The seeds of the garden pea are either yellow or green. A certain cross between pea plants produces progeny where 75% are plants with yellow seeds and 25% are plants with green seeds. What is the minimum number of progeny you would need to grow to have probability no less than 0.99 of obtaining at least 10 plants with green seeds?
I understand how to estimate a required sample size when I have data such as standard deviation, mean, correlation, etc., but I don't even know where to start to estimate it based on the percentage values with a certain probability.
So far I set up this code in SAS:
Proc power;
onesamplefreq test=Z method=normal
sides=1
alpha=.01
nullproportion=.5
proportion=.25
power=.99
ntotal= .;
run;
Running this program resulted in a sample size of 76, but I don't feel like this is correct. I don't know how to specify that I need at least 10 plants with green seeds, and I don't know how to set the nullproportion or if it matters.
It is a Binomial distribution kind problem. Where chance of winning (green plant) is 25%. You want to win at least 10 times, so how many times you need to play (that is, how many seeds you need)?
Mean of binomial distribution will answer this question which is:
np = 10
n*0.25 = 10
n = 40
So required seed is 40. This is purely probabilistic. But we need to consider Type I and Type II error. So sample size 76 seems reasonable to me.
Related
I'm using the ModelChain class to estimate DC and AC values for a fictitious solar plant. Input parameters include module, inverter, number of strings, number of modules, number of inverters, albedo, PVGIS TMY data, etc. I apply simple math to calculate number of modules per string and number of strings per inverter then I create one PVSystem object consisting of a single PVArray, per inverter. The I run the ModelChain model for each inverter and, for simplicity, add up the AC output to estimate the total AC for all arrays like this:
for idx in range(0, num_of_inverters):
array = {
'name': f'pvsystem-{idx+1}-array',
'mount': mount,
'module': module_name,
'module_parameters': module_parameters,
'module_type': module_type,
'albedo': albedo,
'strings': strings_per_inverter,
'modules_per_string': modules_per_string,
'temperature_model_parameters': temperature_model_parameters,
}
pvsystem=pvlib.pvsystem.PVSystem(arrays=[pvlib.pvsystem.Array(**array)], inverter_parameters=inverter_parameters)
mc = pvlib.modelchain.ModelChain(pvsystem, location)
mc.run_model(tmy_weather)
total_ac += mc.results.ac.sum()
According to PVLib documentation, the AC output is yearly in Watts hour.
But now I need to get the DC output as well (yearly in Watts hours) so I can calculate the DC/AC ratio. Running mc.results.dc gives me a Dataframe with several values (columns) that are hard to grasp for a newbie like me:
i_sc : Short-circuit current (A)
i_mp : Current at the maximum-power point (A)
v_oc : Open-circuit voltage (V)
v_mp : Voltage at maximum-power point (V)
p_mp : Power at maximum-power point (W)
i_x : Current at module V = 0.5Voc, defines 4th point on I-V curve for modeling curve shape
i_xx : Current at module V = 0.5(Voc+Vmp), defines 5th point on I-V curve for modeling curve shape
I tried using p_mp and adding it up: mc.results.dc['p_mp'].sum() but the output is much bigger than the estimated AC. I usually expect the DC/AC ratio to be somewhere > 1 and <= 1.5, roughly. However, I'm getting DC values that are like 3-5 times bigger which probably means I'm doing something wrong.
Example: 1 string, 1 inverter, 10 modules per string:
Output (yearly):
AC: 869.61kW
DC: 3326.36kW
Ratio: 3.83
Any help is appreciated.
As for why the total DC and AC generation values are so different, it's because the inverter is way undersized for the array. The inverter is rated for 250 W maximum, which is not much more than what a single module produces at STC (calculate by Impo * Vmpo as below, or noticing the "220" in the module name), and you have ten modules total. So the inverter will be saturated at even very low light, and the total AC production will be severely curtailed as a result. I think if you make a plot (mc.results.ac.plot()) you will see that the daily inverter output curve is clipped at 250 W while the simulated DC power can be nearly 10x higher. It's always a good idea to plot your time series when things aren't making sense!
