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Copy initials attempt

greetings my name is christopher breen and we are gonna walk through cse 535 week two practice question number two on the computation question involving markov chains so the first thing that we need to identify is that while we're presented with the battery charges in a table by day we can really look at this as one continuous string of values monday 7 a.m simply transitions to tuesday 8 a.m so what we've done is we've copied this raw data exactly as it stated there into one single column all the way down notice that we stop at 169 don't count the first one so we have 168 values 24 hours times seven days that adds up 168. now the next step we did because the demonstration that we got that's how they did it right we could choose our states however we wanted because they didn't uh specify that in the problem so we'll stick with uh how the professor did it for now and we simply type in s9 for anything from 90 to 100 s 8 80 to 89 7 seven to eight is 79 and so on and we manually type that in all the way down now the next step is we need to figure out the transitions and we want to do that as painlessly as possible so what we're going to do transitions and we're simply going to concatenate our state so i'm going to do b2 i'm going to get my s9 we'll go ahead and throw a greater than in there just to make it pretty and then we're going to do b3 to get the next state so there you go so s9 goes to s9 and then we're just going to drag this all the way to the bottom okay now notice we have s9 to s9 and then our very last value s9 and there's nothing there we do not know what this 100 is going to go to next that's the whole idea of the markov chain to figure out the probability of what it does next so this just leaving it like that is going to take care of itself when we start doing the math so now just for readability i'm going to enter the labels for our matrix we have state 0 through 9 state 0 through 9. okay all right so now essentially what i want to do here is count all of the transitions from s0 to s0 we're going to create a formula for that that we can then just copy and paste to the rest of the matrix so how do we do this we're going to do a count if for the range we want to look in the entire c column we use the dollar signs to static to c so that as we copy this formula to the right in other columns it doesn't shift that c now for our criteria we need to match the text that we have in that transition column so in our case that is going to be dollar sign e 2 that gives us our s 0. and we have our quotes greater than and and now we want f dollar sign one so this will count the number of times we have s0 greater than s0 and we have none which is to be expected now i can just drag this to the right drag this down and if we were to check our math against what the professor provided we should see that this lines up perfectly now the next step is to convert this into our probability matrix so in order to do that i am going to identify here that we are doing m matrix to the first power this a little bit easier to read and then what i want to do is essentially take this value the number of times we transitioned from s0 to s0 and divided by the total number of transitions out of s0 forget what you learned in the video about separately counting each of the occurrences of a state that gets you into trouble when you end up counting that last state that doesn't transition it's an unnecessary step we don't need to do that all we want to do is sum the total number of transitions now there is a catch we do need to keep this static because we're going to copy this to other cells other columns and we don't want that that range to shift on us so we want the column to remain static and that column to remain static and the row we do want to change so i'll put that there i'm actually going to move this i probably should have put another blank row in there so let's actually move that yeah okay so i just moved this so that i can get some headers we can still see what's going on this is going to be m1 still doing our first one all right so as expected this should be zero now take this across and i can take it down and there we go here is our initial transient probability matrix now depending on what the question says if you need to figure out what is uh the probability of being uh in state two having started in state one after four steps right or in our case each step each transition is an hour so if the question was it's 10 am you have 90 battery at 5 pm actually let's yeah okay we'll go ahead to 5 p.m what is the probability that you have 70 battery so to do that we need to know how many steps so 10 to 5 that would be 1 2 3 4 5 6. so 7 steps so we need to figure out matrix multiplication by 7. so that's what we're going to do here i know this is not actually part of this particular question i just made that up to show you an important feature or question that we may see so how do we do the matrix multiplication the first thing that we want to do to make this a little bit easier is let's name this so we're going to take this matrix formulas define name and i have this from a previous entry and delete that and so this is what you will see and i'm just going to name it m okay so now this is m so now if i want to do the matrix multiplication i can say equals matrix multiplication and then put in both your arrays so if i put m comma m okay i don't think this got named properly okay i didn't have a range when i did it last time so now i'm going to do m there we go so now that's named m and now this matrix multiplication m by m we get this now this is actually just m squared this is not m7 so how do we get to m7 or 27 or 207 unfortunately there is no matrix to the power of in excel without writing code and visual basic so we have to nest many of these together so essentially what we need is just this part right here and so we can put as many of these in as we want now this is going to be two right matrix just the matrix multiplied m m is now matrix to the second power so this is second power third power fourth power five six seven now we have to add that last m in there for that second array and now we can simply close out all of our parentheses if you get the count wrong not a big deal hit enter it will correct it for you and there we go so this would be m to the seven so the probability of going from any state to another state is right here was the example we used going from i don't know when we say it was 10 a.m going from this would be state uh nine to state seven so nine going to seven there is a eighteen point six percent chance that will occur that after seven hours you will go from you will be at seventy percent having started at ninety percent now if we need to get the average time like it says here that's our question we need to get this to the steady state matrix first remember this is is the transient state matrix the only time that we need the steady state matrix is to get that average time if you are simply trying to calculate the probability of being in one state after x number of transitions right and those transitions may be times it could be hours it could be whatever in our case a transition step is one hour to do those questions this is what you need and then you just take it to the power of however many steps you're trying to calculate that probability for average time we need steady state so that's the one we're going to do here i'm going to copy this i'm going to copy this then i'm going to do this again matrix multiplication m comma and then i'm going to come up here and i'm just going to copy that and so remember this is to the power of 2 3 4 5 6 7 8 9 10 11 12 13. now we could try to do multiple iterations of this we could go to the 10 to the 15 to the 20 and see where our values stop changing and that would probably be a good idea to do if you've got the time if you're short on time why not just hold down a control v do this more than enough times to make sure you get to steady state you can't go too many times but you can go too few times all right so that should be enough i'm going to finish with that final m and then i'm going to close my race and again if i don't get all the parentheses that's okay just hit enter [Music] okay 64 levels of nesting so it looks like 64 is as far as you can go in a given matrix so let's call this 64. and comma so that's two three four five six seven eight nine ten one two three four five six seven eight nine twenty one two three four five six seven thirty one two three four five six seven eight nine forty 50 60 one two three four comma close enter fix boom so there we go so this is our steady state matrix now in order to get average time for one state to another we simply take one divided by this probability call this average time equals one divided by make sure i'm doing this right one divided by and then i'm just going to grab that and we don't need to worry about any static columns or rows because this one i do want to carry over and here you have it so here is now these are not probabilities anymore because we did the one divided by the probability so this is now the average number of transitions to go from zero to zero so in our case we want to know how long it would take for a fully charged battery to drain to go from state 9 to state 0 it would take 83.5 transitions or in our case a transition is an hour so that means 83.5 hours to go from state 9 to state 1 would only be 23.85 hours to go to state 2 13.9 and there you have it that is the markov chains in excel thanks for watching