Trend: How to Separate Direction From Noise in a Series

A trend is a direction that persists across a stretch of a record, and it is a description of that stretch rather than a property of the thing being recorded. A straight line through the Nakshatra unit's 72 prices rises Rs 1.8414/- a month from an intercept of Rs 85.5450/-, with an R squared of 0.7755. Over those same 72 months, 28 of them fell.

Most arguments about trends come from taking one half of that answer and dropping the other, so both halves are worth holding on to. The line is real: put it through the 72 prices and it rises, month after month, without a single flat step. The falling months are real too, and there are 28 of them. A reader who has only the first half will treat every fall as a surprise. A reader who has only the second half will say the direction was never there. Neither is what the arithmetic says.

Three things built elsewhere are used directly here: fitting a straight line to a set of points and reading its slope, its intercept and its R squared; the moving average, a second description of the same direction; and the price path of the six year record itself, the same 72 dated observations of the Nakshatra unit used throughout these notes on ordered data.

What is a trend, and what exactly is it a claim about?

Start with a stall selling tea outside one office building. The owner says it has been busier every year for three years. Nothing can be done with that sentence yet, and the reason is not that the owner is doubted. The trouble is that the sentence does not say which three years. If they are the last three, the stall is growing. If they are three years picked out of a run of ten, the other seven were never shown, and the sentence has said almost nothing.

A trend is the claim that across some stated stretch of a record, the series moved in one direction more than it moved in the other, once the movement from one observation to the next has been set aside. The two words that get dropped are the two that carry the weight: stated stretch. A trend with no stretch attached to it is not a weak claim or a rough claim. A trend with no stretch is an unfinished claim, in the same way that the tea stall sentence is unfinished, and no amount of careful arithmetic afterwards repairs the missing half.

The record used here is the six year record of the Nakshatra unit, an invented traded unit built for teaching. The record holds 72 dated monthly observations. The first is dated 31 January 2019 and the last 31 December 2024. Both are read against an opening markA reference height fixed just before a record begins, so that its first observation has something to be measured against. On this record it is Rs 100.00/- dated 31 December 2018. of Rs 100.00/- set at 31 December 2018. Over the whole six years the price finished at Rs 187.4539/-, a rise of 87.4539 per cent. Along the way it fell as low as Rs 96.8923/- in August 2019 and rose as high as Rs 238.6196/- in November 2023, so it did not simply climb.

How is a straight line put through a record, and what does the slope mean?

The arithmetic for this is already in hand. The months are numbered 1 to 72, the price goes on one side and the month number on the other, and the ordinary straight line is fitted. Out comes a slope, an intercept and an R squared, and nothing about the record being dated changes how any of those three are computed. The whole difficulty is what those three figures are then taken to mean.

On the six year record the line rises Rs 1.8414/- a month from an intercept of Rs 85.5450/-. Read that intercept plainly: it is the height the line would sit at one month before the record opens, and it is not the opening mark of Rs 100.00/-, nor is it meant to be. The line then reads Rs 87.3863/- at the first month and Rs 218.1253/- at the last. Its R squared is 0.7755.

The slope says that the line's single best guess is that each month adds Rs 1.8414/- to the price, and not one month of the record did anything so orderly. The largest month added 16.00 per cent and the smallest took away 13.00 per cent, and the line knows about neither. The slope is an average direction laid flat across 72 months that were not flat at all. The flattening is exactly what makes the figure easy to quote and easy to over-read in the same breath.

Rupee figures in this guide are carried to four decimal places so that a reader can reproduce every one of them from the record itself. Rounded to the paisaA rupee splits into a hundred paise, and everyday amounts are written down to that point and no further. Two decimal places, in other words, where this guide uses four., the closing price is Rs 187.45/- and the slope is Rs 1.84/- a month, and nothing in the argument below turns on the extra digits.

There is a second way to look at the same line that makes its flatness impossible to miss. Take the intercept as a starting height and add the slope on 72 times. Twelve of those steps come to Rs 22.0967/-, so each year of the line adds the same amount as every other year, and six identical years stacked on the intercept arrive at Rs 218.1253/-. The record, meanwhile, gained 3.81 per cent in one of its years and lost 19.01 per cent in another.

What does the six year record's straight line actually say?

Here is every figure the fit produces, in one place. The three slopes in the lower half of the table are the same arithmetic run on three different stretches of the same 72 prices, and they are set out together on purpose.