In [23]: pvlib.pvsystem.retrieve_sam('cecinverter')['ABB__MICRO_0_25_I_OUTD_US_208__208V_']['Paco']
Out[23]: 250.0
In [24]: pvlib.pvsystem.retrieve_sam('sandiamod')['Canadian_Solar_CS5P_220M___2009_'][['Impo', 'Vmpo']]
Out[24]:
Impo 4.54629
Vmpo 48.3156
Name: Canadian_Solar_CS5P_220M___2009_, dtype: object
A couple other notes:
Please be careful about units:
Summing (really, integrating) an hourly time series of power (Watts) produces energy (Watt-hours). An annual output in kW doesn't make sense, since kW is for power and power is an instantaneous rate of energy generation. If this is new to you, it might be helpful to think about speed vs distance: a car might be traveling at 60mph at any given time point, but the total distance it travels in a year is measured in miles, not mph. Power is energy per unit time just like speed is distance per unit time.
Summing voltages (num_of_inverters * mc.results.dc['v_mp'].sum()) makes no sense that I can see. Volt-hours doesn't seem like a useful unit to me outside of some very specialized power electronics engineering contexts.
The term "DC/AC ratio" is typically understood to mean the ratio of rated capacities, not annual productions. So for the example in your gist, the DC/AC ratio would be calculated as (220 W/module * 10 modules/string * 2 strings/inverter = 4400 W DC) / (250 W AC) = 17.6 (which is a crazy DC/AC ratio).
I am learning about Two Dimensional Neuron Network so I am facing many obstacles but I believe it is worth it and I am really enjoying this learning process.
Here's my plan: To make a 2-D NN work on recognizing images of digits. Images are 5 by 3 grids and I prepared 10 images from zero to nine. For Example this would be number 7:
Number 7 has indexes 0,1,2,5,8,11,14 as 1s (or 3,4,6,7,9,10,12,13 as 0s doesn't matter) and so on. Therefore, my input layer will be a 5 by 3 neuron layer and I will be feeding it zeros OR ones only (not in between and the indexes depends on which image I am feeding the layer).
My output layer however will be one dimensional layer of 10 neurons. Depends on which digit was recognized, a certain neuron will fire a value of one and the rest should be zeros (shouldn't fire).
I am done with implementing everything, I have a problem in computing though and I would really appreciate any help. I am getting an extremely high error rate and an extremely low (negative) output values on all output neurons and values (error and output) do not change even on the 10,000th pass.
I would love to go further and post my Backpropagation methods since I believe the problem is in it. However to break down my work I would love to hear some comments first, I want to know if my design is approachable.
Does my plan make sense?
All the posts are speaking about ranges ( 0->1, -1 ->+1, 0.01 -> 0.5 etc ), will it work for either { 0 | .OR. | 1 } on the output layer and not a range? if yes, how can I control that?
I am using TanHyperbolic as my transfer function. Does it make a difference between this and sigmoid, other functions.. etc?
Any ideas/comments/guidance are appreciated and thanks in advance
Well, by the description given above, I think that the design and approach taken it's correct! With respect to the choice of the activation function, remember that those functions help to get the neurons which have the largest activation number, also, their algebraic properties, such as an easy derivative, help with the definition of Backpropagation. Taking this into account, you should not worry about your choice of activation function.
The ranges that you mention above, correspond to a process of scaling of the input, it is better to have your input images in range 0 to 1. This helps to scale the error surface and help with the speed and convergence of the optimization process. Because your input set is composed of images, and each image is composed of pixels, the minimum value and and the maximum value that a pixel can attain is 0 and 255, respectively. To scale your input in this example, it is essential to divide each value by 255.
