The expected squared deviation of a random variable from its mean; or, the average squared deviation of data about their mean.

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24 views

What is variance of an election poll?

I was working through Harvard data science course: I came across this question: Assume you have $M$ polls with sample sizes $n_1,\ldots,n_M$. If the polls are independent, what is the average of the ...
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1answer
22 views

Forecasting Skew & Kurtosis

I have seen several models for estimating the expected value and variance of a distribution. I am curious, to learn if anyone has looked at models that extend beyond these first two moments ...
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13 views

Order Statistic Moments, INID Gaussian, Equal Variance, Unequal Means

I've been playing with this for a couple hours now and would like to figure it out if there is a well known solution. I want to calculate the mean of the order statistical distribution of N INID ...
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68 views

Explained variation ($R^2$) from MCMC glmm - Nakagawa & Schielzeth 2013

This is a follow up to a question I asked previously, where I suggested that variance-covariance matrices could be used to derive correlations, which are then usable as estimates of how much each ...
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1answer
49 views

Algebraic manipulation of $Var(Y|X)=E[(Y-E(Y|X))^2|X]$

Q: Show that $Var(Y|X)=E[(Y-E(Y|X))^2|X]$ is equal to $Var(Y|X)=E[Y^2|X]-(E[Y|X)]^2$. Answer: I know I have to use the law of iterated expectation to get to the second statement but I am having ...
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36 views

Derivative of multivariate gaussian wrt to covariance

The derivative of the logarithm of a multivariate Gaussian distribution wrt the covariance matrix is: $$ -\frac{1}{2}\Sigma^{-T} + \frac{1}{2}\Sigma^{-T}(x - \mu)(x-\mu)^T\Sigma^{-T} $$ The derivative ...
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1answer
21 views

Randomized Block ANOVA Model: Intermediate Steps

While reading through my lecture notes, I came across a randomized block ANOVA model and some assumptions. How did the authors get to these assumptions (Expectation and Variance) from the given model? ...
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2answers
57 views

Mean Preserving PDF Spreading

I have a histogram representing the PDF of an unknown discrete RV. The histogram is asymmetrical. To be clear, all I have is the histogram. Is there a known way to increase/decrease the variance of ...
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3answers
91 views

Covariance greater than Variance?

It is straightforward to verify that for two random variables $X$ and $Y$ with variances $\sigma^2_X \neq \sigma^2_Y$, we have that $$\Big|{\rm Cov}(X, Y)\Big| \leq \max\{\sigma^2_X,\, \sigma^2_Y\}$$ ...
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1answer
16 views

Estimating conditional variance y|x

I am building a predictor for $y = f(x)$ using training samples ${(x_i, y_i)}$ (assume) drawn i.i.d from some distribution $p(x,y)$, by optimising the empirical L2-loss: $f(x) = argmin_f \; \sum_i ...
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1answer
20 views

Variance of a linear combination of vectors

Let $A$ and $B$ be two constant matrices and let $x$ and$ y$ be two random vectors, what is the general formula for $Var(Ax+By)$? I know the formula for when $x$ and $y$ are scalar random variables ...
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27 views

Regarding variance of “total number of records”

Here is my problem: Let $\{X_i\}_{i=1}^n$ be a sequence of continuous random variables. A record is said to have occured at time $k$ if $X_k > X_i$ $\forall i=1,...,k-1$. Let $N$ denote the ...
3
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1answer
28 views

Is an F-test for equality of variance appropriate for a very large dataset?

I have a dataset with about 500,000 subjects and I am trying to establish whether the variance is equal. I first performed an F-test but then I realised the data is slightly skewed with kurtosis. So ...
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0answers
16 views

Finding a consistent estimator mathematically

This is my first post on this website so hopefully everything will go smoothly. Let me first ask the question, then go over my problem. Q: Suppose that we are given $({X_{1i},X_{2i}})$ which is a ...
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0answers
28 views

Multiple comparisons for variance structure in R lme fit

How can I compare variances for different levels of a factor in a mixed effect model? I'm fitting a mixed effects model (in R using the ...
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0answers
7 views

Coefficient of variation to identify unique groups across datasets

I have a dataset containing a number of normal and diseased samples, with log2 expression values for ~12k genes . both normal and diseased samples come from different patients and/or diseases and I ...
3
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1answer
21 views

