Mixing
Intuition of interpreting sensors as random variables and combining their estimates
Clash Royale CLAN TAG#URR8PPP
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please bear with me, I'm really new to probability theory and inference, and I got stuck while pondering the following thought experiment. Reading through other similar questions confused me more than it helped. I hope that the notation that I used below is correct, please inform me if it's wrong.
Let's imagine I want to track my cat $C$ with a set of identical proximity sensors $i$ that I distributed uniformly along a 1D track. Each sensors activity depends on the distance of the cat from the sensor. I model this as a random variable $X_i sim mathcalN(0, sigma^2)$: if the cat is on top of a sensor, the sensor's activity should indicate that the cat is there, but this activity drops of with distance from the sensor. To know where my cat is, I then simply look at the sensor $i$ which is maximally active, and look into my list where I wrote down the physical location of $i$, i.e. an inference or decoding problem.
Now, let's assume that I placed the sensors very densely due to their limited proximity range. This would give me a very high resolution of my cat's location. However, often I would like to get only a rough estimate. For instance, sensors $X_1$ and $X_2$ are in front of the fish tank (my cat likes to watch fish) and I'd like to know if she is there watching fish, but don't care about her "high-resolution location".
How would I model the fish tank variable $Z$ to inform me about if she's sitting there? My initial guess would be $Z = X_1 + X_2$, because somehow I need to integrate the sensor responses? I think it's not a mixture, because I don't randomly choose one of the two sensors over the other, but I'm not quite sure. I'm even more lost when thinking about how to infer the location. I'd appreciate any hints.
probability random-variables statistical-inference
asked Jul 28 at 8:20
CatTracker
11
add a comment |Â
up vote
0
down vote
favorite
please bear with me, I'm really new to probability theory and inference, and I got stuck while pondering the following thought experiment. Reading through other similar questions confused me more than it helped. I hope that the notation that I used below is correct, please inform me if it's wrong.
Let's imagine I want to track my cat $C$ with a set of identical proximity sensors $i$ that I distributed uniformly along a 1D track. Each sensors activity depends on the distance of the cat from the sensor. I model this as a random variable $X_i sim mathcalN(0, sigma^2)$: if the cat is on top of a sensor, the sensor's activity should indicate that the cat is there, but this activity drops of with distance from the sensor. To know where my cat is, I then simply look at the sensor $i$ which is maximally active, and look into my list where I wrote down the physical location of $i$, i.e. an inference or decoding problem.
Now, let's assume that I placed the sensors very densely due to their limited proximity range. This would give me a very high resolution of my cat's location. However, often I would like to get only a rough estimate. For instance, sensors $X_1$ and $X_2$ are in front of the fish tank (my cat likes to watch fish) and I'd like to know if she is there watching fish, but don't care about her "high-resolution location".
How would I model the fish tank variable $Z$ to inform me about if she's sitting there? My initial guess would be $Z = X_1 + X_2$, because somehow I need to integrate the sensor responses? I think it's not a mixture, because I don't randomly choose one of the two sensors over the other, but I'm not quite sure. I'm even more lost when thinking about how to infer the location. I'd appreciate any hints.
probability random-variables statistical-inference
asked Jul 28 at 8:20
CatTracker
11
Are those like RSSI measurements? Namely the sensor output is proportional to the distance from the cat?
– Royi
Jul 28 at 8:31
Yes, for instance when the cat's on top of one of those (imaginary) sensors, the sensor would blink 10 times. When the cat's further away, this activity should be lower.
– CatTracker
Jul 28 at 8:40
Number of blinks or the value of signal peak? Usually if you have something like proximity sensor its amplitude is proportional to the distance. Hence something like Localization is done. In your case you should better define the output of the sensor and what is random about it.
– Royi
Jul 28 at 8:52
Ah, I see. I wanted to stay as independent of any real sensor as possible, like saying: When the cat is sitting in an area that the sensor can observe, then the sensor is active (or reports 10 blinks, or turns from red to green). But due to measurement noise, the sensor will work only perfectly when the cat is exactly on top of it, and sometimes misses the cat, wenn she's not exactly there.
– CatTracker
Jul 28 at 9:52
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
please bear with me, I'm really new to probability theory and inference, and I got stuck while pondering the following thought experiment. Reading through other similar questions confused me more than it helped. I hope that the notation that I used below is correct, please inform me if it's wrong.
Let's imagine I want to track my cat $C$ with a set of identical proximity sensors $i$ that I distributed uniformly along a 1D track. Each sensors activity depends on the distance of the cat from the sensor. I model this as a random variable $X_i sim mathcalN(0, sigma^2)$: if the cat is on top of a sensor, the sensor's activity should indicate that the cat is there, but this activity drops of with distance from the sensor. To know where my cat is, I then simply look at the sensor $i$ which is maximally active, and look into my list where I wrote down the physical location of $i$, i.e. an inference or decoding problem.
Now, let's assume that I placed the sensors very densely due to their limited proximity range. This would give me a very high resolution of my cat's location. However, often I would like to get only a rough estimate. For instance, sensors $X_1$ and $X_2$ are in front of the fish tank (my cat likes to watch fish) and I'd like to know if she is there watching fish, but don't care about her "high-resolution location".
