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Maths Tutorial: Seasonal Indices (Seasonal Index)

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Maths Tutorials. Data Analysis/Statistics: Seasonal Indices. How calculate the seasonal index, deasonalise data, convert deseasonalised and actual data back and forth, how to apply trend lines to seasonal data and how to interpret deseasonalised data. For more tutorials, visit www.vcefurthermaths.com

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Including video

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Describes residual analysis in regression. Shows how to use residual plots to evaluate linear regression models. Includes free, video lesson on residuals.

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Great worked examples and plots

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When you run a regression, Statwing automatically calculates and plots residuals to help you understand and improve your regression model. Read below to learn everything you need to know about interpreting residuals (including definitions and examples).

Observations, Predictions, and Residuals

To demonstrate how to interpret residuals, we'll use a lemonade stand dataset, where each row was a day of Temperature and Revenue.

The regression equation describing the relationship between Temperature and Revenue is

Revenue = 2.7 * Temperature - 35

Let's say one day at the lemonade stand it was 30.7 degrees, and Revenue was $50. That 50 is your observed or actual output, the value that actually happened.

So if we insert 30.7 at our value for Temperature ...

Revenue = 2.7 * 30.7 - 35
Revenue = 48

...we get $48. That's the predicted value for that day, the value for Revenue the regression equation would have predicted based on the Temperature.

Your model isn't always perfectly right, of course. In this case, the prediction is off by 2; that difference, the 2, is called the residual, the bit that's left when you subtract the predicted value from the observed value.

Residual = Observed - Predicted

You can imagine that every row of data now has in addition a predicted value and a residual.

We're going to use the observed, predicted, and residual values to assess and improve the model.

Understanding Accuracy with Observed vs Predicted

In a simple model like this, with only two variables, you can get a sense of how accurate the model is just by relating Temperature to Revenue. Here's the same regression run on two different lemonade stands, one where the model is very accurate, one where the model is not:

It's clear that for both lemonade stands, a higher Temperature is associated with higher Revenue. But at a given Temperature, you could forecast the Revenue of the left lemonade stand much more accurately than the right lemonade stand, which means the model is much more accurate.

But most models have more than one explanatory variable, and it's not practical to represent more variables in a chart like that. So instead, let's plot the predicted values versus the observed values for these same datasets.

Again, the model for the chart on the left is very accurate, there's a strong correlation between the model's predictions and its actual results. The model for the chart on the far right is the opposite, the model's predictions aren't very good at all.

Note that these charts look just like the Temperature vs. Revenue charts above them, but the x-axis is predicted Revenue instead of Temperature. That's common when your regression equation only has one explanatory variable. More often, though, you'll have multiple explanatory variables, and these charts will look quite different from a plot of any one explanatory variable vs Revenue.

Examining Predicted vs Residual ("The residual plot")

The most useful way to plot the residuals, though, is with your predicted values on the x-axis, and your residuals on the y-axis.

(Statwing presents residuals as standardized residuals which means every residual plot you look at with any model is on the same standardized y-axis; more details for the curious.)

In this plot (on the right) each point is one day, where the prediction made by the model is on the x-axis, and the accuracy of the prediction is on the y-axis. The distance from the line at 0 is how bad the prediction was for that value.

Since...

Residual = Observed - Predicted

...positive values for the residual (on the y-axis) mean the prediction was too low, and negative values mean the prediction was too high; 0 means the guess was exactly correct.

Ideally your plot of the residuals look like one of these:

That is,
(1) they're pretty symmetrically distributed, tending to cluster towards the middle of the plot
(2) they're clustered around the lower single digits of the y-axis (e.g., 0.5 or 1.5, not 30 or 150)
(3) in general there aren't clear patterns

Here's some residual plots that don't meet those requirements:

These plots aren't evenly distributed vertically, or they have an outlier, or they have a clear shape to them.

If you can detect a clear pattern or trend in your residuals, then your model has room for improvement.

In a second we'll break down why, and what to do about it. But first:

How much does this matter?

"Essentially, all models are wrong, but some are useful"

How concerned should you be if your model isn't perfect, if your residuals look a bit unhealthy? It's up to you.

