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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7 views

Comparing variances between treatments

I have a dataset that comes from a simulated experiment. It looks something like this: ...
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1answer
16 views

Mean Preserving PDF Spreading

I have a univariate discrete random variable and a histogram representing its PDF (which is asymmetrical). Is there a known way to increase/decrease the variance of the distribution (i.e. scaling it ...
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3answers
61 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
15 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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22 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 ...
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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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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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27 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 ...
2
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1answer
10 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
20 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
10 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 ...
4
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3answers
80 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
33 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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0answers
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
42 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
72 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
21 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). ...
7
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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
49 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 $(i,j)$ ...
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1answer
83 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 ...
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1answer
63 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
50 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
35 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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0answers
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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57 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 ...
3
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2answers
94 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
478 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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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 ...
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1answer
23 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
61 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 ...
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1answer
70 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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0answers
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 ...
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0answers
19 views

Squared uncertainties: what may it be used for?

If the squared standard deviation of a set of values is the variance of this sample, then, what is the squared standard error of the mean of this sample ? and what may it be used for ? A quick search ...
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22 views

Variable bandwidth selection with Silverman's rule

In the formula for variable bandwidth selection, we end up with a standard deviation estimation. I have a hard time understanding what is exactly this standard deviation. Can you apply this method for ...
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1answer
26 views

Why is $E(u^2)=Var(y)$? (Binary Response Model)

I'm trying to show some results in binary response models, and a couple of the proofs use the "fact" that $E(u^2)=Var(y)$, but I can't see why this is. The setup is that $y$ takes on the value $0$ or ...
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1answer
24 views

Confidence interval for the variance when we know the mean of the population

We have $n$ random samples from a population which has normal distribution, and the mean of the population is known. How do we change the procedure of finding a confidence interval for the population ...
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1answer
55 views

Getting a meaningful metric for variation in this type of cyclical, panel data? WSS won't exactly cut it!

This is related to a previous question I have asked, but I am not after visualization but rather a meaningful summary statistic. Situation: I have many (150k) customers. Each generates his own ...
3
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3answers
59 views

What is the variance of an MLE for a trinomial distribution?

I am playing with the following trinomial (multinomial) distribution which can get values (a,b,c) with the probabilities: $(\theta^2, 2\theta(1-\theta), (1-\theta)^2)$. Say I have n observations from ...