Bias/variance tradeoff tutorial

I'm looking for a good tutorial about bias/variance tradeoff. In particular, I'd like to find someone that explains how different algorithms in machine learning play in this tradeoff, and possibly how ...
3
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1answer
20 views

Estimating counts from sampled data

I am working on counting events from sampled web logs. To formalize the problem, consider a random process in which we randomly record an event with known probability $r$. Say we have $n$ recorded ...
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1answer
52 views

Interpretation of variance in multilevel logistic regression

Please help me to interpret the findings of my model. The specifications of the model are: Dependent variable: treatment (1) or no-treatment (0). Independent variables: age, number of drugs used, ...
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0answers
13 views

Extension of Marginal R2 to calculate percent variance explained by individual fixed effects

I'm using lmer in R to build a LMM on a large observational dataset. I want to be able to compare the response magnitude between my fixed and random effects, and please forgive me if I'm not ...
5
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3answers
87 views

Computing the Variance of an MLE

Suppose we have i.i.d. $n$ observations $(X_1,X_2,...X_n)$ from a population with density $$f_\theta(x)=\begin{cases}\theta x^{\theta-1}&\text{ if }0\leq x\leq ...
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1answer
35 views

Posterior covariance from GPML toolbox

I am currently using the GPML toolbox to perform regression. Generally, after learning the hyperparameters we can extract the posterior mean and variance by using the function in the toolbox as ...
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13 views

Does minimum variance imply a maximum for ess?

Does minimum of variance of weights $$ \text{var}(x) = \frac{1}{n} \sum_{i }^n {x_i}^2 - \frac{1}{n^2} \left({\sum_{i }^n {x_i}}\right)^2 $$ imply the maximum of Kish's approximate formula for ...
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1answer
43 views

calculation of variance

Hi I have a question regarding calculation of variance of survey results. Consider the following example. If 5 (x) people answered a survey which consists of 3 (y) yes (1) or no (2) questions. You ...
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2answers
74 views

Finding the Mean and Variance from PDF

A random variable $n$ can be represented by its PDF $$p(n) = \frac{(\theta - 1) y^{\theta-1} n}{ (n^2 + y^2)^{(\theta+1)/2}}.$$ $\theta$ is a positive integer and $y$ is a positive parameter. If ...
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1answer
23 views

K-fold validation, how to use MSE and STD for model selection

When using K-fold validation for model selection I'm wondering what's the best approach to select a model using both the mean square error (MSE) and the standard deviation of errors among folds (STD). ...
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1answer
52 views

Is there any required amount of variance captured by PCA in order to do later analyses?

I have a dataset with 11 variables and PCA (orthogonal) was done to reduce the data. Deciding on the number of components to keep it was evident for me from my knowledge about the subject and the ...
2
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1answer
50 views

Variance expressed in Quadratic Form

Given a vector $\pmb x$ of length $n$, $\pmb x=\{x_1,...,x_n\}$ the variance is proportional to $\pmb x^\top \pmb x - \frac{1}{n}{\left(\sum_{i=1}^n x_i\right)}^2$ I'm trying to determine $(i,j)$ ...
4
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1answer
84 views

Why is variance (instead of standard deviation) the default measure of information content in principal components?

The information content of principal components is almost always expressed as a variance (e.g., in scree plots or in statements like "the first three PCs contain 95% of the total data variance"). The ...
4
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1answer
64 views

What is the simple standard error for MCMC?

Simply put: suppose that we have observed $X=\left\{ X_{1},\ldots,X_{n}\right\}$. We then need to calculate some statistic $T$ using MCMC, using $M$ loops (By "loops" I mean the number of times the ...
4
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2answers
54 views

How many components to use in PCA in order to preserve a certain amount of variance?

I want to reduce the dimensionality of my data with PCA, until it preserves $\alpha = 0.99$ of the variance. How do I decide how many eigenvectors I should use? So I'm looking for a function ...
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1answer
37 views

partitioning variances, higher order products and multivariate skewness

If A,B and C all contribute to Z as: $$ Z = A*B*C $$ we can get the contributions of ABC to the variance in Z as: $$ Var(Z) = Var(ABC) $$ getting independent contributions of A, B and C and ...
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26 views

Probability - expected value and variance

"A man is playing versus a machine in the following way: The machine chooses 2 numbers randomly from the set of numbers 1,2,3,4,5, where a number can be chosen twice (with replacement). If the ...
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0answers
46 views

If $Var[X]$ is much larger than $E[X]^2$ is E[X] an unreliable estimate?

