# perObservationLoss

Per observation regression or classification error of incremental drift-aware learner

*Since R2022b*

## Description

returns the per observation regression or classification error for model
`Err`

= perObservationLoss(`Mdl`

,`X`

,`Y`

)`Mdl`

trained using the predictors in `X`

and true
observed values in `Y`

.

`Err`

is an *n*-by-1 vector, where
*n* is the number of observations.

The `fit`

function
internally calls `perObservationLoss`

during incremental drift-aware learning.
`perObservationLoss`

is suitable for using in custom workflows.

specifies additional options using one or more name-value arguments. For example, you can
specify the dimension of the predictor data and the loss function to compute.`Err`

= perObservationLoss(`Mdl`

,`X`

,`Y`

,`Name=Value`

)

## Examples

### Measure and Visualize Model Performance During Incremental Learning

Load the human activity dataset. Randomly shuffle the data.

load humanactivity; n = numel(actid); rng(123) % For reproducibility idx = randsample(n,n);

For details on the data set, enter `Description`

at the command line.

Define the predictor and response variables.

X = feat(idx,:); Y = actid(idx);

Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing.

Dichotomize the response by identifying whether the subject is moving (actid > 2).

Y = Y > 2;

Flip labels for the second half of the dataset to simulate drift.

Y(floor(numel(Y)/2):end,:) = ~Y(floor(numel(Y)/2):end,:);

Initiate a default incremental drift-aware model for classification as follows:

Create a default incremental linear SVM model for binary classification.

Initiate a default incremental drift-aware model using the incremental linear SVM model as the base learner.

BaseLearner = incrementalClassificationLinear(); idaMdl = incrementalDriftAwareLearner(BaseLearner);

`idaMdl`

is an `incrementalDriftAwareLearner`

model. All its properties are read-only.

Preallocate the number of variables in each chunk for creating a stream of data and the variable to store the classification error.

numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ce = array2table(zeros(nchunk,3),VariableNames=["Cumulative" "Window" "Loss"]); PoL = zeros(nchunk,numObsPerChunk); % To store per observation loss values driftTimes = [];

Simulate a data stream with incoming chunks of 50 observations each. At each iteration:

Call

`updateMetrics`

to measure the cumulative performance and the performance within a window of observations. Overwrite the previous incremental model with a new one to track performance metrics.Call

`fit`

to fit the model to the incoming chunk. Overwrite the previous incremental model with a new one fitted to the incoming observations.Call

`perObservationLoss`

to compute classification error on each observation in the incoming chunk of data.Call

`loss`

to measure the model performance on the incoming chunk.Store all performance metrics in

`ce`

to see how they evolve during incremental learning. The`Metrics`

property of`idaMdl`

stores the cumulative and window classification error, which is updated at each iteration. Store the loss values for each chunk in the third column of`ce`

.

for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1)+1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; idaMdl = updateMetrics(idaMdl,X(idx,:),Y(idx)); idaMdl = fit(idaMdl,X(idx,:),Y(idx)); PoL(j,:) = perObservationLoss(idaMdl,X(idx,:),Y(idx)); ce{j,["Cumulative" "Window"]} = idaMdl.Metrics{"ClassificationError",:}; ce{j,"Loss"} = loss(idaMdl,X(idx,:),Y(idx)); if idaMdl.DriftDetected driftTimes(end+1) = j; end end

The `updateMetrics`

function evaluates the performance of the model as it processes incoming observations. The function writes specified metrics, measured cumulatively and within a specified window of processed observations, to the `Metrics`

model property. The fit function fits the model by updating the base learner and monitoring for drift given an incoming batch of data.

Plot the cumulative and per window classification error. Mark the warmup and training periods, and where the drift was introduced.

h = plot(ce.Variables); xlim([0 nchunk]) ylim([0 0.07]) ylabel("Classification Error") xlabel("Iteration") xline(idaMdl.MetricsWarmupPeriod/numObsPerChunk,"g-.","Warmup Period",LineWidth= 1.5) xline(idaMdl.TrainingPeriod/numObsPerChunk,"b-.","Training Period",LabelVerticalAlignment="middle",LineWidth= 1.5) %xline(floor(numel(Y)/2)/numObsPerChunk,"m--","Drift",LabelVerticalAlignment="middle",LineWidth= 1.5) xline(driftTimes,"m--","Drift",LabelVerticalAlignment="middle",LineWidth=1.5) legend(h,ce.Properties.VariableNames) legend(h,Location="best")

The yellow line represents the classification error on each incoming chunk of data. `loss`

is agnostic of the metrics warm-up period, so it measures the classification error for all iterations. After the metrics warm-up period, `idaMdl`

tracks the cumulative and window metrics.

