I'm forecasting a time series using exponential smoothing method.
There is an obvious additive trend but no seasonality observed based on ACF plot and seasonal plot. Initially a Linear Exponential Smoothing (LES) model is considered the best.
However, when I tried to fit by Holt-Winter's additive model with seasonality (no matter 6,12,18..etc), it actually gave me a lower MSE than LES.
What is the possible implication behind? It looks awkward to fit a model with seasonality which is in fact not observable or explainable? Many thanks.
Time series
ACF plot
Im a beginner in time series analyses.
I need help finding the SARIIMA(p,d,q,P,D,Q,S) parameters.
This is my dataset. Sampletime 1 hour. Season 24 hour.
S=24
Using the adfuller test I get p = 6.202463523469663e-16. Therefor stationary.
d=0 and D=0
Plotting ACF and PACF:
Using this post:
https://arauto.readthedocs.io/en/latest/how_to_choose_terms.html
I learn to "start counting how many “lollipop” are above or below the confidence interval before the next one enter the blue area."
So looking at PACF I can see maybe 5 before one is below the confidence interval. Therefor non seasonal p=5 (AR).
But I having a hard time finding the q - MA parameter from the ACF.
"To estimate the amount of MA terms, this time you will look at ACF plot. The same logic is applied here: how much lollipops are above or below the confidence interval before the next lollipop enters the blue area?"
But in the ACF plot not a single lollipop is inside the blue area.
Any tips?
There are many different rules of thumb and everyone has own views. I would say, in your case you probably do not need the MA component at all. The rule with the lollipop refers to ACF/PACF plots that have a sharp cut-off after a certain lag, for example in your PACF after the second or third lag. Your ACF is trailing off which can be an indicator for not using the MA component. You do not have to necessarily use it and sometimes the data is not suited for an MA model. A good tip is to always check what pmdarima’s auto_arima() function returns for your data:
https://alkaline-ml.com/pmdarima/tips_and_tricks.html
https://alkaline-ml.com/pmdarima/modules/generated/pmdarima.arima.auto_arima.html
Looking at you autocorrelation plot you can clearly see the seasonality. Just because the ADF test tells you it is stationary does not mean it necessarily is. You should at least check if you model works better with seasonal differencing (D).
I have some geographical trajectories sampled to analyze, and I calculated the histogram of data in spatial and temporal dimension, which yielded a time domain based feature for each spatial element. I want to perform a discrete FFT to transform the time domain based feature into frequency domain based feature (which I think maybe more robust), and then do some classification or clustering algorithms.
But I'm not sure using what descriptor as frequency domain based feature, since there are amplitude spectrum, power spectrum and phase spectrum of a signal and I've read some references but still got confused about the significance. And what distance (similarity) function should be used as measurement when performing learning algorithms on frequency domain based feature vector(Euclidean distance? Cosine distance? Gaussian function? Chi-kernel or something else?)
Hope someone give me a clue or some material that I can refer to, thanks~
Edit
Thanks to #DrKoch, I chose a spatial element with the largest L-1 norm and plotted its log power spectrum in python and it did show some prominent peaks, below is my code and the figure
import numpy as np
import matplotlib.pyplot as plt
sp = np.fft.fft(signal)
freq = np.fft.fftfreq(signal.shape[-1], d = 1.) # time sloth of histogram is 1 hour
plt.plot(freq, np.log10(np.abs(sp) ** 2))
plt.show()
And I have several trivial questions to ask to make sure I totally understand your suggestion:
In your second suggestion, you said "ignore all these values."
Do you mean the horizontal line represent the threshold and all values below it should be assigned to value zero?
"you may search for the two, three largest peaks and use their location and probably widths as 'Features' for further classification."
I'm a little bit confused about the meaning of "location" and "width", does "location" refer to the log value of power spectrum (y-axis) and "width" refer to the frequency (x-axis)? If so, how to combine them together as a feature vector and compare two feature vector of "a similar frequency and a similar widths" ?
Edit
I replaced np.fft.fft with np.fft.rfft to calculate the positive part and plot both power spectrum and log power spectrum.
code:
f, axarr = plt.subplot(2, sharex = True)
axarr[0].plot(freq, np.abs(sp) ** 2)
axarr[1].plot(freq, np.log10(np.abs(sp) ** 2))
plt.show()
figure:
Please correct me if I'm wrong:
I think I should keep the last four peaks in first figure with power = np.abs(sp) ** 2 and power[power < threshold] = 0 because the log power spectrum reduces the difference among each component. And then use the log spectrum of new power as feature vector to feed classifiers.
I also see some reference suggest applying a window function (e.g. Hamming window) before doing fft to avoid spectral leakage. My raw data is sampled every 5 ~ 15 seconds and I've applied a histogram on sampling time, is that method equivalent to apply a window function or I still need apply it on the histogram data?
Generally you should extract just a small number of "Features" out of the complete FFT spectrum.
First: Use the log power spec.
Complex numbers and Phase are useless in these circumstances, because they depend on where you start/stop your data acquisiton (among many other things)
Second: you will see a "Noise Level" e.g. most values are below a certain threshold, ignore all these values.
Third: If you are lucky, e.g. your data has some harmonic content (cycles, repetitions) you will see a few prominent Peaks.
If there are clear peaks, it is even easier to detect the noise: Everything between the peaks should be considered noise.
Now you may search for the two, three largest peaks and use their location and probably widths as "Features" for further classification.
