A Factor Analysis of Earthquake Activity in the Ridgecrest Area

  The combined factors or stress index used in the last blog didn't correlate well with the earthquake activity defined as the base 10 log of the number of earthquakes in a day.

Modifying the angle factors by adding one and dividing the result by two worked better.

One can use the correlation coefficient to estimate the number of earthquakes per day for a given stress index, SI. When we compare this estimate with the earthquake histogram we get a fairly good agreement for the background activity.

There is a deficit of earthquakes before the M7.1 earthquake and a surplus afterwards that may be associated with aftershocks. The new stress index lacks the peaks for when Moon's declination is negative.

The definition of the modified factors is as follows.

The stress index suggests that we might start to see a noticeable increase in earthquake activity in the next few days. The correlation coefficients for the individual factors with the earthquake activity have values similar to that of the SI.

Forming Hypotheses About the Chances of an Earthquake

  One can study the USGS Ridgecrest area earthquake data and try to come up with some hypotheses about factors indicating the likelihood of the occurrence of an earthquake. The horizontal axis is the day of the year (doy).

When we compare this data with JPL's Horizons Lunar data we encounter some factors that seem to correlate well with the occurrence of the M6.4 and M7.1 and we get a number of coincidental events such as the relative force acting on the Earth's surface, the declination of the Moon and the Moon-Earth-Sun angle (∠M-E-S). The ratio of the average range of the Moon and its value at some time can be represent by the dimensionless factor φ and ϕ² is a measure of the relative force acting on a portion of the Earth's surface. The relative force was near a maximum at the time of the two earthquakes.

The sine of the Moon's declination and the cosine of M-E-S angle were also near maximums.

The product of the first and third factors might be used to indicate the lunar perigee and syzygy alignment which tells us when there will be a Supermoon. The product of the first and second is a measure of the relative torque acting on the Earth's equatorial bulge. If we combine all three factors we also get a good agreement of their maximum with the time of the two earthquakes.

If we look at this indicator over the entire year we see that it was near a global maximum at the times of the two earthquakes.

It will be interesting to see what the plots of the earthquake data look like over the next couple of months or so.

Supplemental (Jul 19): Replaced the histogram above with one that included some missing days at the end. One can modify the definitions of the indices to let them represent relative quantities better such as replacing sin(dec) with μ=sin(dec)/sin(ι) where ι=23.473 deg is the inclination of the ecliptic. This makes the combined indicator a little more meaningful.

The doy of the indicator peaks can be converted into calendar dates.

It should be noted that one cannot accurately evaluate the indices based on a small sample of data over a short period of time. The peaks of the indices are likely to diverge as the time increases since there are different periods associated with each index. We may be able to determine which of the indices best match the pattern of earthquake activity over time.

Using Macros to Assist Plotting USGS Earthquake Data

  The macros that one uses in worksheets can be quite sophisticated. One can use them to record one's actions to simplify doing a complicated repetitive task. For example, one can search the USGS earthquake catalog for the earthquakes that occurred in a specified region in given time span and save the results in a .csv which Excel can open. The macros need to be saved in an otherwise blank worksheet and to which the .csv data can be copied and pasted to. The first macro was designed for eliminating unneeded files and converting the date and time of an earthquake to decimal UTC day of year (doy). The next macro does the data needed for the histogram. The blank worksheet also contains macros to plot earthquake magnitudes vs doy and one to plot the histogram data.

Recording and Modifying Macros in Excel

  One can record macros in Excel to perform repetitive tasks. If you've never done this before you may need to look up "record macro" in Help. To record a macro activate the Developer tab and go to the code section. When one clicks on "Record Macro" one is asked to supply a name and complete a shortcut by adding a letter that will help activate it. Here is the VBA code for a recorded macro that copies a range of numbers and pastes the values into another column.

The " _" at the end of the 4th line indicates that the line is continued on the next line. The update macro used to repeat the copy and paste values operation and increment the tallies a given number of times uses VBA code to modify a similarly recorded macro that was give the name "update."

A macro that will zero the tallies is also useful.

If you've done everything correctly this is the what one can do.

