummy03 0.000 1.000 0.000 0.000 0.000 -1.00 0.000
Dummy04 0.050 0.050 0.050 -0.95 0.050 0.050 0.000
Dummy05 0.050 0.050 0.050 0.050 -0.95 0.050 0.000
Dummy06 0.000 1.000 0.000 0.000 -1.00 0.000 0.000
</code></pre>
<p>Here are a few DMU results when I only implement Dummy02</p>
<pre><code>Summary Results for Wynnewd
Level: 1 Status: Efficient Efficiency Rating: 1
Reference Set Value Multipliers Value Slack
NetCntrb(O) 0.077685
MktShare(I) 0.04682
Salaries(I) 0.070701
Occupan$(I) 0.070701
#_Houses(I) 0.04682
Inc/Hous(I) 1.1268
OtherExp(I) 0.04682
-----------------------------------
Summary Results for CasaLind
Level: 1 Status: Efficient Efficiency Rating: 1
Reference Set Value Multipliers Value Slack
NetCntrb(O) 0.17643
MktShare(I) 0.020472
Salaries(I) 0.020472
Occupan$(I) 0.069489
#_Houses(I) 0.020472
Inc/Hous(I) 1.3244
OtherExp(I) 0.020472
----------------------------------------------
Summary Results for JoseyLn
Level: 1 Status: Inefficient Efficiency Rating: 0.84999
Reference Set Value Multipliers Value Slack
OakCliff 0.33882 NetCntrb(O) 1.3178e-017
FarmBrch 0.57486 MktShare(I) 0.028515
Plsntgrv 0.086311 Salaries(I) 0.059021
Occupan$(I) 0.31351
#_Houses(I) 0
Inc/Hous(I) 0.67949
OtherExp(I) 0.15003
Variables Observed Ideal
NetCntrb(O) 0.5 1.3737
MktShare(I) 0.823 0.67496
Salaries(I) 0.675 0.52583
Occupan$(I) 0.416 0.35359
#_Houses(I) 0.769 0.60187
Inc/Hous(I) 0.972 0.82619
OtherExp(I) 0.523 0.44454
</code></pre>
<p>And Here is Wynnewd when I implement Dummy02 and Dummy03</p>
<pre><code>
Summary Results for Wynnewd
Level: 1 Status: Efficient Efficiency Rating: 1
Reference Set Value Multipliers Value Slack
NetCntrb(O) 0.074264
MktShare(I) 0.049537
Salaries(I) 0.049537
Occupan$(I) 0.079083
#_Houses(I) 0.049537
Inc/Hous(I) 1.1379
OtherExp(I) 0.049537</code></pre>
<p>Also, any tips on Non-discretionary variables as categorical/dummy? </p> | 49,183 | [
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<p>I have large correlation matrix in Excel that I'd like to use to inform my choice of explanatory variables in a multiple linear regression model. One problem is that the initial data was very sparse, and some columns had significantly more zeroes than others. How would I go about choosing the variables with the lowest correlation without falling into the trap of choosing those that have the lowest pairwise correlation solely because they were both very sparse? </p> | 73,530 | [
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] |
<p>I was just reading <a href="http://www.bayesian-inference.com/bayesfactors">this article</a> on the Bayes factor for a completely unrelated problem when I stumbled upon this passage</p>
<blockquote>
<p>Hypothesis testing with Bayes factors is more robust than frequentist hypothesis testing, since the Bayesian form avoids model selection bias, evaluates evidence in favor the null hypothesis, includes model uncertainty, and allows non-nested models to be compared (though of course the model must have the same dependent variable). <strong>Also, frequentist significance tests become biased in favor of rejecting the null hypothesis with sufficiently large sample size.</strong> [emphasis added]</p>
</blockquote>
<p>I've seen this claim before in <a href="http://www.fil.ion.ucl.ac.uk/~karl/Ten%20ironic%20rules%20for%20non-statistical%20reviewers.pdf">Karl Friston's 2012 paper in NeuroImage</a>, where he calls it the <em>fallacy of classical inference</em>.</p>
<p>I've had a bit of trouble finding a truly pedagogical account of why this should be true. Specifically, I'm wondering:</p>
<ol>
<li>why this occurs</li>
<li>how to guard against it</li>
<li>failing that, how to detect it</li>
</ol> | 35,923 | [
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<p>I have a distance matrix for some data I want to cluster. However, I don't just want to assign elements to clusters, but I also want to assign a probability for each element to belong to each cluster. </p>
<p>Can you give me examples of clustering algorithms that can do that?</p>
<p>Is there, for instance, a hierarchical clustering algorithm that returns, for each element, a probability to belong to any cluster? </p> | 73,531 | [
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<p>I have a sample taken from a population that I know to have a highly positive skew, although I know nothing else about the population. </p>
<p>My sample displays a similar skew. Sample size is 120.</p>
<p>Is it incorrect to transform (Box-Cox) my sample to normality to perform simple parametric tests like a mean with a confidence interval?</p>
<p>Should I just stick with medians?</p> | 47,351 | [
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<p>Part of my thesis tries to investigate the extent to which students "test Score" on the topic of electrical circuits in physic can be predicted from how they find the topic to be "interesting" ,Their "enjoyment" of the topic, and their perception of how "difficult" the topic is. I intend to use linear multiple regression analysis to do this, where "test Score" will be my independent variable and the predictors will be the "Student's enjoyment", "Students perception of difficulty" and how "interesting" student find the topic. I think student's test score is a continues variable. However the suspected predictors were obtained thus: </p>
<p>1. Working with electric circuits in physics is not interesting</p>
<p>Strongly Agree[ ] Agree[ ] Undecided[ ] Disagree[ ] Strongly Disagree[ ]</p>
<p>2. Working with electric circuits in physics is not difficult</p>
<p>Strongly Agree[ ] Agree[ ] Undecided[ ] Disagree[ ] Strongly Disagree[ ]</p>
