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5
votes
1answer
1k views

Linear MMSE estimate of MMSE estimator

This question is prompted by a recent discussion about the relationship between conditional expectation and covariance. Suppose that $X$ and $Y$ are zero-mean unit-variance random variables with ...
4
votes
1answer
327 views

Matlab code for finding the curvature of a curve using given data points

I have data points $(x,y)$ for a plane curve, and I would like to find its curvature. Wwhile I was googling to check how could I start, I found this matlab code: ...
4
votes
1answer
220 views

Straight line through data by eye - least squares? [closed]

I heard an interesting fact a while ago about how people draw a line through a cloud of points on a scatter plot. Usually, when calculating lines of best fit, we use the minimal the sum of squares of ...
4
votes
1answer
111 views

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
3
votes
1answer
583 views

Minimize combined variance of multiple measurements with known (but varying) variance

I have multiple measurements for the same property but with different but known uncertainty (variance). And I would like to combine that measurements in a way that I get as close to the real value as ...
3
votes
3answers
331 views

Why root mean “square” error?

Root-mean-square error is frequently used in for calculating the error between a predicted value and actual value. The formula for RMSE is given below: $\mathrm{RMSE} = ...
3
votes
0answers
229 views

root mean square distance between two simplices

As the title says, I want to compute the root mean square distance between two n-dimensional simplices. Say I have two surfaces $S$ and $S'$, the mean error is $$ d_m(S,S') = \frac{1}{|S|} ...
2
votes
1answer
462 views

Minimizing Mean Squared Error for Exponential Function

I have a function that I'm trying to model using an exponential function and I'm trying to determine the constants for the exponential. I know I could optimize it using trial-and-error in R or another ...
2
votes
0answers
27 views

minimizing mean square error with type 1 and 2 error weights

Suppose we have a random variable $X$ with a pmf that puts strictly positive probability only on integer values $0,1,2,\dots,n$. The objective is to choose a $z\in\mathbb{Z}$ that minimizes ...
2
votes
0answers
83 views

Measuring Model Bias

If given the choice between two statistical models (for argument's sake, let's say Model 1 is $y = \beta_0 + \beta_1 x_1 + \epsilon$ and Model 2 is $y = \beta_0 + \beta_1 x^2_1 + \epsilon$), is there ...
1
vote
2answers
341 views

Value minimizing mean absolute percentage error

What value for $c$ would minimize the formula: $$\frac{1}{n}\;\sum^{n}_{i=1}\left | \frac{y_i-c}{y_i}\right|$$ given the values $y_1, ..., y_n$. For example in the mean squared error we have the ...
1
vote
2answers
3k views

What is the purpose of subtracting the mean from data when standardizing?

What is the purpose of subtracting the mean from data when standardizing? and What is the purpose of dividing by the standard deviation?
1
vote
2answers
470 views

Given a function, calculate MMSE and LMMSE

Let $X = \frac{1}{1+U}$ where $U$ is uniformly distributed over $[0,1]$. I need to evaluate $E[X\mid U]$ and $\hat{E}[X\mid U]$ and the calculate the MSE, $E[(X-E[X\mid U])^2]$ and ...
1
vote
2answers
280 views

Least squares fit with a trick

Fitting a circle with least squares is easy once you get the trick for $c = r^2 - a^2 -b^2$ and you got a linear set of equations. my problem is as follows: $$z = \alpha x + \beta y + \alpha \beta$$ ...
1
vote
2answers
119 views

Linear Regression to quadratic function

What is the optimal linear regression (w and w/o y-intercept) for a quadratic curve w.r.t. mean square error. Mathematically speaking: Given, $$y = x^2$$ for $$x = [-a,a]$$. What is the best ...
1
vote
1answer
76 views

On normalized error measures

I have function values $f_1,\ldots,f_n$ that are approximated by data $y_1,\ldots,y_n$. I am looking for a measure that describes the error in the data $y_1,\ldots,y_n$ and I want the measure to take ...
1
vote
1answer
128 views

comparing MSE of estimations of binomial random variables

$X$ is a binomial random variable defined over 12 Bernoulli trials with a success probability of $p$ in each (i.e. $X\sim\operatorname{Bin}(12,p)$. Consider $\hat p=\frac X{10}$ Determine the range ...
1
vote
1answer
259 views