If you're publishing your thesis in particle physics, you probably want to make sure your model is as accurate as humanly possible. If you're trying to run a quick and dirty analysis of your nephew's lemonade stand, a less-than-perfect model might be good enough to answer whatever questions you have (e.g., whether Temperature appears to affect Revenue).

Most of the time a decent model is better than none at all. So take your model, try to improve it, and then decide whether the accuracy is good enough to be useful for your purposes.
If yours looks like one of the below, click that residual to understand what's happening and learn how to fix it.
If you're not sure what a residual is, take five minutes to read the above, then come back here.

Below is a gallery of unhealthy residual plots. Your residual may look like one specific type from below, or some combination.
Show details about this plot, and how to fix it.
(Throughout we'll use a lemonade stand's Revenue vs. that day's Temperature as an example dataset.)

Y-axis Unbalanced

Problem

Imagine that for whatever reason, your lemonade stand typically has low revenue, but every once and a while you get very high-revenue days, such that Revenue looked like this...

...instead of something more symmetrical and bell-shaped like this...

So Temperature vs Revenue might look like this, with most of the data bunched at the bottom...

The black line represents the model equation, the model's prediction of the relationship between Temperature and Revenue. Look above at each prediction made by the black line for a given Temperature (e.g., at Temperature 30, Revenue is predicted to be about 20). You can see that the majority of dots are below the line (that is, the prediction was too high), but a few dots are very far above the line (that is, the prediction was far too low).

Translating that same data to the diagnostic plots, most of the equation's predictions are a bit too high, and then some would be way too low.

Implications

This almost always means your model can be made significantly more accurate. Most of the time you'll find that the model was directionally correct but pretty inaccurate relative to an improved version. It's not uncommon to fix an issue like this and consequently see the model's r-squared jump from 0.2 to 0.5 (on a 0 to 1 scale).

Show details about this plot, and how to fix it.

How to fix

Heteroscedasticity

Problem

These plots exhibit "heteroscedasticity", meaning that the residuals get larger as the prediction moves from small to large (or from large to small).

Imagine that on cold days, the amount of revenue is very consistent, but on hotter days sometimes revenue is very high, and sometimes it's very low.

You'd see plots like these:

Implications

This doesn't inherently create a problem, but it's often an indicator that your model can be improved.

The only exception here is that if your sample size is less than 250, and you can't fix the issue using the below, your p-values may be a bit be a bit higher or lower than they should be, so possibly a variable that is right on the border of significance may end up erroneously on the wrong side of that border. Your regression coefficients (the number of units Revenue changes when Temperature goes up one) will still be accurate, though.

Show details about this plot, and how to fix it.

How to Fix

Nonlinear

Problem

Imagine that it's hard to sell lemonade on cold days, easy to sell it on warm days, and hard to sell it on very hot days (maybe because no one leaves their house on very hot days).

That plot would look like this:

The model, represented by the line, is terrible. The predictions would be way off, meaning your model doesn't accurately represent the relationship between T emperature and R evenue.

Accordingly, residuals would look like this:

Implications

If your model is way off, as in the example above, your predictions will be pretty worthless (and you'll notice a very low r-squared, like the 0.027 r-squared for the above).

Other times a slightly suboptimal fit will still give you a good general sense of the relationship, even if it's not perfect, like the below:

That model looks pretty accurate. If you look closely (or if you look at the residuals) you can tell that there's a bit of a pattern here, that the dots are on a curve that the line doesn't quite match.

Does that matter? It's up to you. If you're getting a quick understanding of the relationship, your straight line is a pretty decent approximation. If you're going to use this model for prediction and not explanation, the most accurate possible model would probably account for that curve.

Show details about this plot, and how to fix it.

How to Fix

Outliers

Problem

What if one of your datapoints had a Temperature of 80 instead of the normal 20s and 30s? Your plots would look like this:

This regression has an outlying datapoint on an input variable, Temperature (outliers on an input variable are also known as "leverage points").

What if one of your datapoints had $160 in revenue instead of the normal $20 - $60? Your plots would look like this:

This regression has an outlying datapoint on an output variable, Revenue.