<p>3. I do not enjoy practical work involving electrical circuits in physics</p>
<p>Strongly Agree[ ] Agree[ ] Undecided[ ] Disagree[ ] Strongly Disagree[ ]</p>
<p>the options are given these values:</p>
<pre><code>Strongly Agree ---- 1
Agree ------------- 2
Undecided --------- 3
Disagree ---------- 4
Strongly Disagree - 5
</code></pre>
<p>Now since this is an ordinal scale, is it proper to use linear multiple regression
analysis ?</p> | 73,532 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://stats.stackexchange.com/questions/30275/giving-more-details-to-my-last-post-im-very-lost-and-dont-know-which-tests-to">Giving more details to my last post: I'm very lost and don't know which tests to use or how to enter data in SPSS</a> </p>
</blockquote>
<p>I conducted an experiment using a within subjects design and 2x2 categorical fixed factors design with a categorical DV (every subject was in every condition). I have no idea what test to use. </p>
<p>Thanks for your help,
Emily</p> | 49,329 | [
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<p>I ran a Canonical Correlation Analysis on about 845 cases with 1000 variables each. (It originally started with 1000 cases and 400 variables but by using a kernel I got a 1000x1000 matrix)</p>
<p>As a result I got mainly eigenvalues very close to 1 (e.g. the highest: 0.9999987616532545), then there were a couple with E-8, five with 0 and some negative ones.</p>
<p>I find especially the nearly 1000 1's quite strange or is that a result that is possible/makes sense? I didn't quite get the results that I was hoping for, so I was wondering if all those nearly 1's could be the cause and if anyone has an idea what might have caused those?</p>
<p>I use EigenDecomposition from Commons Math 3.0 (<a href="http://commons.apache.org/math/apidocs/index.html" rel="nofollow">http://commons.apache.org/math/apidocs/index.html</a>).
I also tried some other matrices earlier and then got "MaxCountExceededException: illegal state: convergence failed" error. Not sure if that has anything to do with it, but I thought I'd add it.</p> | 73,533 | [
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<p>I was reading in "A Guide to Econometrics" that given $Y = X \beta + \epsilon$, the variance covariance matrix of $\beta^\text{OLS}$ is given by $\sigma^2 (X' X)^{-1}$ where $\sigma^2$ is the variance of the error term...</p>
<p>it then says that in the case of a single regressor $y = \beta_1 + \beta_2 x$, that this simplifies to $\sigma^2 / \sum(x-\bar{x})^2$. I don't quite see this. Isn't the matrix $X$ in this case still a two by two matrix? Namely:</p>
<p>$$
X =
\left(
\begin{array}{cc}
1 & x_1 \\\
1 & x_2
\end{array}
\right)
$$</p>
<p>How are we getting something like $\sum(x-\bar{x})^2$ ?</p> | 47,357 | [
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<p>I know I can subdivide a dataset by socio-demographic factors such as gender, age, income level etc. and analyse the responses for each set separately and then compare the results.</p>
<p>For example, I am testing the relationship between Variable A (hours of study) and Variable B (exam marks), and I have a dataset made up of 400 students. I have subdivided the dataset by gender and have 220 male students and 180 female students. I can run Pearson correlation on hours of study and exam marks separately for male students and for female students.</p>
<p>While the above is fairly straight forward, my difficulty is in dividing my dataset by a factor that is not a socio-economic factor. I want to divide my dataset by two locations as follows:</p>
<ul>
<li>School (hours of study vs. exam marks)</li>
<li>Home (hours of study vs. exam marks)</li>
</ul>
<p>To do this, I included a simple statement in my questionnaire that asked "where do you generally do most of your assignments?". The answer is either "home" (coded as "1") or "school" (coded as "2"). </p>
<p>This is a forced statement and I have deliberately made this nominal coding to avoid correlating it with the two variables I am testing. In my mind (which is not very statistically-orientated!), this statement functions like the socio-demograhic factors as above i.e. is a constant because I have deliberately coded it that way. </p>
<p>At the end of the day, and depending on my findings, I want to say something like "The relationship between hours of study and exam marks is stronger for students who completed their assignments in school".</p>
<p>Is this a risky approach?</p> | 73,534 | [
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<p>I am completely new to R, just downloaded and installed it today. I am familiar with SAS and Stata; I am using R because I have found out that in survey regression analysis, R is capable of using data that have stratum with one PSU. However, I cannot figure out how to write the code at all.</p>
<p>Here is what I have done so far: read a Stata dataset and save the .RData file. I have also put in the MASS, pscl, and survey (for svyglm) packages.</p>
<p>Here's what I need to do:
1) I am using survey data, so I have a "weight" variable, a "strata" variable, and a "PSU" variable. I need to incorporate those; I know how to use svyset in Stata, but no idea in R.
2) I have stratum with singleton PSUs. I need to use an option called survey.lonely.psu I believe, and I have no idea where to even begin with that. This is the reason why I am using R instead of Stata as I do not want to collapse stratum or delete observations.
3) The types of regression models I have to run: survey negative binomial, survey zero-inflated negative binomial (need to also determine the predictors of zeros), survey logistic, and survey OLS regression.
