Zaman Serisi Analizi: Trend, Mevsimsellik ve Tahminleme

What is time series analysis?

A time series consists of values observed in order at specific points or intervals in time. Time series analysis is the process of examining the structure, dependencies and change in those observations in order to explain, monitor or forecast.

Daily sales, hourly electricity consumption, monthly inflation, website visits, call centre demand and social media mention volume are all time series. The order of the observations matters here: today's sales are rarely independent of yesterday's, or of the day of the week.

The components of a time series

To understand a series, it helps to think about which patterns the observed value may be made of. Classical decomposition considers trend, seasonality and an irregular component; in some cases cyclical movements are treated separately.

Additive and multiplicative structure

In an additive model the components sum to the observation. In a multiplicative model the size of the seasonal swing grows with the level of the series.

# Additive decomposition
Y[t] = Trend[t] + Seasonal[t] + Remainder[t]

# Multiplicative decomposition
Y[t] = Trend[t] * Seasonal[t] * Remainder[t]

If the extra summer sales grow in absolute terms as sales rise overall, a multiplicative structure or a log transform makes sense. If the series contains zeros or negative values, the multiplicative approach cannot be applied directly — the logarithm is undefined.

How do you analyse trend?

Trend shows the long-term direction of the series. But not every rise means lasting growth; calendar effects, campaigns, measurement changes and outliers easily distort the impression of a trend.

Moving average

A moving average smooths short-term fluctuation so the broader tendency becomes visible. On daily sales, a 7-day moving average largely removes day-of-week variation.

import pandas as pd
import matplotlib.pyplot as plt

df = pd.read_csv("gunluk_satis.csv", parse_dates=["tarih"])
df = df.sort_values("tarih").set_index("tarih")

df["ortalama_7g"] = df["satis"].rolling(7, min_periods=7).mean()

df[["satis", "ortalama_7g"]].plot()
plt.title("Daily sales and 7-day moving average")
plt.tight_layout()
plt.show()

A moving average is not a forecasting model; it is a visualisation and smoothing tool. The wider the window, the smoother the series — and the slower it reacts to sudden change. Writing min_periods=7 prevents misleading values computed from a half-filled window over the first six days.

Linear trend

Simple linear regression works when the relationship between time and value is roughly linear. In real series the trend need not be linear; breaks and saturation are common.

import numpy as np
from sklearn.linear_model import LinearRegression

y = df["satis"].dropna()
t = np.arange(len(y)).reshape(-1, 1)

model = LinearRegression().fit(t, y)
trend = model.predict(t)

This example assumes observations arrive at equal intervals. If the calendar has gaps, the index produced by np.arange no longer represents real time; you then need to model the actual time index or handle the missing periods explicitly.

How do you analyse seasonality?

Seasonality is a pattern that repeats over a particular calendar cycle. Daily data shows weekly seasonality, monthly data yearly seasonality. What separates it from a cycle is that seasonality has a known, fixed period.

Examples: weekend peaks in restaurants, winter rises in heating consumption, the summer season in tourism, year-end campaigns in retail. When examining seasonal effects, calendar effects such as public holidays, moving religious holidays and campaigns need separate thought — in Turkish data, holidays shifting every year make the fixed-period assumption insufficient on its own.

Decomposition with STL

STL (Seasonal-Trend decomposition using Loess) is a flexible method for splitting a series into trend, seasonal and remainder components. It suits cases where the seasonal pattern changes over time better than classical decomposition.

from statsmodels.tsa.seasonal import STL

# Weekly seasonality on a daily series: period=7
sonuc = STL(df["satis"].asfreq("D").interpolate(),
            period=7, robust=True).fit()

sonuc.plot()
plt.tight_layout()
plt.show()

Here period=7 assumes weekly seasonality on a daily series; for a yearly cycle in monthly data you would use period=12. robust=True limits how much outliers distort the seasonal component. The interpolate() call is for demonstration only: in production the reason for missing time points and the appropriate filling method need separate consideration.

What is stationarity and why does it matter?

A time series is considered stationary if its statistical properties are roughly unchanging over time. Under weak stationarity, the mean and variance are expected to be constant and the covariance to depend only on the lag between two observations.