Improving the mean squared error of the ordinary least squares solution

I have the following relationship that holds: $y = a x_1 + b x_2$ Now I would like to determine constants $a$ and $b$ by performing tests. Each test results in 10 values for $y$, $x_1$ and $x_2$. ...
1
vote
1answer
889 views

Minimizing mean squared error

I want to find a $d$ that minimizes the value of the expression below. I think the first step is to find the derivative w.r.t. $d$ (is that correct? If not, what is the first step?). If so, I'm having ...
1
vote
1answer
27 views

Calculate the Mean earnings of a person

There are $12$ employees who earn $\$750$ collectively. I need to calculate the mean salary earned by a single employee. My workings so far: I tried to divide $750 / 12$, but I am not sure if this ...
1
vote
1answer
64 views

Expected mean squared error and MSR

In a small-scale regression study, five observations on $Y$ were obtained corresponding to $X = 1,4,10, 11$, and $14$. Assume that $\sigma=0.6,B_0=5,B_1=3$ a. What are the expected values ...
1
vote
0answers
26 views

Can the error term variance ever be estimated without fitting a regression line in a basic linear regression model?

Can the error term variance ever be estimated without fitting a regression line in a basic linear regression model? I don't understand how this would be possible and why. Because wouldn't you always ...
1
vote
0answers
41 views

Mean squared error consistency of estimator

Given is the following distribution: $f_\theta(x)=\frac{1}{\theta}$ if $0<x\leq\theta$, and $0$ otherwise; $\theta<0$. I need to show that the maximum likelihood estimator of $\theta$, ...
1
vote
1answer
27 views

Standard error of RMSE?

If I want to calculate the RMSE between an estimated value $\hat{x}$ and its reference value $x_{\textrm{ref}}$, let \begin{equation} y_i = \hat{x}_i-x_{i,\textrm{ref}} \end{equation} Since ...
1
vote
0answers
179 views

Calculating MSE for two different size matrixes

I have two $2$-column matrixes, one of the has $467$ rows while the other one has $61468$ rows. Both them are trajectory paths of same robot, the big matrix is kind of raw data and the smaller one is ...
1
vote
0answers
39 views

Can convergence in distribution say anything about mean-square convergence rate?

Suppose I have a sequence $\{x_n\}$ that I already know converges in the mean-square-sense ($\lim_n E |x_n|^2\to 0$). Suppose further I know that the sequence $\{x_n\}$ converges in distribution to ...
1
vote
0answers
37 views

Geometric accuracy analysis of 2d rectangular models

I have reconstructed set of rectangular objects lie on a 2D plane (for ex. ABCD). All these objects are in a one coordinate system. On the other hand, I have reference models for all of them ...
1
vote
0answers
73 views

MSE For a Single Calculation (intel processor errors)

This is the question, from a practice final for a stat course: The Intel Pentium Processor chip has been discovered to make small errors occasionally; that is, errors of +1 or –1 (in $10^{-4}$ ...
1
vote
0answers
192 views

Unbiased estimators of theta

Suppose $\hat\theta_1$ and theta $\hat\theta_2$ are both uncorrelated and unbiased estimators of $\theta$, and that $\text{var}\hat\theta_1=2\cdot \text{var}(\hat\theta_2)$. a) Show that for any ...
1
vote
0answers
842 views

What is the difference between RMS and RMSE?

I did not unserstand what is the difference between root mean square (RMS) and root mean square error (RMSE). In some sources RMS term is used for error analysis, in others RMSE. Can you explain ...
1
vote
0answers
902 views

Hypergeometric Distribution Probability (mean, variance, Std Deviation)

The distribution of the number of children per household for households receiving aid to dependent children (ADC) in a large eastern city is as follows: 5% of ADC households have one child, 35% of ADC ...
1
vote
0answers
60 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
1
vote
0answers
171 views

Getting the correct error for a mean calculation

A constant k needs to be calculated including its gaussian error. $k = f_{(u,t)}$ $k_i$ can be calculated with the values and errors of $u_i$ and $t_i$ and their respective errors. Main issue is ...
0
votes
2answers
1k views

Step by step correlation calculation

I must understand how I can calculate the correlation for the following probability variables. ...
0
votes
1answer
36 views

Negative Mean Square Error

For simple random sampling, I have calculated somemean square errors for ratio-type estimators such as Isaki estimator, and Prasad Singh estimator. But, Mean Square Errors i obtained are negative. ...
0
votes
1answer
39 views