Implications

Statwing runs a type of regression that generally isn't affected by output outliers (like the day with $160 revenue) but is affected by input outliers (like a Temperature in the 80s). In the worst case, your model can pivot to try to get closer to that point at the expense of being close to all the others, and end up being just entirely wrong, like this:

The blue line is probably what you'd want your model to look like, and the red line is the model you might see if you have that outlier out at Temperature 80.

How to fix

It's possible that this is a measurement or data entry error, where the outlier is just wrong, in which case you should delete it.

It's possible that what appears to be just a couple outliers is in fact a power distribution. Consider transforming the variable if one of your variables has an asymmetric distribution (that is, it's not remotely bell-shaped).

If it is indeed a legitimate outlier, you should assess the impact of the outlier.

Show details about this plot, and how to fix it.

Large Y-axis Datapoints

Problem

Imagine that there are two competing lemonade stands nearby. Most of the time only one is operational, in which case your revenue is consistently good. Sometimes neither is active, and revenue soars; at other times, both are active, and revenue plummets.

Revenue vs Temperature might look like this...

...with that top row being days when no other stand shows up, and the bottom row being days when both other stands are in business.

That'd result in these residual plots

That is, there's quite a few datapoints on both sides of 0 that have residuals of 10 or higher, which is to say that the model was way off.

Now if you'd collected data every day for a variable called Number of active lemonade stands, you could add that variable to your model, and this problem would be fixed. But often you don't have the data you need (or even a guess as to what kind of variable you need).

Implications

Your model isn't worthless, but it's definitely not as good as if you had all the variables you needed. You could still use it, and you might say something like "This model is pretty accurate most of the time, but then every once and a while it's way off." Is that useful? Probably, but that's your decision, and it depends on what decisions you're trying to make based on your model.

Show details about this plot, and how to fix it.

How to fix

X-axis Unbalanced

Problem

Imagine that Revenue is driven by nearby Foot traffic, in addition to or instead of just Temperature. Imagine that for whatever reason, your lemonade stand typically has low revenue, but every once and a while you get extremely high-revenue days, such that your revenue looked like this...

...instead of something more symmetrical and bell-shaped like this:

So Foot traffic vs Revenue might look like this, with most of the data bunched on the left side:

The black line represents the model equation, the model's prediction of the relationship between Foot traffic and Revenue. You can see that the model can't really tell the difference between Foot traffic of 0 and of, say, 100, or 1,000; for each of those values it would predict revenue near $53.

Translating that same data to the diagnostic plots:

Implications

Sometimes there's actually nothing wrong with your model. In the above example, it's quite clear that this isn't a good model; but sometimes the residual plot is unbalanced and the model is quite good.

The only ways to tell are to (1) experiment with transforming your data and see if you can improve it and (2) look at the Predicted vs Actual plot and see if your prediction is wildly off for a lot of datapoints as in the above example (but unlike this below example:)

But while there's no explicit rule that says your residual can't be unbalanced and still be accurate, and indeed this model is quite accurate, it's more often the case that an x-axis unbalanced residual means your model can be made significantly more accurate. Most of the time you'll find that the model was directionally correct but pretty inaccurate relative to an improved version. It's not uncommon to fix an issue like this and consequently see the model's r-squared jump from 0.2 to 0.5 (on a 0 to 1 scale).

How to fix

The solution to this is almost always to transform your data, typically an explanatory variable. (Note that the example shown below will reference transforming your reponse variable, but the same process will be helpful here.)

It's also possible that your model lacks a variable.

Assessing the impact of an outlier

Let's assume that you have an outlying datapoint that is legitimate, not a measurement or data error. To decide how to move forward, you should assess the impact of the datapoint on the regression.
(Go back to the gallery of residuals)
The easiest way to do this is to note the coefficients of your current model, then filter out that datapoint from the regression. If the model doesn't change much, then you don't have much to worry about.

If that changes the model significantly, examine the model, and particularly Actual vs Predicted, and decide which one feels better to you. It's okay to ultimately discard the outlier as long as you can theoretically defend that, saying "In this case we're not interested in outliers, they're just not of interest," or "That was the day Uncle Jerry came buy and tipped me $100; that's not predictable, and it's not worth including in the model."

