# Fit an Autoregression Model to the Tidal Depth Data

This example shows how to fit an autoregression (AR) model for data in your ThingSpeak™ channel and calculate the regression parameters along with their uncertainties. Autoregression models are used to represent a time-dependent process in nature.

### Read Data from the Ockway Bay Real-Time Tide Gauge

ThingSpeak channel 50289 contains data about tidal depth at Ockway Bay. The data is collected once every 5 minutes. Field 1 of the channel contains tidal depth data.

```% Read the data using the |thingSpeakRead| function from channel 50289 on a particular day, for example, July 01, 2016. ```
```startDate = datetime('July 1, 2016 12:01:00 AM'); endDate = datetime('July 2, 2016 12:01:00 AM'); dateRange = startDate:endDate; [data,timestamps] = thingSpeakRead(50289,'DateRange',dateRange,'Fields',1); ```

### Fit an AR Model to the Data

Use the `iddata` function to create an iddata object of the tidal depth data. Use `detrend` to ensure the data has a zero mean.

```sampleTime = 5; IDdata = iddata(data,[],sampleTime,'OutputName',{'Tidal Depth'},'TimeUnit','minutes') IDdata = detrend(IDdata,0); ```
```IDdata = Time domain data set with 288 samples. Sample time: 5 minutes Outputs Unit (if specified) Tidal Depth ```

### Fit Model to Data

Since the tidal depth varies with time, use the `ar` function to fit a discrete-time autoregressive model to the data.

```modelOrder = 8; sys = ar(IDdata,modelOrder) ```
```sys = Discrete-time AR model: A(z)y(t) = e(t) A(z) = 1 - 1.154 z^-1 - 0.1668 z^-2 + 0.2144 z^-3 + 0.2974 z^-4 - 0.4227 z^-5 + 0.1509 z^-6 - 0.1612 z^-7 + 0.2491 z^-8 Sample time: 5 minutes Parameterization: Polynomial orders: na=8 Number of free coefficients: 8 Use "polydata", "getpvec", "getcov" for parameters and their uncertainties. Status: Estimated using AR ('fb/now') on time domain data "IDdata". Fit to estimation data: 98.5% FPE: 0.04741, MSE: 0.04485 ```

### Show Parameters

Use the `getpvec` function to show the estimated parameters along with their uncertainties.

```[Parameters,Uncertainties] = getpvec(sys) ```
```Parameters = -1.1543 -0.1668 0.2144 0.2974 -0.4227 0.1509 -0.1612 0.2491 Uncertainties = 0.0580 0.0918 0.0932 0.0918 0.0921 0.0970 0.0962 0.0647 ```

The output shows the estimated AR model parameters and the one standard deviation value of the estimated parameters.

### Write Parameters to ThingSpeak

Use the `thingSpeakWrite` function to write the array of values to ThingSpeak, with one vaue per field. Transpose the data to be 8 x 1. Change the `channelID` and the `writeAPIKey` to send data to your channel.

```channelID=17504; writeAPIKey='23ZLGOBBU9TWHG2H'; response = thingSpeakWrite(channelID,'Values',Parameters','WriteKey',writeAPIKey) ```
```response = struct with fields: Field1: '-1.154266029802091' Field2: '-0.1668388400729965' Field3: '0.2143807521019717' Field4: '0.2973816840220466' Field5: '-0.4226981725238166' Field6: '0.1509427726183032' Field7: '-0.1612303290788889' Field8: '0.2490548535561231' Latitude: [] Longitude: [] ChannelID: 17504 Created: 10-Jan-2019 15:10:41 LastEntryID: 20736 Altitude: [] ```