Why can a null measurement be more accurate than one using standard voltmeters and ammeters?

What factors limit the accuracy of null measurements?

What factors limit the accuracy of null measurements?

zi2lalZ
2020-12-25
Answered

What factors limit the accuracy of null measurements?

You can still ask an expert for help

asked 2022-06-20

I want to implement a Kalman Filter for predicting x/y positions. I have a sensor which gives me the current position (noisy). Now I want to smooth and predict the position. Thus Kalman Filter came to my mind.

How do I have to design the filter taking the time in regards? I want to predict the next state which is ahead of the upcoming measurement. Moreover since i do not have a velocity, I'd like to estimate the velocity by the kalman as well.

State := [xpos,ypos,xvelocity,yvelocity]

Measurement := [xpos,ypos]

ControlInput := predictionTime

I would run the following algorithm, whenever i get a measurement:

1. Measure

2. Update kalman gain

3. Predict

4. Get State estimate

Now, when i use a predictionTime which is newer than the next measurement, the next measurement does not fit to the predicted state.

Is there a strategy to solve this issue? My predicted state and my next measurement do not fit, how could I fix this?

How do I have to design the filter taking the time in regards? I want to predict the next state which is ahead of the upcoming measurement. Moreover since i do not have a velocity, I'd like to estimate the velocity by the kalman as well.

State := [xpos,ypos,xvelocity,yvelocity]

Measurement := [xpos,ypos]

ControlInput := predictionTime

I would run the following algorithm, whenever i get a measurement:

1. Measure

2. Update kalman gain

3. Predict

4. Get State estimate

Now, when i use a predictionTime which is newer than the next measurement, the next measurement does not fit to the predicted state.

Is there a strategy to solve this issue? My predicted state and my next measurement do not fit, how could I fix this?

asked 2022-05-16

Any measurement, say length of any object, will have some errors. The random errors that are present in the measurement can be reduced if we take mean of a large number of samples. This is because the standard deviation of mean is $m/sqrt(n)$ where $m$ is the mean of the n values we measured.

Can anyone give a visual or intuitive reason why would uncertainty in mean be low?

Can anyone give a visual or intuitive reason why would uncertainty in mean be low?

asked 2022-05-14

Is there a kind of "formalism" which define how unit measures come out from integration?

An example: given a point $P(x,y,z)\in {\mathbb{R}}^{3}$ there is the concept of mass m associated to this point. Mass is measured in $\text{kg,g,lb,...}$ I indicate the generic unite measure of the mass $[m]$. Now, there is also the concept of density of mass $\rho (x,y,z)$ which is a (scalar) function which represents "mass per unit volume", or also $\frac{[m]}{[s{]}^{3}}$ (where $[s]$ is the unit measure of space), e.g. if we take a constant density over a volume, to know the mass of the volume it suffices to multiply density $\rho $ with volume $V$. The effect of this multiplication is coherent with unit measures involved.

Now, in general for non-constant density function, one needs to integrate the function over the volume to know mass:

$m={\int}_{V}\rho (x,y,z)\text{d}\tau $ where $\tau $ is the volume element.

Now, integrals are pure mathematical objects, how can I relate the fact that an integral is not only (naively) "a product of the integrand for the measure of the space of integration" with the fact that, in the end, there will be $[m]=\frac{[m]}{[s{]}^{3}}[s{]}^{3}$

Now, naively I can argument something like this ${\int}_{[{s}^{3}]}\frac{[m]}{[s{]}^{3}}\text{d}([{s}^{3}])$, but the integrand is constant so $\frac{[m]}{[s{]}^{3}}{\int}_{[{s}^{3}]}\text{d}([s{]}^{3})=[m]$. But here there is no mathematical formalism, only a naive thought about such an "integral of unit measures" sounds to me.

Is there actually an ad-hoc formal argument for this problem?

An example: given a point $P(x,y,z)\in {\mathbb{R}}^{3}$ there is the concept of mass m associated to this point. Mass is measured in $\text{kg,g,lb,...}$ I indicate the generic unite measure of the mass $[m]$. Now, there is also the concept of density of mass $\rho (x,y,z)$ which is a (scalar) function which represents "mass per unit volume", or also $\frac{[m]}{[s{]}^{3}}$ (where $[s]$ is the unit measure of space), e.g. if we take a constant density over a volume, to know the mass of the volume it suffices to multiply density $\rho $ with volume $V$. The effect of this multiplication is coherent with unit measures involved.

Now, in general for non-constant density function, one needs to integrate the function over the volume to know mass:

$m={\int}_{V}\rho (x,y,z)\text{d}\tau $ where $\tau $ is the volume element.