4)I also really can't make much sense in R of how to write the model in R code. In Stata, I can simply write the model as:</p>
<p>svy: nbreg dependent_var independent_var1 independent_var2 independent_var3</p>
<p>I can't figure out how to do that at all in R.</p>
<p>Any and all help will be greatly appreciated.</p> | 35,931 | [
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] |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://stats.stackexchange.com/questions/577/is-there-any-reason-to-prefer-the-aic-or-bic-over-the-other">Is there any reason to prefer the AIC or BIC over the other?</a> </p>
</blockquote>
<p>Can anyone please interpret each term in AIC, BIC and KIC. And the difference between the three.</p>
<p>Thanks in advance,</p> | 49,615 | [
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<p>There have been good questions on handling imbalanced data in the <em>classification</em> context, but I am wondering what people do to sample for regression.</p>
<p>Say the problem domain is very sensitive to the sign but only somewhat sensitive to the magnitude of the target. However the magnitude is important enough that the model should be regression (continuous target) not classification (positive vs. negative classes). And say in this problem domain that any set of training data will have 10x more negative than positive targets.</p>
<p>In this scenario, I might oversample the positive-target examples to match the count of negative-target examples, and then train a model to differentiate the two cases. Obviously the training approach does badly on imbalanced data, so I need to do sampling of some sort. What would be a decent way to "undo" this oversampling when making predictions? Perhaps translating by the (negative) mean or median of the target of the natural training data?</p> | 73,535 | [
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<p>In <a href="http://en.wikipedia.org/wiki/Girsanov_theorem" rel="nofollow">Girsanov</a> theorem, the change of probability measure variable $Z_t = \frac{dQ}{dP}|_{\mathcal{F}_t}$, why does it need to be a martingale with respect to measure $P$ for the change of measure $\frac{dQ}{dP}$ to exist?</p>
<p>I am having trouble understanding this. Anyone familiar with this?</p> | 73,536 | [
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<p>I have same data and I would like to choose a model for it. To start with I fit an exponential distribution and a gamma distribution. Now I wanted to do a simple <a href="http://en.wikipedia.org/wiki/Likelihood-ratio_test" rel="nofollow">likelihood ratio test</a> . However, I am told that to do this properly the two models have to be nested (which they are) and the parameter space of one has to be in the interior of the other, not on the boundary. This second condition doesn't seem to hold unfortunately. </p>
<p>What can I do?</p> | 73,537 | [
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<p>In my data, some individuals have missing data on the central predictor (father missed the intake assessment). Comparing the DVs' means for those with a missing/non-missing predictor yielded some sizeable effects.</p>
<p>Now I want to find out whether the systematic missings may have led me to underestimate the size of the OLS regression coefficients. What's a good way to do this?</p>
<p>Simply comparing the variances of the DVs in the group without missings to the group with missings is easy to do?<br>
But conceptually I want to know whether the whole sample has significantly less variability when I leave the group with missings (and significantly-lower-than-average-scores) out, not whether the two groups (with missings and without missings) have different variances.<br>
All that I found so far was about independent samples, not about subsets.</p>
<p>Also, just to bring me up to speed: heteroscedasticity is usually used in the context of residual variance, right? What's a good term for constricted variance that would give me better luck with google?</p> | 73,538 | [
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<p>By the definition</p>
<p>(1) $S(x)=\begin{cases}
S_0 = a_0x^3 + b_0x^2 + c_0x + d_0, & \text{if }t_0\le x\le t_1\\
.....\\
S_{k} = a_kx^3 + b_kx^2 + c_kx + d_k, & \text{if }t_{k-1}\le x\le t_k\\
\end{cases}
$</p>
<p>Using the necessary conditions which ensure that $S(x)$ is twice differentiable at the knots $t_0,...,t_k$ yields:</p>
<p>(2) $ \ S_i(x) = \frac{z_i}{6h_i}(t_{i+1}-x)^3 + \frac{z_{i+1}}{6h_i}(x-t_{i})^3 + (\frac{y_{i+1}}{h_i}-\frac{z_{i+1}h_i}{6})(x-t_i) + (\frac{y_{i}}{h_i}-\frac{z_{i}h_i}{6})(t_{i+1}-x)$</p>
<p>where $z_i = S_i''(t_i)$, $y_i = S_i(t_i)$ and $h_i = t_{i+1}-t_i$.</p>
<p>According to <em>Wood - Generalized Additive Models an Introduction with R (page 145, 146)</em>, one can rewrite (1) and (2) to</p>
<p>(3) $S(X) = a_j^-(x)\beta_j + a_j^+(x)\beta_{j+1} + c_j^-(x)\boldsymbol{F}_j\boldsymbol{\beta} + c_j^+(x)\boldsymbol{F}_{j+1}\boldsymbol{\beta}$</p>
<p>Since this is just a reformulation of (2) I'm really interested in how the exact formulation of $b_i(x)$ will look like since</p>
<p>$S(x) = \sum_{i=1}^k b_i(x)\beta_i$</p>
<p>is actually used to fit the data.</p> | 73,539 | [
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<p>I run two univariate linear regressions within one sample (same subjects, two different conditions). Now I would like to know how to compare the regression coefficients in order to find out whether they differ significantly. All of the methods I find apply to a case with two different independent samples. How do I do this in SPSS?</p> | 73,540 | [
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<p>Let's say I have a file A1.txt and a file A2.txt. I have written the statements</p>
<pre><code>filename in1 'A1.txt';
filename in2 'A2.txt';
</code></pre>
<p>Now, I want to re-do this using B1 and B2 (and eventually C1, C2, D1, D2), and just rename the variable in one place. So, I want a statement like</p>
<pre><code>%let prefix = 'B';
</code></pre>
<p>and then I want to put that prefix in the filename statements. </p>
<pre><code>filename in1 '&prefix1.txt';
filename in2 '&prefix2.txt';
</code></pre>
<p>I'm not doing this in a data statement. I can almost get it. filename in1 &prefix'.txt' tries to read B'.txt but that extra quote is pesky. There's something vague in my memory banks about using an _ but I can't pull it out. </p>
<p>Thanks. </p> | 73,541 | [
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<p>I have written a paired t-test and I am looking for some standard unit tests that I can make sure my code is correct. Do such tests exist? How does R take care of unit testing for its t.test()?</p> | 35,940 | [
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<p>I am new to bootstrapping. Assume I have some non-normal data, can be any distribution, it doesn't matter, and I want to find a confidence interval for the mean, median and standard deviation. For the median this question is relevant also for the normal case.
