I've fit a 4th order polynomial curve to my data like so:
y<-c(-13,16,35,40,28,36,43,33,40,33,22,-5,-27,-31,-29,-25,-26,-31,-26,-24,-25,-29,-23,4)
x<-1:24
#4th order polynomial fit
fit<-lm(y~poly(x,4,raw=TRUE))
plot(x,y,ylim=c(min(y)-10,max(y)+10))
lines(x,predict(fit,data.frame(x=x)),col="red")
abline(h=0,lty=2)
My final goal would be to calculate the 3 points of this curve where it meets the zero line.
So first, I need to extend the end of the curve fit so it passes beyond the zero line for a third time. Once I have done this, I would want to calculate the 3 points where this equation passes through the zero line.
You can use the predict function to get values from your fitted model. For example
pred <- function(x) predict(fit, newdata=data.frame(x=x))
Then if you want multiple roots, you can use a function like uniroot.all from the rootSolve package
rootSolve::uniroot.all(pred, c(0,30))
# 1.428859 11.990087 24.420745
which will find the roots between 0 and 30 from your model. You could also call the base function uniroot multiple times.
Related
I am writing a code for estimating the parameter of a GPD using weighted nonlinear least square(WNLS) method.
The WNLS method consist of 2 steps
step 1: $(\hat{\xi_1} , \hat{b_1}) = arg\ \min_{(\xi,b)} \sum_{i=1}^{n} [\log(1-F_n(x_i)) - log(1-G_{\xi,b}(x_i))]$,
here $F_n$ is the ECDF and $1-G_{\xi,b}$ is the generalized pareto distribution.
Can anyone let me know how to calculate EDF function $F_n$ for a data "X" in R?
Does ecdf(X)(X) will calculate the ECDF? If so then, what is the need for ecdf(X) other than plotting? Also it would be really helpful if someone share some example code which involves the calculation of ECDF for data.
The ecdf call creates a function. That is, you can apply ecdf(X) to other data, as your ecdf(X)(X) call does. However, you might want to apply ecdf(X) to something other than X itself. If you want to know the empirical quantile to which three numbers a, b, and c_ correspond, an easy way to do that is to call ecdf(X)(c(a, b, c_)).
So I've got a data set that I want to parameterise but it is not a Gaussian distribution so I can't parameterise it in terms of it's mean and standard deviation. I want to fit a distribution function with a set of parameters and extract the values of the parameters (eg. a and b) that give the best fit. I want to do this exactly the same as the
lm(y~f(x;a,b))
except that I don't have a y, I have a distribution of different x values.
Here's an example. If I assume that the data follows a Gumbel, double exponential, distribution
f(x;u,b) = 1/b exp-(z + exp-(z)) [where z = (x-u)/b]:
#library(QRM)
#library(ggplot2)
rg <- rGumbel(1000) #default parameters are 0 and 1 for u and b
#then plot it's distribution
qplot(rg)
#should give a nice skewed distribution
If I assume that I don't know the distribution parameters and I want to perform a best fit of the probability density function to the observed frequency data, how do I go about showing that the best fit is (in this test case), u = 0 and b = 1?
I don't want code that simply maps the function onto the plot graphically, although that would be a nice aside. I want a method that I can repeatedly use to extract variables from the function to compare to others. GGPlot / qplot was used as it quickly shows the distribution for anyone wanting to test the code. I prefer to use it but I can use other packages if they are easier.
Note: This seems to me like a really obvious thing to have been asked before but I can't find one that relates to histogram data (which again seems strange) so if there's another tutorial I'd really like to see it.
I'm new to R. Having a set of samples along with the target, I want to fit a numeric function to solve the target of new samples. My sample is time in seconds indicating the duration of a user's staying at this place:
>b <- c(101,25711,13451,19442,26,3083,133,184,4403,9713,6918,10056,12201,10624,14984,5241,
+21619,44285,3262,2115,1822,11291,3243,12989,3607,12882,4462,11553,7596,2926,12955,
+1832,3539,6897,13571,16668,813,1824,10304,2508,1493,4407,7820,507,15866,7442,7738,
+5705,2869,10137,11276,12884,11298,...)
Firstly, I convert them to hours dividing by 3600, and I want to fit a function as pdf of the duration:
> b <- b/3600
> hist(c,xlim=c(0,13),prob=T,breaks=seq(0,24,by=0.5))
> lines(density(x), col=red)
I want to fit the red line on the figure, and interpolate new values to find the probability of the specific duration on this place say p(duration = 1.5hours).
Thanks for your attention!
As suggested above, you can fit a distribution with fitdistr in MASS package.
If you use a continuous distribution you will have the probability that the time is within an interval. If you use a discrete distribution, you may compute the probability of a certain time (in hours).