Now, integrals are pure mathematical objects, how can I relate the fact that an integral is not only (naively) "a product of the integrand for the measure of the space of integration" with the fact that, in the end, there will be $[m]=\frac{[m]}{[s{]}^{3}}[s{]}^{3}$

Now, naively I can argument something like this ${\int}_{[{s}^{3}]}\frac{[m]}{[s{]}^{3}}\text{d}([{s}^{3}])$, but the integrand is constant so $\frac{[m]}{[s{]}^{3}}{\int}_{[{s}^{3}]}\text{d}([s{]}^{3})=[m]$. But here there is no mathematical formalism, only a naive thought about such an "integral of unit measures" sounds to me.

Is there actually an ad-hoc formal argument for this problem?

asked 2022-07-01

Let ${I}_{1},{I}_{2}...$ be an arrangement of the intervals $[\frac{i-1}{{2}^{n}},\frac{i}{{2}^{n}})$ in a sequence. If ${X}_{n}=n$ on ${I}_{n}$ and 0 elsewhere then $su{p}_{n}E{X}_{n}<\mathrm{\infty}$ but $P(\underset{n}{sup}{X}_{n}<\mathrm{\infty})=0$. My basic space is [0,1] with Lebesgue measure.

$A:=\{\underset{n}{sup}{X}_{n}<\mathrm{\infty}\}=\{\omega \in \mathrm{\Omega}:\mathrm{\exists}M>0,\mathrm{\forall}n\in \mathbb{N},{X}_{n}(\omega )\le M\}$

hence the complement is

$\begin{array}{rl}\overline{A}& =\{\omega \in \mathrm{\Omega}:\mathrm{\forall}M\in \mathbb{N},\mathrm{\exists}n\in \mathbb{N},{X}_{n}(\omega )>M\}\\ & =\bigcap _{M\in \mathbb{N}}\bigcup _{n\in \mathbb{N}}\{{X}_{n}>M\}\\ & =\bigcap _{M\in \mathbb{N}}\bigcup _{n\ge M}\{{X}_{n}>M\}\end{array}$

How do i go on from here?

$A:=\{\underset{n}{sup}{X}_{n}<\mathrm{\infty}\}=\{\omega \in \mathrm{\Omega}:\mathrm{\exists}M>0,\mathrm{\forall}n\in \mathbb{N},{X}_{n}(\omega )\le M\}$

hence the complement is

$\begin{array}{rl}\overline{A}& =\{\omega \in \mathrm{\Omega}:\mathrm{\forall}M\in \mathbb{N},\mathrm{\exists}n\in \mathbb{N},{X}_{n}(\omega )>M\}\\ & =\bigcap _{M\in \mathbb{N}}\bigcup _{n\in \mathbb{N}}\{{X}_{n}>M\}\\ & =\bigcap _{M\in \mathbb{N}}\bigcup _{n\ge M}\{{X}_{n}>M\}\end{array}$

How do i go on from here?

asked 2022-07-30

After Julia had driven for half an hour, she was 155 miles from Denver. After driving 2 hours, she was 260 miles from Denver. Assume that Julia drove at a constant speed. Let f be a function that gives Julia's distance in miles from Denver after having driven for t hours.

a. Determine a rule for the function f.

b. Interpret f'(500). Calculate f'(500).

c. Determine a rule for f'.

a. Determine a rule for the function f.

b. Interpret f'(500). Calculate f'(500).

c. Determine a rule for f'.

asked 2020-11-22

What does a z-score of –1.5 mean?

A.The mean is 1.5 standard deviations less than the measurement.

B.There is an error in the measurement.

C.The measurement is 1.5 standard deviations less than the mean.

D.The variance cannot be calculated because the z-score is negative.

A.The mean is 1.5 standard deviations less than the measurement.

B.There is an error in the measurement.

C.The measurement is 1.5 standard deviations less than the mean.

D.The variance cannot be calculated because the z-score is negative.

asked 2022-04-08

Trihalomethanes (or THM’s) refers to a broad class of organic compounds, including chloroform and bromoform. Many THM’s are known or suspected human carcinogens, and, because they are a biproduct of disinfection in water treatment facilities, their concentrations in drinking water are regulated by federal law. Four measurements of THM levels in drinking water yielded the following results, in parts per billion (ppb): 48, 65, 72, and 70. Assume that we know, from other drinking water quality surveys, that THM concentrations of this magnitude tend to be roughly normally distributed.

(a) What is the mean THM concentration for this sample of four measurements?

(b) What is the standard deviation of the sample and what is the standard error of the mean?

(c) What is the 95% (two-tailed) confidence interval for the mean?

(d) What is the 95% prediction interval for the individual measurements? That is, what interval encloses 95%

of all possible data, given the mean and standard deviation that you have calculated?

(a) What is the mean THM concentration for this sample of four measurements?

(b) What is the standard deviation of the sample and what is the standard error of the mean?

(c) What is the 95% (two-tailed) confidence interval for the mean?

(d) What is the 95% prediction interval for the individual measurements? That is, what interval encloses 95%

of all possible data, given the mean and standard deviation that you have calculated?