Now, with respect to the training problems, Have you tried checking if your gradient calculation routine is correct? i.e., by using the cost function, and evaluating the cost function, J? If not, try generating a toy vector theta that contains all the weight matrices involved in your neural network, and evaluate the gradient at each point, by using the definition of gradient, sorry for the Matlab example, but it should be easy to port to C++:
perturb = zeros(size(theta));
e = 1e-4;
for p = 1:numel(theta)
% Set perturbation vector
perturb(p) = e;
loss1 = J(theta - perturb);
loss2 = J(theta + perturb);
% Compute Numerical Gradient
numgrad(p) = (loss2 - loss1) / (2*e);
perturb(p) = 0;
end
After evaluating the function, compare the numerical gradient, with the gradient calculated by using backpropagation. If the difference between each calculation is less than 3e-9, then your implementation shall be correct.
I recommend to checkout the UFLDL tutorials offered by the Stanford Artificial Intelligence Laboratory, there you can find a lot of information related to neural networks and its paradigms, it's worth to take look at it!
http://ufldl.stanford.edu/wiki/index.php/Main_Page
http://ufldl.stanford.edu/tutorial/
If it is needed to generate randoms in [N, M] range, but with more numbers close to avg (N <= avg <= M), which is better to use:
poisson_distribution or
normal_distribution?
Seeing at examples at cppreference pages (at bottom of the pages), they both generate what is needed:
poisson_distribution at point 4:
0 *
1 *******
2 **************
3 *******************
4 *******************
5 ***************
6 **********
7 *****
8 **
9 *
10
11
12
13
normal_distribution at point 5 with standard deviation 2:
-2
-1
0
1 *
2 ***
3 ******
4 ********
5 **********
6 ********
7 *****
8 ***
9 *
10
11
12
What to choose? May be something else?
Neither choice is great if you need the outcomes on a bounded range. The normal distribution has infinite tails at both ends, the Poisson distribution has an infinite upper tail. At a minimum you'd want a truncated form of one of them. If you're not truncating, note that the normal is always symmetric about its mean while a Poisson can be quite skewed. The two distributions also differ in the fact that the normal is continuous, the Poisson is discrete, although you can discretize continuous distributions by binning the results.
If you want a discrete set of outcomes on a bounded range, you could try a scaled and shifted binomial distribution. A binomial with parameters n and p counts how many "successes" you get out of n trials when the trials are independent and all yield success with probability p. Make n = M - N and shift the outcome by N to get outcomes in the range [N,M].
If you want a continuous range of outcomes, consider a beta distribution. You can fudge the parameters to get a wide variety of distribution shapes and dial in the mean to where you want it, and scale+shift it to any range you want.
You can center both distributions in a point that suits your needs.
But if M is small, then the Poisson distribution has a 'fat tail', that is, the probability of getting a number above M is higher compared to the normal distribution.
In the normal case, you can control this chance via the variance parameter (it can be as small as you want).
The other, rather obvious difference is that Poisson will onli give you positive integers, whreas a Normal Distribution will give any number in the [N,M] range.
Plus, when [N,M] are large enough, the Poisson converges to a Normal distribution. So even if the Poisson is the right model, the normal approximation won't be so inaccurate.
With this in mind, if the numbers do not simulate a counting process, I would go for the Normal.
If you need distribution which is within range (not an infinite or semi-infinite one like normal or Poisson), but have clear maximum, you may try Irwin-Hall one with several degrees of freedom. Say IH(16) will have minimum at 0, maximum at 16 and peak at 8, see http://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution
Very easy to sample, easy to scale, and you could play with n to get peak wider or narrower
I prefer Normal distribution, because it is closer to real life problems, while Poisson distribution is used for special cases only. Choosing N.D makes your problem more general.
I'm trying to implement a RBM and I'm testing it on MNIST dataset. However, it does not seems to converge.
I've 28x28 visible units and 100 hidden units. I'm using mini-batches of size 50. For each epoch, I traverse the whole dataset. I've a learning rate of 0.01 and a momentum of 0.5. The weights are randomly generated based on a Gaussian distribution of mean 0.0 and stdev of 0.01. The visible and hidden biases are initialized to 0. I'm using a logistic sigmoid function as activation.
