determine time of significant increase of curve - r

I have the following data, plotted in R, depicting the angular rotation of an object vs. time. I would like to determine the time point at which the movement begins by calculating when the y-value significantly depart from the baseline (the start of the rise phase of the curve). I would love any ideas for how I should approach this problem using R.

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Defining temporal resolution in Comsol post-processing

I have a time dependent heat conduction simulation and need to plot the average temperature of some area over time. However, the exported table data apparently uses only a few data points and interpolates in between.
More specifically, I have some block of material (aluminum) that is heated periodically at some surface. I am now interested in temperature peaks at exactly this surface over time. I have defined the heating function, the surface, and have calculated the average temperature of the surface under observation over time. However, when I plot the exported data
the temperature data is really, REALLY coarse. The heating data however is very fine. Comsol seems to interpolate between very few points. Calculating with a finer temporal resolution won't fix it.
How can I tell Comsol to evaluate the temperature at every step?
OK, I found the answer:
https://www.comsol.com/support/knowledgebase/1254
Turns out the timesteps the solver chooses are completely separate from the ones the user can define for the simulation. This honestly makes me question the usefulness of the initial definition of the timesteps. It really seems to be just an additional hoop for people to be dependent on support....
Solution:
Turn the maximum timestep in Solution/Time Dependent Solver to an acceptable minimal value.

segmenting lat/long data graph into lines/vectors

I have lat/lng data of multirotor UAV flights. There are alot of datapoints (~13k per flight) and I wish to find line segments from the data. They give me flight speed and direction. I know that most of the flights are guided missons meaning a point is given to fly to. However the exact points are unknown to me.
Here is a graph of a single flight lat/lng shifted to near (0,0) so they are visible on the same time-series graph.
I attempted to generate similar data, but there are several constraints and it may take more time to solve than working on the segmenting.
The graphs start and end nearly always at the same point.
Horisontal lines mean the UAV is stationary. These segments are expected.
Beginning and and end are always stationary for takeoff and landing.
There is some level of noise in the lines for the gps accuracy tho seemingly not that much.
Alot of data points.
The number of segments is unknown.
The noise I could calculate given the segments and least squares method to the line. Currently I'm thinking of sampling the data (to decimate it a little) and constructing lines. Merging the lines with smaller angle than x (dependant on the noise) and finding the intersection points of the lines left.
Another thought is to try and look at this problem in the frequency domain. The corners should be quite high frequency. Maybe I could make a custom filter kernel that would enable me to use a window function and win in efficency.
EDIT: Rewrote the question for more clarity and less rambling.

Graphite - offset series by standard deviation of itself

I'm using Graphite and Grafana and I'm trying to plot a series against a time shifted version of itself for comparison.
(I.e. is the current value similar to this time last week?)
What I'd like to do is plot;
the 5 minute moving average of the series
a band consisting of the 5 minute moving average of the series timeshifted by 7 days, bounded above and below by the standard deviation of itself
That way I can see if the current moving average falls within a band limited by the standard deviation of the moving average from a week ago.
I have managed to produce a band based on the timeshifted moving average, but only by offsetting either side by a constant amount. I can't work out any way of offsetting by the standard deviation (or indeed by any dynamic value).
I've copied a screenshot of the sort of thing I'm trying to achieve. The yellow line is the current moving average, the green area is bounded by the historical moving average offset either side by the standard deviation.
Is this possible at all in Grafana using Graphite as the backend?
I'm not quite on the latest version, but can easily upgrade (and will do so shortly anyway).
Incidentally, I'm not a statistician, if what I'm doing actually makes no sense mathematically, I'd love to know! ;-) My overall goal is to explore better alternatives, instead of using static thresholds, for highlighting anomalous or problematic server performance metrics - e.g. CPU load, disk IOPS, etc.

How do I do waveform analysis in R or Excel?

I'm trying to get some information out of a couple of waveforms, which I currently have in the format of a CSV table with the columns: Time, Volume, Pressure, Flow. (The data is the flow/pressure/volume data obtained from a healthcare ventilator.)
I want to get excel / R / another-programme-that-I've-not-yet-thought-of to do the following with the waveform:
Break it up into individual breaths - starting with when the pressure starts to rise above a baseline (shortly followed by the flow rising above 0)
For each breath, extract:The highest pressure that occurs, the pressure just before the start of the next breath, the lowest pressure that occurs
Does anyone know how to do that, without doing it manually? (It'd mean trawling through a few hours-worth of ventilator data for the project I'm trying to do)
I've attached a copy of the shapes of the waves I'm dealing with, to try to help make more sense.Pressure & Volume against time
My intention is to work out the breath-to-breath variability in maximum, minimum, and baseline pressures.
Ultimately, I've come up with my own fix using excel.
For each pressure value, I've taken the mean of that value and the two either side of it. I've then compared this to the previous mean. For my data, a change of this mean of more than 1% in either direction identifies the beginning and end of the up- and down-stroke of the pressure curve, and I can identify where in the curve I am based on the absolute value for pressure (the peak should never be lower than the highest baseline for the data I've collected), and for the slopes, the first direction of change of the mean (this smooths out occasionally inconsistent slopes).
I'm then graphing the data, along with the transition points, the calculated phase of the breath, and the breath count, so I can sense-check it visually.
Ged

Calculate volume from crossections

I have an irregularly shaped 3d object. Of this object I know the areas of the crossections in regular intervals. How can I calculate the volume of this object?
You can only approximate the volume. Just add up all the areas and then multiply by the distance between intervals.
Obviously the smaller the distance between intervals, the more accurate the volume. It is just integration (calculus).
Discretize it using tetrahedra or bricks and add up their volumes, a la finite element methods. Integrate using Gaussian quadrature and sum.
You're estimating a Riemann integral. There are many methods to do this, of varying complexity. Simpson's rule is reasonably straightforward and will be pretty accurate as long as the cross-sectional area varies in a smooth enough fashion, however it requires that the number of intervals be even.
Ed Heal's answer is a Riemann sum that approaches the (volume) integral in the limit. Depending on where the cross-sections are located with respect to the extent of the object, it might be viewed as an application of the midpoint rule.
Assuming the cross-section area varies smoothly with distance (twice continuously differentiable along the axis perpendicular to the cross-sections), the midpoint rule and trapezoid rule have accuracy that improves with the square of the interval width (here assumed regular). Averaging the midpoint and trapezoid rule approximations amounts to an application of Simpson's rule, outlined in Peter Milley's answer, with higher order accuracy (improving with the fourth power of the interval width) provided the integrand is sufficiently smooth (continuous 4th derivative of cross-section area with respect to distance).
Of course many real world figures will not have such smoothness (too many corners, holes, etc.), so it is prudent not to expect exceptional accuracy from making more sophisticated approximations.

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