What I don't know is (and asking you to help with) is:</p>
<p>I understand the idea of bootstrapping, the re-sampling idea of generating n samples with replacement, and therefore getting n means, medians or standard deviations. What I don't understand, is how the confidence interval is being calculated. I mean, given a sample of n medians, which aren't normally distributed, how do I calculate it, I don't have Z's or t's values.</p> | 28,353 | [
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<p>I am fitting a linear model in R with many variables:</p>
<p><code>lm(Y~X1+X2+...+X100)</code></p>
<p>I want to check also for all pairwise interactions. Writing <code>X1*X2*...*X100</code> is not good since it checks the full model (more than pairwise). Writing all pairs explicitly is annoying (100 over 2). Is there a way to do this in R?</p> | 73,542 | [
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<p>I am performing quantile regressions in R using the package quantreg. My dataset includes 12,328 observations ranging from 0.12 to 330. The timepoints for my data are not exactly continuous; all data fall into one of a few dozen bins ranging from 73 to 397.</p>
<p>When I performed a linear regression on this data using the lm() function, I was able to do this with polynomials up to 4:</p>
<pre><code>lm(Y~poly(X,3,raw=TRUE),data=mydata)
</code></pre>
<p>However, with the package quantreg and the rq() command, I cannot use any polynomials. A simple regression works just fine:</p>
<pre><code>rq(Y~X,data=mydata,tau=.15)
</code></pre>
<p>But as soon as I get into polynomials, no dice. When I enter this:</p>
<pre><code>rq(Y~poly(X,2,raw=TRUE),data=mydata,tau=.15)
</code></pre>
<p>I get the following error message:</p>
<pre><code>Error in rq.fit.br(x, y, tau = tau, ...) : Singular design matrix
</code></pre>
<p>I am very new to all of this. I've read up on singular matrices as much as I can and I think there might be two reasons for this: (1) I only have one variable on each axis, or (2) my data are binned/the Y variable isn't truly continuous.</p>
<p>Can anyone tell me why I'm getting this error?</p>
<p>Thank you!</p>
<p>PS - This is how the graph looks:</p>
<p><img src="http://i.stack.imgur.com/htUdA.png" alt="enter image description here"></p> | 73,543 | [
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<p>Setting: Large N, short T panel dataset. Very few 1's (probably 1 percent), most zeroes. I estimate a logistic regression and find a negative significant coefficient on the IV of interest. Reassuringly, it's negatively correlated with the indicator variable.</p>
<p>But when I run this in a LPM (with OLS), I find a positive significant coefficient.</p>
<p>What's driving this? Which (the negative or positive) is more likely correct?</p>
<p>Thanks!</p> | 73,544 | [
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<p>In baseball statistics, there is a statistic called "luck" which is the difference between a team's win-loss record and their <a href="http://en.wikipedia.org/wiki/Pythagorean_expectation" rel="nofollow">Pythagorean win-loss record</a>. This statistic is supposed to measure how lucky or unlucky a team was to win however many games they did in a season. </p>
<p>Suppose one has a big data set which, for each year n, includes </p>
<ol>
<li>team winning percentage $P(n)$</li>
<li>team winning percentage the previous year $P(n-1)$</li>
<li>team luck the previous year $L(n-1)$</li>
</ol>
<p>and wants to create a linear regression model using $P(n-1)$ and $L(n-1)$ to estimate $P(n)$. </p>
<p>There's no apparent relationship between $L(n-1)$ and $P(n)$, but it seems as though we could use $L(n-1)$ in conjunction with $P(n-1)$ to better predict $P(n)$ based on how "flukey" $P(n-1)$ was and in what way. </p>
<p>So, the question is, how could one incorporate a luck-type measure into a linear regression model like I've discussed? I'm not concerned with this particular luck-type measure, but rather any measure which does something similar to what this one is supposed to do. </p> | 35,943 | [
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<p>Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$.</p>
<p>definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted
process if there exist times $0=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=T $ and random variables $\eta_{0},\eta_{1},\cdots,\eta_{n}$ such that $\eta_{0}$ is a constant,$\eta_{i}$ is $\mathcal F_{i}$-measurable,For simple adapted processes Ito integral $\int X dB$ is defined as a sum $$\int_{0}^{T}X(t)dB(t)=\sum_{i=0}^{n-1}\eta_{i}(B(t_{i+1}-B(t_{i}))$$ </p> | 73,545 | [
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<p>When doing a paired t-test, is it appropriate to use the paired t value to calculate $\eta^2$?<br>
I ask because it is not appropriate to calculate Cohen's d based on the paired t value.</p>
<p>Thanks.</p> | 73,546 | [
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<p>SPSS uses the Levene test to evaluate homogeneity of variances in the independent group t-test procedure. </p>
<p>Why is the Levene test better than a simple F ratio of the ratio of the variances of the two groups?</p> | 73,547 | [
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<p>Suppose you have some measurement series with quite different variances, see e.g., figure below.</p>
<p><img src="http://i.stack.imgur.com/sxuOB.png" alt="enter image description here"></p>
<p>The problem is that the differences between a, b, and d are barely recognizable, because the y-scale is rather large, because of the first measurement (these are no measurement errors, they are quite essential and can not be ignored).