Series with a clear trend or a shifting seasonal level are not stationary. Stationarity matters particularly for the ARMA family; not every forecasting method demands a stationary input in the same way. Exponential smoothing and tree-based models do not require the assumption directly.

Assessing stationarity

• Look at the time plot to see whether the level and variance change over time.
• Examine the behaviour of correlation across lags with an ACF plot.
• Use tests such as ADF as supporting evidence; do not decide on a p-value alone.

from statsmodels.tsa.stattools import adfuller

seri = df["satis"].dropna()
adf_sonuc = adfuller(seri, autolag="AIC")

print("ADF statistic:", adf_sonuc[0])
print("p-value:", adf_sonuc[1])

The null hypothesis of the ADF test is that a unit root is present. A small p-value provides evidence against that null under the test's assumptions — that is, it weakens the claim that the series is not stationary. The test is affected by breaks, seasonality and sample length, so read it together with the plot and domain knowledge.

Differencing

df["satis_fark"] = df["satis"].diff()

Differencing models the change between consecutive observations and removes some trend structures. Seasonal differencing compares observations a fixed period apart — diff(12) on a monthly series, for example. Differencing more than necessary reduces interpretability and distorts the model's behaviour.

The main forecasting methods

Build a simple benchmark first

The only way to know whether a complex model is worth it is to compare it against a naive or seasonal naive forecast. A model that cannot beat the baseline cannot justify its complexity — and this happens far more often than people expect.

import numpy as np

# Assuming weekly seasonality on a daily series
mevsim = 7
train = df["satis"].iloc[:-28]
test  = df["satis"].iloc[-28:]

tahmin = train.iloc[-mevsim:].to_numpy()
tahmin = np.resize(tahmin, len(test))

This repeats the values of the final training week across the test period. Simple as it looks, it performs surprisingly well on strongly seasonal series. For a fuller evaluation, pay attention to the forecast horizon, missing days and the weekday/weekend structure.

How do you measure forecast accuracy?

Evaluating a model with a random train-test split is wrong for time series: future observations leak into training and the model looks better than it is. Use a chronological split, or rolling-origin time series cross-validation instead.

import numpy as np
from sklearn.metrics import mean_absolute_error, mean_squared_error

mae  = mean_absolute_error(test, tahmin)
rmse = np.sqrt(mean_squared_error(test, tahmin))

print(f"MAE: {mae:.2f}")
print(f"RMSE: {rmse:.2f}")

One test period is not enough when choosing a model. Evaluating the same horizon from several different start dates shows how consistent the model is across periods. Prediction intervals matter at least as much as the point forecast: they are what communicates the size of the uncertainty to a decision-maker.

Four real-world scenarios

In social media analysis, a rise in mentions on a particular day is not lasting growth. It may come from a viral post, a news event or a campaign. The model should look not only at past counts but, where possible, at an events calendar and explanatory variables.

An end-to-end time series workflow

• Frequency: check whether daily, weekly or monthly observations are regular.
• Missing time points: is an absent record a zero, or was no data collected? The distinction changes the forecast directly.
• Outliers: bad data or a real event — do not delete before investigating.
• Calendar effects: identify known effects such as holidays, weekends and campaigns in advance.
• Validation: split from past to future, and measure over the same horizon you will actually forecast.
• Monitoring: compare realised values against forecasts and keep watching the error and any drift in the data.

Mistakes to avoid

• Fitting a model without putting the observations in time order.
• Filling missing dates with zero automatically, conflating zero with unobserved.
• Letting information about the future into the training data (leakage).
• Choosing a model purely on its fit to the training data.
• Ignoring seasonality and adding only a linear trend.
• Failing to report forecast uncertainty and possible structural breaks.
• Assuming that past performance guarantees future performance.

Time series analysis starts with understanding patterns

A successful time series project does not start by picking a complex algorithm. First you understand the frequency of the data, the missing time points, the trend and the seasonality; then you assess stationarity and the dependency structure. Simple forecasting methods are a strong starting point; more complex models should only be preferred when they deliver measurable benefit.

Whether you work on sales, energy, finance, operations or social media data, building the right time index, validating chronologically and monitoring errors regularly are the fundamental steps that improve forecast quality.