Probability and Statistics in Geology

Probability and Statistics in Geology Probability and statistics are an important aspect of Earth Science.  Understanding the details, population of a data sample  How rounded are these pebbles ? Where did they come from ?  How likely is an earthquake here in Northridge ?  Probability and Statistics in Geology Statistics  Histograms  Probability  Error Analysis  Regression  ] Discuss next week Are Statistics Always Right ? Can They be Misleading ? Toss a coin 6 times....”Heads or tails ? ”  What is most unlikely ? .....six tails  What is more likely ? .....3 heads and 3 tails   So is HTHTHT more likely than TTTTTT ? ...No, both are equally unlikely! Are Statistics Always Right ? Can They be Misleading ? The result 3 heads and 3 tails is more likely only because  There are many combinations where this can occur (e.g. HHTHTT, or HTHTTH, or HHHTTT...)  Let's try it...  What is a Statistic ?  Is this a statistic ? “ In 1970, the oil refining capacity of Belgium was 32.6 million tonnes per year”  This is actually, just a fact – not a statistic What is a Statistic ? Consider a pebbly beach  How could you determine the composition, mass, length, shape of these particular pebbles ?  Would these sizes be the same on every beach ?  What is a Statistic ? - Specimen Let's pick up a pebble and look at it – this is a specimen  This pebble could probably give us the composition but would it be inclusive of all the pebbles ? Is it typical ?  How could be improve this specimen ?  What is a Statistic ? - Sample We could pick up 100 pebbles, this is a sample from the beach  This should give you a much better idea of your beach rocks  Could we do any better ?  What is a Statistic ? - Population Or we could sample ALL the pebbles on the beach!  This is the population of all pebbles  Now measure the composition, size, shape, of each  Is this a realistic plan ?  What is a Statistic ? - Population Specimen:  Sample:  Population:  One object A subset number of objects All the objects These terms are often misused in science and literature. Faults in Southern California Above is a map of faults found in southern California  If we just study the San Jacinto fault, what is this called statistically ?  If we study the system, San Jacinto, Elsinor, and San Andreas what is this called statistcally ?  So What is a Statistic ? Is the average mass of a pebble a statistic ?  This depends on whether this average is determine From a sample of pebbles or the total population...  If we take the average of the total population – this considered a parameter and is now a simple fact  The average of a sample, however, is a statistic.  So What is a Statistic ?   A statistic is an attempt to estimate the average mass of all the pebbles by calculating the average mass of some of the pebbles Statistics are generally based on a sample of the population Election Polls Polling question: “ Who did the best job in the debate ?” Obama 54% McCain 30%  Estimates of voter intentions obtained before an election are statistics...a sample of the population Obama 365 McCain 162 Election Polls Obama 66,882,230 McCain 58,343,671 The final result of an election, however, is an election parameter  The final result is a fact, a measure of the entire voting population  Back to the Pebbly Beach Average, Mean, and Median Pebble# 1 2 3 4 5 6 7 8 9 10  Mass (g) 374 389 395 364 224 250 378 376 330 310 The typical mass of pebbles on a particular beach can be described by the mean, w (same as the average) N w = 1/N wi i=1  The mean is the “total mass of the sample” divided by The “number of pebbles” - What is mean of these pebbles ? Back to the Pebbly Beach Average, Mean, and Median Pebble# 1 2 3 4 5 6 7 8 9 10 11 Mass (g) 225 250 310 330 364 374 376 378 389 395 399 Another way of finding the typical mass of pebbles is to use the median value.  Median means “middle” and is the weight of the middle Pebble if all are lined up (ranked) from lightest to heaviest.  You must have an odd number of pebbles to get the median  In the above example, pebble #6 has a mass of 374 g which gives the median value of this pebble sample  Back to the Pebbly Beach Average, Mean, and Median Pebble# 1 2 3 4 5 6 7 8 9 10 11   Mass (g) 225 250 310 330 364 374 376 378 389 395 399 Will the median always be the same as the mean ? With an even number of pebbles (100), you can average The 50th and 51st pebbles. Back to the Pebbly Beach - Dispersion Pebble# 1 2 3 4 5 6 7 8 9 10 11 Mass (g) 225 250 310 330 364 374 376 378 389 395 399 What about other aspects of the distribution of pebbles ?  How can we tell if the pebbles are similar in size  (i.e. well or poorly sorted)  We could give the total range of sizes – known as the dispersion  But how much does this tell us about all the sample pebbles ?  Back to the Pebbly Beach - Dispersion Pebble# 1 2 3 4 5 6 7 8 9 10 11 Mass (g) 225 250 310 330 364 374 376 378 389 395 399 The heaviest and lightest pebbles may not be “typical”  One way to get an accurate measure of how similar your Pebbles are is to use the mean square of the standard deviation The standard deviation is the 2 2  = (mass - w) square root of this value.  This measures the deviation from the mean – also known as the variance - the bar indicates the average of all calculations  Back to the Pebbly Beach - Dispersion Pebble# 1 2 3 4 5 6 7 8 9 10 11  2 Mass (g) 225 250 310 330 364 374 376 378 389 395 399 = (mass - w)  Why do we square this difference ?  Some will be negative, we just want the deviation of each From the average value.  