Location is the x-value of the peak i.e. the 'frequency'. It says something how "fast" your cycles are in the input data.
If your cycles don't have constant frequency during the measuring intervall (or you use a window before caclculating the FFT), the peak will be broader than one bin. So this widths of the peak says something about the 'stability' of your cycles.
Based on this: Two patterns are similar if the biggest peaks of both hava a similar frequency and a similar widths, and so on.
EDIT
Very intersiting to see a logarithmic power spectrum of one of your examples.
Now its clear that your input contains a single harmonic (periodic, oscillating) component with a frequency (repetition rate, cycle-duration) of about f0=0.04.
(This is relative frquency, proprtional to the your sampling frequency, the inverse of the time beetween individual measurment points)
Its is not a pute sine-wave, but some "interesting" waveform. Such waveforms produce peaks at 1*f0, 2*f0, 3*f0 and so on.
(So using an FFT for further analysis turns out to be very good idea)
At this point you should produce spectra of several measurements and see what makes a similar measurement and how differ different measurements. What are the "important" features to distinguish your mesurements? Thinks to look out for:
Absolute amplitude: Height of the prominent (leftmost, highest) peaks.
Pitch (Main cycle rate, speed of changes): this is position of first peak, distance between consecutive peaks.
Exact Waveform: Relative amplitude of the first few peaks.
If your most important feature is absoulute amplitude, you're better off with calculating the RMS (root mean square) level of our input signal.
If pitch is important, you're better off with calculationg the ACF (auto-correlation function) of your input signal.
Don't focus on the leftmost peaks, these come from the high frequency components in your input and tend to vary as much as the noise floor.
Windows
For a high quality analyis it is importnat to apply a window to the input data before applying the FFT. This reduces the infulens of the "jump" between the end of your input vector ant the beginning of your input vector, because the FFT considers the input as a single cycle.
There are several popular windows which mark different choices of an unavoidable trade-off: Precision of a single peak vs. level of sidelobes:
You chose a "rectangular window" (equivalent to no window at all, just start/stop your measurement). This gives excellent precission of your peaks which now have a width of just one sample. Your sidelobes (the small peaks left and right of your main peaks) are at -21dB, very tolerable given your input data. In your case this is an excellent choice.
A Hanning window is a single cosine wave. It makes your peaks slightly broader but reduces side-lobe levels.
The Hammimg-Window (cosine-wave, slightly raised above 0.0) produces even broader peaks, but supresses side-lobes by -42 dB. This is a good choice if you expect further weak (but important) components between your main peaks or generally if you have complicated signals like speech, music and so on.
Edit: Scaling
Correct scaling of a spectrum is a complicated thing, because the values of the FFT lines depend on may things like sampling rate, lenght of FFT, window, and even implementation details of the FFT algorithm (there exist several different accepted conventions).
After all, the FFT should show the underlying conservation of energy. The RMS of the input signal should be the same as the RMS (Energy) of the spectrum.
On the other hand: if used for classification it is enough to maintain relative amplitudes. As long as the paramaters mentioned above do not change, the result can be used for classification without further scaling.
I have used ARIMA(0,1,0)(0,1,1)12 where seasonal lag is 12. The ACF plot & PACF plot of residuals suggests pattern where as box test for 12 lags, 24 lags etc shows p value in the range of 20% to 40% indicating randomness.
Is it really possible to have contradictory results OR there might be a problem in the way i modeled it.
I am using Arima function of R.
The original series has strong seasonality and upward trend.
Regards
Lakshmi
Assume a sequence of numbers (wave-like data). I perform then the DFT (or FFT) transform. Next step I want to achieve is to find the frequencies, that correspond to the real frequencies that are included in data. As we know, DFT output has real and imaginary part a[i] and b[i]. If we look at spectrum (sqrt(a[i]^2+b[i]^2) then the maximum in it corresponds to the frequency that is included to the data. The question is how to find all frequencies from DFT? The problem arises when there are many other peaks that can be falsely selected.
I had a similar problem when doing spectral analysis processing of data when I was writing my honours thesis.
You are right: To find dominant frequencies you generally only need to look at the magnitude of the complex value in the DFT.
Unfortunately, you pretty much have to write some sort of intelligent algorithm which will identify the peaks (frequencies). The way the algorithm works is highly dependent on what the DFT looks like for your application. My DFTs all had similar characteristics, so it wasn't too difficult to put together a heuristic algorithm. If your DFT can take on any form, then you will probably get a lot of false positives and/or false negatives.
The way I did it was to identify regions in the DFT with high magnitude (peaks) which were surrounded by low magnitude (troughs). You can define the minimum difference between peaks and troughs (the sensitivity) as a constant times the standard deviation of the data. Additionally, you can say that any peaks that fall below a certain magnitude (threshold) are ignored altogether, as they are just noise.
Of course, the above technique will only really work if you have relatively well defined frequencies in your data. If your DFT is highly random, then you will need to take extra care to set the sensitivity and threshold carefully.
Don't forget that the magnitude of your data is symmetric, so you only need to look at half of it.
Once you have identified the frequencies in your DFT, don't forget to convert it into the units you want. From memory, if you have n samples taken with time discretisation dt, then if you have a peak at data point 5 (for example), where the first data point is 1, then the frequency is 1/(n*dt) radians per time unit. (I haven't done this in a while, so that formula might be off by a factor of Pi or something)