After each epoch, I compute the average reconstruction error of all mini-batches, here are the errors I get:
epoch 0: Reconstruction error average: 0.0481795
epoch 1: Reconstruction error average: 0.0350295
epoch 2: Reconstruction error average: 0.0324191
epoch 3: Reconstruction error average: 0.0309714
epoch 4: Reconstruction error average: 0.0300068
I plotted the histograms of the weights to check (left to right: hiddens, weights, visibles. top: weights, bottom: updates):
Histogram of the weights after epoch 3
Histogram of the weights after epoch 3 http://baptiste-wicht.com/static/finals/histogram_epoch_3.png
Histogram of the weights after epoch 4
Histogram of the weights after epoch 4 http://baptiste-wicht.com/static/finals/histogram_epoch_4.png
but, except for the hidden biases that seem a bit weird, the remaining seems OK.
I also tried to plot the hidden weights:
Weights after epoch 3
Weights after epoch 3 http://baptiste-wicht.com/static/finals/hiddens_weights_epoch_3.png
Weights after epoch 4
Weights after epoch 4 http://baptiste-wicht.com/static/finals/hiddens_weights_epoch_4.png
(they are plotted in two colors using that function:
static_cast<size_t>(value > 0 ? (static_cast<size_t>(value * 255.0) << 8) : (static_cast<size_t>(-value * 255.)0) << 16) << " ";
)
And here, they do not make sense at all...
If I go further, the reconstruction error falls a bit more, but do no go further than 0.025. Even if I change the momentum after sometime, it goes higher and then goes down a bit but not interestingly. Moreover, the weights do no make more sense after more epochs. In most example implementations I've seen, the weights were making some sense after iterating through the complete data set two or three times.
I've also tried to reconstruct an image from the visible units, but the results seems almost random.
What could I do to check what goes wrong in my implementation ? Should the weights be within some range ? Does something seems really strange in the data ?
Complete code: https://github.com/wichtounet/dbn/blob/master/include/rbm.hpp
You are using a very small learning rate. In most NNs trained by SGD you start out with a higher learning rate and decay it over time. Search for learning rate or adaptive learning rate to find more information on that.
Second, when implementing a new algorithm I would recommend finding the paper that introduced it and reproducing their results. A good paper should include most of the settings used - or the method used to determine the settings.
If a paper is unavailable, or it was tested on a data set you don't have access to - go find a working implementation and compare the outputs when using the same settings. If the implementations are not feature compatible, turn off as many features as you can that are not shared.
How can I measure this area in C++?
(update: I posted the solution and code as an answer rather than edit the question again)
The ideal line (dashed red) is the plot from starting point with the average rise added with each angle of measurement; this I obtain via average. I measured the test data in black. How can I quantify the area of the dip in blue? X-axis is unitized, so slopes and math are simplified.
I could determine a cutoff for the size of areas like this and then flag this part for retesting or failure. Rarely, there is another dip that appears closer to the right, but setting a cutoff value for standard deviation usually fails those parts.
Update
Diego's answer helped me visualize this. Now that I can see what I'm trying to do, I'll work on the algorithm to implement the "homemade dip detector". :)
Why?
I created a test bench to test throttle position sensors I'm selling. I'm trying to programatically quantify how straight the plot is by analyzing the data collected. This one particular model is vexing me.
Sample plot of a part I prefer not to sell:
The X axis are evenly spaced angles of throttle opening. The stepper motor turns the input shaft, stopping every 0.75° to measure the output on a 10 bit ADC, which gets translated to the Y axis. The plot is the translation of data[idx] to idx,value mapped to (x,y) bitmap coordinates. Then I draw lines between the points within the bitmap using Bresenham's algorithm.
My other TPS products produce amazingly linear output.
The lower (left) portion of the plot is crucial to normal usage of any motor vehicle; it's when you're driving around town, entering parking lots, etc. This particular part has a tendency to develop a dip around 15° opening and I wish to use the program to quantify this "dip" in the curve and rely less upon the tester's intuition. In the above example, the plot dips but doesn't return to what an ideal line might be.