What is the most elegant way to plot the data without losing too much information.</p>
<p>Two possibilities come to my mind:</p>
<p>a) Leave the first measurement out, and supply it otherwise (e.g., in the text or as a table).</p>
<p>b) Make the y-scale exponential (inverse of log). However, this is quite uncommon (how do you do this with R?).</p>
<p>What would you suggest?</p> | 73,548 | [
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<p>I am applying binary logistic regression as my dependent variable is a dichotomous variable with 740 sample size. I have used enter method to input my variables and have designed two blocks. In block 1 I have tried to see the impact of only demographic variables and in block 2 I have added 10 new predictors. Chi-Square value has increased significantly from block 1 to block 2, Hosmer and lemeshow test has value greater than 0.05 (It was 0.000 for block 1)and -2 log likelihood has also decreased drastically from block 1 to block 2, indicating good model fit. But In block 1 I can see significant impact of some of the demographic variables, however in block 2 there is no impact of any of the predictors. Please help me with my output..what does it signify? Am i mistaking somewhere?</p> | 26,740 | [
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<p>I have six variables measuring hotels' facilities by a group of managers. In the end, I want to compare mean scores for these six variables to determine which facility is actually favored in the hotel industry. </p>
<p>Can I use a one-way repeated measures ANOVA to check if there are statistical differences between mean scores for these variables?</p> | 73,549 | [
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] |
<p>In <em>Optimal Hedging Using Cointegration</em>, C. Alexander, 1999 (<a href="http://www.icmacentre.ac.uk/pdf/cointegration.pdf" rel="nofollow">http://www.icmacentre.ac.uk/pdf/cointegration.pdf</a>), page 3, footnote #2:</p>
<blockquote>
<p>[...] a PCA of cointegrated variables will yield the common
stochastic trend as the first Principal Component [..]</p>
</blockquote>
<p>I am struggling to work out a proof of this statement.
I'd be thankful if you could give me a hand.</p>
<p>Thanks!</p> | 35,946 | [
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<p>I am having trouble interpreting the z values for categorical variables in logistic regression. In the example below I have a categorical variable with 3 classes and according to the z value, CLASS2 might be relevant while the others are not. </p>
<p>But now what does this mean?</p>
<p>That I could merge the other classes to one? <br>
That the whole variable might not be a good predictor?</p>
<p>This is just an example and the actual z values here are not from a real problem, I just have difficulties about their interpretation. </p>
<pre><code> Estimate Std. Error z value Pr(>|z|)
CLASS0 6.069e-02 1.564e-01 0.388 0.6979
CLASS1 1.734e-01 2.630e-01 0.659 0.5098
CLASS2 1.597e+00 6.354e-01 2.514 0.0119 *
</code></pre> | 49,483 | [
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] |
<p>I want to use the inverse normal distribution in winbugs but I don't find it there. Please help me to find a solution.</p>
<p>I have here truncated normal distribution for y</p>
<pre><code>for (j in 1:P) {
y1[i,j] ~ dnorm(mu1[i,j], psi1[j])I(thd[j,z1[i,j]], thd[j,z1[i,j]+1])
}
</code></pre>
<p>where <code>y</code> is treated as a missing variables and augmented with the posterior analysis.
Variable <code>z</code> are ordered categorical variables and <code>thd</code> set of thresholds. </p> | 73,550 | [
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<p>Suppose that we have an $M/GI/\infty$ queue, that is, we have infinitely many servers, a Poisson arrival process with rate $\lambda$ (i.e., random arrival times $0=t_0 < t_1 < t_2 < \dots < t_n < \dots$, with inter-arrival times $\theta_n = t_n - t_{n-1}$ being independent exponentially distributed with mean $1/\lambda$ ), and i.i.d delays (independent of arrivals) with finite mean $\tau$. We know (e.g., from the book of Takacs[1962]) that there exists a stationary distribution such that the size $q(t)$ of queue at time $t$ (in the steady-state) follows a Poisson distribution with mean $\lambda \alpha$, that is,</p>
<p>$ P(q(t) = k ) = \dfrac{e^{-\lambda \tau}(\lambda \tau)^k}{k!}, \qquad $ for all $t \ge 0$.</p>
<p>Now suppose that we want the distribution of the size of queue at the time of the $n$ arrival, assuming that it is in the steady state. That is, we want $p_k = P(q_n = k)$, where $q_n = \lim_{t \to t_n^-} q(t)$. How can we obtain $p_k$?</p> | 35,948 | [
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<p>I'm a programer, but I know very little about statistics and am not even sure where or how to ask this. Lets say you have 2 variables about people in general, var A and var B, that are tangible characterists of these people. People either possess A or B. </p>
<p>I then take 11 different measurements about the person and use those to determine if they are actually A or B without looking at them. The program successfully determines if someone is A or B in a group of 10 people. But as I test more and more people, I find that some people have slight differences or exceptions in their variables that I have to account for.</p>
<p>Example: All A people have the first variable in a range of 12 to 13, the 2nd variable in a range of 5 to 6, but then I find an A person who has a range of 1 for the 2nd variable. So I add to the formula that if the 2nd variable = 1, then the person has A. </p>
<p>My question - How many people would I have to test out to get an accuracy rating above 80% of the program, or is that even possible. As I add more and more subjects that fit that equation, does that translate into an increase in accuracy of the program when used on the general population?</p> | 73,551 | [
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<p>I am searching for a good measure to capture clustering in time based events. Say that for a given interval of 5 minutes, cars randomly pass in front of house ... say one car at every 10 seconds (for a total of 30 cars). For another interval of 5 minutes, 3 cars passes at every thirty seconds, and the time between the three cars is 0 (again 30 cars over 5 minutes). The difference between both intervals is that for the second intervals, the arrival of cars "clusters". What can be a good measure to capture clustering? I would like a measure that indicates that there was more clustering in the second interval than the first one. </p> | 73,552 | [
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<p>I am testing a self-implemented PCA on a set of image data ($N \times d$, $N = 10000$ sample size, $d = 28 \times 28$ feature size). When I observed the classification accuracy, I found the accuracy increase somehow with the increased <em>target reduced dimension</em> $k$ and decresed from then on. See the image below:</p>