A More Detailed Look at the t-test for Averages

  I did the last blog more carefully and got significantly improved statistics for the relative error of the test results. I also cleaned up the notation somewhat. A script "l," ℓ, is now used for the level of significance of 0.05. Subscripts are used to distinguish the various t-values. The spread of the t̃_σ only depends on the spread of the x̄, the average value of x, since σ is a constant and can be computed from the t-values shown in the following plot.

The t-value t̃_s also depends of the spread of the estimated standard deviation s which makes it broader so more values are rejected. The two hypotheses tested were H₁: |t̃_σ|≤t_ℓ and H₂: |t̃_s|≤t_ℓ which are either accepted (A) or rejected (R). The level of significance determines the t_ℓ which is used in the hypotheses.

A FOR loop was added to the update macro to permit the collection of data for a fixed number of trials so one did not have to hold down the keys for the macro shortcut. It transferred the values of the set of random numbers the x column and did the updates for the decision tallies. Each pair of binary decisions for the tests can be represented by a pair of letters consisting of the letter designating the H₁ decision followed by that designating the H₂ decision.

The formulas used are contained in the following figure. Using n-1 for the estimated standard deviation s gives better results since dividing by n for the rms error is not quite correct.

With the FOR loop I was able to increase the number of trials to 250,000. The calculation took about 3½ hrs. The changes gave surprisingly good results for the rates. The rejection rate for hypothesis H₂ involving the estimated standard deviation s agreed very well with the 5% level of significance.

The t-test for Averages

  One can use the Excel t-test functions to check a sample of data to see if it is representative of a given population with known mean and standard deviation. Let's say we have a sample drawn from a population with a normal distribution of errors and with mean μ=0 and standard deviation σ=20 for which we want to check the average x̄ using a significance level SL=0.05.

For the data above we get x̄=-5.7149 and an estimated standard deviation s=18.2921. The estimate, s, is just the rms error for the sample. The t-value for the sample computed using σ is t̃(σ)=√n̄|x̄-μ|/σ and if σ is unknown we can use its estimate s to get t̃(s)=√n̄|x̄-μ|/s.

For SL=0.05 we find a 2-tails limiting value of t̃(SL)=T.INV.2T(SL,n-1)=2.2010. To check x̄ we use the "null hypothesis" t̃(s)<t̃(SL). If this is true we accept the sample and if not we reject it. Using t̃(σ)<t̃(SL) we get a more accurate test.

We can repeat the process by again using an update macro with shortcut ctrl-shft-U to transfer the values of a set of random numbers to the x column and evaluate the resulting t-values. Either both t-values are accepted or rejected or we get a Type I error when the t̃(s) value rejects an acceptable t̃(σ) value or we get a Type II error when a t̃(s) value accepts an unacceptable t̃(σ) value. We can record a set of macros to keep a tally on the outcomes for a set of trials. The trick for the tally macros is to augment a count in the cell below it, paste the augmented count in the original cell and then clear the contents of the second cell. We can then use the statistics collected to estimates the rates at which each event occurs.

Note that the sum of the rejected rates is approximately equal to the significance level α so the test doesn't take the mistakes into account.

Supplemental (Jul 6): I rewrote the update macro to increment the tallies in addition to transferring the random values to the x column. Now one just has to hold down ctrl-shft-U to increase the number of trials so there's less chance of human error but it can be rather tedious. With 10,000 trials one would expect the error in the rates to be about 1 percent. Here are the results for two runs.

Edit (Jul 7): Changed the variable α used to represent the significance level to SL to avoid possible confusion with other uses.

Additional Student t Distribution Functions in Excel

  Excel has two additional functions related to the cumulative Student t distribution for use in hypothesis testing, T.DIST.RT and T.DIST.2T. The values in the table below are calculated two ways, one using an Excel function and the second using the cumulative t distribution. The function T.DIST.RT computes values of the single tailed probability, p(t̃<t), that is, the probability that Student's t will be greater than t̃ which is seen to be equal to the probability that t is not less than t̃ which can be computed using the cumulative distribution.The two tailed function T.DIST.2T gives the probability
p(not -t̃<t<t̃) or the probability that t is not in the interval bounded by -t̃ and t̃. Note that this function is only defined for 0<t̃. Note that if B=not A then p(B)=1-p(A) since the probability of both p(A)+p(B)=1.

As t̃ increases the probabilities decrease since the area under the curve for the t probability density function decreases.

The following plots help to visualize the probabilities associated with one tail and two tails.