# Sinusoidal data

The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions.

In Section 2. In Progress Check 2. A mathematical model is a function that describes some phenomenon. For objects that exhibit periodic behavior, a sinusoidal function can be used as a model since these functions are periodic.

However, the concept of frequency is used in some applications of periodic phenomena instead of the period. The frequency of a sinusoidal function is the number of periods or cycles per unit time. A typical unit for frequency is the hertz.

### How do I find the equation of a sinusoidal graph?

One hertz Hz is one cycle per second. This unit is named after Heinrich Hertz — Since frequency is the number of cycles per unit time, and the period is the amount of time to complete one cycle, we see that frequency and period are related as follows:. The volume of the average heart is milliliters mland it pushes out about one-half its volume 70 ml with each beat. In addition, the frequency of the for a well-trained athlete heartbeat for a well-trained athlete is 50 beats cycles per minute.

We will model the volume, V. Our function is. For example:. Suppose that we want to know at what times after the heart is full that there will be milliliters of blood in the heart.

Although we will learn other methods for solving this type of equation later in the book, we can use a graphing utility to determine approximate solutions for this equation. To solve the equation, we need to use a graphing utility that allows us to determine or approximate the points of intersection of two graphs.

This can be done using most Texas Instruments calculators and Geogebra. We really only need to find the coordinates of one of those points since we can use properties of sinusoids to find the others.

The summer solstice in was on June 21 and the winter solstice was on December A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sineof which it is the graph. It occurs often in pure and applied mathematicsas well as physicsengineeringsignal processing and many other fields. Its most basic form as a function of time t is:. The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude.

It is the only periodic waveform that has this property.

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This property leads to its importance in Fourier analysis and makes it acoustically unique. This equation gives a sine wave for a single dimension; thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, and their product as a dot product.

For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. This wave pattern occurs often in nature, including wind wavessound waves, and light waves. Because of this head startit is often said that the cosine function leads the sine function or the sine lags the cosine.

The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics. To the human ear, a sound that is made of more than one sine wave will have perceptible harmonics; addition of different sine waves results in a different waveform and thus changes the timbre of the sound.

Presence of higher harmonics in addition to the fundamental causes variation in the timbre, which is the reason why the same musical note the same frequency played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves which are periodicthen the sound will be perceived to be noisy, as noise is characterized as being aperiodic or having a non-repetitive pattern.

InFrench mathematician Joseph Fourier discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform, including square waves. Fourier used it as an analytical tool in the study of waves and heat flow. It is frequently used in signal processing and the statistical analysis of time series. Since sine waves propagate without changing form in distributed linear systems[ definition needed ] they are often used to analyze wave propagation.

Sine waves traveling in two directions in space can be represented as.The horizontal displacement means the number of units right or left in from the x axis. The vertical displacement is the displacement up or down from the y axis. First, let's note the amplitude. The amplitude is given by "equation of max" - "equation of normal". What does this signify? It signifies there has been a reflection over the x axis, which means parameter a is negative. Hence, parameter a is Note that the amplitude can never be negative, so it's given by a.

Next, let's determine the period. Look back at the definition above of "period". It is the distance between two maximums or two minimums. In the graph above, the distance between any two maximums or minimums is pi. We know the period now, all that remains is to find the value of b. As for horizontal displacements, there are none, since the minimum is on the y axis; it hasn't been moved left or right.

This is a little more complicated. We first note that a vertical displacement has occurred. This graph has undergone no reflection over the x axis, so parameter a is positive in this scenario. As for the period, the distance between all two maximums and minimums is 1so the period is 1.

We must determine the value of b :. Finally, we need to determine the factor of the horizontal displacement. We find that it is 1 unit to the right. How do I find the equation of a sinusoidal graph? Noah G. Jul 12, I will provide you with two examples. Explanation: Before we get to problems, I would like to go through a little bit of vocabulary. This being done, we can now look at a few applications to these particular words. Example 1: What is a cosine equation for the following graph?

Example 2: Determine the equation of the following graph. Hopefully this helps! Related questions What are common mistakes students make with sinusoidal graphs?

How do I graph sinusoidal functions? What are sinusoidal graphs? What is the amplitude of a sinusoidal graph? How do I find the frequency of a sinusoidal graph?By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. Although I read this post, I still have no idea how to apply this to my own data and hope that someone can help me out. A better fit still would perhaps account for the outliers in that series in some way, reducing their influence. As Stefan suggested, different starting values do seem to improve the fit dramatically.

