rrdgraph_rpn
RPN expression:=vnameoperatorvalue[,RPN expression]
If you have ever used a traditional HP calculator you already know RPN (Reverse Polish Notation).
The idea behind RPN is that you have a stack and push your data onto this stack.
Whenever you execute an operation,
it takes as many elements from the stack as needed.
Pushing is done implicitly,
so whenever you specify a number or a variable,
it gets pushed onto the stack automatically.
At the end of the calculation there should be one and only one value left on the stack.
This is the outcome of the function and this is what is put into the vname.
For CDEF instructions,
the stack is processed for each data point on the graph.
VDEF instructions work on an entire data set in one run.
Note,
that currently VDEF instructions only support a limited list of functions.
Example: VDEF:maximum=mydata,MAXIMUM
This will set variable "maximum" which you now can use in the rest of your RRD script.
Example: CDEF:mydatabits=mydata,8,*
This means: push variable mydata,
push the number 8,
execute the operator *.
The operator needs two elements and uses those to return one value.
This value is then stored in mydatabits.
As you may have guessed,
this instruction means nothing more than mydatabits = mydata * 8.
The real power of RPN lies in the fact that it is always clear in which order to process the input.
For expressions like a = b + 3 * 5 you need to multiply 3 with 5 first before you add b to get a.
However,
with parentheses you could change this order: a = (b + 3) * 5 .
In RPN,
you would do a = b,
3,
+,
5,
* without the need for parentheses.
 Boolean operators

LT,
LE,
GT,
GE,
EQ,
NE
Less than,
Less or equal,
Greater than,
Greater or equal,
Equal,
Not equal all pop two elements from the stack,
compare them for the selected condition and return 1 for true or 0 for false.
Comparing an unknown or an infinite value will result in unknown returned ...
which will also be treated as false by the IF call.
UN,
ISINF
Pop one element from the stack,
compare this to unknown respectively to positive or negative infinity.
Returns 1 for true or 0 for false.
IF
Pops three elements from the stack.
If the element popped last is 0 (false),
the value popped first is pushed back onto the stack,
otherwise the value popped second is pushed back.
This does,
indeed,
mean that any value other than 0 is considered to be true.
Example: A,B,C,IF should be read as if (A) then (B) else (C)
 Comparing values

MIN,
MAX
Pops two elements from the stack and returns the smaller or larger,
respectively.
Note that infinite is larger than anything else.
If one of the input numbers is unknown then the result of the operation will be unknown too.
MINNAN,
MAXNAN
NANsafe version of MIN and MAX.
If one of the input numbers is unknown then the result of the operation will be the other one.
If both are unknown,
then the result of the operation is unknown.
LIMIT
Pops two elements from the stack and uses them to define a range.
Then it pops another element and if it falls inside the range,
it is pushed back.
If not,
an unknown is pushed.
The range defined includes the two boundaries (so: a number equal to one of the boundaries will be pushed back).
If any of the three numbers involved is either unknown or infinite this function will always return an unknown
Example: CDEF:a=alpha,0,100,LIMIT will return unknown if alpha is lower than 0 or if it is higher than 100.
 Arithmetics

+,
,
*,
/,
%
Add,
subtract,
multiply,
divide,
modulo
ADDNAN
NANsafe addition.
If one parameter is NAN/UNKNOWN it'll be treated as zero.
If both parameters are NAN/UNKNOWN,
NAN/UNKNOWN will be returned.
SIN,
COS,
LOG,
EXP,
SQRT
Sine and cosine (input in radians),
log and exp (natural logarithm),
square root.
ATAN
Arctangent (output in radians).
ATAN2
Arctangent of y,x components (output in radians).
This pops one element from the stack,
the x (cosine) component,
and then a second,
which is the y (sine) component.
It then pushes the arctangent of their ratio,
resolving the ambiguity between quadrants.
Example: CDEF:angle=Y,X,ATAN2,RAD2DEG will convert X,Y components into an angle in degrees.
FLOOR,
CEIL
Round down or up to the nearest integer.
DEG2RAD,
RAD2DEG
Convert angle in degrees to radians,
or radians to degrees.
ABS
Take the absolute value.
 Set Operations