How would I model the fish tank variable $Z$ to inform me about if she's sitting there? My initial guess would be $Z = X_1 + X_2$, because somehow I need to integrate the sensor responses? I think it's not a mixture, because I don't randomly choose one of the two sensors over the other, but I'm not quite sure. I'm even more lost when thinking about how to infer the location. I'd appreciate any hints.
probability random-variables statistical-inference
asked Jul 28 at 8:20
CatTracker
11
please bear with me, I'm really new to probability theory and inference, and I got stuck while pondering the following thought experiment. Reading through other similar questions confused me more than it helped. I hope that the notation that I used below is correct, please inform me if it's wrong.
Let's imagine I want to track my cat $C$ with a set of identical proximity sensors $i$ that I distributed uniformly along a 1D track. Each sensors activity depends on the distance of the cat from the sensor. I model this as a random variable $X_i sim mathcalN(0, sigma^2)$: if the cat is on top of a sensor, the sensor's activity should indicate that the cat is there, but this activity drops of with distance from the sensor. To know where my cat is, I then simply look at the sensor $i$ which is maximally active, and look into my list where I wrote down the physical location of $i$, i.e. an inference or decoding problem.
Now, let's assume that I placed the sensors very densely due to their limited proximity range. This would give me a very high resolution of my cat's location. However, often I would like to get only a rough estimate. For instance, sensors $X_1$ and $X_2$ are in front of the fish tank (my cat likes to watch fish) and I'd like to know if she is there watching fish, but don't care about her "high-resolution location".
How would I model the fish tank variable $Z$ to inform me about if she's sitting there? My initial guess would be $Z = X_1 + X_2$, because somehow I need to integrate the sensor responses? I think it's not a mixture, because I don't randomly choose one of the two sensors over the other, but I'm not quite sure. I'm even more lost when thinking about how to infer the location. I'd appreciate any hints.
probability random-variables statistical-inference
asked Jul 28 at 8:20
CatTracker
11
edited Jul 28 at 8:37
asked Jul 28 at 8:20
CatTracker
11
asked Jul 28 at 8:20
CatTracker
11
asked Jul 28 at 8:20
CatTracker
11
11
Are those like RSSI measurements? Namely the sensor output is proportional to the distance from the cat?
– Royi
Jul 28 at 8:31
Yes, for instance when the cat's on top of one of those (imaginary) sensors, the sensor would blink 10 times. When the cat's further away, this activity should be lower.
– CatTracker
Jul 28 at 8:40
Number of blinks or the value of signal peak? Usually if you have something like proximity sensor its amplitude is proportional to the distance. Hence something like Localization is done. In your case you should better define the output of the sensor and what is random about it.
– Royi
Jul 28 at 8:52
Ah, I see. I wanted to stay as independent of any real sensor as possible, like saying: When the cat is sitting in an area that the sensor can observe, then the sensor is active (or reports 10 blinks, or turns from red to green). But due to measurement noise, the sensor will work only perfectly when the cat is exactly on top of it, and sometimes misses the cat, wenn she's not exactly there.
– CatTracker
Jul 28 at 9:52
add a comment |Â
Are those like RSSI measurements? Namely the sensor output is proportional to the distance from the cat?
– Royi
Jul 28 at 8:31
Yes, for instance when the cat's on top of one of those (imaginary) sensors, the sensor would blink 10 times. When the cat's further away, this activity should be lower.
– CatTracker
Jul 28 at 8:40
Number of blinks or the value of signal peak? Usually if you have something like proximity sensor its amplitude is proportional to the distance. Hence something like Localization is done. In your case you should better define the output of the sensor and what is random about it.
– Royi
Jul 28 at 8:52
Ah, I see. I wanted to stay as independent of any real sensor as possible, like saying: When the cat is sitting in an area that the sensor can observe, then the sensor is active (or reports 10 blinks, or turns from red to green). But due to measurement noise, the sensor will work only perfectly when the cat is exactly on top of it, and sometimes misses the cat, wenn she's not exactly there.
– CatTracker
Jul 28 at 9:52
Are those like RSSI measurements? Namely the sensor output is proportional to the distance from the cat?
– Royi
Jul 28 at 8:31
Are those like RSSI measurements? Namely the sensor output is proportional to the distance from the cat?
– Royi
Jul 28 at 8:31
Yes, for instance when the cat's on top of one of those (imaginary) sensors, the sensor would blink 10 times. When the cat's further away, this activity should be lower.
– CatTracker
Jul 28 at 8:40
Yes, for instance when the cat's on top of one of those (imaginary) sensors, the sensor would blink 10 times. When the cat's further away, this activity should be lower.
– CatTracker
Jul 28 at 8:40
Number of blinks or the value of signal peak? Usually if you have something like proximity sensor its amplitude is proportional to the distance. Hence something like Localization is done. In your case you should better define the output of the sensor and what is random about it.