Plot the per observation loss.

```
figure()
plot(PoL,'b.');
```

`perObservationLoss`

computes the classification loss for each observation in the incoming chunk of data.

### Compute Custom Loss on Incoming Chunks of Data

Create the random concept data and the concept drift generator using the helper functions `HelperRegrGenerator`

and `HelperConceptDriftGenerator`

, respectively.

concept1 = HelperRegrGenerator(NumFeatures=100,NonZeroFeatures=[1,20,40,50,55], ... FeatureCoefficients=[4,5,10,-2,-6],NoiseStd=1.1); concept2 = HelperRegrGenerator(NumFeatures=100,NonZeroFeatures=[1,20,40,50,55], ... FeatureCoefficients=[4,7,10,-1,-5],NoiseStd=1.1); driftGenerator = HelperConceptDriftGenerator(concept1,concept2,15000,1250);

`HelperRegrGenerator`

generates streaming data using features and feature coefficients for regression specified in the call to the function. At each step, the function samples the predictors from a normal distribution. Then, it computes the response using the feature coefficients and predictor values and adding a random noise from a normal distribution with mean zero and specified noise standard deviation.

`HelperConceptDriftGenerator`

establishes the concept drift. The object uses a sigmoid function `1./(1+exp(-4*(numobservations-position)./width))`

to decide the probability of choosing the first stream when generating data [3]. In this case, the position argument is 15000 and the width argument is 1250. As the number of observations exceeds the position value minus half of the width, the probability of sampling from the first stream when generating data decreases. The sigmoid function allows a smooth transition from one stream to the other. Larger width values indicate a larger transition period where both streams are approximately equally likely to be selected.

Configure an incremental drift-aware model for regression as follows:

Create an incremental linear model for regression: Track the mean absolute deviation (MAD) to measure the performance of the model. Create an anonymous function that measures the absolute error of each new observation. Create a structure array containing the name

`MeanAbsoluteError`

and its corresponding function. Specify a metrics warm-up period of 1000 observations. Specify a metrics window size of 500 observations.Initiate an incremental concept drift detector for continuous data. Use the Hoeffding's Bounds Drift Detection Method with moving average (HDDMA).

Using the incremental linear model and the concept drift detector, instantiate an incremental drift-aware model. Specify the training period as 1000 observations.

maefcn = @(z,zfit,w)(abs(z - zfit)); % Mean absolute deviation function maemetric = struct(MeanAbsoluteError=maefcn); baseMdl = incrementalRegressionLinear(MetricsWarmupPeriod=1000,MetricsWindowSize=400,Metrics=maemetric,EstimationPeriod=0); dd = incrementalConceptDriftDetector("hddma",Alternative="greater",InputType="continuous"); idaMdl = incrementalDriftAwareLearner(baseMdl,DriftDetector=dd,TrainingPeriod=2000);

Generate an initial sample of 20 observations and configure the model to predict responses by fitting it to the initial sample.

```
initobs = 20;
rng(1234); % For reproducibility
[driftGenerator,X,Y] = hgenerate(driftGenerator,initobs);
idaMdl = fit(idaMdl,X,Y);
```

Preallocate the number of variables in each chunk and number of iterations for creating a stream of data, the variables to store the classification error, drift status, and drift time(s).

numObsPerChunk = 50; numIterations = 500; mae = array2table(zeros(numIterations,3),VariableNames=["Cumulative" "Window" "Chunk"]); PoL = zeros(numIterations,numObsPerChunk); % Per observation loss values driftTimes = []; dstatus = zeros(numIterations,1); statusname = strings(numIterations,1);

Simulate a data stream with incoming chunks of 50 observations each and perform incremental drift-aware learning. At each iteration:

Simulate predictor data and labels, and update the drift generator using the helper function

`hgenerate`

.Call

`updateMetrics`

to compute cumulative and window metrics on the incoming chunk of data. Overwrite the previous incremental model with a new one fitted to overwrite the previous metrics.Call

`loss`

to compute the MAD on the incoming chunk of data. Whereas the cumulative and window metrics require that custom losses return the loss for each observation,`loss`

requires the loss on the entire chunk. Compute the mean of the absolute deviation.Call

`perObservationLoss`

to compute the per observation regression error.Call

`fit`

to fit the incremental model to the incoming chunk of data.Store the cumulative, window, and chunk metrics and per observation loss to see how they evolve during incremental learning.

