# How to find out if a set of daily measurements are random or not?

There is a set of daily measurements. Time and measured values are both discrete. I want to find out whether measured values depend on the day the measurement was taken, or whether measurements are completely random. In other words, I want to find out if it is possible to predict measured values of a certain day or not.

• What subjects of statistics should I study to be able to solve this problem?

Please give me some keywords or direction.

Edit
Some more information. Imagine the following situation. A machine chooses each day a letter (simply a byte) and displays it on a screen. The process which is used to choose daily letter is unknown, but it is clearly algorithmic (not measure the wind speed or count people in the room or similar). Someone collected part of daily letter over a period of time and now wants to understand if it is possible to produce the next (or any) daily letter. Some methods possibly employed by the machine are considered "hard" (or random). For example use a secret key to encrypt the current date (predicting the next letter will be equivalent in most cases to braking the encryption). Other are considered "easy", for example xor of all bytes in date's representation. If the method is random, then there is no hope to produce the next daily letter.

• First graph your data to see if there are any obvious patterns. Do you have any theory suggesting the data might be non-random? That may guide your choice of statistical model. Jul 13, 2011 at 1:54
• @Michael: Unfortunately graphing your data doesn't always disclose the "unusual" due to background variability. In trivial cases the human eye is as good as a good statistical program whose focus is separating signal from noise and partitioning the noise into randomness and the exceptional. In non-trivial cases my bet would be on good/superior analytics and not on "tired/confused eyes". Save your eyes ! Use Statistic Methods to do what it was designed to do ! Jul 13, 2011 at 14:27
• @Irish I don't think @Michael was suggesting that the analysis be limited to graphing. It would be foolhardy at best to trust the output of "good/superior analytics" without ever looking at the data. Given a choice between some statistical output and a good graph of a dataset (but not both), I will take the graph every time.
– whuber
Jul 13, 2011 at 16:03
• @whuber:When one is faced with analyzing/characterizing a large number of problems,one needs to use "productivity aids". Good luck with relying on graphs when one has noisy data with large background variability.In trivial cases where the eye can catch the anomaly one doesn't need to acquire "expensive software" as one can use the human brain/eyesight as a proxy for the "expensive software".Where I come from "human time" is more expensive than software but perhaps lacking software they can use their eyes.Academics are often unable to acquire "better eyes" & that's why they often need glasses. Jul 13, 2011 at 16:41
• Are you measuring a single thing, or multiple things each day? (E.g. are you measuring temperature, or are you measuring temperature, humidity, rainfall, etc?) Are you making only one measurement of each thing, or multiple measurements of each thing per day? If you are measuring one thing, you have a univariate time series, otherwise multivariate, for example. Jul 13, 2011 at 17:06

In my opinion you want to focus on time series analysis which deals directly with the subject you raise. When dealing with time series data one can use memory models (ARIMA/Autoprojective Structure) to capture the importance of previous values in predicting future values. Google Box-jenkins or ARIMA for more. An equally interesting approach is to use a fixed-effects (X) approach which might incorporate day-of-the-week effects, weekly effects, Holiday/Event effects, Particular days-of-the-month effects. What is even more powerful is to incorporate both the ARIMA component and the X structure into an ARMAX model or a Transfer Function Model. Care should also be taken to identify unusual data via Intervention Detection to accomodate Pulses, Level Shifts, Seasonal Pulses and even Local Time Trends. It also might be important to validate a Gaussian Error Process and to take steps to ensure same. I have made a number of comments on this board about such things. You might review my posts and other posts that you may find equally informative.

Modified my answer to deal with the need for models to detect anomalies that if untreated inflate the variance of the errors causing incorrect acceptance of the hypothesis of randomness. Prof.J.K.Ord has referred to this as "the Alice in wonderland effect". The problem is that you can't catch an outlier without a model (at least a mild one) for your data. Else how would you know that a point violated that model? In fact, the process of growing understanding and finding and examining outliers must be iterative. This isn't a new thought. Bacon, writing in Novum Organum about 400 years ago said: "Errors of Nature, Sports and Monsters correct the understanding in regard to ordinary things, and reveal general forms. For whoever knows the ways of Nature will more easily notice her deviations; and, on the other hand, whoever knows her deviations will more accurately describe her ways."

• Though you did not mention them directly, I found out through your post about autocorrelation and ARMA and as these are more basic I think I will read about them first. Regarding the second paragraph. I am not sure I understood. Do you mean that I always must have a guess about the model of the series? Jul 14, 2011 at 0:09
• @Artium" If you want to contact me directly please do so and I will try and help. Essentially the values that represent unusual activity increase the error sum of squares thus reducing the probability of finding anything non-random. Please refer to my email address and send me your contact info and I will try and help. Essentially the characteristics of the data can suggest a starting model which can then be iteratively re-constructed which can then be used to detect errant observations which can then result in a model that may be useful in separating signal and noise. Jul 14, 2011 at 0:16

Your edited question sheds new light on the issue, thanks!

My first thought is that you'd want to do some kind of Chi-squared test to determine if the output is random. If the test indicates that is is not random (actually not pseudo-random, of course), then you'd have to attempt to model it as a time series or in other ways, perhaps trying to model possible letter generation algorithms. I'm not expert enough to go beyond this, and am not sure if a time series approach (ARIMA, etc) will work without external data.