For the continuous case, you can use a Gamma distribution: fitdistr(b, "Gamma") will give you the parameter estimates, and then you can use pgamma with those estimates and an interval.
For the discrete case, you can use a Poisson distribution: fitdistr(b, "Poisson") and then the dpois function with the estimate and the value you want.
To decide which one to use, I'd just plot the pdf with the histogram and take a look.
I am attempting to understand how the predict.loess function is able to compute new predicted values (y_hat) at points x that do not exist in the original data. For example (this is a simple example and I realize loess is obviously not needed for an example of this sort but it illustrates the point):
x <- 1:10
y <- x^2
mdl <- loess(y ~ x)
predict(mdl, 1.5)
[1] 2.25
loess regression works by using polynomials at each x and thus it creates a predicted y_hat at each y. However, because there are no coefficients being stored, the "model" in this case is simply the details of what was used to predict each y_hat, for example, the span or degree. When I do predict(mdl, 1.5), how is predict able to produce a value at this new x? Is it interpolating between two nearest existing x values and their associated y_hat? If so, what are the details behind how it is doing this?
I have read the cloess documentation online but am unable to find where it discusses this.
However, because there are no coefficients being stored, the "model" in this case is simply the details of what was used to predict each y_hat
Maybe you have used print(mdl) command or simply mdl to see what the model mdl contains, but this is not the case. The model is really complicated and stores a big number of parameters.
To have an idea what's inside, you may use unlist(mdl) and see the big list of parameters in it.
This is a part of the manual of the command describing how it really works:
Fitting is done locally. That is, for the fit at point x, the fit is made using points in a neighbourhood of x, weighted by their distance from x (with differences in ‘parametric’ variables being ignored when computing the distance). The size of the neighbourhood is controlled by α (set by span or enp.target). For α < 1, the neighbourhood includes proportion α of the points, and these have tricubic weighting (proportional to (1 - (dist/maxdist)^3)^3). For α > 1, all points are used, with the ‘maximum distance’ assumed to be α^(1/p) times the actual maximum distance for p explanatory variables.
For the default family, fitting is by (weighted) least squares. For
family="symmetric" a few iterations of an M-estimation procedure with
Tukey's biweight are used. Be aware that as the initial value is the
least-squares fit, this need not be a very resistant fit.
What I believe is that it tries to fit a polynomial model in the neighborhood of every point (not just a single polynomial for the whole set). But the neighborhood does not mean only one point before and one point after, if I was implementing such a function I put a big weight on the nearest points to the point x, and lower weights to distal points, and tried to fit a polynomial that fits the highest total weight.
Then if the given x' for which height should be predicted is closest to point x, I tried to use the polynomial fitted on the neighborhoods of the point x - say P(x) - and applied it over x' - say P(x') - and that would be the prediction.
Let me know if you are looking for anything special.
To better understand what is happening in a loess fit try running the loess.demo function from the TeachingDemos package. This lets you interactively click on the plot (even between points) and it then shows the set of points and their weights used in the prediction and the predicted line/curve for that point.
Note also that the default for loess is to do a second smoothing/interpolating on the loess fit, so what you see in the fitted object is probably not the true loess fitting information, but the secondary smoothing.
Found the answer on page 42 of the manual:
In this algorithm a set of points typically small in number is selected for direct
computation using the loess fitting method and a surface is evaluated using an interpolation
method that is based on blending functions. The space of the factors is divided into
rectangular cells using an algorithm based on k-d trees. The loess fit is evaluated at
the cell vertices and then blending functions do the interpolation. The output data
structure stores the k-d trees and the fits at the vertices. This information
is used by predict() to carry out the interpolation.
I geuss that for predict at x, predict.loess make a regression with some points near x, and calculate the y-value at x.
Visit https://stats.stackexchange.com/questions/223469/how-does-a-loess-model-do-its-prediction
I attached image:
(source: piccy.info)
So in this image there is a diagram of the function, which is defined on the given points.
For example on points x=1..N.
Another diagram, which was drawn as a semitransparent curve,
That is what I want to get from the original diagram,
i.e. I want to approximate the original function so that it becomes smooth.
Are there any methods for doing that?
I heard about least squares method, which can be used to approximate a function by straight line or by parabolic function. But I do not need to approximate by parabolic function.
I probably need to approximate it by trigonometric function.
So are there any methods for doing that?
And one idea, is it possible to use the Least squares method for this problem, if we can deduce it for trigonometric functions?
One more question!
If I use the discrete Fourier transform and think about the function as a sum of waves, so may be noise has special features by which we can define it and then we can set to zero the corresponding frequency and then perform inverse Fourier transform.
So if you think that it is possible, then what can you suggest in order to identify the frequency of noise?