Even though this is an embedded application, printing the report takes 10 seconds, thus I do not consider stepping through an array of 120 points of data multiple times a waste of cycles. Also, since I'm using a uC32 PIC32 microcontroller, there's plenty of memory, so I have the luxury of being able to ponder this problem within the controller.
What I'm trying already
Array of rise between test points: I dismiss the X-axis entirely, considering it unitized, and then make an array of change from one reading to the next. This array is what contributes to the report's "Min rise between points: 0 Max: 14". I call this array deltas.
I've tried using standard deviation on deltas, however, during testing I have found that a low Std Dev is not a reliable measure for this part. If the dip quickly returns to the original line implied by early data points, the Std Dev can be deceptively low (observed to be as low as 2.3) but the part is still something I wouldn't want to use. I tried setting a cutoff at 2.6, but it failed too many parts with great plots. The other, more linear part linked to above can reliably count on Std Dev for quality.
Kurtosis seems not to apply for this situation at all. I learned of Kurtosis today and found a Statistics Library which includes Kurtosis and Skewness. During continued testing, I found that of these two measures, there was not a trend of positive, negative, or amplitude which would correspond to either passing or failing. That same gentleman has shared a linear regression library, but I believe Lin Reg is unrelated to my situation, as I am comfortable with the assumption of the AVG of deltas being my ideal line. Linear Regression and R^2 are more for finding a line from less ideal data or much larger sets.
Comparing each delta to AVG and Std Dev I set up a monitor to check each delta against final average of the deltas's data. Here, too, I couldn't find a reliable metric. Too many good parts would not pass a test restricting any delta to within 2x Std Dev away from the Average. Ultimately, the only variation from AVG I could settle on is to be within AVG+Std Dev difference from the AVG itself. Anything more restrictive would fail otherwise good parts. And the elusive dip around 15° opening can sneak through this test.
Homemade dip detector When feeding deltas to the serial monitor of the computer, I observed consecutive negative deltas during the dip, so I programmed in a dip detector, but it feels very crude to me. If there are 5 or more negative deltas in a row, I sum them. I have seen that if I take that sum the dip's differences from AVG then divide by the number of negative deltas, a value over 2.9 or 3 could mean a fail. I have observed dips lasting from 6 to 15 deltas. Readily observable dips would have their differences from AVG sum up to -35.
Trending accumulated variation from the AVG The above made me think watching the summation of deltas as it wanders away from AVG could be the answer. Meaning, I step through the array and sum the differences of each delta from AVG. I thought I was on to something until a good part blew this theory. I was seeing a trend of the fewer times the running sum varied from AVG by less than 2x AVG, the more straight the line appeared. Many ideal parts would only show 8 or less delta points where the sumOfDiffs would stray from the AVG very far.
float sumOfDiffs=0.0;
for( int idx=0; idx<stop; idx++ ){
float spread = deltas[idx] - line->AdcAvgRise;
sumOfDiffs = sumOfDiffs + spread;
...
testVal = 2*line->AdcAvgRise;
if( sumOfDiffs > testVal || sumOfDiffs < -testVal ){
flag = 'S';
}
...
}
And then a part with a fantastic linear plot came through with 58 data points where sumOfDiffs was more than twice the AVG! I find this amazing, as at the end of the ~120 data points, sumOfDiffs value is -0.000057.
During testing, the final sumOfDiffs result would often register as 0.000000 and only on exceptionally bad parts would it be greater than .000100. I found this quite surprising, actually: how a "bad part" can have accumulated great accuracy.
Sample output from monitoring sumOfDiffs This below output shows a dip happening. The test watches as the running sumOfDiffs is more than 2x the AVG away from the AVG for the whole test. This dip lasts from deltas idx of 23 through 49; starts at 17.25° and lasts for 19.5°.