The six year record was written down as an arithmetic recipe, and every rupee figure here was produced by running that recipe again rather than by copying a number across from anywhere else. No maintained record is read at any point, so these figures carry no as-of date.

How can a record trend upward while 28 of its 72 months fell?

The answer is short enough to be suspicious of. Of the six year record's 72 months, 28 fell, 38 rose and 6 did not move at all. There is a stretch from May to August 2019 where four consecutive months fell, taking the price from Rs 105.0804/- down to Rs 96.8923/-, a fall of Rs 8.1881/- and the only time in six years the record spent any real time below its opening mark. The fitted line was rising through every one of those four months and through all 28 of the falling ones.

A trend is a statement about a whole stretch, and it makes no claim at all about any single observation inside it. A falling month is therefore not evidence against a rising trend, and a rising month is not evidence for one. Read that twice, because it is the sentence that gets skipped. The two claims are not in tension and they never were. The two claims answer two different questions: what did the stretch do overall, and what did this particular month do.

The household version is the one to keep. A shop adds up its takings once a year. Five years running the yearly total goes up. Inside those five years there were slow weeks, a fortnight when the road outside was dug up, and a month when a neighbouring shop opened and took half the customers. None of those weeks is an argument against the five year picture, and none of them was predicted by it either. Had somebody said in year two that the takings were on a rising path, that sentence would still have been true during the fortnight the road was closed, and it would still have been useless for deciding what to do that fortnight.

What happens to the slope when the stretch changes?

Take the same 72 prices, change nothing about them, and fit the line three times over. Fitted to the first three years alone the slope is Rs 1.1227/- a month. Fitted to the last three years alone it is Rs 1.0395/- a month. Fitted to all six it is Rs 1.8414/- a month. No observation was added, none was removed, and none was corrected between the three fits.

The whole record's slope is 1.6402 times the first three years' slope and 1.7714 times the last three years', and the only thing that changed between the three fits is which months were shown to the arithmetic. The reason is not mysterious once the picture is set out: each half sits at its own level, the second half sitting well above the first, and a single line across both is forced to climb from one level to the other on top of climbing within each. The extra climb belongs to the join between the halves and not to either half.

The fit quality moves too, and it moves further than the slope does. The R squared is 0.7755 across all six years, 0.7558 across the first three and 0.1547 across the last three. Same record, same method, and a fit that looked reasonably close over the long stretch is barely describing the short one. Both figures are correct. Only one of them is usually quoted.

What is the difference between the straight line and the twelve month moving average?

The straight line fit is one description of a direction. The twelve month moving average is another, and it is built quite differently: at each month, that month and the eleven before it are averaged, twelve observations in all. The average cannot say anything until the twelfth month has arrived, so on this record it starts at December 2019 and gives 61 readings rather than 72.

On the six year record the twelve month moving average reads Rs 101.2600/- at the close of 2019 and Rs 196.8859/- at the close of 2024. The straight line reads Rs 87.3863/- and Rs 218.1253/- at the two ends of the same record. Both are saying the record rose. The two descriptions are not saying it in the same way.

The straight line commits to one direction for the whole record, and the moving average commits to nothing, which makes the line far easier to quote and the average far harder to be wrong with. The line has two numbers in it and they apply everywhere. The average has 61 numbers in it and each one applies to twelve months and no others; it bends when the record bends, dips when the record dips, and says nothing at all about a month it has not reached. The refusal to commit is the point. No single figure carries the average into a sentence, so what it gains in honesty it gives up in portability.

One more thing separates them, and it is easy to miss. The moving average is a rolling windowA calculation redone at every observation over a fixed number of the most recent observations, so the window slides forward as time passes. The effect of that sliding on a figure is covered separately., so its value at December 2024 was computable at December 2024 and its value at December 2019 was computable at December 2019. Fitting the straight line used all 72 months at once, so its slope was not computable at any month except the last. Neither fact makes either one better. The two descriptions are simply not available at the same times, and a comparison that passes over that difference has set beside one another two things that were never on the desk together.

Why does a line with an R squared of 0.7755 still miss by Rs 30.6714/-?

An R squared of 0.7755 sounds close, and it is the figure that travels. Here is what the same line does at a month it was fitted to. At December 2024 the line reads Rs 218.1253/- and the price was Rs 187.4539/-, so the line is out by Rs 30.6714/-. The gap is not a forecast going wrong. December 2024 is inside the stretch; the line had already been shown that month when it was fitted.