When I put that into nls 's start list, I got a curve that was much more reasonable, although it still has some systematic biases. Depending on what your goal is with this data set, you could try to improve the fit by adding additional terms or using a nonparametric approach like a Gaussian process with a periodic kernel.

If you want to pick the dominant frequency, you can use a fast Fourier transform FFT. This is way out of my area of expertise, so I'll let other folks fill in the details if they'd like especially about steps 2 and 3but the R code below should work.

You can also plot abs truncated. I'm not sure. In any case, knowing the starting values for the other parameters shouldn't be nearly as important as for omega if they're in the right ballpark. You could probably get decent estimates of the other parameters from the FFT, but I'm not certain how that would work. As an alternative to what has already been said, it may be worth noting that an AR 2 model from the class of ARIMA models can be used to generate forecasts with a sine wave pattern.

A stochastic cycle pattern can be thought of a distorted sine wave pattern in the forecast pattern: It is a sine wave with a stochastic probabilistic period, amplitude, and phase angle. To see if such a model could be fitted to the data I used the auto. It turns out that the auto. That is, we can still check the aforementioned condition to see if sine wave forecasts will be produced. The plot below shows the original series, y, the fit of the ARMA 2,2 model, and 14 out-of-sample forecasts.

As can be seen, the out-of-sample forecasts follow a sine-wave pattern. Keep in mind two things. The current methods to fit a sin curve to a given data set require a first guess of the parameters, followed by an interative process. This is a non-linear regression problem.

A different method consists in transforming the non-linear regression to a linear regression thanks to a convenient integral equation. Then, there is no need for initial guess and no need for iterative process : the fitting is directly obtained. If you know the lowest and highest point of your cosine-looking data, you can use this simple function to compute all cosine coefficients:.

Below it is used to simulate the variation of temperature throughout the day with a cosine function, by entering the hours and temperature values for the lowest and warmest hour:. The output is below:. Another option is using the generic function optim or nls. I've tried both none of them is completely robust. Sign up to join this community.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Fit a sinusoidal term to data Ask Question.In statisticssignal processingand time series analysisa sinusoidal model to approximate a sequence Y i is:. This sinusoidal model can be fit using nonlinear least squares ; to obtain a good fit, nonlinear least squares routines may require good starting values for the constant, the amplitude, and the frequency.

Fitting a model with a single sinusoid is a special case of least-squares spectral analysis.

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A good starting value for C can be obtained by calculating the mean of the data. If the data show a trendi.

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That is, the model becomes. The starting value for the frequency can be obtained from the dominant frequency in a periodogram. A complex demodulation phase plot can be used to refine this initial estimate for the frequency. The root mean square of the detrended data can be scaled by the square root of two to obtain an estimate of the sinusoid amplitude.

A complex demodulation amplitude plot can be used to find a good starting value for the amplitude. In addition, this plot can indicate whether or not the amplitude is constant over the entire range of the data or if it varies. If the plot is essentially flat, i. However, if the slope varies over the range of the plot, one may need to adjust the model to be:.

A linear fit is specified in the model above, but this can be replaced with a more elaborate function if needed. As with any statistical modelthe fit should be subjected to graphical and quantitative techniques of model validation.

For example, a run sequence plot to check for significant shifts in location, scale, start-up effects and outliers. A lag plot can be used to verify the residuals are independent. The outliers also appear in the lag plot, and a histogram and normal probability plot to check for skewness or other non- normality in the residuals. A different method consists in transforming the non-linear regression to a linear regression thanks to a convenient integral equation.

## 2.3: Applications and Modeling with Sinusoidal Functions

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### Sinusoidal model

This lead may be surprisingly small to some people (Democrats) or surprisingly large to others (believers in fundamental models). A lot more to follow in the next 6 months and 4 days. It has been so much fun, but there is a really good chance that the primary effectively ends in 30 minutes feeling nostalgic yet. A quick post on the state of the election at 191 until Election Day. Also, just added Newt Gingrich into the Republican VP slot should keep an eye on him.

We pushed some updates to the site this morning. Musings will be brief updates on data, so I can reserve blog posts for more complete thoughts. Our polling reflects bleeding-edge research into cost-effective, fast, flexible, and accurate polling with an emphasis on politics and finance. We infuse raw polling data with machine-learning-based post-polling analytics along with a host of other data. The backbone of predictions on this site are market-based, generated from real-money markets that trade contracts on upcoming events.

PredictWise is run by David Rothschild an innovative, and stylish economist at Microsoft Research in New York City, but should in no way be construed as representing the views or predictions of Microsoft or any of its entities. How do these match. Prior to the end of the primary strategic or upset party voters will state they will not vote for the other possible nominees.

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