If 2 is small – then the masses are similar and well sorted 2  If  is large – then the masses are widely varying and are poorly sorted 2 Visualizing Distribution of Data   How can you display graphically the distribution of a large number of pebbles ? Which sizes occur most often ? Which are fairly rare ? Visualizing Distribution of Data: Histogram Frequency Distribution 200-235 236-260 261-285 286-315 316-335 336-365 366-385 386-415 416-435 436-465 Number 1 3 7 9 16 22 19 14 6 2 Frequency Range(g) Pebble mass (g) A histogram displays the pebble mass count in bins (10 bins shown)  We first count the number of occurences (frequency) in each bin and list them in a table called the frequency distribution  Then plot this frequency as a bar chart against mass  Histograms in Matlab (or Octave) Pebble# 1 2 3 4 5 6 7 8 9 10 11 Mass (g) 225 250 310 330 364 374 376 378 389 395 399 Frequency Count of all pebbles Pebble mass (g)  To plot histograms in Matlab: >> x = 200:25:500 >> y = pebblefile(:,2) >> hist(y,x) % set bin range and increment, here 25 % read column 2 of file of pebble masses % plots histogram shown above for data (y) and bins (x) Visualizing Distribution of Data Marine seismic study, Weeraratne et al., 2007   We're interested in earthquake paths which come from every possible azimuth within 360o (the back azimuth). How can we graphically represent the distribution of cyclical data or direction ? Visualizing Distribution of Data: Rose Diagrams A rose diagram is like plotting a histogram on a polar graph.  The direction is represented by The angle around the plot and The frequency is proportional To distance from the center.  Here frequency ranges from o 0 to 6 and an angle of 30 is the most frequent occuring 6 times.   A list of fault dip angles could be plotted in this way. Plotting Rose Diagrams in Matlab (or Octave)  To plot rose diagrams in Matlab: >> dip = faultdipfile(:,1) >> dipradians = dip.*pi./180 >> bins = 100 >> rose(dipradians,bins) % reads first column of data input % converts angles to radians % specify the number of bins % plot the rose diagram Probability     If I measure a large number of data points, how often do I obtain a particular result ? For the pebbles masses measured here, the most probably mass is 350 grams This mass value occurs in 22 (frequency) out of 100 cases or 22% of the time. Thus the estimated probability of picking up a pebble in this area with a mass of 350 grams is 22%. Frequency  What is Probability ? Pebble mass (g) Probability     If I measure a large number of data points, how often do I obtain a particular result ? For the pebbles masses measured here, the most probably mass is 350 grams This mass value occurs in 22 (frequency) out of 100 cases or 22% of the time. Thus the estimated probability of picking up a pebble in this area with a mass of 350 grams is 22%. Frequency  What is Probability ? Pebble mass (g) Probability     If I measure a large number of data points, how often do I obtain a particular result ? For the pebbles masses measured here, the most probably mass is 350 grams This mass value occurs in 22 (frequency) out of 100 cases or 22% of the time. Thus the estimated probability of picking up a pebble in this area with a mass of 350 grams is 22%. Frequency  What is Probability ? Pebble mass (g) Probability Frequency Distribution & Probability 200-235 236-260 261-285 286-315 316-335 336-365 366-385 386-415 416-435 436-465   Number 1 3 7 9 16 22 19 14 6 2 Probability .01 .03 .07 .09 .16 .22 .19 .14 .06 .02 Probability Range(g) Pebble mass (g) We can then add another column to the data which shows the probability for each bin size You can now plot probability in a histogram Probability: What is Normal ? Frequency Distribution & Probability 200-235 236-260 261-285 286-315 316-335 336-365 366-385 386-415 416-435 436-465  Number 1 3 7 9 16 22 19 14 6 2 Probability .01 .03 .07 .09 .16 .22 .19 .14 .06 .02 Probability Range(g) Pebble mass (g) You can compare your data distribution to theoretical estimates The most common distribution used is a normal distribution also known as a Gaussian distribution.  Gaussian Distribution P(x) = e [-(x-x)2/22] 2 sqrt(2 )   The Gaussian distribution is written as above and describes the relative probability of obtaining the value, x. Here  is the standard deviation and x is the average of all x Gaussian Distribution P(x) = e [-(x-x)2/22] 2 sqrt(2 ) P(x) x This is a Gaussian distribution for xmean= 5.0 and  = 2.0  You are more likely to obtain a value between 4-6 where the graph is high  And less likely to obtain a value between 1-2, or 9-10  Gaussian Distribution P(x) x  We can quantify this by looking at the area under the curve, the total area under the curve is 1.0  The area under the curve between 1 - 2 is shown in gray.  This area is much smaller than the dark gray block between 4 - 7. Gaussian Distribution To quantify these “areas” we use established values for multiples of the standard deviation from the mean P(x)  1.0  2.0  x The area under the curve between 3-7 is 0.683 and is termed 1.0  (this is known as the 68% confidence limit)  The area under the curve between 1-9 is 0.954 and is termed (this is known as the 95% confidence limit)  Linear Regression: How to Fit a Line to Scattered Data Now that we've learned statistical analysis of a single variable We can also consider statistical analysis of two related variables.  We may be able to approximate this relationship by a straight line.  How do we find this line ? Which line is best ?  Distance from shore (m)  Pebble diameter Linear Regression: How to Fit a Line to Scattered Data The line draw to the right is one possibility. How can we determine whether this line is better than another – in a quantitative way ?  We can calculate the mean square deviation by looking the distance each point is from the predicted line  Distance from shore (m)  y Pebble diameter The devation of one point is shown by y and is estimated in the “y direction” only.  Linear Regression: How to Fit a Line to Scattered Data This gives you the deviation of one point from the line. To obtain the mean square deviation, we take the average ofy for all points  We calculate this using the same equation for standard deviation which we used before.   2 = 2 (y - y) Distance from shore (m)  y Pebble diameter The line with the smallest  will have the best fit to the data 