December 2024 is not even the worst of it. The line's largest miss anywhere in the record is at October 2024, where it reads Rs 214.4425/- against a price of Rs 158.1089/-, out by Rs 56.3336/-. Across all 72 months the line is out by Rs 14.9621/- on average. Set that against an average price across the record of Rs 152.7558/-, and the size of the ordinary miss stops being a rounding matter.

If a description is out by Rs 30.6714/- on a month it has already seen, and by Rs 56.3336/- on its worst such month, it is not a device for reaching months it has not seen. Extending the line past December 2024 is the single commonest thing done with a trend line, and the two misses above are the reason it fails. No arithmetic turns a description of 72 months into a statement about a seventy third.

What can a trend not be used for?

Three things, and each of them is refused for a different reason. Taking them in order stops the trend quietly growing into something larger while nobody is looking.

A trend cannot say what the next observation will do. The refusal follows straight from the misses just shown: the line is out by Rs 30.6714/- and Rs 56.3336/- at months it was fitted to, so its readings at months it was not fitted to have nothing behind them at all.

A trend cannot say whether the direction it found is a repeating calendar pattern or a genuine drift. A different test entirely tells the two apart, one that asks whether the movement adds up to nothing over one full turn of the calendar and comes back, and seasonal adjustmentStripping the month of year effect out of a record before its months are set beside one another, so that a strong January is not read as a strong stretch. Which part of a record survives the operation is covered separately. together with that test is covered separately. A line has no way of asking whether a movement comes back over one turn of the calendar, so the fitted line cannot be made to tell the two apart.

And it cannot say whether the record will come back to the line. A record that wanders off with nothing pulling it back is said to carry a unit rootA feature of some records: nothing tugs them towards any particular height, so what happens in one month stays in the running total instead of being reversed in a later one. Whether a given record has one is settled by its own test, covered separately., and whether this record has one is settled by a test that is covered separately and is not the fitted line. Drawing a line through a record and watching the price sit above and below it is not that test and never was.

The trend is one of the smallest claims anywhere in these notes and one of the most heavily over-read, and the gap between those two facts is where the damage lives. It says: over these stated months, in this direction, by about this much a month, with this much left over. The trend says nothing else, and every extra thing it seems to say was added by the reader.

What has to be printed beside any trend figure?

A trend figure becomes usable on a Monday only once it is written down in full. A trend travels as one number, and one number is never enough, so four fields go with it every time. An analyst reading somebody else's work, a lender reading a borrower's own summary of its takings, or a household looking at three years of its own electricity bills all need the same four.

First, the stretch it was fitted to. Second, whether it was fitted to a level, meaning the thing itself, or to a change, meaning the month to month movement. The two produce entirely different figures from one record. Third, how well it fits, as a number. Fourth, its largest miss inside the record.

The fourth field is almost never given, and it is the one that tells a reader how much the first three are worth. The first three all flatter the line. A named stretch sounds careful, a level sounds precise, and 0.7755 sounds close. The fourth field is the only one of the four that reports what the line got wrong on a month it had already been shown. On this fit that miss is Rs 56.3336/- at October 2024. Read it first and the other three arrive in proportion.

There is a fifth habit worth having, and it costs one extra run of the same arithmetic: refit on at least two stretches before quoting either number. On this record that habit turns Rs 1.8414/- a month into three figures that disagree by nearly a factor of two. Three disagreeing figures are a great deal more useful than one that looks settled. The habit also guards against a quieter error: dating a description to a month at which it could not have been computed. The error is called lookaheadUsing information that did not exist yet at the moment a figure is dated to, usually by accident rather than design. How it gets into an ordinary computed column is covered separately., it is covered separately, and the habit of writing down which months went into a figure is what catches it in practice.

Which of these figures rests on somebody else's word?

None of them. Numbering the observations 1 to 72, fitting the ordinary straight line to them and taking the result away from the record is arithmetic and nothing else. There is no body that issues arithmetic, no country it applies in and no date on which it was last revised, so not one figure above is the kind of thing anybody could go and confirm. The record stands in for no published series either.

None of this exactness transfers to a record somebody measured instead of writing down. The counts of 28, 38 and 6 are precise here only because every observation came out of the recipe above; on a measured record those same counts arrive with gaps in them, with revisions behind them and with observations that turned up late, and every one of those quietly moves the fit without announcing that it has. The procedure travels between the two situations. The tidiness stays behind, and it belongs to this record alone.

The Nakshatra unit and its six year record are invented.
Educational material. Not advice on any investment, tax, budget or market position.