Avg rise: 6.75 Std dev: 2.577
idx: delta diff from avg sumOfDiffs Flag
23: 5 -1.75 -14.05 S
24: 6 -0.75 -14.80 S
25: 7 0.25 -14.55 S
26: 5 -1.75 -16.30 S
27: 3 -3.75 -20.06 S
28: 3 -3.75 -23.81 S
29: 7 0.25 -23.56 S
30: 4 -2.75 -26.31 S
31: 2 -4.75 -31.06 S
32: 8 1.25 -29.82 S
33: 6 -0.75 -30.57 S
34: 9 2.25 -28.32 S
35: 8 1.25 -27.07 S
36: 5 -1.75 -28.82 S
37: 15 8.25 -20.58 S
38: 7 0.25 -20.33 S
39: 5 -1.75 -22.08 S
40: 9 2.25 -19.83 S
41: 10 3.25 -16.58 S
42: 9 2.25 -14.34 S
43: 3 -3.75 -18.09 S
44: 6 -0.75 -18.84 S
45: 11 4.25 -14.59 S
47: 3 -3.75 -16.10 S
48: 8 1.25 -14.85 S
49: 8 1.25 -13.60 S
Final Sum of diffs: 0.000030
RunningStats analysis:
NumDataValues= 125
Mean= 6.752
StandardDeviation= 2.577
Skewness= 0.251
Kurtosis= -0.277
Sobering note about quality: what started me on this journey was learning how major automotive OEM suppliers consider a 4 point test to be the standard measure for these parts. My first test bench used an Arduino with 8k of RAM, didn't have a TFT display nor a printer, and a mechanical resolution of only 3°! Back then I simply tested deltas being within arbitrary total bounds and choosing a limit of how big any single delta could be. My 120+ point test feels high class compared to that 30 point test from before, but that test had no idea about these dips.
Premises
the mean of a set of data has the mathematical property that the sum of the deviations from the mean is 0.
this explains why both bad and good datasets alwais give almost 0.
basically the result when differs from zero is essentially an accumulations of rounding errors in the diffs and that's why unfortunately cannot hold useful informations
the thing that most clearly define what you're looking for is your image: you're looking for an AREA and this is why you're not finding the solution in this ways:
looking to a metric in the single points is too local to extract that information
looking to global accumulations or parameters (global standard deviation) is too global and you lose the data among too much information and source of variations
kurtosis (you've already told I know but is for completeness) is out of its field of applications since this is not a probability distribution
in the end the more suitable approach of your already tryied ones is the "Homemade dip detector" because thinks in a way that is local but not too much.
Last but not least:
Any Algorithm you're going to choose has its tacit points on which it stands.
So maybe one is looking for a super clever algorithm that with no parametrization and tuning automatically adapts to the problem and self define thereshods and other.
On the other side there is an algorithm that will stand on the knowledge by the writer of the tipical data behavior (good and bad) and that is itself stupid in the way that if there is another different and unespected behavior the results are unpredictable
Ok, the right way is one of this two or is in-between them depending on the application. So if it works also the "Homemade dip detectors" can be a solution. There is not reason to define it crude but it could be that is not sufficient based on applicaton needs and that's an other thing.
How to find the area
Once you have the data the first thing is to clearly define the "theoretical straight line". I give some options:
use RANSAC algorithm (formally the best option IMHO)
this give you the best fit to the aligned points disregarding the not aligned ones
it is quite difficult and maybe oversized for this work (IMHO)
consider the line defined by the first and last point
you told that the dip is almost always in the same position that is not near boundaries so first and last points can be thought as affordable
very easy to implement
this is an example of using the knowledge about expected behaviors as I told before so you need to think if and how much confidence you give to this assumption
consider a linear fit to the first 10 points and last 10 points
is only a more affordable version of previous since using more points you can be less worried that maybe just the first point or the last were affected by any measure problem and so all fails because of this
also quite easy to implement
if I were you I will use this or something inspired to this
calculate the Y value given by the straight line for each X
calculate the area between the two curves (or the areas under the function Y_dev = Y_data - Y_straight that is mathematically the same) with this procedure:
PositiveMax = 0; NegativeMax = 0;
start from first point (value can be positive or negative) and put in a temporary area accumulator tmp_Area
for each next point
if the sign is the same then accumulate the value
if it is different
stop accumulating
check if the accumulated value is the greater than PositiveMax or below NegativeMax and if it is than store as new PositiveMax or NegativeMax
in any case reset the accumulator with tmp_Area = Y_dev; to the current value starting this way a new accumulation
in the end you will have the values of the maximum overvalued contiguous area and maximum undervalued contiguous area that I think are the scores you're looking for.
