How to fit time series to the curve-linear trend

Hi,

I have fluctuating data with its own curve-linear trend(2) and by polynomial fitting I got curve-linear trend(1). How can I fit fluctuating data(2) to the curve-linear trend(1), so that data are extended by symetric reflection or any kind of data extension?

Like in the figure below.

(Thank you Adam Danz)

[Adam Danz]

No one can see the line "1" since it's covered by the data "2". It would have been easier to see if the line were plotted on top of the noisy data.

About your question, do you want to vertically shift your data points so that they minimize error between those points and the trend line? Is that the goal?

[Zain]

Line "1" is not covered, it is just polynomial fit trend extension of line "2" trend. I don't want to do any vertical shifting, I want instead of curve (1) to have similar fluctuations as in curve (2). So I tried to multiply 1st 15000* points of data from (2) with the curve (1) and take square root

sqrt(data2(15000:-1:1).*data1)

, but than I get this (around 2 times narrower characteristic than original):

* number of points for which data set is expanded.

[Adam Danz]

Ahhh, I see. I'll continue in the answer section below.

Answer 1

[Adam Danz]

Since your data look approximately normally distributed, you can measure the standard deviation of your data and use that to produce random numbers from a normal distrubution with a mean of your trendline. Here's general instructions.

Step 1: standardize your data by subtracting the mean (which is the trend line): dataStandard = data - trendline

Step 2: calcuate the standard deviation of the standardized data: sd = std(dataStandard)

Step 3: produce random-normal noise with standard deviation calucated in step 2 and the mean will be your red trend line.: newData = normrnd(0, sd, 1, length(trendline)) + trendline.

Step 4: plot the new data: plot(trendline, newData)

[Zain]

Well, it is kind of solution, but I would prefer more if it is possible to just make symetric reflection of existing data, instead of producing new random data.

[Adam Danz]

Where would you pull your data from? There are no y-values for the data between x=0 to x=20000 and that section of the trend line is a lot lower than the rest of your data.

Maybe you could standardize your data as I showed in step 1 (above), then take a section of those data that matches the size needed to fill in the red part of the trend line. Then you would change the x values of your selection and for the y values, you would just add the red trendline value.

Is that what you're looking for? I'd be glad to help further but your goal is not clear to me.

[Zain]

Your last solution, I understood as:

dataStandard = data - trendline;

new_data = dataStandard + trendline; new_data = data?

To answer your question - yes, there are no y values between let's say x = 0:15000, that is why I want to symetrically reflect data from x=15001:30001 in a way x(30001:-1:150001) and fit it to follow the trend line.

[Adam Danz]

To clarify my solution (which seems like it would work in your explanation above),

dataStandard = data - trendlineBlue;

new_data = dataStandard + trendlineRed;