<p><img src="http://i.stack.imgur.com/upcc3.png" alt="y: accuracy out of 10k, x: reduced dimension"></p>
<p>The maximum accuracy corresponded to $PoV = 89\%$.</p>
<p>I also tried reducing the data to $k = d$, the accuracy was still much lower than the maximum showed in the graph.</p>
<p>Is these what is expected for <strong>PCA</strong>? Or my <strong>classifier</strong> is not working?</p>
<p><em>How can I interpret the "information lost" for different $k$ in PCA?</em></p>
<p>ps: I even tried the "princomp' built-in function, it yield similar result.</p>
<hr>
<p>UPDATE: </p>
<p>Sorry for not stating clearly. For both training set itself and a separate test set as test
set, I got similar result as in the graph.</p> | 35,950 | [
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<p>I hope this is the right forum to ask this question.</p>
<p>Sequential filtering frameworks perform 2-steps essentially:</p>
<p>Process model (forecast) : $x_{t+1} = f_t(x_t) + v_t$</p>
<p>Observation model (update): $z_t = h_t(x_t) + w_t$</p>
<p>Here, $v_t, w_t$ are uncorrelated zero-mean noise processes, and $f_t, h_t$ are functions (may be linear or non-linear).</p>
<p>For the particular case of Kalman filters, $f_t, h_t$ are <em>linear</em> and the noise processes are <em>Gaussian</em>.</p>
<p>In my application, the user <em>learns</em> the observation model using historical data (linear regression, for example). Since that is the case, the user might as well learn an <em>inverse observation model</em>, such that $$x_t = g_t(z_t) + \eta_t$$</p>
<p>Here, the underlying state vector is expressed as a function of observations $z_t$. Assume that $\eta_t$ is Gaussian.</p>
<p>With the inverse observation model, the update step is simplified. Apply $g_t$ to $z_t$ and then simply <em>assimilate</em> the resulting vector (call it $\hat{x}_t$) to $x_t$. The assimilation here is a weighted addition of two Gaussian distributed variables where the weights are in inverse proportion of their covariance matrices (for Kalman optimal solution).</p>
<p>So my question is: is there anything wrong with this approach of using an inverse observation model instead of an observation model? The advantage is that I can define the weighted addition as an operator in my data structure.</p> | 73,553 | [
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<p>I want to train a model
but the number of examples of the categories differ a lot
1 category : 100 examples
2 category : 2000 examples
3 category : 50
4 category : 4500</p>
<p>if I take the 50% of the examples of each category as
training and the other 50% as test set, the fact that I
have very few examples for the category 1 and 3
will affect my model?</p> | 73,554 | [
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<p>I started evaluating and comparing some methods in forecasting. I used Price of dozen eggs in US, 1900–1993, in constant dollars in the R software FMA package. I held out the last 10 years for assessment of forecast. Below are the results:</p>
<p>I used auto arima method in the R software. Obviously the results are way off. Am I doing something incorrect ? Below is the forecast. It does not recognize the declining trend. </p>
<p><img src="http://i.stack.imgur.com/KIM9O.jpg" alt="auto arima"></p>
<p>I also used an unobserved components model (UCM) and obtained a good forecast, as below.</p>
<ol>
<li>Without outliers/level shifts there are very large standard errors and therefore wide confidence bands. <img src="http://i.stack.imgur.com/dlIXM.png" alt="UCM without outliers level shifts"></li>
<li>After some iterative work, below is the output with outliers/level shifts (I know I'm overfitting here) but it did a pretty good job in forecasting; there are also narrow confidence bands. <img src="http://i.stack.imgur.com/5SrYJ.png" alt="UCM with outliers level shifts"></li>
</ol>
<p>In looking at just this example the UCM seems to predict the hold-out sample more accurately than auto.arima.</p>
<p><strong>Why is auto.arima not providing a reasonable forecast?</strong></p>
<p><strong>Are state space models/UCMs better for forecasting long range?</strong></p>
<p><strong>Are there any benefits of using one method over other?</strong></p> | 49,926 | [
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<p>What are some Goodness-of-Fit tests or indices for the case of continuous variables?</p>
<p>For example, I am looking at the Kolmogorov–Smirnov test. What I don't get is how one gets the emprical CDF in the first place? What I mean is, let's say I do a regression analysis with Gaussian errors. I have the maximum-likelihood estimate of the parameters. Now I also need to do a density estimation for the empirical CDF? Aren't they the same thing? Isn't my likelihood already giving me a goodness of fit? Why do I need K–S? Really confused.</p> | 35,952 | [
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<p>I'm doing one-way ANOVA power calculations (in R) for an RCBD trial with 15 groups, 3 blocks, and n = 3 per group. How is it possible that two different ANOVA models have exactly the same power even if the response variables are different? Even if the responses are somewhat related, I don't understand why the power is the same even to 7 significant figures. The fact that the power is very low makes sense. </p>
<p>Here are the calculations for two possible responses:</p>
<pre><code> groups = 15
n = 3
between.var = 0.3422844
within.var = 1.026853
sig.level = 0.01
power = 0.2162478
groups = 15
n = 3
between.var = 1.759788
within.var = 5.279365
sig.level = 0.01
power = 0.2162478
</code></pre> | 28,476 | [
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<p>I'm reading up on the Guass-Markov theorem on <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem" rel="nofollow">wikipedia</a>, and I was hoping somebody could help me figure out the main point of the theorem.</p>
<p>We assume a linear model, in matrix form, is given by:
$$ y = X\beta +\eta $$
and we're looking for the BLUE, $ \widehat\beta $.</p>
<p>In accordance with <a href="http://en.wikipedia.org/wiki/Errors_and_residuals_in_statistics" rel="nofollow">this</a>, I would label $\eta = y - X\beta$ the "residual" and
$\varepsilon = \widehat\beta - \beta$ the "error". (I.e the opposite of the usage on the Gauss-Markov page).</p>
<p>The OLS (ordinary least-squares) estimator may be derived as the argmin of $||\text{residual}||_2^2 = ||\eta||_2^2$. </p>
<p>Now, let $\mathbb{E}$ denote the expectation operator. To my understanding, what the Gauss-Markov theorem tells us is that, if $\mathbb{E}(\eta) = 0$ and $\text{Var}(\eta) = \sigma^2 I $, then the argmin, over all linear, unbiased estimators, of $\mathbb{E}(||\text{error}||_2^2) = \mathbb{E} (||\varepsilon||_2^2)$ is given by the same expression as the OLS estimator. </p>
<p>I.e.
