Maths Tutorial: Patterns and Trends in Time Series Plots (statistics)

the first thing to look for is whether your time series has a positive or negative trend to it perhaps so it could be going up the page like this and you can see that if you were putting a line on this you could kind of see that the graph is going upwards like that in that positive linear sort of direction so this one has a trend going up the page it's got a positive trend and sometimes you'll see the word secular as well so it would have a positive secular trend or it might have an upward secular trend for a trend like this to emerge you have to have enough data you have to have enough time going along here so for example if you've only got seven days worth of data that's probably not enough to talk about whether there's a trend if you've got a year's worth with something measured weekly for example and you've got 52 dots on there that's going to be enough time to call it a trend so just watch out for there being enough data and then look for the pattern going up the page the same works going down the page if you had something that looked a bit like this that one you can see is decreasing as we go across the page as time goes by this is going down like this so here we have a negative trend or a negative secular trend the next thing you look for is the variation and that's talking about peaks and troughs so high points and low points in the data so here is a high point each time it's like the top of the mountain the peak of the mountain and here's a low point each time it's the bottom of the mountain it's like a valley so we're going to look at where the time series goes up and where it goes down again a seasonal pattern or seasonality as they it's sometimes referred to has regular peaks and troughs and by regular I mean you could almost predict when they happen based on when they've happened in the past you can see on this graph we've got these peaks happening but they happen at the same time each year so this is quarters and here we've got the first quarter it's down then it's up then it's down then it's a peak in the fourth and then it's down up down up a peak in the eighth and then it's down up down up and a peak in the twelfth so these Peaks are happening every four quarters they're happening at a regular interval and so that we might attribute to the seasons for example four quarters of the year that's seasons Auto summer winter spring so perhaps this is ice cream sales or something like that in fact it is look at that sales of ice creams so ice cream sales go right up in the summer and they go all the way back down again in the winter because who wants to buy ice cream in winter now seasonality doesn't have to be tied to seasons exactly so it doesn't have to be summer autumn winter spring it might be if you have some weekly data for example your sales of something might always peak on a Monday and it might always trough on a Friday and then if you put daily if you create a daily time series we'd be able to see Monday is up and then Tuesdays down Wednesday Thursday Friday and then we've got Monday and then we've got Tuesday Wednesday Thursday Friday and then we've got Monday and then Tuesday Wednesday Thursday Friday and then we've got Monday and then Tuesday Wednesday Thursday all Fridays worked on their Monday and so on so it's not necessarily seasons it's just that it's regular peaks and troughs this trough is this peak is always occurring on a Monday this trough is always occurring on a Friday here's a couple of other seasonal examples you can see here we've got some Peaks happening there's a peak there's a peak peak peak peak and we've got some troughs down here as well where are these peak happening Peaks happening this is happening right on the dash of the one year and this one's happening right on the dash of the two year this one's happening right on the dash of the three four and five so if we had six here we could pretty reasonably predict that that might be higher up here as well it would be another peak because that always happens at this dash the in between sections here we have a trough and then a high and then a off so it goes down up down in between and can you see how that's repeating down up down down up down for this interim part of the scale so this is seasonal because we can see regular peaks and troughs and they're occurring at sort of expected times in the time series they repeat their pattern over and over at the same time each year or at the same time each month or at the same time each week within the cycle they repeat over and over with consistency this one's seasonal over here as well we can see this peak is happening at the fourth quarter this peak at the eighth this peak at the 12th this peak at the 16th this peak at the twentieth this peak at the 24th so every four quarters there's a peak and these troughs are pretty regular too if you have a look at those that's occurring at five that's occurring at nine that's occurring at 13 that's occurring at 17 that's occurring at 21 so there's a little dip there's a trough there's a valley at the same time in each of these time series cycles as if you like each time the other thing that this graph has is it actually has an upward sort of a trend can you see that so this is seasonal and it also has an upward or a positive secular trend with seasonality so a seasonal time series has peaks and troughs that are at predictable times cyclical or a cyclic time series plot has peaks and troughs but they're at unpredictable times so this one will still have a shape where you can see mountains and valleys significant ups and downs but you can't predict when they're going to happen they don't happen at regular time intervals they just happen whenever so for example a share price might go up like this when you hire a new CEO and then it might go down like this when you sell off a part of the business and then it might go up like this because you know you acquire some new business something like that this peak didn't happen because it's August you know this trough didn't happen because it's January this peak didn't happen because it's you know February the peaks and troughs happen for other reasons and for because of that you can't predict according to time when they're going to happen there's still a trend that goes like that up and down generally but the time gap between