if you want you can only manage the NegativeMax based on observed and expected data behaviors
you may find useful to put a thereshold so that if a value Y_dev is lower than the thereshold you do not accumulate it.
this in order to not obtain large accumulations from many points close to the straight line that can be similar to the accumulations of few points far from the line
the need of this and and the proper thereshold needs to be evaluated on some sample data
you need to find an appropriate thereshold for this contiguous area and you can have it only from observation of sample data.
again: it can be you observing and deciding the thereshold or you can build a repository of good and bad samples and write a program that automatically learn which thereshold to use. But his is not the algorithm, this is how to find its operative parameters and there is nothing wrong to do by human brain.. ..it only depends if we're looking for a method to separate bad and good things or if we're looking for and autoadaptive algorithm that does this.. ..you decide the target.
It turns out the result of my gut feeling and Diego's method is an average of the integral. I still don't like that name, so I have described the algorithm and have asked on Math.SE what to call this, which got migrated to "Cross Validated", Stats.SE .
I Updated graphs after a massive edit of my Math.SE question. It turns out I'm taking the average of a closed integral of the derivative of the data. :P First, we gather the data:
Next is the "derivative": step through the original data array to form the deltas array which is the rise of ADC values from one 0.75° step to the next. "Rise" or "slope" is what the derivative is: dy/dx.
With the "slope" or average leveled out, I can find multiple negative deltas in a row, sum them, then divide by the count at the end of the dip. The sum is an integral of the area between average and the deltas and when the dip goes back positive, I can divide the sum by the count of the dips.
During testing, I came up with a cutoff value for this average of the integral at 2.6. That was a great measure of my "gut instinct" looking at the plot thinking a part was good or bad.
In case someone else finds themselves trying to quantify this, here's the code I implemented. Note that it is only looking for negative dips. Also, dipCountLimit is defined elsewhere as 5. In addition to the dip detector/accumulator (ie Numerical Integrator) I also have a spike detector that arbitrarily flags the test as bad if any data points stray from the average by the amount of average + standard deviation. AVG+STD DEV as a spike limit was chosen arbitrarily based on the observed plots of the parts it would fail.
int dipdx=0;
// inDipFlag also counts the length of this dip
int inDipFlag=0;
float dips[140] = { 0.0 };
for( int idx=0; idx<stop; idx++ ){
const float diffFromAvg = deltas[idx] - line->AdcAvgRise;
// state machine to monitor dips
const int _stop = stop-1;
if( diffFromAvg < 0 && idx < _stop ) {
// check NEXT data point for negative diff & set dipFlag to put state in dip
const float nextDiff = deltas[idx+1] - line->AdcAvgRise;
if( nextDiff < 0 && inDipFlag == 0 )
inDipFlag = 1;
// already IN a dip, and next diff is negative
if( nextDiff < 0 && inDipFlag > 0 ) {
inDipFlag++;
}
// accumulate this dip
dips[dipdx]+= diffFromAvg;
// next data point ends this dip and we advance dipdx to next dip
if( inDipFlag > 0 && nextDiff > 0 ) {
if( inDipFlag < dipCountLimit ){
// reset the accumulator, do not advance dipdx to next entry
dips[dipdx]=0.0;
} else {
// change this entry's value from dip sum to its ratio
dips[dipdx] = -dips[dipdx]/inDipFlag;
// advance dipdx to next entry
dipdx++;
}
// Next diff isn't negative, so the dip is done
inDipFlag = 0;
}
}
}