$$ \text{argmin}_{\text{} \widehat\beta(y)} \, ||\eta||_2^2
\;=\; (X'X)^{-1}X'y \;=\;
\text{argmin}_{\text{linear, unbiased } \widehat\beta(y)} \, \mathbb{E}(||\varepsilon||_2^2)$$</p>
<p>Is my understanding correct? And if so, would you say that it deserves more prominent emphasis in the article?</p> | 45,561 | [
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<p>Seismic exploration involves the excitation of seismic waves using the industrially made explosive charges. </p>
<p>What could be the distribution of explosion energy of such a charges? </p>
<p>My guess is that it could be gaussian with mean that can be derived from the charge specs and with very small dispersion. However, the acquisition of the explosion wavefield on the earth surface suggests that the distribution has very strong skewness (right tail is very long).</p> | 73,555 | [
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<p>Let's say you wanted to map the intersections between n different online communities. For example:</p>
<ul>
<li>80% of CrossValidated users also have a Stack Overflow account</li>
<li>50% of Hacker News users also have a Reddit account</li>
<li>etc.</li>
</ul>
<p>What would be a good way to visualize this? Any examples of existing diagrams?</p>
<p>I've seen <a href="http://stats.stackexchange.com/questions/6234/visualizing-the-intersections-of-many-sets">this question</a>, which provides an answer. But I'm curious to know if there are other visualization techniques that would specifically lend themselves to comparing the relationships between communities. </p> | 73,556 | [
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<p>I'm wrestling with a question regarding random effects that I haven't been able to figure out with my regular resources. I am examining the effects of two treatments (heat and water) on plant biomass data that I've collected for 4 years in 20 plots (each plot has 4 quadrants that are not independent but are measured separately).</p>
<p>My <code>lme</code> model (using <code>nlme</code> in <code>R</code>) has two nested categorical random effects: year is nested within plot and quad such that my R script is the following:</p>
<pre><code>lme(biomass ~ heat*water + ... + elevation, data, random=~1|plot/quad/year)
</code></pre>
<p>There are no error messages given and the model appears to run smoothly. I only have 4 levels for year (80 for plot/quad) and have read that that a low number of levels will make it hard to get accurate estimates of variance. I tried adding year as a fixed effect and get completely different results (there is a lot of year-to-year variability in my dependent variable). </p>
<p>My questions: </p>
<ol>
<li><p>Is the small size of the second-level random effect much of a problem? Is it only the number of levels of the main random effect that I should worry about?</p></li>
<li><p>If you think having only 4 levels is a problem, how can I test to see if it is messing up my model? Collecting data for several more years just to correct this issue is not possible.</p></li>
</ol>
<p>Below are the summary output of the model:</p>
<pre><code>> summary(model)
Linear mixed-effects model fit by REML
Data: data
AIC BIC logLik
2327.316 2359.081 -1154.658
Random effects:
Formula: ~1 | plot
(Intercept)
StdDev: 13.89338
Formula: ~1 | quad %in% plot
(Intercept)
StdDev: 11.32595
Formula: ~1 | year %in% quad %in% plot
(Intercept) Residual
StdDev: 18.00256 7.39362
Fixed effects: forbs ~ h * w + elevation
Value Std.Error DF t-value p-value
(Intercept) 1290.2549 2576.2021 183 0.5008360 0.6171
hy -10.6171 10.5874 15 -1.0028063 0.3319
wy -13.6224 10.3395 15 -1.3175078 0.2074
elevation -0.3382 0.7265 15 -0.4654946 0.6483
hy:wy 37.1865 15.0869 15 2.4648273 0.0263
Correlation:
(Intr) hy wy elevtn
hy -0.222
wy -0.189 0.503
elevation -1.000 0.220 0.187
hy:wy 0.275 -0.728 -0.713 -0.274
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-0.76090000 -0.23913835 -0.02857524 0.19955752 1.39996632
Number of Observations: 257
Number of Groups:
plot quad %in% plot year %in% quad %in% plot
20 74 257
</code></pre>
<p>Thanks for your help, and please let me know if I can provide any additional info. I'd be happy to send you my data and script if it helps.</p> | 73,557 | [
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<p>I have an integral equation of the form
$$
T_1(x) = \int_0^x g(T_1(y)) \ d\hat{F}_n(y)
$$
where $\hat{F}_n$ is the empirical cdf and $g$ is a function.