Least-squares solution to a linear matrix equation

Let $\\A$ be a matrix of size $\\(m, n)$, $\\b$ a column vector of size $\\m$, $\\x$ a column vector of size $\\n$ and $\\a$ a real number. If $\begin{bmatrix} x \\ a \end{bmatrix}$ is the ...
0
votes
1answer
43 views

mean square error comparison

Do you have any idea about how i can solve the question below? $X_1$ and $X_2$ are random variables that satisfy $E[X_1]=E[X_2]=\mu$ and $Var[X_1]=Var[X_2]=1$. Show that when $|\mu - 10| \leq ...
0
votes
1answer
30 views

Error of function with errors on arguments and size of arguments

EDIT: my question is not how to derive the formula below (I think the derivation is more or less what I guessed, like the answer below supports), but whether it can be made valid for the case where ...
0
votes
1answer
259 views

MMSE (Minimum min square estimate) problem

I have a problem as follows. As of now, I cannot provide the definition of X and Y but can anyone provide a rough overview of what needs to be done ? An experiment consist of rolling a single ...
0
votes
0answers
5 views

Determining forecast error of realtime prediction of binary outcomes [migrated]

Given datasets consisting of a daily prediction and confidence percentage for each of a small number of binary outcomes, what is the proper way to calculate the forecast error of each series and of ...
0
votes
0answers
18 views

error calculation when the error is not constant

I have to calculate the error on the following quantity: $$f(\epsilon^M,\epsilon^S)= \sqrt{ \frac{1}{N}\sum_{i=1}^N (\log{\epsilon_i^M} - \log{\epsilon_i^S})^2 } $$ Usually I would use this standard ...
0
votes
0answers
13 views

Minimizing error of estimation in a differential equation system

I have a system of equations which describe dynamic nature of a system. There is no closed form solution for this system. System of equations is as follow: $$I_c=I_1 + I_2 + I_3$$ $$R_3 = ...
0
votes
0answers
22 views

How to solve this vector MSE equation?

Let's assume an error at time $k$ is: $e_k = \mathbf{c}^T \mathbf{r}_k - a_{k-d} - \mathbf{b}^T \mathbf{a}_k$, where $\mathbf{c} = [c_0, ..., c_{N_c-1}]^T$, $\mathbf{r}_k = [r_0, ..., r_{N_c-1}]^T$, ...
0
votes
0answers
10 views

Is the weighted mean of residuals over another variable equal to $0$?

I understand how residual errors must sum to zero around in a random sample (e.g. $y$-axis price of diamond predicted by x-axis weight of diamond). However, why must the weighted sum of residuals with ...
0
votes
0answers
12 views

What is the best formula to calculate percentage area of low-volume data (i.e. values close to 0)

I initially used the MAPE to calculate the percentage error between the actual data, and the data I got based on a model (forecast). However, near the values close to 0, the graph/percentage error got ...
0
votes
1answer
27 views

A way to calculate the error of a model?

I am currently making a model for a set of raw data of sea levels from the NOAA data base. On the site, the sea level is recorded every 6 minutes. Because I wouldn't have time to copy data every 6 ...
0
votes
0answers
16 views

Finding estimator with the smallest MSE

There is an estimator $\hat{\theta}$ of $\theta$ which has expectated value $\frac{3n}{3n+1} \theta$ and $E(\hat{\theta}^2)=\frac{3n}{3n+2} \theta^2$. Now I need to pick another estimator $X$, such ...
0
votes
1answer
70 views

How to minimize the minimum mean square error of this difference

I am trying to minimize the mean square error. More precisely, I am trying to minimize the following optimization problem $$\arg \min _{\bf{w_1},\bf{w_2}}\mathbb{E} \,\,[\|{\bf s} - {\bf Wy}\|^2 ]$$ ...
0
votes
0answers
18 views

Finding the closest vector to an observation

I have a collection of vectors (a codebook in hand) which are presented within a matrix $A$ $$ A = [ a (\theta_1), \, a (\theta_2), \, a (\theta_3), \, a (\theta_4), \,\cdots]$$ We have obtained ...
0
votes
0answers
15 views

MMSE detector for elliptical distribution.

Suppose we have $Y=HX+W$ where dimension of $Y$ is $N$ and $W$ is elliptically distributed $H$ is also elliptically distributed $X$ is uniformly distributed. We want to estimate $\hat{X}$ using MMSE ...