Ask a Statistician

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• We're just grad students
• Statistics can be a contrvercial field, these are our opinions – loosely based in reality.
• We hope this stimulates furher questions and discussion

We are comparing time series solar observations to find long term trends. Currently, we take a 28 day mean to compare between observations (e.g. compare emerging flux and filament tilt). The comparison indicates a few possible correlated features by eye. However, we cannot confirm the correlation by standard regression techniques because outside the features the time series observations show significant noise. Therefore, we have two questions.

• First, are running means the best way to correlate time series observations or do more advanced statistical techniques exist?
• Second, is there statistical way to match features across different time series observations when the noise is large?

• We see no significant evidence of signals in the Lomb-Scargle Periodogram.
  
• There are some models of universe expansion that predict some sort of oscillatory signals (though not generally pure sine modes), modulated by various parameters.
• We would like to say something along the lines of: “given this model with these free parameters, our analysis shows that values of the parameters outside of this range are ruled out with such and such certainty.”“

Your data looks like this: \[ X_{ij} \sim N (\mu_{i}, 1) \] for \( i = 1, ..., k \) and \( j = 1, ...n \)

• We know that a good estimate of \( \mu_{i} \) is the MLE \[ \bar{X}_i = \frac{1}{n} \sum_{j = 1}^n X_{ij} \]
• Turns out, an “overall” better estimate when \( k>2 \) is \[ \mu_i = \left(1 - \frac{n(k-2)}{\sum_{i = 1}^k \bar{X}_i^2}\right)\bar{X}_i \]

\[ (\bar{X}_1, \bar{X}_2, \bar{X}_3) \sim N((\mu_1,\mu_2,\mu_3), I_3/n) \]

We consider two estimators of the vector \( (\mu,\mu,\mu) \) for this visualization:

• Mean: \( X \)
• James-Stein: \( JS = (1-\frac{k-2}{||X||^2})X \)

• Assume there exists a very wide field X-ray telescope with 2 x 10⁶ cells of position.
• The background level in each cell is 4.2 x 10^-4 counts/sec, Poisson distributed.
• An X-ray source appears in only a single cell.
• Its strength is S1.
• How long an exposure is needed to be have 99.9% confidence that the source exists?

Discussion Topics:

• Scenario 1: If we know which cell the source is in.
• Scenario 2: We don't know.
• Multiple Hypothesis Testing
• Correlated Hypotheses – Modeling?