Exploring exponential models

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Exploring exponential models

The underlying purpose for employing these techniques eluded me for too long. That is, until I came to understand this: I thought translating some of his work to Python could help others who are less familiar with R.

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I have also adapted code from other bloggers as well. This article is a living document. I will update it with corrections as needed and more useful information as time passes.

A time series is a series of data points indexed or listed or graphed in time order. I find the pictures very intuitive. Why do we care about stationarity? Most of the models we use in TSA assume covariance-stationarity 3 above.

This means the descriptive statistics these models predict Exploring exponential models. And if the mean and variance of a series are not well-defined, then neither are its correlations with other variables. Therefore a large part of TSA involves identifying if the series we want to predict is stationary, and if it is not we must find ways to transform it such that it is stationary.

More on that later Serial Correlation Autocorrelation Essentially when we model a time series we decompose the series into three components: The random component is called the residual or error.

It is simply the difference between our predicted value s and the observed value s. Serial correlation is when the residuals errors of our TS models are correlated with each other.

Recall that the residuals errors of a stationary TS are serially uncorrelated by definition! If we fail to account for this in our models the standard errors of our coefficients are underestimated, inflating the size of our T-statistics.

The result is too many Type-1 errors, where we reject our null hypothesis even when it is True! By definition a time series that is a white noise process has serially UNcorrelated errors and the expected mean of those errors is equal to zero.

Exploring exponential models

Another description for serially uncorrelated errors is, independent and identically distributed i. This is important because, if our TSM is appropriate and successful at capturing the underlying process, the residuals of our model will be i. Therefore part of TSA is literally trying to fit a model to the time series such that the residual series is indistinguishable from white noise.

Below I introduce a convenience function for plotting the time series and analyzing the serial correlation visually. This code was adapted from the blog Seanabu.

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Series y with plt. Below that we can see the QQ and Probability Plots, which compares the distribution of our data with another theoretical distribution. In this case, that theoretical distribution is the standard normal distribution.

If the TS we are modeling is a random walk it is unpredictable. Random Walk without a drift np. Thus the first differences of our random walk series should equal a white noise process! We can use the "np. However, notice the shape of the QQ and Probability plots.

This means that there should be better models to describe the actual price change process. Linear Models Linear models aka trend models represent a TS that can be graphed using a straight line.

The basic equation is: The distribution is approximately normal. Before using this model to make predictions we would have to account for and remove the obvious autocorrelation present in the series.Learn pros and cons of seven standard multi-channel attribution models, and how to create a powerful custom model.

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Exploring exponential models

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science investigates cause-and-effect relationships by seeking the mechanisms that underlie them. The next concept—scale, proportion, and quantity—concerns the sizes of things and the mathematical relationships among disparate elements.

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