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

Question

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.

Answer 1

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."

Answer 2

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.