them isn't the same each time so the difference between seasonal and cyclical is the predictability of the peaks and troughs cyclic time series plots might look a bit like this here's some peaks and troughs peak trough trough peak so this one's here's going up and then down and then up down and then up and then down and then down down down up and then down and up so we've got peaks and troughs we've got a mountain range going on here but they don't occur at regular intervals this ones are carrying it what's that three maybe and then the trough is at four and then this peak is at 10 and then this trough this trough this downward part of the trend is longer than this one and then this downward part of the trend is much longer so there isn't a regular gap between all of these peaks and troughs over here again we've got a clear sort of shape we've got our mountain range but the time that they're occurring isn't regular having a look at this example do you think it's seasonal or cyclical cyclic or let's have a look at where these peaks and troughs are occurring we've got months going along here 12 24 36 so that's at the end of the year that's at the end of the year and that's at the end of the year and what do you know the same time each year there's a bit of a trough what about where these Peaks are occurring that peak looks like it's happening about halfway through the year what happens about halfway through this year what do you know there's another peak what happens about halfway through this year oh there's that peak again so this is seasonal because these peaks and troughs are happening at the same time each year the other thing that this graph has is a sort of upward kind of trend even though these troughs down here are pulling it down this way over all over time it's going up the page so this one has a positive trend and it's seasonal another way that would have been useful to work that out if you have a look at what we're actually measuring dinner sales it makes sense before you even look at the graph think about the sales of dinners what would they be like they probably have a high point right in the middle of the year when it's winter and they probably don't sell very many of them at the end of the year when it's summer when it's hot which makes sense here to these little troughs so with before you even look at the time series plot sometimes you can predict whether something might be seasonal cyclical or random a random time series plot doesn't have any obvious peaks and troughs happening for particular reasons it's basically what random sounds like it's just all over the shop seasonal you can predict peaks and troughs cyclic all it has peaks and troughs and they're just they occur for different reasons other than time and random doesn't really have an obvious sort of pattern of the variation to it the variation is just all over the shop there could be a trend upward or downward with the variation being random but in general you don't see a pattern of peak and then trough and then peak and then trough here are two examples of random time series plots this one here doesn't really have any significant peace Peaks or troughs until about here and that doesn't repeat at any other point in this time series so it's random because it's kind of all wavy along there and then there's this going on there isn't an obvious up down up down up down to it this one here has a peak and then a peak but then it doesn't repeat again over here it sort of breaks its own pattern so this variation is random because there isn't something that's really repeating itself in any significant way what can we say about this time series is there any sort of pattern to it can we see an obvious pattern of peaks and troughs well here there's a peak and here there's a peak but it doesn't really look like there's any sort of pattern to it there's no rhyme or reason for why things are going up and then down and road fatalities in Victoria you wouldn't expect to have regular peaks and troughs either but this one does have a sort of downward trend as time goes by this is all generally decreasing so there's no pattern to the variation but there is a negative secular trend with random variation here our fourth time series plots have a look and see if you can work out what the pattern or trend to these might be so this first one here you can see that it's got a peak and a trough peak trough peak trough so it seems like it does have a pattern to that variation it's doing the same thing over and over but it's not doing it at regular intervals so that would make this one cyclic what about this one down here it's going down up up down up up down down up down down up down down up it looks like the distance between each peak is about the same and the distance between each trough is about the same it's repeating with a certain amount of regularity there's predictability to this pattern so this is seasonal what about this one up here it's going up down up down up down up down up down up down up down up down up it's doing something over and over it's got a pattern to it but there isn't an even space between each one there isn't predictability to when these ups and downs are going to occur so this one is cyclic what about this one it's going down up nothing really significant with the down up stuff here the peaks and troughs aren't really standing out to me which means there isn't much variation going on with that the variation there is probably just random but there is a definite trend going down the page so to describe this one I would say it has a negative secular trend this one up here also actually has a trend generally going up the page so that would be something worth talking about this one here has an ever so slightly negative trend and this one here doesn't have a trend it's not really going up or down it's basically staying the same and have a look at these for time series plots and see if you can work out what's going on with them let's start with this top left one is there any significant pattern happening with peaks and troughs there is one peak here and then a bit of a trough here but there isn't something that's repeating because we can only see one peak and one trough so the variation of that one is pretty random and is there a positive trend going up the page or a negative trend if anything it's maybe slightly negative but only just what about