My question is simple and I got this confusion when going through a Markov chains text book.If $Var[X]$ is much larger than $E[X]^2$ is $E[X]$ an unreliable estimate?
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58 views

Bayes Linear regression- logarithmic transformation of prior distribution of the variance

I have a Bayesian version of a linear regression with 3 covariates. The model is given by \begin{align*} Y\sim N(\mu,\tau)\end{align*} \begin{align*} \mu=\alpha + \sum\beta_{i}x_{i}\end{align*} where ...
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2answers
97 views

How can I measure fluctuation?

For a certain product I have a data series with its price for each day over a long period of time (20 years). I want to investigate how the fluctuation of the price changes over the whole time. ...
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4answers
488 views

Why does increasing the sample size lower the variance?

Big picture: I'm trying to understand how increasing the sample size increases the power of an experiment. My lecturer's slides explain this with a picture of 2 normal distributions, one for the ...
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18 views

Overestimating variance with MCMC

I'm working with a very specific type of proposal distribution in MCMC algorithm. To validate it I use a simple multivariave Gaussian with $\mu=0$ and $\Sigma$ an identity matrix. The proposal ...
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1answer
24 views

Is this problem convex ? (regularization term on xTw)

Suppose we want to solve the following: $$ \min_{w} f(x^Tw, y) + \lambda g(x^Tw) $$ with $f$ a (logistic) loss and $g$ something like a variance. Is this a convex optimization problem ? What are ...
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0answers
18 views

How to add a fixed variance structure do a GAM

I am using GAM to fit a smooth line to represent the recovery of timber stocks following forest harvest. The data is heterogenous and I do not want to transform it. I understand that a nice way to ...
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1answer
93 views

Variance of two correlated variables

Suppose X and Y are two correlated random variables, and var(X), cov(X,Y) are known. Is there a unique solution for var(Y) and can we determine var(Y) by only using var(X) and cov(X,Y)?
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18 views

Bootstrapping - Variance of Time Series with Micro-level Data

I have micro-level (individuals) time series data and I am able to calculate some aggregate statistic for each time period. The data is not a panel, so each month is a different cross-section of ...
2
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0answers
8 views

Measure of variance between two populations

I have a set of genes, each of which composed of $10$ values, $5$ coming from population $a$ and $5$ coming from population $b$. I would like to define a measure of the variation between the two ...
0
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1answer
24 views

How exactly can I determine if customer ratings are based on (employee) vs. just random variation?

We have customer satisfaction surveys, and I can tell at least SOME portion of the variance is due to the employee that helped them. The surveys are all phrased -- how would you rate EMPLOYEE on ...
2
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1answer
46 views

Radial distance?

I'm looking at some code that calls $$\sqrt{(\sigma_x^2)^2+(\sigma_y^2)^2+(\sigma_z^2)^2}$$ the "radial distance", where $\sigma$ is the standard deviation. What is the significance of this measure?
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13 views

Genetic algorithm for factor selection

I'm planning to implement analysis of variance for different levels of factors, the problem is that I have 20 independent factors. Of course, the best model should include only significant factors. Is ...
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1answer
25 views

Relate $Var(y)$ with $Var(y)_{(i)}$

How can I relate $Var(y)$ with $Var(y)_{(i)}$ where $Var(y)_{(i)}$ is de variance of the data with the ith item removed. It is necesary first relate $\bar{y}$ with $\bar{y}_{(i)}$ and it complicates ...
3
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1answer
71 views

Residual variance for glmer

I am running a glmer model and I want to determine the total variance. My data is for survival and it is coded as 0 and 1, where 1 represents that the individual survived and 0 represents that the ...
5
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1answer
71 views

Error bars on log of big numbers

I am calculating a quantity of the following form: $\mu = \log( \frac{1}{n} \sum_{i=1}^{n} e^{\phi(X_i)} )$ via MC. $X_i$ are iid and I can sample them. I want to give error bars\ confidence ...
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8 views

Random process variance estimation

I need to estimate a variance (ensemble average) of a stationary random process (Vp). My measuring device has "internal" white noise, with its own averaged variance(Vn, which is known by the device ...