Transforming variables

Overview

The most common way to improve a model is to transform one or more variables, usually using a "log" transform.

Transforming a variable changes the shape of its distribution. Typically the best place to start is a variable that has an asymmetrical distribution, as opposed to a more symmetrical or bell-shaped distribution. So find a variable like this to transform:

In general, regression models work better with more symmetrical, bell-shaped curves. Try different kinds of transformations until you hit upon the one closest to that shape. It's often not possible to get close to that, but that's the goal. So let's say you take the square root of Revenue as an attempt to get to a more symmetrical shape, and your distribution looks like this:

That's good, but it's still a bit asymmetrical. Let's try taking the log of revenue instead, which yields this shape:

That's nice and symmetrical. You're probably going to get a better regression model with log(Revenue) instead of Revenue. Indeed, here's how your equation, your residuals, and your r-squared might change:

Statwing shows a small version of the variable's distribution inline with the regression equation:

Select the transformation fx button to the left of the variable...

...then select a transformation, most often log(x)...

...then examine the histogram to see if it's more centered, as this one is after transformation:

After transforming a variable, note how its distribution changes, the r-squared of the regression changes, and the patterns of the residual plot changes. If those improve (particularly the r-squared and the residuals), it's probably best to keep the transformation.

If a transformation is necessary, you should start by taking a log transformation because the results of your model will still be easy to understand. Note that you'll run into issues if the data you're trying to transform includes zeros or negative values, though. To learn why taking a log is so useful, or if you have non-positive numbers you want to transform, or if you just want to get a better understanding of what's happening when you transform data, read on through the details below.

Details

If you take the log ₁₀() of a number, you're saying "10 to what power gives me that number." For example, here's a simple table of four datapoints, including both Revenue and Log( Revenue)

Note that if we plot Temperature vs Revenue, and Temperature vs Log( Revenue), the latter model fits much better.

The interesting thing about this transformation is that your regression is no longer linear. When Temperature went from 20 to 30, Revenue went from 10 to 100, a 90-unit gap. Then when Temperature went from 30 to 40, Revenue went from 100 to 1000, a much larger gap.

If you've taken a log of your response variable, it's no longer the case that a one-unit increase in Temperature means a X-unit increase in Revenue, now it's a X-percent increase in Revenue (in this case, a ten-unit increase in Temperature is associated with a 1000% increase in Y-that is, a one-unit increase in Temperature is associated with a 26% increase in Revenue.
(Go back to the gallery of residuals)
Note also that you can't take the log of 0 or of a negative number (there is no X where 10 X = 0 or 10 X = -5), so if you do a log transform, you'll lose those datapoints from the regression. There's 4 common ways of handling the situation:

1. Take a square root, or a cube root. Those won't change the shape of the curve as dramatically as taking a log, but they allow 0s to remain in the regression.
2. If it's not too many rows of data that have a zero, and those rows aren't theoretically important, you can decide to go ahead with the log and lose a few rows from your regression.
3. Instead of taking log(y), take log(y+1), such that zeros become 1s and can then be kept in the regression. This biases your model a bit and is somewhat frowned upon, but in practice its negative side effects are typically pretty minor.

Missing variable

Probably the most common reason that a model fails to fit is that not all the right variables are included. This particular issue has a lot of possible solutions.

Adding a New Variable

Sometimes the fix is as easy as adding another variable to the model. For example if lemonade stand Revenue was much larger traffic was much larger on weekends than week days, your Predicted vs Actual plot might look like the below (r-squared of 0.053), since the model is just taking the average of weekend days and weekdays:

If the model includes a variable called Weekend?, then the Predicted vs Actual plot might look like this (r-squared of 0.974):

The model makes far more accurate predictions because it's able to take into account whether a day of the week is a weekday or not.

Note that sometimes you'll need to create variables in Statwing to improve your model in this fashion. For example, you might have had a Date variable (with values like "10/26/2014") and you might need to create a new variable called Day of Week ("Sunday") or Weekend? ("Weekend").

Unavailable Omitted Variable

It's rarely that easy, though. Quite frequently the relevant variable isn't available because you don't know what it is, or it was difficult to collect. Maybe it wasn't a weekend-vs-weekday issue, but instead something like Number of Competitors in the Area that you failed to collect at the time.

