# Time Series Analysis Using Max/Min... and some Neuroscience.

### Introduction

Time series have **maximum** and **minimum** points as general patterns. Sometimes the noise present on it causes problems to spot general behavior.

In this post, we will **smooth** time series -reducing noise- to maximize the story that data has to tell us. And then, an easy formula will be applied to find and plot max/min points thus characterize data.

### What we have

```
# reading data sources, 2 time series
t1=read.csv("ts_1.txt")
t2=read.csv("ts_2.txt")
# plotting...
plot(t1$ts1, type = 'l')
plot(t2$ts2, type = 'l')
```

As you can see there are many peaks, but intuitively you can imagine a more smoother line crossing in the middle of the points. This can achieved by applying a **Seasonal Trend Decomposition** (STL).

### Smoothing the series

```
# first create the time series object, with frequency = 50, and then apply the stl function.
stl_1=stl(ts(t1$ts1, frequency=50), "periodic")
stl_2=stl(ts(t2$ts2, frequency=50), "periodic")
```

*Important*: If you don't know the `frequency`

beforehand, play a little bit with this parameter until you find a result in which you are comfortable.

### Finding max and min

Creating the functions...

```
ts_max<-function(signal)
{
points_max=which(diff(sign(diff(signal)))==-2)+1
return(points_max)
}
ts_min<-function(signal)
{
points_min=which(diff(sign(diff(-signal)))==-2)+1
return(points_min)
}
```

### Visualizing the results!

```
trend_1=as.numeric(stl_1$time.series[,2])
max_1=ts_max(trend_1)
min_1=ts_min(trend_1)
## Plotting final results
plot(trend_1, type = 'l')
abline(v=max_1, col="red")
abline(v=min_1, col="blue")
```

With the line: `stl_1$time.series[,2]`

we are accessing the time series `trend`

component. This is the smoothing method we will use, but there are others.

This first series has 3 maximums *(red line)* and 2 minimums *(blue line)* in the following places:

```
# When the max points occurs:
max_1
```

```
# When the min points occurs:
min_1
```

### Comparing two time series

```
trend_2=as.numeric(stl_2$time.series[,2])
max_2=ts_max(trend_2)
min_2=ts_min(trend_2)
# create two aligned plots
par(mfrow=c(2,1))
## Plotting series 1
plot(trend_1, type = 'l')
abline(v=max_1, col="red")
abline(v=min_1, col="blue")
## Plotting series 2
plot(trend_2, type = 'l')
abline(v=max_2, col="red")
abline(v=min_2, col="blue")
```

Some **conclusions** from both plots:

`Series 2`

starts with a`min`

while 1 does with a`max`

`Series 1`

has 3`max`

and 2`min`

, just the opposite to the other series

Why is this important? Because of the nature of the data, which is in next section.

### What is this data about?

`ts1`

and `ts2`

are two typical responses to a brain stimulus, in other words: what happens with the brain when a person looks at a picture / move a finger / think in a particular thing, etc... Electroencephalography.

Some studies in **neuroscience** focus on averaging several responses to one stimulus -for example, to look at one particular picture. They present several times a particular image to the person. Averaging all of these signal/time series, you get the **typical response**.

Then you can **predict** based on the similarity between this **typical response** and the **new image** (stimulus) that the person is looking at.

#### Typical response (or Event Related Potential)

It's important to get the **when** the positive peaks occur. In this case they are: `P1`

, `P2`

and `P3`

. The same goes for the negative ones.

Wiki: Event related potential.

*Note: It´s a common practice to invert negative and positive values.*

#### Finally...

Typically the signal time length for this kind of studies last for **400ms**, thus 1 point per millisecond, just the displayed plots. And the amplitude is in **volts**, *(actually micro-volts)*. The same unit of measurement used by the notebook you are using now ;)