for j = 1:numIterations % Generate data [driftGenerator,X,Y] = hgenerate(driftGenerator,numObsPerChunk); % Perform incremental fitting and store performance metrics idaMdl = updateMetrics(idaMdl,X,Y); PoL(j,:) = perObservationLoss(idaMdl,X,Y,'LossFun',@(x,y,w)(maefcn(x,y))); mae{j,1:2} = idaMdl.Metrics{"MeanAbsoluteError",:}; mae{j,3} = loss(idaMdl,X,Y,LossFun=@(x,y,w)mean(maefcn(x,y,w))); idaMdl = fit(idaMdl,X,Y); statusname(j) = string(idaMdl.DriftStatus); if idaMdl.DriftDetected driftTimes(end+1) = j; dstatus(j) = 2; elseif idaMdl.WarningDetected dstatus(j) = 1; else dstatus(j) = 0; end end

`idaMdl`

is an `incrementalDriftAwareLearner`

model object trained on all the data in the stream. During incremental learning and after the model is warm, `updateMetrics`

checks the performance of the model on the incoming observations, and the `fit`

function fits the model to those observations.

Plot the drift status.

gscatter(1:numIterations,dstatus,statusname,'gmr','*',4,'on',"Iteration","Drift Status") xlim([0 numIterations])

Plot the performance metrics to see how they evolved during incremental learning.

figure h = plot(mae.Variables); xlim([0 numIterations]) ylim([0 4]) ylabel("Mean Absolute Deviation") xlabel("Iteration") xline(idaMdl.MetricsWarmupPeriod/numObsPerChunk,"g-.","Warmup Period",LineWidth= 1.5) xline(idaMdl.TrainingPeriod/numObsPerChunk,"b-.","Training Period",LabelVerticalAlignment="middle",LineWidth= 1.5) xline(driftTimes,"m--","Drift",LabelVerticalAlignment="middle",LineWidth=1.5) legend(h,mae.Properties.VariableNames)

The plot suggests the following:

`updateMetrics`

computes the performance metrics after the metrics warm-up period only.`updateMetrics`

computes the cumulative metrics during each iteration.`updateMetrics`

computes the window metrics after processing 400 observationsBecause

`idaMdl`

was configured to predict observations from the beginning of incremental learning,`loss`

can compute the MAD on each incoming chunk of data.

Plot the per observation loss.

```
figure()
plot(PoL,'b.');
```

`perObservationLoss`

computes the classification loss for each observation in the incoming chunk of data after the metrics warm-up period.

## Input Arguments

`Mdl`

— Incremental drift-aware learning model

`incrementalDriftAwareLearner`

model object

Incremental drift-aware learning model fit to streaming data, specified as an `incrementalDriftAwareLearner`

model object. You can create
`Mdl`

using the `incrementalDriftAwareLearner`

function. For more details, see the object reference page.

`X`

— Chunk of predictor data

floating-point matrix

Chunk of predictor data used to compute per observation loss, specified as a
floating-point matrix of *n* observations and
`Mdl.BaseLearner.NumPredictors`

predictor variables.

When `Mdl.BaseLearner`

accepts the
`ObservationsIn`

name-value argument, the value of
`ObservationsIn`

determines the orientation of the variables and
observations. The default `ObservationsIn`

value is
`"rows"`

, which indicates that observations in the predictor data are
oriented along the rows of `X`

.

The length of the observation responses (or labels) `Y`

and the
number of observations in `X`

must be equal;
`Y(`

is the response (or label) of
observation * j*)

*j*(row or column) in

`X`

.**Note**

`perObservationLoss`

supports only floating-point
input predictor data. If your input data includes categorical data, you must prepare an encoded
version of the categorical data. Use `dummyvar`

to convert each categorical variable
to a numeric matrix of dummy variables. Then, concatenate all dummy variable matrices and any
other numeric predictors. For more details, see Dummy Variables.

**Data Types: **`single`

| `double`

`Y`

— Chunk of responses or labels

floating-point vector | categorical array | character array | string array | logical vector | cell array of character vectors

Chunk of responses or labels used to compute per observation loss, specified as one of the following.

Floating-point vector of

*n*elements for regression models, where*n*is the number of rows in`X`

.Categorical, character, or string array, logical vector, or cell array of character vectors for classification models. If

`Y`

is a character array, it must have one class label per row. Otherwise, it must be a vector with*n*elements.

The length of `Y`

and the number of observations in
`X`

must be equal; `Y(`

is the response (or label) of observation * j*)

*j*(row or column) in

`X`

.For classification problems:

If

`Y`

contains a label that is not a member of`Mdl.BaseLearner.ClassNames`

,`perObservationLoss`

issues an error.The data type of

`Y`

and`Mdl.BaseLearner.ClassNames`

must be the same.