If the variable you ned is unavailable, or you don't even know what it would be, then your model can't really be improved, and you have to assess it and decide how happy you are with it, whether it's useful or not even though it's flawed. LINK TO ASSESSING THE MODEL

Interactions Between Variables
(Go back to the gallery of residuals)
Perhaps on weekends the lemonade stand is always selling at 100% of capacity, so regardless of the temperature, revenue is high. But on weekdays, the lemonade stand is much less busy, so Temperature is an important driver of Revenue. If you ran a regression that included Weekend? and Temperature, you might see a Predicted vs Actual plot like this, where the row along the top are the weekend days.

We would say that there's an "interaction" between Weekend? and Temperature, the effect of one of them on Revenue is different based on the value of the other. If we create an interaction variable, we get a much better model, where Predicted vs Actual looks like this:

Fixing Nonlinearity

Let's say you have a relationship that looks like this:

You might notice that the shape there is that of a parabola, which you might recall is typically associated with formulas that look like this:

By default regression uses a linear model that looks like this:

y = x + 1

In fact the line in the plot above has this formula:

y = 1.7x + 51

But it's a terrible fit. So if we add an x ² term, our model has a better chance of fitting the curve. In fact, it creates this:

The formula for that curve is:

y = -2x² +111x - 1408

That means our diagnostic plots change from this...

...to this...

Note that these are healthy diagnostic plots, even though the data appears to be unbalanced to the right side of it. (Learn more about that in the X-axis Unbalanced section.)

The above approach can be extended to other kinds of shapes, particularly an S-shaped curve, by adding an x ³ term. That's relatively uncommon, though.
(Go back to the gallery of residuals)
A few cautions:

Generally speaking if you have an x² term because of a nonlinear pattern in your data, you want to have an plain-old-x-not-x² term. You may find that your model is perfectly good without it, but you should definitely try both to start.

The regression equation may be difficult to understand. For the linear equation at the beginning of this section, for each additional unit of Temperature, , Revenue went up 1.7 units. When you have both x² and x in the equation, it's not easy to say "When Temperature goes up one degree, here's what happens." Sometimes for that reason it's easier to just use a linear equation, assuming that equation fits well enough, as in this example.

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Interpreting residual plots to improve your regression

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When you run a regression, Statwing automatically calculates and plots residuals to help you understand and improve your regression model. Read below to learn everything you need to know about interpreting residuals (including definitions and examples). To demonstrate how to interpret residuals, we'll use a lemonade stand dataset, where each row was a day of Temperature and Revenue.

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extrapolation

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xkcd.com is best viewed with Netscape Navigator 4.0 or below on a Pentium 3±1 emulated in Javascript on an Apple IIGS at a screen resolution of 1024x1. Please enable your ad blockers, disable high-heat drying, and remove your device from Airplane Mode and set it to Boat Mode.

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Not reliable because it is predicting outside the existing data

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Interpolation is INside the data, extrapolation is EXternal to the data

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Video with examples

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Definition of coefficient of determination, from the Stat Trek dictionary of statistical terms and concepts. This statistics glossary includes definitions of all technical terms used on Stat Trek website.

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Video

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In this video we discuss one of the best methods psychologists have for predicting behaviors, the correlation. But does that mean that a behavior is absolutely going to happen? Let's find out!

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Bivariate Data Scatter Plots and Associations

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Using a scatter plot to look for correlation between two variables. Calculating the correlation coefficient. Calculating and drawing the linear regression line. Making predictions using interpolation.

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Useful video by Udacity

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Normal distribution - Intro to Psychology

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Only used for normal distributions

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Tutorial

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Want to do some practice questions now? Visit http://mathsonline.com.au to access a bank of questions. You can also access the maths lesson summary and worksheet. If you have any questions just email us info@mathsonline.com.au PH: 1300 88 20 45 Online math program online math tutor math tuition

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rand(1,6) to get a number between 1 and 6. And randlist(1, 6, 3) to get three numbers between 1 and 6.

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Forgot your CAS? Use this site.

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Distilling the world's data, information & knowledge into beautiful infographics & visualizations