SORT,
REV
Pop one element from the stack.
This is the count of items to be sorted (or reversed).
The top count of the remaining elements are then sorted (or reversed) in place on the stack.
Example: CDEF:x=v1,v2,v3,v4,v5,v6,6,SORT,POP,5,REV,POP,+,+,+,4,/ will compute the average of the values v1 to v6 after removing the smallest and largest.
AVG
Pop one element (count) from the stack.
Now pop count elements and build the average,
ignoring all UNKNOWN values in the process.
Example: CDEF:x=a,b,c,d,4,AVG
MEDIAN
pop one element (count) from the stack.
Now pop count elements and find the median,
ignoring all UNKNOWN values in the process.
If there are an even number of nonUNKNOWN values,
the average of the middle two will be pushed on the stack.
Example: CDEF:x=a,b,c,d,4,MEDIAN
TREND,
TRENDNAN
Create a "sliding window" average of another data series.
Usage: CDEF:smoothed=x,1800,TREND
This will create a halfhour (1800 second) sliding window average of x.
The average is essentially computed as shown here:
+!!!!!!!!>
now
delay t0
<>
delay t1
<>
delay t2
<>
Value at sample (t0) will be the average between (t0delay) and (t0)
Value at sample (t1) will be the average between (t1delay) and (t1)
Value at sample (t2) will be the average between (t2delay) and (t2)
TRENDNAN is  in contrast to TREND  NANsafe. If you use TREND and one source value is NAN the complete sliding window is affected. The TRENDNAN operation ignores all NANvalues in a sliding window and computes the average of the remaining values.
PREDICT, PREDICTSIGMA, PREDICTPERC
Create a "sliding window" average/sigma/percentil of another data series, that also shifts the data series by given amounts of time as well
Usage  explicit stating shifts: CDEF:predict=<shift n>,...,<shift 1>,n,<window>,x,PREDICT CDEF:sigma=<shift n>,...,<shift 1>,n,<window>,x,PREDICTSIGMA CDEF:perc=<shift n>,...,<shift 1>,n,<window>,<percentil>,x,PREDICTPERC
Usage  shifts defined as a base shift and a number of time this is applied CDEF:predict=<shift multiplier>,n,<window>,x,PREDICT CDEF:sigma=<shift multiplier>,n,<window>,x,PREDICTSIGMA CDEF:sigma=<shift multiplier>,n,<window>,<percentil>,x,PREDICTPERC
Example: CDEF:predict=172800,86400,2,1800,x,PREDICT
This will create a halfhour (1800 second) sliding window average/sigma of x, that average is essentially computed as shown here:
+!!!!!!!!!!!!!!!!!>
now
shift 1 t0
<>
window
<>
shift 2
<>
window
<>
shift 1 t1
<>
window
<>
shift 2
<>
window
<>
Value at sample (t0) will be the average between (t0shift1window) and (t0shift1)
and between (t0shift2window) and (t0shift2)
Value at sample (t1) will be the average between (t1shift1window) and (t1shift1)
and between (t1shift2window) and (t1shift2)
The function is by design NANsafe. This also allows for extrapolation into the future (say a few days)  you may need to define the data series whit the optional start= parameter, so that the source data series has enough data to provide prediction also at the beginning of a graph...
The percentile can be between [100:+100]. The positive percentiles interpolates between values while the negative will take the closest.
Example: you run 7 shifts with a window of 1800seconds. Assuming that the rrdfile has a step size of 300 seconds this means we have to do the percentile calculation based on a max of 42 distinct values (less if you got NAN). that means that in the best case you get a step rate between values of 2.4 percent. so if you ask for the 99th percentile, then you would need to look at the 41.59th value. As we only have integers, either the 41st or the 42nd value.
With the positive percentile a linear interpolation between the 2 values is done to get the effective value.
The negative returns the closest value distance wise  so in the above case 42nd value, which is effectively returning the Percentile100 or the max of the previous 7 days in the window.
Here an example, that will create a 10 day graph that also shows the prediction 3 days into the future with its uncertainty value (as defined by avg+4*sigma) This also shows if the prediction is exceeded at a certain point.
rrdtool graph image.png imgformat=PNG \
start=7days end=+3days width=1000 height=200 altautoscalemax \
DEF:value=value.rrd:value:AVERAGE:start=14days \
LINE1:value#ff0000:value \
CDEF:predict=86400,7,1800,value,PREDICT \
CDEF:sigma=86400,7,1800,value,PREDICTSIGMA \
CDEF:upper=predict,sigma,3,*,+ \
CDEF:lower=predict,sigma,3,*, \
LINE1:predict#00ff00:prediction \
LINE1:upper#0000ff:upper\ certainty\ limit \
LINE1:lower#0000ff:lower\ certainty\ limit \
CDEF:exceeds=value,UN,0,value,lower,upper,LIMIT,UN,IF \
TICK:exceeds#aa000080:1 \
CDEF:perc95=86400,7,1800,95,value,PREDICTPERC \
LINE1:perc95#ffff00:95th_percentile
Note: Experience has shown that a factor between 3 and 5 to scale sigma is a good discriminator to detect abnormal behavior. This obviously depends also on the type of data and how "noisy" the data series is.
Also Note the explicit use of start= in the CDEF  this is necessary to load all the necessary data (even if it is not displayed)
This prediction can only be used for short term extrapolations  say a few days into the future.
 Special values