– Royi
Jul 28 at 8:52
Number of blinks or the value of signal peak? Usually if you have something like proximity sensor its amplitude is proportional to the distance. Hence something like Localization is done. In your case you should better define the output of the sensor and what is random about it.
– Royi
Jul 28 at 8:52
Ah, I see. I wanted to stay as independent of any real sensor as possible, like saying: When the cat is sitting in an area that the sensor can observe, then the sensor is active (or reports 10 blinks, or turns from red to green). But due to measurement noise, the sensor will work only perfectly when the cat is exactly on top of it, and sometimes misses the cat, wenn she's not exactly there.
– CatTracker
Jul 28 at 9:52
Ah, I see. I wanted to stay as independent of any real sensor as possible, like saying: When the cat is sitting in an area that the sensor can observe, then the sensor is active (or reports 10 blinks, or turns from red to green). But due to measurement noise, the sensor will work only perfectly when the cat is exactly on top of it, and sometimes misses the cat, wenn she's not exactly there.
– CatTracker
Jul 28 at 9:52
add a comment |Â
1 Answer
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So you have a sensor which is a multiplication of Bernoulli Trial and something which says the probability of the cat being close to the sensor.
If the sensor detected the Cat - Bernoulli Trial with parameter $ p $.
The parameter $ p $ value is a function of the cat distance from the sensor (Which you can simulate by Gaussian Random Variable).
Now given measurements from few sensors you'll be able to build the PDF of the Cat location.
Then you can chose to chose the Mean (MMSE Estimation) of the Maximum of the PDF (MAP Estimation).
answered Jul 28 at 11:08
Royi
2,93012046
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
So you have a sensor which is a multiplication of Bernoulli Trial and something which says the probability of the cat being close to the sensor.
If the sensor detected the Cat - Bernoulli Trial with parameter $ p $.
The parameter $ p $ value is a function of the cat distance from the sensor (Which you can simulate by Gaussian Random Variable).
Now given measurements from few sensors you'll be able to build the PDF of the Cat location.
Then you can chose to chose the Mean (MMSE Estimation) of the Maximum of the PDF (MAP Estimation).
answered Jul 28 at 11:08
Royi
2,93012046
add a comment |Â
up vote
0
down vote
So you have a sensor which is a multiplication of Bernoulli Trial and something which says the probability of the cat being close to the sensor.
If the sensor detected the Cat - Bernoulli Trial with parameter $ p $.
The parameter $ p $ value is a function of the cat distance from the sensor (Which you can simulate by Gaussian Random Variable).
Now given measurements from few sensors you'll be able to build the PDF of the Cat location.
Then you can chose to chose the Mean (MMSE Estimation) of the Maximum of the PDF (MAP Estimation).
answered Jul 28 at 11:08
Royi
2,93012046
add a comment |Â
up vote
0
down vote
up vote
0
down vote
So you have a sensor which is a multiplication of Bernoulli Trial and something which says the probability of the cat being close to the sensor.
If the sensor detected the Cat - Bernoulli Trial with parameter $ p $.
The parameter $ p $ value is a function of the cat distance from the sensor (Which you can simulate by Gaussian Random Variable).
Now given measurements from few sensors you'll be able to build the PDF of the Cat location.
Then you can chose to chose the Mean (MMSE Estimation) of the Maximum of the PDF (MAP Estimation).
answered Jul 28 at 11:08
Royi
2,93012046
So you have a sensor which is a multiplication of Bernoulli Trial and something which says the probability of the cat being close to the sensor.
If the sensor detected the Cat - Bernoulli Trial with parameter $ p $.
The parameter $ p $ value is a function of the cat distance from the sensor (Which you can simulate by Gaussian Random Variable).
Now given measurements from few sensors you'll be able to build the PDF of the Cat location.
Then you can chose to chose the Mean (MMSE Estimation) of the Maximum of the PDF (MAP Estimation).
answered Jul 28 at 11:08
Royi
2,93012046
answered Jul 28 at 11:08
Royi
2,93012046
answered Jul 28 at 11:08
Royi
2,93012046
answered Jul 28 at 11:08
Royi
2,93012046
2,93012046
add a comment |Â
add a comment |Â
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Are those like RSSI measurements? Namely the sensor output is proportional to the distance from the cat?
– Royi
Jul 28 at 8:31
Yes, for instance when the cat's on top of one of those (imaginary) sensors, the sensor would blink 10 times. When the cat's further away, this activity should be lower.
– CatTracker
Jul 28 at 8:40
Number of blinks or the value of signal peak? Usually if you have something like proximity sensor its amplitude is proportional to the distance. Hence something like Localization is done. In your case you should better define the output of the sensor and what is random about it.
– Royi
Jul 28 at 8:52
Ah, I see. I wanted to stay as independent of any real sensor as possible, like saying: When the cat is sitting in an area that the sensor can observe, then the sensor is active (or reports 10 blinks, or turns from red to green). But due to measurement noise, the sensor will work only perfectly when the cat is exactly on top of it, and sometimes misses the cat, wenn she's not exactly there.
– CatTracker
Jul 28 at 9:52