this one down here the bottom left we've got trough peak trough peak trough peak trough peak trough peak trough peak okay so there is a pattern going up and down there's definitely this repeating up/down business is it happening with regularity not really it doesn't seem to have the same pattern occurring it just goes up and down at different times which means that this one is cyclic cyclic and what's the trend is it going up the page or down Paige it's kind of hard to tell if anything probably slightly upward what about this one on the top right I can see peaks and troughs peaks and troughs peaks and troughs and what do you know it's going down up down down up a little bit up a lot down a little bit down a lot up a little bit up a lot down a little bit down a lot and if you have a look at the distance between these Peaks that's occurring at regular intervals and the distance between these troughs is occurring at regular intervals as well so this one is seasonal and if anything there's a slightly upward trend to it but that's not too obvious and what about this last one any obvious peaks and troughs yeah obvious peak obvious trough obvious peak obvious trough so there's a pattern to it now how regularly is that occurring is that occurring because of time or because of something else so this peak here and then it stays high for another three sort of time periods and then we go low for three time periods so there's four periods in that peak and there's only three in this peak and then there's two down here it doesn't look like that's a repeating pattern due to time it looks like those peaks and troughs are occurring because of something else so this one's cyclical it goes in cycles a couple more examples let's see what's going on with these this one here has got some obvious peaks and troughs happening and where's this peak happening that's happening in winter we've got summer autumn winter spring here then we've got a trough happening in summer and then we have another peak happening in winter again okay and then another trough happening in summer and a peak let me guess Oh what do you know what's happening in winter so probably what would happen over here we would be able to fill this in a little bit and say well there's probably going to be a trough in summer somewhere down here like this and then there'll be another peak in that winter it'll get higher again maybe up there so this one is seasonal and what about this one on the right is there any regularity I can see the peaks and troughs are occurring over and over and when are these Peaks occurring that's at four then that's at eight then that's at twelve okay it sounds like it's happening every four years and these troughs are occurring immediately after that peak as well so this one is seasonal as well and the other one that this thing has is that trend can you see how it's going up the page as we progress it's got that positive secular trend and lastly here's how this gets tested on the exams this is from the 2005 exam one multiple choice and it was questioned thirteen the question said which one of the following statements best describes the time series plot for the period shown so looking at this time series plot okay the share price shows no trend and no change in variability well it's not going to be that because I can already see there's this trendline on there it's got a positive trend to it see that positive going up the page along here that trend so it's not going to be this one that says no trend and it's not going to be this one which says no trend so it's going to be one of these three that says the share price shows an increasing trend the other options are constant variability decreasing variability or increasing variability so let's have a look at the variability the variability is these peaks and troughs so along here it's conforming to the line it's not moving around very much it doesn't have really fluctuating peaks and troughs and then it starts to get a bit more variable it's got a peak trough peak trough and then it's got a huge peak huge trough huge peak huge trough so while it was little variability here it started to get bigger and bigger as we went along so it shows an increasing linear trend and the variability is increasing over time so the answer here is e and finally this question appeared on the 2008 exam one multiple choice and it was question eleven and the question said this the time series plot below shows the number of users each month for an online help service over a 12-month period so what do you notice about this time series to me it looks like it's going up the page like that it's got a bit of a positive trend to it that's the first thing I notice then I noticed some significant peaks and troughs happening here so I'll investigate that now what does the question say the time series plot has a no trend well it's not going to be that because there's clearly a trend going across the page so let's rule that one out B no variability I can see peaks and troughs and peaks and troughs and peaks and troughs so there's variability going on there so it's not that one now our other options are seasonality only an increasing trend with seasonality or an increasing trend only so let's have a look and see if there's seasonality there first of all we can probably rule out C because C is about there being only seasonality so let's rule that out because we've got this increasing trend now do we have an increasing trend with seasonality or just an increasing trend let's check this one is happening at 5 and then this one is happening at 9 and then this one is happening at 11 okay so there was one two three four months between that one and then there was two months between that one it doesn't really seem like there's a particular time series reason for those peaks and troughs they're not occurring because of the month of the year they're occurring because of something else they're not repeating in a regular sort of way here we've got 1 2 3 4 down within a peak and then we've got 1 2 3 down and then a peak and then we've only got one down and then we've got a peak so the pattern isn't regular it's changing each peak and trough so that's not seasonal so the answer here is an increasing trend only

Channel: Further Maths

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Length: 21min 12sec (1272 seconds)

Published: Fri Apr 01 2011