UNKN
Pushes an unknown value on the stack
INF, NEGINF
Pushes a positive or negative infinite value on the stack. When such a value is graphed, it appears at the top or bottom of the graph, no matter what the actual value on the yaxis is.
PREV
Pushes an unknown value if this is the first value of a data set or otherwise the result of this CDEF at the previous time step. This allows you to do calculations across the data. This function cannot be used in VDEF instructions.
PREV(vname)
Pushes an unknown value if this is the first value of a data set or otherwise the result of the vname variable at the previous time step. This allows you to do calculations across the data. This function cannot be used in VDEF instructions.
COUNT
Pushes the number 1 if this is the first value of the data set, the number 2 if it is the second, and so on. This special value allows you to make calculations based on the position of the value within the data set. This function cannot be used in VDEF instructions.
 Time

Time inside RRDtool is measured in seconds since the epoch. The epoch is defined to be Thu Jan 1 00:00:00 UTC 1970 .
NOW
Pushes the current time on the stack.
TIME
Pushes the time the currently processed value was taken at onto the stack.
LTIME
Takes the time as defined by TIME, applies the time zone offset valid at that time including daylight saving time if your OS supports it, and pushes the result on the stack. There is an elaborate example in the examples section below on how to use this.
 Processing the stack directly

DUP, POP, EXC
Duplicate the top element, remove the top element, exchange the two top elements.
DEPTH
pushes the current depth of the stack onto the stack
a,b,DEPTH > a,b,2
n,COPY
push a copy of the top n elements onto the stack
a,b,c,d,2,COPY => a,b,c,d,c,d
n,INDEX
push the nth element onto the stack.
a,b,c,d,3,INDEX > a,b,c,d,b
n,m,ROLL
rotate the top n elements of the stack by m
a,b,c,d,3,1,ROLL => a,d,b,c
a,b,c,d,3,1,ROLL => a,c,d,b
These operators work only on VDEF statements. Note that currently ONLY these work for VDEF.
 MAXIMUM, MINIMUM, AVERAGE

Return the corresponding value, MAXIMUM and MINIMUM also return the first occurrence of that value in the time component.
Example: VDEF:avg=mydata,AVERAGE
 STDEV

Returns the standard deviation of the values.
Example: VDEF:stdev=mydata,STDEV
 LAST, FIRST

Return the last/first nonnan or infinite value for the selected data stream, including its timestamp.
Example: VDEF:first=mydata,FIRST
 TOTAL

Returns the rate from each defined time slot multiplied with the step size. This can, for instance, return total bytes transferred when you have logged bytes per second. The time component returns the number of seconds.
Example: VDEF:total=mydata,TOTAL
 PERCENT, PERCENTNAN

This should follow a DEF or CDEF vname. The vname is popped, another number is popped which is a certain percentage (0..100). The data set is then sorted and the value returned is chosen such that percentage percent of the values is lower or equal than the result. For PERCENTNAN Unknown values are ignored, but for PERCENT Unknown values are considered lower than any finite number for this purpose so if this operator returns an unknown you have quite a lot of them in your data. Infinite numbers are lesser, or more, than the finite numbers and are always more than the Unknown numbers. (NaN < INF < finite values < INF)
Example: VDEF:perc95=mydata,95,PERCENT VDEF:percnan95=mydata,95,PERCENTNAN
 LSLSLOPE, LSLINT, LSLCORREL

Return the parameters for a Least Squares Line (y = mx +b) which approximate the provided dataset. LSLSLOPE is the slope (m) of the line related to the COUNT position of the data. LSLINT is the yintercept (b), which happens also to be the first data point on the graph. LSLCORREL is the Correlation Coefficient (also know as Pearson's Product Moment Correlation Coefficient). It will range from 0 to +/1 and represents the quality of fit for the approximation.
Example: VDEF:slope=mydata,LSLSLOPE
rrdgraph gives an overview of how rrdtool graph works. rrdgraph_data describes DEF,CDEF and VDEF in detail. rrdgraph_rpn describes the RPN language used in the ?DEF statements. rrdgraph_graph page describes all of the graph and print functions.
Make sure to read rrdgraph_examples for tips&tricks.
Program by Tobias Oetiker <tobi@oetiker.ch>
This manual page by Alex van den Bogaerdt <alex@vandenbogaerdt.nl> with corrections and/or additions by several people