**Data Types: **`single`

| `double`

| `categorical`

| `char`

| `string`

| `logical`

| `cell`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

**Example: **`ObservationsIn="columns",LossFun="hinge"`

specifies that the
observations are in columns and the loss function is the built-in hinge loss.

`ObservationsIn`

— Orientation of data in `X`

`"rows"`

(default) | `"columns"`

Orientation of the data in `X`

, specified as
`"rows"`

or `"columns"`

.
`perObservationLoss`

supports `ObservationsIn`

only
if `Mdl.BaseLearner`

supports the `ObservationsIn`

name-value argument.

**Example: **`ObservationsIn="columns"`

**Data Types: **`char`

| `string`

`LossFun`

— Loss function

`"squarederror"`

| `"epsiloninsensitive"`

| `"classiferror"`

| `"binodeviance"`

| `"exponential"`

| `"hinge"`

| `"logit"`

| `"quadratic"`

| `"mincost"`

| function handle

Loss function, specified as a built-in loss function name or a function handle.

The following table lists the built-in loss function names.

For Regression Models:

Name Description `"squarederror"`

Squared error `"epsiloninsensitive"`

Epsilon-insensitive error Default is

`"squarederror"`

for regression models.For Classification Models:

Name Description `"binodeviance"`

Binomial deviance `"classiferror"`

Misclassification error rate `"exponential"`

Exponential `"hinge"`

Hinge `"logit"`

Logistic `"quadratic"`

Quadratic `"mincost"`

Minimal expected misclassification cost (for `incrementalClassificationNaiveBayes`

only)Default is

`"mincost"`

for`incrementalClassificationNaiveBayes`

model object and`"classiferror"`

for other classification objects.**Note**You can only specify

`"classiferror"`

for`incrementalClassificationECOC`

.

To specify a custom loss function, use function handle notation. The function must have one of these forms:

For Regression Models:

lossval =

*lossfcn*(Y,YFit)The output argument

`lossval`

is a floating-point scalar.You specify the function name (

).`lossfcn`

`Y`

is a length*n*numeric vector of observed responses.`YFit`

is a length*n*numeric vector of corresponding predicted responses.

For Classification Models:

lossval =

*lossfcn*(C,S)The output argument

`lossval`

is an*n*-by-1 floating-point vector, where*n*is the number of observations in`X`

. The value in`lossval(`

is the classification loss of observation)`j`

.`j`

You specify the function name (

).`lossfcn`

`C`

is an*n*-by-*K*logical matrix with rows indicating the class to which the corresponding observation belongs.`K`

is the number of distinct classes (`numel(Mdl.BaseLearner.ClassNames)`

), and the column order corresponds to the class order in the`Mdl.BaseLearner.ClassNames`

property. Create`C`

by setting`C(`

=,`p`

)`q`

`1`

, if observation

is in class`p`

, for each observation in the specified data. Set the other elements in row`q`

to`p`

`0`

.`S`

is an*n*-by-*K*numeric matrix of predicted classification scores.`S`

is similar to the`Score`

output of`predict`

, where rows correspond to observations in the data and the column order corresponds to the class order in the`Mdl.BaseLearner.ClassNames`

property.`S(`

is the classification score of observation,`p`

)`q`

being classified in class`p`

.`q`

**Example: **`LossFun="logit"`

**Example: **`LossFun=@`

`lossfcn`

**Data Types: **`char`

| `string`

| `function_handle`

## References

[1] Barros, Roberto S.M. , et al.
"RDDM: Reactive drift detection method." *Expert Systems with
Applications*. vol. 90, Dec. 2017, pp. 344-55. https://doi.org/10.1016/j.eswa.2017.08.023.

[2] Bifet, Albert, et al. "New
Ensemble Methods for Evolving Data Streams." *Proceedings of the 15th ACM SIGKDD
International Conference on Knowledge Discovery and Data Mining*. ACM Press,
2009, p. 139. https://doi.org/10.1145/1557019.1557041.

[3] Gama, João, et al. "Learning with
drift detection". *Advances in Artificial Intelligence – SBIA 2004*, edited
by Ana L. C. Bazzan and Sofiane Labidi, vol. 3171, Springer Berlin Heidelberg, 2004, pp. 286–95.
https://doi.org/10.1007/978-3-540-28645-5_29.

## Version History

**Introduced in R2022b**

## See Also

`fit`

| `incrementalDriftAwareLearner`

| `loss`

| `updateMetrics`

| `updateMetricsAndFit`

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