I have a contraction mapping and so I am trying to solve the integral equation by using the Banach Fixed Point theorem sequence. </p>
<p>However, this runs very slowly in R and I am thinking it is because I am integrating using the sum() function for $x \in \hat{F}_n$ over and over again.</p>
<p>Is there a faster way to integrate using the empirical distribution with a function such as integrate()? </p> | 35,956 | [
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<p>I have been trying to find good resources on the internet describing what Wisard Neural Networks are and their differences with traditional Neural Networks but failed to find anything substantial. Can anybody please explain their main characteristics please? </p> | 6,783 | [
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<p>I have a data set with a small number of samples (322) and a large number of features (318.976). My data consists of images, and I want to train a binary classifier.</p>
<p>Since I have such a small amount of data available, I have employed the following strategy:</p>
<ul>
<li>Reduce dimensionality using a few hand-written rules (from 318.976 to 12.000 features)</li>
<li>Train Sparse Filter to reduce dimensionality further (from 12.000 to 1000 features
<ul>
<li>To train the sparse filter, I construct additional data by shifting my images horizontally which my problem is invariant under. This gives me 19.320 samples instead of 322.</li>
</ul></li>
<li>Now my data set has 19.320 samples and 1000 features</li>
<li>In the training set, only around 2% of the samples are positives</li>
<li>Now I train a random forest on the data set weighting the positive samples up so they in total achieves the same weight as the negative samples</li>
</ul>
<p>Here my problem arises. I get a quite good OOB-estimate from my random forest, but it is overly confident. The problem is that when I shifted my data to obtain more samples and then trained the sparse filter, I get some some sets of features which are just shifted versions of each other. When I then train my random forest, these features have a very high correlation, and the OOB-estimate does not take into account that some of my features are very highly correlated, consequently over-estimating the predictive power.</p>
<p>Since I have so few samples originally, I have no good way to do cross-validation, especially since I have a highly imbalanced data set. Is there any way to get slightly more accurate error-estimates?</p> | 73,558 | [
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<p>In a multiple regression, the regression coefficient of the two subsamples is opposite to that of the full sample. Both coefficients are significant. The sum of the number of obs in the two subsamples is equal to that of the full sample. Is this normal? What are the possible causes? Thanks.</p> | 49,779 | [
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<p>I have this confusion about which transformation to use in my data. The histogram of my original data looks like this </p>
<p><img src="http://i.stack.imgur.com/ANDSY.jpg" alt="enter image description here"></p>
<p>Now I have seen at most of the places to take log transformation in case the data is positively skewed. But when I take log transformation I get something like this
which is negatively skewed, not what I desire.</p>
<p><img src="http://i.stack.imgur.com/aorf0.jpg" alt="enter image description here"></p>
<p>If I take square root transformation and cube root transformation I get like this </p>
<p><img src="http://i.stack.imgur.com/hwls7.jpg" alt="enter image description here">
<img src="http://i.stack.imgur.com/uJ70C.jpg" alt="enter image description here"></p>
<p>Now the data is pretty much close to normal. But I didn't get the intuition behind this. Why didn't it work with the log transformation when I see at so many places people mentioning about log transformation useful when data is positively skewed. Here square root and cube root worked.</p>
<p>I want to know at which condition we should use log transformation and at which condition to use cube root transformation. Suggestions?</p> | 73,559 | [
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<p><strong>Is it appropriate to include intermediate outcomes in a predictive model?</strong> </p>
<p>It is quite clear that one should not control for post-treatment variables / intermediate outcomes when the goal is causal inference, but I wasn't sure if the same advice should hold when one's goal is to build a model for prediction.</p>
<p>Here is some context for my question: I'm trying to build a model that predicts if a college student will earn a bachelor's degree within 6 years of high school graduation using a large observational data set. I have data on students' high school variables (HS GPA, test scores, number activities participated in, etc.), some data on the students' college experiences (delayed enrollment in college, full-time / part-time status, transferred within two years of enrolling), as well as data on the characteristics of the college they attend. In other words, I have student level and institutional level data. I would account for the nesting of students within a particular institution. </p>
<p>Some have told me that the student level data on college experiences are intermediate outcomes and I shouldn't include them in the model. It isn't clear to me if I should / could include the college experience variables (which could be considered intermediate outcomes) in the predictive model, and if I do include them, how they should be treated. </p> | 73,560 | [
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<p>I have a dataset with ~90000 observations and less than 10 features (all continuous). The problem is that the response variable has ~300 categories. Currently I would try to fit a multinomial linear model or a random forest. I have to predict probabilities for the 300 categories. I need some inspiration and would like to know which model others would use?</p> | 73,561 | [
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<p>I have some observed (i.e., not generated by any hypothesized distribution, but generated by real processes) data that I believe is bi-modal (it may have more modes than two).</p>
<p>In this dataset, there are 241,792 observations. The deciles are the following values:</p>
<pre><code>MIN - 0.1472
10th - 0.3072
20th - 0.4094
30th - 0.6267
40th - 0.9396
50th - 1.4678
60th - 2.4125
70th - 4.2814
80th - 7.8145
90th - 16.7443
MAX - 241.0211
</code></pre>
<p>How can I identify modes using R?</p>
<p>Once identified and assuming that each mode is the location of central tendency for a local, somewhat Gaussian distribution, how should I determine where one tail ends and the next begins?</p>
<p>The end result is that I'm trying to treat the set of data as the composition of partitions, where each partition can be analyzed on its own. </p> | 73,562 | [
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