Unfortunately many solutions here presented don't solve the problem and/or they are plain wrong.
There are many approaches and they are specifically built to solve conditions and requirements you must be aware of !
a) Approximation theory: If you have a very sharp defined function without errors (given by either definition or data) and you want to trace it exactly as possible, you are using
polynominal or rational approximation by Chebyshev or Legendre polynoms, meaning that you
approach the function by a polynom or, if periodical, by Fourier series.
b) Interpolation: If you have a function where some points (but not the whole curve!) are given and you need a function to get through this points, you can use several methods:
Newton-Gregory, Newton with divided differences, Lagrange, Hermite, Spline
c) Curve fitting: You have a function with given points and you want to draw a curve with a given (!) function which approximates the curve as closely as possible. There are linear
and nonlinear algorithms for this case.
Your drawing implicates:
It is not remotely like a mathematical function.
It is not sharply defined by data or function
You need to fit the curve, not some points.
What do you want and need is
d) Smoothing: Given a curve or datapoints with noise or rapidly changing elements, you only want to see the slow changes over time.
You can do that with LOESS as Jacob suggested (but I find that overkill, especially because
choosing a reasonable span needs some experience). For your problem, I simply recommend
the running average as suggested by Jim C.
http://en.wikipedia.org/wiki/Running_average
Sorry, cdonner and Orendorff, your proposals are well-minded, but completely wrong because you are using the right tools for the wrong solution.
These guys used a sixth polynominal to fit climate data and embarassed themselves completely.
http://scienceblogs.com/deltoid/2009/01/the_australians_war_on_science_32.php
http://network.nationalpost.com/np/blogs/fullcomment/archive/2008/10/20/lorne-gunter-thirty-years-of-warmer-temperatures-go-poof.aspx
Use loess in R (free).
E.g. here the loess function approximates a noisy sine curve.
(source: stowers-institute.org)
As you can see you can tweak the smoothness of your curve with span
Here's some sample R code from here:
Step-by-Step Procedure
Let's take a sine curve, add some
"noise" to it, and then see how the
loess "span" parameter affects the
look of the smoothed curve.
Create a sine curve and add some noise:
period <- 120 x <- 1:120 y <-
sin(2*pi*x/period) +
runif(length(x),-1,1)
Plot the points on this noisy sine curve:
plot(x,y, main="Sine Curve +
'Uniform' Noise") mtext("showing
loess smoothing (local regression
smoothing)")
Apply loess smoothing using the default span value of 0.75:
y.loess <- loess(y ~ x, span=0.75,
data.frame(x=x, y=y))
Compute loess smoothed values for all points along the curve:
y.predict <- predict(y.loess,
data.frame(x=x))
Plot the loess smoothed curve along with the points that were already
plotted:
lines(x,y.predict)
You could use a digital filter like a FIR filter. The simplest FIR filter is just a running average. For more sophisticated treatment look a something like a FFT.
This is called curve fitting. The best way to do this is to find a numeric library that can do it for you. Here is a page showing how to do this using scipy. The picture on that page shows what the code does:
(source: scipy.org)
Now it's only 4 lines of code, but the author doesn't explain it at all. I'll try to explain briefly here.
First you have to decide what form you want the answer to be. In this example the author wants a curve of the form
f(x) = p0 cos (2π/p1 x + p2) + p3 x
You might instead want the sum of several curves. That's OK; the formula is an input to the solver.
The goal of the example, then, is to find the constants p0 through p3 to complete the formula. scipy can find this array of four constants. All you need is an error function that scipy can use to see how close its guesses are to the actual sampled data points.
fitfunc = lambda p, x: p[0]*cos(2*pi/p[1]*x+p[2]) + p[3]*x # Target function
errfunc = lambda p: fitfunc(p, Tx) - tX # Distance to the target function
errfunc takes just one parameter: an array of length 4. It plugs those constants into the formula and calculates an array of values on the candidate curve, then subtracts the array of sampled data points tX. The result is an array of error values; presumably scipy will take the sum of the squares of these values.
Then just put some initial guesses in and scipy.optimize.leastsq crunches the numbers, trying to find a set of parameters p where the error is minimized.
p0 = [-15., 0.8, 0., -1.] # Initial guess for the parameters
p1, success = optimize.leastsq(errfunc, p0[:])
The result p1 is an array containing the four constants. success is 1, 2, 3, or 4 if ths solver actually found a solution. (If the errfunc is sufficiently crazy, the solver can fail.)
This looks like a polynomial approximation. You can play with polynoms in Excel ("Add Trendline" to a chart, select Polynomial, then increase the order to the level of approximation that you need). It shouldn't be too hard to find an algorithm/code for that.
Excel can show the equation that it came up with for the approximation, too.